Identification-Robust Inference on Risk Premia of Mimicking Portfolios of Non-traded Factors

Frank Kleibergen; Zhaoguo Zhan

doi:10.1093/jjfinec/nby005

Identification-Robust Inference on Risk Premia of Mimicking Portfolios of Non-traded Factors

Kleibergen, Frank;Zhan, Zhaoguo 2018-04-01 00:00:00 Abstract Mimicking portfolios of economic (non-traded) factors are commonly constructed by projecting the factors on a set of base assets. When these factors are associated with small betas, the beta-estimator using their mimicking portfolios has non-standard limit behavior. This jeopardizes inference on risk premia in the commonly used Fama and MacBeth (1973) two-pass procedure. Using sorting or the average excess returns on the mimicking portfolios to estimate the risk premia leads to similar non-standard behavior. We therefore propose a novel test for the risk premia on mimicking portfolios. Its validity does not depend on the magnitude of the betas. Simulation evidence suggests that it performs well in terms of size and power. We use it to analyze the risk premium on the leverage factor of Adrian, Etula, and Muir (2014). Our results indicate that the leverage factor is a weak factor which leads to substantially different results for its risk premium. An intersection of economics and finance is that a large group of economic factors are found useful for financial asset pricing. These factors include, for example, consumption growth in Breeden, Gibbons, and Litzenberger (1989), labor income growth in Jagannathan and Wang (1996), consumption–wealth ratio in Lettau and Ludvigson (2001), GDP growth in Vassalou (2003), investment growth in Li, Vassalou, and Xing (2006), among many others. The prevalence of these factors has recently led to a sizeable and growing literature that scrutinizes their usefulness. One major concern for economic factors is that their minor correlation with asset returns (see, e.g., Bai and Ng, 2006) invalidates the conventional inference methods used in asset pricing studies, such as the t-test on risk premia in the Fama and MacBeth (1973) (FM) two-pass methodology. Consequently, empirical support for economic factors based on conventional inference methods is up to careful scrutiny. An early contribution along this line is Kan and Zhang (1999), who show that the t-test in the FM methodology can spuriously support useless factors that are independent of asset returns. More recently, Kleibergen (2009) further warns that when factors are only weakly correlated with asset returns, so their betas are small, traditional inference on the risk premia based on the FM two-pass procedure is also spurious. Instead of using economic factors themselves, their mimicking portfolios are also widely used to replace these non-traded factors in asset pricing studies. The theoretical support for such practice is provided by Breeden (1979) and Huberman, Kandel, and Stambaugh (1987), who establish that factors can be replaced by their mimicking portfolios for asset pricing tests. In terms of economic factors, the norm for constructing their mimicking portfolios is to project these factors on a set of base assets that span the asset space. This projection is implemented by regressing factors on the returns of base assets in a time-series regression. The resulting mimicking portfolios after projection are also called maximum correlation portfolios. See, for example, Breeden, Gibbons, and Litzenberger (1989); Lamont (2001); Vassalou (2003); Avramov and Chordia (2006); and Adrian, Etula, and Muir (2014). Alternatively, mimicking portfolios are also commonly constructed after sorting the assets using their betas or other characteristics of the assets. On the one hand, besides other potential advantages,1 using mimicking portfolios instead of non-traded factors appears able to bypass the weak statistical correlation issue studied in Kleibergen (2009), etc. For instance, by projecting factors on base assets, it is natural that the resulting mimicking portfolios exhibit improved correlation with asset returns so their betas are amplified. In existing studies (e.g., Vassalou, 2003; Adrian, Etula, and Muir, 2014), this projection is commonly interpreted as a way to remove the noise in factors while keeping only their relevant information for asset pricing. Because of this interpretation, inference on risk premia using mimicking portfolios is believed to be more informative compared with that using the original non-traded factors. On the other hand, using mimicking portfolios instead of factors has its own costs which are rarely discussed in the existing literature. In this paper, we reveal the consequences of using mimicking portfolios in the FM two-pass procedure when their background factors are only minorly correlated with asset returns. We show that although these mimicking portfolios may appear relevant for asset pricing, the risk premia estimator in the FM two-pass procedure has a non-standard distribution which jeopardizes the t-test on the risk premia. This results since when the betas of the non-traded factors are small, the beta-estimator of their mimicking portfolios has non-standard limiting behavior which makes the t-test unreliable. Similarly, we find that the excess returns of such mimicking portfolios also fail to appropriately reflect the risk premia when the quality of the corresponding factors is poor. Our findings therefore suggest that using mimicking portfolios instead of economic factors for asset pricing does not necessarily imply improved inference on risk premia or more generally, improved performance of asset pricing tests. Ironically, since economic factors are almost indistinguishable from useless factors in terms of their minor correlation with asset returns in finite samples, empirical findings based on mimicking portfolios could be driven by the noise occurred in the construction of such portfolios rather than by the relevant pricing information contained in economic factors and preserved in the construction of their mimicking portfolios.2 Since our result puts the validity of the conventional t-test on risk premia of mimicking portfolios resulting from economic factors under doubt, we propose a novel test. Unlike the t-test, the asymptotic size of our test does not depend on the magnitude of the betas so it remains trustworthy when economic factors have small betas. The suggested test is in line with the robust tests proposed by Kleibergen (2009) for the risk premia on factors but the extension to a mimicking portfolios setting is new. The performance of the test is examined using Monte-Carlo simulation. For illustrative purposes, we apply the proposed test to the leverage factor model from Adrian, Etula, and Muir (2014). Following Adrian, Etula, and Muir (2014), we construct mimicking portfolios by regressing the leverage factor on seven base assets: excess returns of the six Fama–French portfolios on size and book-to-market, plus the momentum factor. By inverting the proposed test, we obtain confidence intervals for the risk premium of the constructed mimicking portfolios, with the usage of various test assets. We show that these confidence intervals differ substantially from those that result from inverting the conventional t-test, which is consistent with the fact that the leverage factor is associated with a small beta. Related to our paper, there exists a sizeable literature that examines identification and inference issues in asset pricing. Besides those related articles we cite above, see, for example, Burnside (2016) and Balduzzi and Robotti (2008). Unlike this paper, Burnside (2016) does not focus on mimicking portfolios but shows that when factors are weak, lack of identification exists in linear stochastic discount factor models. The rank test of Kleibergen and Paap (2006) is consequently suggested in Burnside (2016) as a model diagnostic. Balduzzi and Robotti (2008) do not focus on the FM two-pass procedure but argue that the average excess return on the mimicking portfolios provides an estimator for the risk premium. We show that this argument no longer applies when the correlation between factors and asset returns is small. The paper proceeds as follows. In Section 1, we describe a linear factor model with observed factors and present the limiting behavior of the beta-estimator using mimicking portfolios. In Section 2, we propose the robust test on the risk premia of mimicking portfolios and provide simulation evidence for its size and power. In Section 3, we re-evaluate the pricing performance of the leverage factor using the mimicking portfolios as in Adrian, Etula, and Muir (2014). Section 4 concludes the paper. Proofs, technical details, and additional results are provided in the Appendix. Throughout the paper, we use the following notation: μ is used for mean, while V is for covariance; μ^ and V^ are the sample analogs of μ and V, respectively; ιN is the N×1 dimensional vector of ones, vec(A) stands for the column vectorization of a matrix A, PA=A(A′A)−1A′, MA=I−PA,I is the identity matrix; “ →p”and “ →d” stand for convergence in probability and convergence in distribution, respectively. 1 Weak Identification with Mimicking Portfolios 1.1 Linear Factor Model and Observed Factors There is ample empirical evidence that asset returns are to a large extent explained by a small number of factors, see, for example, Merton (1973), Ross (1976), Roll and Ross (1980), Chamberlain and Rothschild (1983), and Connor and Korajczyk (1988, 1989). We therefore use a linear factor model for asset returns Rt (N×1) with k potentially unobserved factors Ft (k×1) to reflect the factor structure: Rt=ιNλ0+β(F¯t+λF)+vt, (1) where F¯t=Ft−μF, vt is the mean zero error term, t=1, … , T, β is the N × k full-rank matrix. Equation (1) implies the moment condition for the risk premia, that is, E(Rt)=ιNλ0+βλF, (2) with λ0: the zero-β return, λF: the k×1 vector of factor risk premia. Moreover, there exists a partition of Rt=(R1t′,R2t′)′, with R1t: N1×1, R2t: N2×1, N1+N2=N. From now on, we consider R1t as the returns on test assets, and R2t as the returns on base assets that are used for constructing the mimicking portfolios.3 Correspondingly, β=(β1′,β2′)′ with β1: N1×k, β2: N2×k, and vt=(v1t′,v2t′)′ with v1t: N1×1, v2t: N2×1. To accommodate that the factors are potentially unobserved, we use an m×1 vector of observed proxy factors Gt. We relate Ft to Gt by F¯t=δG¯t+ut, (3) where G¯t=Gt−μG and ut is the error term, see Kleibergen and Zhan (2015). The quality of the approximation of the true factors by the proxy factors is reflected by the k × m matrix δ. Ideally, Ft is observed so Gt coincides with it and δ=Ik,ut = 0. On the other hand, if δ is approximately equal to zero then Ft is poorly approximated by Gt. Plugging Equation (3) into Equation (1), and if the vector of risk premia of Gt, denoted by λG, is defined by the moment condition E(Rt)=ιNλ0+βδλG, then λF and λG are also related by δ since λF=δλG.4 In addition, Equation (1) is re-written as, Rt=ιNλ0+βδ(G¯t+λG)+et, (4) where et=(e1t′,e2t′)′=βut+vt and Σ=βΛβ′+Ω with Σ=var(et), Λ=var(ut), Ω=var(vt). The beta corresponding to Gt is thus βδ, which is small if δ is close to zero. We make the following assumptions for the model described above. Assumption 1. 1T∑t=1T((1Ft)⊗(Rt−ιNλ0−β(F¯t+λF)))→d(ϕRϕF), where (ϕRϕF)∼N(0,QF⊗Ω), QF=(1μF′μFVFF+μFμF′), Ω=var((v1t′,v2t′)′)=(Ω11Ω12Ω21Ω22). Assumption 2. All the covariance of R1t and R2t is captured by the factors so Ω12=Ω21′=0 and cov (R1t,R2t)=β1VFFβ2′. Assumption 3. 1T∑t=1T((1Gt)⊗(Rt−ιNλ0−β(F¯t+λF)))→d(ϕRϕG), where (ϕRϕG)∼N(0,QG⊗Ω), QG=(1μG′μGVGG+μGμG′). Assumption 1 is a central limit theorem that holds under rather mild conditions, like, for example, Assumptions 1 and 2 from Shanken (1992). Because of the independence of the disturbances over time and their finite variance, Assumption 1 holds under the linear factor model (1) with a constant covariance matrix as well. Assumption 2 is a necessary condition for the beta of the mimicking portfolios to be a nonsingular transformation of β1 which we further show below in Footnotes 5 and 6.5 Assumption 3 is a corollary of Assumption 1, with the observed Gt replacing the unknown Ft. 1.2 Beta of Mimicking Portfolios In accordance with common practice, we first construct mimicking portfolios by projecting Gt on base assets R2t. We also briefly discuss mimicking portfolios constructed by sorting on betas later on. The feasible mimicking portfolios that result from projection are then specified by: V^GR2V^R2R2−1R2t, (5) where V^GR2V^R2R2−1 is the sample counterpart of the infeasible VGR2VR2R2−1, and results from regressing Gt on R2t, t=1,…,T. With R1t as test assets, the beta-estimator for the mimicking portfolios in Equation (5) reads6: β˜^1=V^R1R2V^R2R2−1V^R2G(V^GR2V^R2R2−1V^R2G)−1. (6) Theorem 1. When Assumptions 1–3 hold, the limiting behavior of β˜^1 is described by: When δ is fixed and the number of elements of Gt equals the number of elements of Ft, so δ is a square invertible matrix: β˜^1→pβ1VFFδ−1′VGG−1,so when Gt=Ft, δ=Ik, and β˜^1→pβ1. If the proxy factors Gt are weak factors so δ=d/T, with d a fixed full-rank matrix, then regardless of k = m: T−12β˜^1→dβ1VFFβ2′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G) [(β2(dVGG+ψuG)+ψv2G)′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G)]−1,where 1T∑t=1T(G¯t⊗ut)→dvec (ψuG), 1T∑t=1T(G¯t⊗v2t)→dvec (ψv2G). Proof. See Appendix A. ▪ The first part of Theorem 1 shows that the beta-estimator of the mimicking portfolios converges to a non-singular transformation of the beta of the factors when the mimicking portfolios are constructed from good proxies of the underlying unobserved factors. The proof of Theorem 1 in Appendix A also shows that the covariance of test assets and base assets must be fully captured by the factors to render the beta-estimator consistent as implied by Assumption 2. Put differently, if Assumption 2 is not satisfied, then the estimated beta of the mimicking portfolios does not converge to the transformed beta of the factors, even when the true underlying factors are used to construct the mimicking portfolios. The motivation of this paper comes from the second part of Theorem 1, that is, when mimicking portfolios result from observed factors that are only poor (weak) proxies for underlying factors. It is well known that economic factors commonly exhibit minor correlation with asset returns, so they are likely to be poor proxies for the underlying factors, which we reflect by the weak factor specification: δ=d/T. (7) This drifting specification for δ, where it decreases with the sample size, is taken from the weak-instrument assumption made in econometrics (see, e.g., Staiger and Stock, 1997), in order to reflect that Gt is a poor proxy for Ft. A similar treatment can be found in Kleibergen (2009) and Kleibergen and Zhan (2015). The weak factor specification should not be taken literally but captures the often observed setting where an F-test of δ in Equation (3) equal to zero is rather small. When δ is fixed this F-test should be proportional to the sample size. The weak factor specification in Equation (7) allows for such small values of the F-test, see Kleibergen and Zhan (2015). For the weak factor setting, Theorem 1 shows that the beta-estimator of the mimicking portfolios is increasing with the sample size and has a non-standard limiting distribution. In other words, when the betas of the economic factors are small, the mimicking portfolios of such factors can be associated with betas that are spuriously large. Theorem 1 does not articulate a mixed setting where some of the unobserved factors are well approximated by the observed factors and some are not. In this case, δ would have both elements that drift to zero according to Equation (7) and elements that remain fixed. We left out this mixed case because it results in a combination of the limiting behaviors discussed under 1 and 2 in Theorem 1 which makes it notationally burdensome without altering the main conclusion from Theorem 1 that the magnitude of δ affects the limiting behavior of the beta-estimator. It is also worth noting that if k > m so βδ does not span the column space of β, then misspecification of factor pricing occurs. Our statistic that we propose lateron in the paper also indicates if such misspecification is present alongside deviations of the risk premia from the hypothesized one. In the commonly used FM two-pass procedure, the risk premia of the mimicking portfolios are obtained by regressing the sample average of the test assets μ^R1=1T∑t=1TR1t on β˜^1 and an intercept, that is, (λ˜^0λ˜^G)=[(ιN1⋮β˜^1)′(ιN1⋮β˜^1)]−1(ιN1⋮β˜^1)′μ^R1, (8) where λ˜^G (m×1) denotes the estimated risk premia for the mimicking portfolios that are constructed from Gt. If β˜^1 has a non-standard distribution, it is natural to expect that the behavior of λ˜^G is also non-standard as stated in the corollary below. Corollary 1. When Assumptions 1–3 hold, the limiting behavior of λ˜^G can be described by: When δ is fixed and the number of elements of Gt equals the number of elements of Ft, so δ is a square invertible matrix: λ˜^G→pVGGδ′VFF−1λF, so when Gt=Ft, δ=Ik, and λ˜^G→pλF. When δ=d/T: Tλ˜^G→d(Ψ′β1MιN1Ψβ1)−1Ψ′β1MιN1μR1+1T(Ψ′β1MιN1Ψβ1)−1Ψ′β1MιN1ψR1,where ψR1 is from Assumption 1 with ψR=(ψ′R1,ψ′R2)′, MιN1 is a projection matrix with MιN1=IN1−ιN1(ι′N1ιN1)−1ι′N1, Ψβ1≡β1VFFβ2′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G) [(β2(dVGG+ψuG)+ψv2G)′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G)]−1,see Theorem 1. Proof. See Appendix B. ▪ In line with Theorem 1, Corollary 1 also contains two cases. In the ideal first case, where accurate proxies of the underlying factors are used for constructing mimicking portfolios, λ˜^G is a consistent risk premia estimator. On the other hand in the second case where weak factors are observed, as reflected by δ=d/T, because of the non-standard behavior of β˜^1, λ˜^G also has a non-standard limiting distribution. Due to the non-standard behavior of λ˜^G in Corollary 1 under mimicking portfolios of weak factors, the conventional FM t-statistic for testing risk premia does not have an asymptotic standard normal distribution. Consequently, Corollary 1 implies that the conventional FM t-test on risk premia is under doubt when it is used for mimicking portfolios of economic factors. 1.3 Covariance Specification Instead of the beta specification above, the so-called covariance specification, see, for example, Kan, Robotti, and Shanken (2013) (KRS) suggests that for the mimicking portfolios in Equation (5), we can also use as an estimator for β1: β^1=V^R1R2V^R2R2−1V^R2G. Different from β˜^1, the inverted matrix (V^GR2V^R2R2−1V^R2G)−1 from β˜^1 is omitted in β^1. Theorem 2. When Assumptions 1–3 hold, the limiting behavior of β^1 can be described by: When δ is fixed and the number of elements of Gt equals the number of elements of Ft, so δ is a square invertible matrix: β^1→pβ1VFFβ2′(β2VFFβ2′+Ω22)−1β2δVGG. When δ=d/T: T12β^1→dβ1VFFβ2′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G).The specifications of ψuG and ψv2G are stated in Theorem 1. Proof. See Appendix C. ▪ Unlike Theorem 1, the second part of Theorem 2 shows that the large sample behavior of the weak factor mimicking portfolio’s beta-estimator is now comparable to that of the pure weak factor’s beta-estimator, see Kleibergen (2009) and Kleibergen and Zhan (2015). Put differently, under the covariance specification, the beta-estimator of the mimicking portfolios (denoted by β^1) does not suffer from the exaggeration problem of its counterpart (denoted by β˜^1) which exists under the beta specification, as stated in Theorem 1. However, it is known that small betas jeopardize risk premia estimation in the FM methodology, see for example, Kleibergen (2009). Consequently, a malfunction of the FM two-pass procedure also exists in the covariance specification, albeit for reasons opposite to those of the beta specification: if mimicking portfolios are constructed from weak factors, the beta-estimator is large in magnitude in the beta specification, and small in the covariance specification, both could induce failure of risk premia estimation in the FM methodology. In order to resolve this issue, we develop a new inference method for risk premia of mimicking portfolios, which is presented later on. 1.4 Simulation Study To further illustrate the results in Theorems 1 and 2, we conduct a simple simulation experiment. Asset returns are generated from the factor model (1), with T=1000 and k = 1. Specifically, Ft∼NID (0,VFF), vt∼NID (0,Ω), where VFF is calibrated from the returns on the market portfolio in Fama–French (1993), Ω is calibrated from a regression of the returns on N1 industry portfolios, and N2 size- and book-to-market-sorted portfolios on the returns on the market portfolio. We consider N1=N2 for convenience in the data generation process (d.g.p.) and set them equal to 1, 2, 3, 4, and 5, as reported in the first column of Table 1. The values of the parameters λ0, λF, and β used in the d.g.p. result from the FM two-pass procedure using the described portfolios.7 Table 1. Rejection frequencies of the rank test by Monte Carlo Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.051 0.107 0.060 0.112 1.000 1.000 0.050 0.106 0.059 0.112 1.000 1.000 2. 0.053 0.105 0.014 0.031 0.936 0.949 0.051 0.102 0.013 0.030 0.935 0.949 3. 0.050 0.104 0.005 0.012 0.898 0.916 0.048 0.101 0.005 0.012 0.895 0.914 4. 0.052 0.102 0.003 0.011 0.879 0.899 0.050 0.097 0.002 0.009 0.878 0.897 5. 0.050 0.090 0.001 0.002 0.841 0.864 0.046 0.085 0.000 0.002 0.837 0.860 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.051 0.107 0.060 0.112 1.000 1.000 0.050 0.106 0.059 0.112 1.000 1.000 2. 0.053 0.105 0.014 0.031 0.936 0.949 0.051 0.102 0.013 0.030 0.935 0.949 3. 0.050 0.104 0.005 0.012 0.898 0.916 0.048 0.101 0.005 0.012 0.895 0.914 4. 0.052 0.102 0.003 0.011 0.879 0.899 0.050 0.097 0.002 0.009 0.878 0.897 5. 0.050 0.090 0.001 0.002 0.841 0.864 0.046 0.085 0.000 0.002 0.837 0.860 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. Table 1. Rejection frequencies of the rank test by Monte Carlo Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.051 0.107 0.060 0.112 1.000 1.000 0.050 0.106 0.059 0.112 1.000 1.000 2. 0.053 0.105 0.014 0.031 0.936 0.949 0.051 0.102 0.013 0.030 0.935 0.949 3. 0.050 0.104 0.005 0.012 0.898 0.916 0.048 0.101 0.005 0.012 0.895 0.914 4. 0.052 0.102 0.003 0.011 0.879 0.899 0.050 0.097 0.002 0.009 0.878 0.897 5. 0.050 0.090 0.001 0.002 0.841 0.864 0.046 0.085 0.000 0.002 0.837 0.860 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.051 0.107 0.060 0.112 1.000 1.000 0.050 0.106 0.059 0.112 1.000 1.000 2. 0.053 0.105 0.014 0.031 0.936 0.949 0.051 0.102 0.013 0.030 0.935 0.949 3. 0.050 0.104 0.005 0.012 0.898 0.916 0.048 0.101 0.005 0.012 0.895 0.914 4. 0.052 0.102 0.003 0.011 0.879 0.899 0.050 0.097 0.002 0.009 0.878 0.897 5. 0.050 0.090 0.001 0.002 0.841 0.864 0.046 0.085 0.000 0.002 0.837 0.860 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. In the simulation exercise, we consider two different types of observed factors which are encountered in empirical studies. The first case is the one of observed factors that are good/strong proxies for the underlying factors Ft. Here, we use Gt=Ft. The second case is the one of observed factors that are poor/weak proxies of the underlying factors. We therefore use observed factors Gt that are independently generated as N(0,VFF) distributed random variables so the observed factor is completely useless for asset returns. In Panel A of Table 1, we present the outcome of the simulation study for the strong factor case, while the useless factor case is in Panel B. Both the beta specification and the covariance specification for constructing the beta-estimator are considered in our simulation study. In addition, for each specification, we consider three different manners of using the simulated factors. Under “Fac” in Table 1, the simulated factor is directly used for estimating β1 in the time-series regression using the simulated asset returns R1t. Under “MP” in Table 1, we construct mimicking portfolios of simulated factors as in Equation (5); with the constructed mimicking portfolios, we proceed to compute their beta-estimator β˜^1 (beta specification) or β^1 (covariance specification); furthermore, the asymptotic variances of such estimators are derived by first-order asymptotics taking into account the estimation error contained in the mimicking portfolios (see Appendix D for details). Under “MP as Fac” in Table 1, we treat the constructed mimicking portfolios in the same manner as the observed factors for constructing the variance of the beta-estimator (hence, the estimation error contained in the mimicking portfolios is ignored when computing the variance of their beta-estimator), which is commonly done in existing empirical studies. We apply the rank test of Kleibergen and Paap (2006) for the estimator of β1 in the various scenarios described above. In particular, we test whether the estimand has reduced rank k − 1 and document the rejection frequency of the null at the nominal 5% and 10% level using standard χ2 critical values. Since β1 has full rank under strong factors, we expect that the rank test will strongly reject the null. This is in line with the rejection frequencies reported in Panel A of Table 1. In fact, we show that the rejection frequencies based on 2000 Monte-Carlo replications are all equal to 1, so the outcome of the rank test supports the strong factor, as expected. On the contrary, under useless factors, we expect the rejection frequency of the rank test to be close to the nominal size, since useless factors are associated with zero beta’s so the null hypothesis holds. This is in line with the rejection frequencies reported under “Fac” in Panel B of Table 1. Under “MP,” the rank test appears conservative, since the rejection frequencies quickly decrease from nominal sizes as N1 and N2 increase.8 What is astonishing in Panel B of Table 1 lies in the columns of “MP as Fac”, that is, mimicking portfolios are naively treated as alternatives of factors and used for rank tests. Panel B shows that the rejection frequencies under “MP as Fac” are very large (albeit decrease as N1 and N2 increase). Consequently, mimicking portfolios of useless factors may signal strong factor pricing in rank tests, if they are treated in the same manner as factors. Theorems 1 and 2 show that such treatment is improper, since the beta-estimator under mimicking portfolios does not have the same limiting distribution as when using observed factors. Overall, Table 1 suggests that the rank test can serve as a diagnostic tool for the quality of the factors, if implemented properly so correcting for the estimation error that results from using mimicking portfolios. When factors are strong (weak), the null is rejected (accepted with the probability close to the nominal size), if these factors are used for beta estimation. When mimicking portfolios are used for estimating the betas, Table 1 suggests that the rank test outcome needs to be taken with caution. In particular, if mimicking portfolios are improperly treated as observed factors, so ignoring their estimation error, then the rank test may spuriously favor strong factors.9 1.5 Average Excess Returns and Sorting Risk premia can also be estimated using the average excess returns of the mimicking portfolios, see, for example, Balduzzi and Robotti (2008). For the mimicking portfolios obtained by projection, V^GR2V^R2R2−1R2t, the risk premia estimator then equals the time-series average: V^GR2V^R2R2−1R¯2, (9) with R¯2=1T∑t=1TR2t. The limiting behavior of the risk premia estimator (9) becomes non-standard when the observed factors are weakly correlated with the true underlying factors. Theorem 3. When Assumptions 1–3 hold, the limiting behavior of V^GR2V^R2R2−1R¯2 under excess returns with λ0=0 is described by: When δ is fixed: V^GR2V^R2R2−1R¯2→pVGGδ′β2′VR2R2−1β2λF. When δ=d/T: TV^GR2V^R2R2−1R¯2→d[β2(dVGG+ψuG)+ψv2G]′VR2R2−1β2λF. Proof. It results straightforwardly from the proof of Theorem 1. The specifications of ψuG and ψv2G are stated in Theorem 1. ▪ Theorem 3 shows that when δ is sizeable, the average excess returns of the mimicking portfolios converge to a linear combination of the risk premia λF. The risk premia estimator (9) thus provides an adequate estimator of the risk premia when the observed factors are good proxies of the underlying true factors. In contrast, when the proxy factors only weakly correlate with the true underlying factors, so δ is local to zero, the risk premia estimator V^GR2V^R2R2−1R¯2 is no longer a valid estimator of the risk premia. On top of this, its limiting behavior differs considerably from the one under strong factors, which further complicates inference. We have so far focused on mimicking portfolios constructed by projection. The challenge induced by weak factors, however, also applies to mimicking portfolios constructed by sorting which we briefly discuss next. To facilitate the analysis, consider a single observed factor Gt, so m=1. We construct so-called high-minus-low mimicking portfolios of Gt based on sorting the N2×1 vector V^R2GV^GG−1 of estimated betas. We denote the N2×N2 matrix with the sorting (WLOG in descending order) indicators of V^R2GV^GG−1 by S(V^R2GV^GG−1) so: S(V^R2GV^GG−1)×V^R2GV^GG−1=[max(V^R2GV^GG−1)⋮min(V^R2GV^GG−1)], (10) where max(V^R2GV^GG−1) is the maximal element of V^R2GV^GG−1 and min(V^R2GV^GG−1) the minimal element. Correspondingly, the N2×1 returns of R2t after sorting by V^R2GV^GG−1 read: S(V^R2GV^GG−1)×R2t=[R2t,(1)⋮R2t,(N2)], (11) where the (estimated) beta of R2t,(1) is the largest, the beta of R2t,(N2) is the smallest, and the betas are obtained from V^R2GV^GG−1. The return on the high-minus-low mimicking portfolios is then R2t,(1)−R2t,(N2), if only one asset is taken as “high” and one asset is taken as “low.” In accordance with empirical practice, we consider R2t,(1), … , R2t,(λN2) as the first λN2 asset returns after sorting, where 0<λ<1. Similarly, R2t,((1−λ)N2+1), … , R2t,(N2) are the last λN2 asset returns after sorting. We take the first λN2 assets as “high,” and take the last λN2 assets as “low.” The return on the high-minus-low mimicking portfolios (equally weighted) then reads: 1λN2∑i=1λN2R2t,(i)−1λN2∑i=(1−λ)N2+1N2R2t,(i); in particular, when λ=1/N2, this expression simplifies to R2t,(1)−R2t,(N2). In matrix notation, the return on the high-minus-low portfolios then reads: 1λN2[1,...,1,0,...,0,−1,...,−1]×S(V^R2GV^GG−1)×R2t, (12) where both the number of 1s and −1s above are equal to λN2. If the average returns of the high-minus-low portfolio are used to estimate the risk premia, the risk premia estimator reads: ΔλN2×S(V^R2GV^GG−1)×R¯2, (13) with ΔλN2=1λN2[1,...,1,0,...,0,−1,...,−1] a fixed constant row vector. The sorting operator S(V^R2GV^GG−1) is therefore at the center of our analysis. We distinguish two different settings for its limiting behavior which result from the quality of the approximation of the true underlying factors by the proxy factors. When the observed factors Gt provide good proxies so δ is sizeable: V^R2GV^GG−1→pβ2δ. (14) Hence, in the limit S(V^R2GV^GG−1) effectively sorts β2δ which is a fixed vector. Consequently, as T→∞, the limiting behavior of the risk premia estimator is characterized by ΔλN2×S(V^R2GV^GG−1)×R¯2→pΔλN2×S(β2δ)×β2λF. (15) Since R¯2→pιN2λ0+β2λF. It is worth noting that assets with large elements in β2δs tend to be in “high,” while the assets with small elements in β2δs tend to be in “low,” since sorting is effectively based on β2δ as T gets large. The risk premia estimator is therefore equal to a linear combination of the risk premia λF. Sorting using high-minus-low portfolios thus provides an estimator of the risk premia when the observed factors are good proxies. In contrast, when the observed factors are poor proxies of the true factors so δ=d/T: TV^R2GV^GG−1→dβ2d+(β2ψuG+ψv2G)VGG−1≡[Z1⋮ZN2], (16) where Z1, … , ZN2 are N2 (non-identical) correlated normal random variates. Each of these N2 variates has a non-zero probability of being the largest one in realizations; similarly, each of them also has a non-zero probability of being the smallest one in realizations. Since the scaling by T does not affect the sorting, the limiting behavior of the risk premia estimator is now characterized by ΔλN2×S(V^R2GV^GG−1)×R¯2→dΔλN2×S(β2d+(β2ψuG+ψv2G)VGG−1)×β2λF, (17) which shows that when Gt is a poor proxy, the sorting is conducted on a random vector even when T is large. The risk premia estimator now converges to a random function of the risk premia since the sorting on the beta-estimator is arbitrary as it converges to a random variable. This also shows that identical to the other risk premia estimators, the behavior of the risk premia estimator based on sorting becomes non-standard when the observed factors are poor proxies of the true factors. 2 Robust Inference The previous section discussed issues associated with the conventional approaches for inference on the risk premia when mimicking portfolios result from non-traded factors which are poor proxies of the true factors. To resolve this problem, we propose a robust inference procedure for testing the risk premia of mimicking portfolios. 2.1 Scaled Risk Premia under Mimicking Portfolios Specifically, we suggest to test the risk premium λG,cov on mimicking portfolios using the moment condition: E(R1t)=ιN1λ0+VR1R2VR2R2−1VR2GVGG−1λG,cov (18) where VR1R2VR2R2−1VR2G is the covariance of the test assets R1t with the (infeasible) mimicking portfolios VGR2VR2R2−1R2t, so λG,cov is the scaled risk premium in the covariance specification (scaled by VGG−1).10 If the base assets in R2t span the test assets in R1t, then VR1R2VR2R2−1VR2G reduces to VR1G and λG,cov equals λG in Equation (4). For instance, in the special case that R1t=R2t, so test and base assets coincide, λG,cov reduces to λG. If so, inference on λG,cov reduces to inference on λG, which has been discussed by Kleibergen (2009). Since it is common that R2t does not fully span R1t, λG,cov is consequently not necessarily equal to λG. We thus proceed to consider inference on λG,cov defined in Equation (18). As a starting point, we remove λ0 since our interest lies in λG,cov. This is done by, without loss of generality, removing the return on the N1-th test asset and taking all other test asset returns in deviation from the return on the N1-th asset.11Equation (18) is then re-written as: E(R1t)=VR1R2VR2R2−1VR2GVGG−1λG,cov=Γβ2δλG,cov, (19) where Γ=VR1R2VR2R2−1, R1t=R1t,1:(N1−1)−ιN1−1R1t,N1, with R1t=(R′1t,1:(N1−1),R′1t,N1)′, and VR2GVGG−1=β2δ as in Equation (4). 2.2 Mimicking Portfolio Anderson–Rubin Test To conduct inference on λG,cov, we state the joint behavior of R¯1, B^2, and Γ^ in Theorem 4, with R¯1=1T∑t=1TR1t, B^2=V^R2GV^GG−1, and Γ^=V^R1R2V^R2R2−1. Theorem 4. When Assumptions 1–3 hold: T(R¯1−Γβ2δλG,covvec(B^2−β2δ)vec(Γ^−Γ))→d(ψ1vec(ψ2)vec(ψ3))∼N(0,W), (20)and W=(VR1R1000VGG−1⊗Σ22C′0CVR2R2−1⊗(VR1R1−VR1R2VR2R2−1VR2R1))with C=K(N1−1)N2(VR1FVFF−1δ⊗VR2R2−1Σ22)+(VR2R2−1β2δ⊗Σ12) −(VR2R2−1⊗Γ)(IN22+KN2N2)(β2δ⊗Σ22),K is the commutation matrix and Σ12=cov(e1t,e2t) with e1t=R1t−VR1FVFF−1δ(G¯t+λG). Proof. See Appendix F. ▪ Theorem 4 implies that the limiting distribution of T(R¯1−Γ^B^2λG,cov) is normal. Corollary 2. When Assumptions 1–3 hold and Γ has a full-rank value: T(R¯1−Γ^B^2λG,cov)→dψ1−Γψ2λG,cov−ψ3β2δλG,cov∼N(0,V), (21)with V=VR1R1+(λ′G,covVGG−1λG,cov⊗ΓΣ22Γ′)+((β2δλG,cov)′VR2R2−1(β2δλG,cov)⊗(VR1R1−VR1R2VR2R2−1VR2R1))−{[(β2δλG,cov)′⊗IN1−1]C(λG,cov⊗Γ′)}−{[(β2δλG,cov)′⊗IN1−1]C(λG,cov⊗Γ′)}′. Proof. See Appendix G. ▪ Since the unobserved factor Ft affects both R1 and R2, Γ has a full-rank value. We do not make a full-rank assumption on δ so we allow for weak correlation between observed and unobserved factors. Using Corollary 2, it is straightforward to obtain an asymptotic result to test H0:λG,cov=λG,cov,0. Corollary 3 (MPAR). Under H0:λG,cov=λG,cov,0, MPAR(λG,cov,0)=T(R¯1−Γ^B^2λG,cov,0)′V^−1(R¯1−Γ^B^2λG,cov,0)→dχN1−12 (22)with V^=V^R1R1+(λ′G,cov,0V^GG−1λG,cov,0⊗Γ^Σ^22Γ^′)+((B^2λG,cov,0)′V^R2R2−1(B^2λG,cov,0)⊗(V^R1R1−V^R1R2V^R2R2−1V^R2R1))−[(B^2λG,cov,0)′⊗IN1−1]C^(λG,cov,0⊗Γ^′)−(λ′G,cov,0⊗Γ^)C^′[(B^2λG,cov,0)⊗IN1−1] C^=K(N1−1)N2(B^1⊗V^R2R2−1Σ^22)+(V^R2R2−1B^2⊗Σ^12)−(V^R2R2−1⊗Γ^)(IN22+KN2N2)(B^2⊗Σ^22)and Σ^12=1T∑t=1Te^1te^′2t, Σ^22=1T∑t=1Te^2te^′2t, e^2t=R2t−R¯2−B^2(Gt−μ^G), e^1t=R1t−R¯1−B^1(Gt−μ^G), B^1=V^R1GV^GG−1. Proof. Results from Corollary 2, with V^ a consistent estimator for V. ▪ We refer to this statistic as the Mimicking Portfolio Anderson–Rubin (MPAR) statistic, since it is in line with the Anderson and Rubin (1949) statistic for robust inference and it is proposed using mimicking portfolios. Similar tests in the factor rather than mimicking portfolios setting include the Factor Anderson–Rubin (FAR) test proposed in Kleibergen (2009) and the Hotelling-type statistic proposed in Beaulieu, Dufour, and Khalaf (2013).12 Because of Corollary 3, a test of H0:λG,cov=λG,cov,0 that rejects H0, if MPAR(λG,cov,0) exceeds the 1−α quantile of the χ2 distribution with N1−1 degrees of freedom has an asymptotic size that equals α irrespective of the value of δ. Values of λG,cov,0 that are not rejected by this test thus constitute a 100×(1−α)% confidence set of λG,cov. 2.3 Size of the MPAR Statistic in Simulations To examine the size of the proposed MPAR test, we conduct a simple simulation experiment. The d.g.p. is similar to the one used in Section 1. Specifically, asset returns are generated from Equation (1), with T = 1000 and k = 1. In addition, Ft and vt are generated as independent NID (0,VFF) and NID (0,Ω) random variables, with VFF calibrated from the returns on the market portfolio in Fama–French (1993) and Ω calibrated from the regression of the returns on N1 industry and N2 size- and book-to-market-sorted portfolios on the return on the market portfolio. We consider N1=N2 for convenience and set them equal to 5, 10, 15, 20, and 25, as reported in the first column of Table 2. The values of λ0, λF, and β used in d.g.p. result from the FM two-pass procedure using the portfolios described above. Table 2. Actual sizes of the MPAR test by Monte Carlo, T = 1000 δ=0.01 δ=0.25 δ=0.50 δ=0.75 δ=0.99 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% N1, N2=5 0.050 0.105 0.059 0.105 0.058 0.103 0.057 0.104 0.057 0.104 N1, N2=10 0.020 0.051 0.052 0.108 0.051 0.108 0.050 0.108 0.051 0.108 N1, N2=15 0.018 0.047 0.064 0.115 0.063 0.116 0.063 0.116 0.063 0.116 N1, N2=20 0.002 0.008 0.063 0.127 0.064 0.128 0.063 0.126 0.062 0.126 N1, N2=25 0.001 0.002 0.064 0.124 0.064 0.127 0.063 0.127 0.063 0.126 δ=0.01 δ=0.25 δ=0.50 δ=0.75 δ=0.99 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% N1, N2=5 0.050 0.105 0.059 0.105 0.058 0.103 0.057 0.104 0.057 0.104 N1, N2=10 0.020 0.051 0.052 0.108 0.051 0.108 0.050 0.108 0.051 0.108 N1, N2=15 0.018 0.047 0.064 0.115 0.063 0.116 0.063 0.116 0.063 0.116 N1, N2=20 0.002 0.008 0.063 0.127 0.064 0.128 0.063 0.126 0.062 0.126 N1, N2=25 0.001 0.002 0.064 0.124 0.064 0.127 0.063 0.127 0.063 0.126 Notes: The reported sizes are rejection frequencies of the MPAR test for H0:λG,cov=λG,cov,0 at the nominal 5% and 10%, respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000. Table 2. Actual sizes of the MPAR test by Monte Carlo, T = 1000 δ=0.01 δ=0.25 δ=0.50 δ=0.75 δ=0.99 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% N1, N2=5 0.050 0.105 0.059 0.105 0.058 0.103 0.057 0.104 0.057 0.104 N1, N2=10 0.020 0.051 0.052 0.108 0.051 0.108 0.050 0.108 0.051 0.108 N1, N2=15 0.018 0.047 0.064 0.115 0.063 0.116 0.063 0.116 0.063 0.116 N1, N2=20 0.002 0.008 0.063 0.127 0.064 0.128 0.063 0.126 0.062 0.126 N1, N2=25 0.001 0.002 0.064 0.124 0.064 0.127 0.063 0.127 0.063 0.126 δ=0.01 δ=0.25 δ=0.50 δ=0.75 δ=0.99 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% N1, N2=5 0.050 0.105 0.059 0.105 0.058 0.103 0.057 0.104 0.057 0.104 N1, N2=10 0.020 0.051 0.052 0.108 0.051 0.108 0.050 0.108 0.051 0.108 N1, N2=15 0.018 0.047 0.064 0.115 0.063 0.116 0.063 0.116 0.063 0.116 N1, N2=20 0.002 0.008 0.063 0.127 0.064 0.128 0.063 0.126 0.062 0.126 N1, N2=25 0.001 0.002 0.064 0.124 0.064 0.127 0.063 0.127 0.063 0.126 Notes: The reported sizes are rejection frequencies of the MPAR test for H0:λG,cov=λG,cov,0 at the nominal 5% and 10%, respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000. The observed Gt is simulated as follows: Gt=δ·Ft+1−δ2·NID(0,VFF), so δ coincides with the specification in Equation (3) and reflects the quality of Gt for approximating Ft. When δ is close to zero, Gt is a weak factor which becomes stronger for an increasing value of δ. We consider a sequence of values for δ: δ∈{0.01,0.25,0.50,0.75,0.99}, which covers a wide range of settings of Gt as the observed proxy for Ft. With the simulated asset returns and Gt, we conduct the MPAR test on the risk premia of the mimicking portfolios as described in Corollary 3. The resulting sizes of the test are reported in Table 2. Table 2 shows that the MPAR test does not severely over reject in any of the settings. Its rejection frequency approximately equals the size of the test in many instances. Table 2 shows that the MPAR test is conservative when the factors are weak. It indicates that the covariance matrix estimator V^ is on average too large in this simulation setting. Using the properties of the partitioned inference and that Γ^ does not depend on the strength of the factors, the too large value of V^ can be attributed to the realized values of B^1 and B^2 being away from their expected values (zero). This explains also why the under rejection increases when N1 and N2 get larger. When we increase the sample size, the estimates of B^1 and B^2 become more precise and the under rejection of the MPAR test disappears.13 Table 2 shows that the behavior of the MPAR test accords with Corollary 3 since the under rejection reported in Table 2 results from the estimation of the covariance matrix V. The size of a test is defined as the maximal rejection frequency under the null hypothesis so under rejection does not indicate size distortion. 2.4 Power of the MPAR Test To illustrate the power of the MPAR test, we consider a sequence of values of λG,cov (between −5 and 5) in the d.g.p. described above with T = 1000 and N1=N2=5.14 We then test H0:λG,cov=0 at the 5% level using the generated data. The resulting rejection frequencies of the MPAR test are in Figure 1. Figure 1. View largeDownload slide Power plots of the MPAR test. Notes: This figure presents the power plot of the MPAR test for H0:λG,cov=0 at the 5% level, with δ=0.01 (plus), δ=0.25 (dotted), δ=0.50 (dash-dot), δ=0.75 (dashed), and δ=0.99 (solid line). The d.g.p. is described in the main text, with T=1000 and N1=N2=5. Figure 1. View largeDownload slide Power plots of the MPAR test. Notes: This figure presents the power plot of the MPAR test for H0:λG,cov=0 at the 5% level, with δ=0.01 (plus), δ=0.25 (dotted), δ=0.50 (dash-dot), δ=0.75 (dashed), and δ=0.99 (solid line). The d.g.p. is described in the main text, with T=1000 and N1=N2=5. We use a range of values of δ to reflect the different qualities of the observed factors. Figure 1 has five power plots of the MPAR test, corresponding to δ=0.01 (plus), δ=0.25 (dotted), δ=0.50 (dash-dot), δ=0.75 (dashed), and δ=0.99 (solid line), respectively. Figure 1 shows that the MPAR test has good power when δ gets large. This is as expected since a larger value of δ implies more informative factors. Consequently, when confidence sets of risk premia are to be constructed by inverting the MPAR test, wider confidence sets indicate that the corresponding factors are less informative, while narrower sets signal that the factors are stronger for asset pricing. Note that when δ=0.01 so the factor is close to being useless, Figure 1 shows that the corresponding power plot is close to the nominal 5%. We further compare the MPAR test with the FAR test in Kleibergen (2009). As stated above, in the special case that test assets and base assets coincide so R1t=R2t, the mimicking portfolio risk premia λG,cov coincides with the factor risk premia λG. In this scenario, both MPAR and FAR tests are applicable, and we plot their power curves in Figure 2. As shown in Figure 2, the power of the MPAR test is almost identical to that of the FAR test. Figure 2. View largeDownload slide Power plots of the MPAR test and the FAR test. Notes: This figure presents the power plots of the MPAR test (solid) and the FAR test (dashed) for H0:λG,cov=0 at the 5% level, with δ=0.01 in (a), δ=0.25 in (b), δ=0.50 in (c), δ=0.75 in (d), and δ=0.99 in (e). The d.g.p. is described in the main text, with T = 1000 and N1=N2=5. Figure 2. View largeDownload slide Power plots of the MPAR test and the FAR test. Notes: This figure presents the power plots of the MPAR test (solid) and the FAR test (dashed) for H0:λG,cov=0 at the 5% level, with δ=0.01 in (a), δ=0.25 in (b), δ=0.50 in (c), δ=0.75 in (d), and δ=0.99 in (e). The d.g.p. is described in the main text, with T = 1000 and N1=N2=5. 3 Re-evaluating the Pricing Performance of the Leverage Factor We illustrate the practical usage of the MPAR test by employing it to the leverage factor model proposed by Adrian, Etula, and Muir (2014). Specifically, Adrian, Etula, and Muir (2014) define the leverage level as the ratio of total assets over the difference between total assets and total liabilities of security broker–dealers. The resulting log change of the leverage level makes up their leverage factor. Adrian, Etula, and Muir (2014) find that the leverage factor prices the cross-section of various test portfolios remarkably well, outperforming the standard benchmarks in the literature. Their empirical study uses data for the leverage factor “LevFac” from 1968Q1–2009Q4, so T = 168 and k = 1. For illustrative purposes, we adopt the same data set.15 For base assets, we follow Adrian, Etula, and Muir (2014) who consider seven assets, so N2=7: the excess returns on the six Fama–French portfolios on size and book-to-market (“BL”, “BM”, “BH”, “SL”, “SM”, “SH”), plus the momentum factor “Mom.” These seven assets are widely acknowledged for their ability to span the asset space.16 To construct the mimicking portfolios, we project the leverage factor on these seven base assets. We obtained estimates for the coefficients (denoted by V^GR2V^R2R2−1 in Section 1) by projecting the leverage factor “LevFac” on the base assets (“BL”, “BM”, “BH”, “SL”, “SM”, “SH”, “Mom”) equal to: (−0.22,−0.10,0.56,−0.57,1.24,−0.43,0.43). These coefficients (after normalizing their sum to one) are almost identical to those in Adrian, Etula, and Muir (2014) for the reported weights of the mimicking portfolio.17 Adrian, Etula, and Muir (2014) show that the mimicking portfolio of the leverage factor performs well in various asset pricing tests, for example, it is associated with large R-squareds and low intercepts in cross-sectional regressions. Adrian, Etula, and Muir (2014) do not study the risk premium associated with the mimicking portfolio in cross-sectional regressions while we focus on it. Specifically, we are interested in the risk premium of the mimicking portfolio “LevMP” that results from projecting “LevFac” on the seven base assets. We use as test assets the commonly used 25 Fama–French portfolios on size and book-to-market (25 FF). Since both test assets and base assets are based on size and book-to-market, they are likely driven by the same underlying factors, as in the model setup in Equation (1). In addition, since T = 168 is relatively small, while our simulation study suggests that the size of the MPAR test is more reliable under small N1 and N2 for small T, we divide the 25 FF portfolios into five equal-sized groups (denoted by I–V in the first column of Table 3, by market capitalization), and use each group as test assets with N1=5.18 Table 3. Risk premium of the leverage factor (LevFac) and its mimicking portfolio (LevMP) LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I 22.35 4.02 2.30 (− ∞,−126.02]∪[13.96,∞) 4.11 4.11 1.74 (− ∞,−104.78]∪[16.82,∞) II 11.78 2.72 2.23 [−1.23,98.81] 2.73 2.72 1.62 (− ∞,−25.57]∪[−2.43,∞) III 16.59 2.57 1.79 (− ∞,−43.11]∪[1.67,∞) 2.32 2.38 1.52 (− ∞,−191.49]∪[3.08,∞) IV 8.32 1.31 1.25 (− ∞,∞) 1.01 1.21 1.03 (− ∞,∞) V 4.85 0.92 0.86 (− ∞,∞) 0.80 0.86 0.79 (− ∞,∞) LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I 22.35 4.02 2.30 (− ∞,−126.02]∪[13.96,∞) 4.11 4.11 1.74 (− ∞,−104.78]∪[16.82,∞) II 11.78 2.72 2.23 [−1.23,98.81] 2.73 2.72 1.62 (− ∞,−25.57]∪[−2.43,∞) III 16.59 2.57 1.79 (− ∞,−43.11]∪[1.67,∞) 2.32 2.38 1.52 (− ∞,−191.49]∪[3.08,∞) IV 8.32 1.31 1.25 (− ∞,∞) 1.01 1.21 1.03 (− ∞,∞) V 4.85 0.92 0.86 (− ∞,∞) 0.80 0.86 0.79 (− ∞,∞) Notes: LevFac stands for the leverage factor suggested in Adrian, Etula, and Muir (2014), while LevMP stands for the constructed mimicking portfolio of the leverage factor. Test assets are from the 25 Fama–French portfolios on size and book-to-market (25 FF) in the sample period of 1968Q1–2009Q4, and we divide them into five groups of test assets, I–V, so each group contains five portfolios. Three tests on risk premia are employed, namely, the conventional t-test for the FM methodology, the FAR test of Kleibergen (2009), and the proposed MPAR test in this paper. The KRS and JKZ corrections are also considered for the t-test, LevFac, and LevMP, respectively. Table 3. Risk premium of the leverage factor (LevFac) and its mimicking portfolio (LevMP) LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I 22.35 4.02 2.30 (− ∞,−126.02]∪[13.96,∞) 4.11 4.11 1.74 (− ∞,−104.78]∪[16.82,∞) II 11.78 2.72 2.23 [−1.23,98.81] 2.73 2.72 1.62 (− ∞,−25.57]∪[−2.43,∞) III 16.59 2.57 1.79 (− ∞,−43.11]∪[1.67,∞) 2.32 2.38 1.52 (− ∞,−191.49]∪[3.08,∞) IV 8.32 1.31 1.25 (− ∞,∞) 1.01 1.21 1.03 (− ∞,∞) V 4.85 0.92 0.86 (− ∞,∞) 0.80 0.86 0.79 (− ∞,∞) LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I 22.35 4.02 2.30 (− ∞,−126.02]∪[13.96,∞) 4.11 4.11 1.74 (− ∞,−104.78]∪[16.82,∞) II 11.78 2.72 2.23 [−1.23,98.81] 2.73 2.72 1.62 (− ∞,−25.57]∪[−2.43,∞) III 16.59 2.57 1.79 (− ∞,−43.11]∪[1.67,∞) 2.32 2.38 1.52 (− ∞,−191.49]∪[3.08,∞) IV 8.32 1.31 1.25 (− ∞,∞) 1.01 1.21 1.03 (− ∞,∞) V 4.85 0.92 0.86 (− ∞,∞) 0.80 0.86 0.79 (− ∞,∞) Notes: LevFac stands for the leverage factor suggested in Adrian, Etula, and Muir (2014), while LevMP stands for the constructed mimicking portfolio of the leverage factor. Test assets are from the 25 Fama–French portfolios on size and book-to-market (25 FF) in the sample period of 1968Q1–2009Q4, and we divide them into five groups of test assets, I–V, so each group contains five portfolios. Three tests on risk premia are employed, namely, the conventional t-test for the FM methodology, the FAR test of Kleibergen (2009), and the proposed MPAR test in this paper. The KRS and JKZ corrections are also considered for the t-test, LevFac, and LevMP, respectively. To gauge the statistical quality of the leverage factor, we conduct the Kleibergen and Paap (2006) rank test, which is commonly used as a diagnostic tool in the asset pricing literature. The null of the rank test is that the leverage factor beta has reduced rank, and the resulting p-values are found to be 0.09, 0.01, 0.26, 0.27, and 0.08 for I–V, respectively.19 Since most p-values exceed 5%, the leverage factor appears to be weakly correlated with asset returns and its beta is small. Consequently, the conventional t-test on the risk premium of the leverage factor is under doubt. Table 3 starts out with the FM two-pass methodology. When the leverage factor LevFac is tested using the FM procedure (in this scenario, its risk premium corresponds to that of mimicking portfolios with base assets equal test assets), Table 3 reports that the estimated risk premium is positive and the conventional FM t-statistics are significant in I, II, and III at the 5% significance level. Similarly, when the mimicking portfolio LevMP is tested in the FM procedure, Table 3 shows positive risk premium, associated with FM t-statistics comparable to the factor counterparts. These results thus appear to support the leverage factor for asset pricing. If the KRS and Jiang, Kan, and Zhan (2015) (JKZ) corrections are adopted to account for estimation uncertainty and model misspecification, however, t-statistics only remain significant in I and II under LevFac. It is now known that the FM methodology and its associated t-test are doubtful when the factor is associated with a small beta. As an alternative to the t-test for the factor risk premia, the FAR test is proposed by Kleibergen (2009). Unlike the FM t-test, the FAR test is size-correct since its limiting distribution does not depend on the quality of the observed factors. Table 3 presents the 95% confidence sets of the risk premium that results from inverting the FAR test. These sets are found to be substantially different from those obtained by inverting the FM t-test. Such differences put the quality of the leverage factor under doubt, see also Kleibergen (2009) for a further comparison of the FM t-test and the FAR test. Furthermore, in Appendix H, we present the p-value plots of the FAR test which show how we obtained the 95% FAR confidence sets reported in Table 3. The last column of Table 3 shows the 95% confidence sets that result from the MPAR test. In line with the conventional FM t-test, it shows rejection of the null hypothesis of a zero risk premium for LevMP for I and III; however, unlike the FM t-test, it does not reject a zero risk premium in II. In IV and V, identical to the FAR test but unlike the FM t-test, we find that no information about the risk premium is contained in the leverage factor or its mimicking portfolio. In Appendix H, we also present the p-value plots of the MPAR test, which helps to explain the 95% MPAR confidence sets reported in Table 3. Overall, Table 3 indicates that inference on the risk premium of the leverage factor may substantially change, when robust tests (FAR or MPAR) are employed.20 This is consistent with the fact that the leverage factor is only weakly correlated with asset returns and thus likely to be a weak proxy for the underlying factor(s). Similar findings are presented in Appendix I, where we adopt alternative test assets. 4 Conclusions We document the threats involved in using mimicking portfolios of non-traded factors in the FM two-pass procedure. When these factors have small betas, we show that their mimicking portfolios have betas that are spurious. These spurious betas induce non-standard behavior of the risk premia estimator so conventional t-tests on risk premia become unreliable. A rank test on beta is used in the literature to serve as a diagnostic tool for the quality of the factors. We, however, find that the outcome of the rank test needs to be taken with caution when using mimicking portfolios. This results from the estimation error in the mimicking portfolio which we have to account for in the rank test. It implies a more challenging expression for the covariance matrix estimator employed in the rank test. When we do not account for this estimation error, the rank test performs poorly. When we account for it, the rank test still has some issues when the covariance matrix becomes of large dimension but generally works well. Instead of gauging the quality of factors or mimicking portfolios, inference methods are available for analyzing risk premia that are reliable irrespective of the quality of the factors. These methods are robust in the sense that their limiting distributions do not depend on the quality of factors as reflected by the magnitude of the betas. To the best of our knowledge, the method we propose here is the first one which deals with mimicking portfolios. Robust methods do exist for tests on risk premia in the standard factor pricing setting, see Kleibergen (2009) and Beaulieu, Dufour, and Khalaf (2013). This clearly indicates the need for our developed methods for which the empirical relevance is further emphasized by our application to the risk premium on the leverage factor from Adrian, Etula, and Muir (2014). APPENDIX A. Proof of Theorem 1 Proof. Let us start with V^R1R2, V^R2R2, and V^R2G. Assumptions 1 and 2 imply that: V^R1R2→pβ1VFFβ2′V^R2R2→pβ2VFFβ2′+Ω22, where we used that V^RR→pβVFFβ′+Ω and Ω12=0. The convergence of V^R2G is a bit more tricky. Note that: Rt=ιNλ0+β[(δG¯t+ut)+λF]+vt=ιNλ0+βλF+βδG¯t+βut+vt. When δ is fixed and the number of elements of G equals the number of elements of F, so δ is a square invertible matrix: V^R2G→pβ2δVGG, so β˜^1→pβ1VFFβ2′(β2VFFβ2′+Ω22)−1β2δVGG[VGGδ′β2′(β2VFFβ2′+Ω22)−1β2δVGG]−1=β1VFFδ−1′VGG−1, and when Gt=Ft, δ=Ik, β˜^1→pβ1. When δ=d/T: TV^R2G→dβ2(dVGG+ψuG)+ψv2G, where 1T∑t=1T(G¯t⊗ut)→dvec (ψuG), 1T∑t=1T(G¯t⊗v2t)→dvec (ψv2G), so T−12β˜^1→dβ1VFFβ2′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G) [(β2(dVGG+ψuG)+ψv2G)′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G)]−1. ▪ B. Proof of Corollary 1 Proof. (λ˜^0λ˜^G)=[(ιN1⋮β˜^1)′(ιN1⋮β˜^1)]−1(ιN1⋮β˜^1)′μ^R1=((ι′N1Mβ˜^1ιN1)−1ι′N1Mβ˜^1μ^R1(β˜^1′MιN1β˜^1)−1β˜^1′MιN1μ^R1), where from Assumption 1: μ^R1=μR1+ψR1T+op(1/T), while the limiting behavior of β˜^1 is provided by Theorem 1. The strong factor case: β˜^1→pβ1VFFδ−1′VGG−1, μ^R1→pιN1λ0+β1λF, so λ˜^G=(β˜^1′MιN1β˜^1)−1β˜^1′MιN1μ^R1→p((β1VFFδ−1′VGG−1)′MιN1(β1VFFδ−1′VGG−1))−1(β1VFFδ−1′VGG−1)′MιN1(ιN1λ0+β1λF)=((β1VFFδ−1′VGG−1)′MιN1(β1VFFδ−1′VGG−1))−1(β1VFFδ−1′VGG−1)′MιN1β1λF=VGGδ′VFF−1λF. The weak factor case with δ=d/T: for convenience, we write T−12β˜^1→dΨβ1, where Ψβ1 is the non-standard distribution in Theorem 1, so λ˜^G=(β˜^1′MιN1β˜^1)−1β˜^1′MιN1μ^R1≃ 1T(Ψβ1′MιN1Ψβ1)−1Ψβ1′MιN1(μR1+ψR1T) which implies Tλ˜^G−T(Ψ′β1MιN1Ψβ1)−1Ψ′β1MιN1μR1→d(Ψ′β1MιN1Ψβ1)−1Ψ′β1MιN1ψR1. ▪ C. Proof of Theorem 2 Proof. To derive the limiting behavior of β^1, we first study V^R1R2, V^R2R2, and V^R2G: V^R1R2→pβ1VFFβ2′V^R2R2→pβ2VFFβ2′+Ω22. When δ is fixed and the number of elements of G equals the number of elements of F, so δ is a square invertible matrix: V^R2G→pβ2δVGG so β^1→pβ1VFFβ2′(β2VFFβ2′+Ω22)−1β2δVGG. When δ=d/T: TV^R2G→dβ2(dVGG+ψuG)+ψv2G so T12β^1→dβ1VFFβ2′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G) ▪ D. Asymptotic Variance of Beta Estimators for Rank Testing D.1. Joint Behavior of V^R2G, V^R1R2, and V^R2R2 Here, we present the joint behavior of V^R2G, V^R1R2, and V^R2R2, since they make the beta-estimator of mimicking portfolios. We use the following notation: 1T∑t=1T(G¯t⊗e1tG¯t⊗e2te1t⊗e2t−vec(Σ12)e2t⊗e2t−vec(Σ22)G¯t⊗G¯t−vec(VGG))→d(vec(ψe1G)vec(ψe2G)vec(ψe1e2)vec(ψe2e2)vec(ψGG)), V^R2G, V^R1R2, and V^R2R2 can be rewritten as follows: V^R2G=1T∑t=1TR¯2tG¯t′=1T∑t=1T(β2δG¯t+e2t)G¯t′=β2δVGG+β2δ(1T∑t=1TG¯tGt′−VGG)+1T∑t=1Te2tG¯t′ V^R1R2=1T∑t=1TR¯1tR¯2t′=1T∑t=1T(β1δG¯t+e1t)(β2δG¯t+e2t)′=β1δVGGδ′β2′+β1δ(1T∑t=1TG¯tG¯t′−VGG)δ′β2′+β1δ(1T∑t=1TG¯te2t′)+(1T∑t=1Te1tG¯t′)δ′β2′+1T∑t=1Te1te2t′−Σ12+Σ12 V^R2R2=1T∑t=1TR¯2tR¯2t′=1T∑t=1T(β2δG¯t+e2t)(β2δG¯t+e2t)′=β2δVGGδ′β2′+β2δ(1T∑t=1TG¯tG¯t′−VGG)δ′β2′+β2δ(1T∑t=1TG¯te2t′)+(1T∑t=1Te2tG¯t′)δ′β2′+1T∑t=1Te2te2t′−Σ22+Σ22 The convergence of V^R2G, V^R2R1, and V^R2R2 is thus characterized by: T(V^R2G−β2δVGG)→dβ2δψGG+ψe2Gdenoted by UR2GT(V^R1R2−β1δVGGδ′β2′−Σ12)→dβ1δψGGδ′β2′+β1δψe2G′+ψe1Gδ′β2′+ψe1e2denoted by UR1R2T(V^R2R2−β2δVGGδ′β2′−Σ22)→dβ2δψGGδ′β2′+β2δψe2G′+ψe2Gδ′β2′+ψe2e2denoted by UR2R2 and in our simulation setup, their variances read: vec(UR2G)∼N(0,VGG⊗(VR2R2+β2δVGGδ′β2′)≡W1,1) vec(UR1R2)∼N(0,VR2R2⊗(VR1R1+VR1R2VR2R2−1VR2R1)≡W2,2) vec(UR2R2)∼N(0,2VR2R2⊗VR2R2≡W3,3) In addition, their covariances read: cov(vec(UR1R2), vec(UR2G))=2β2δVGG⊗β1δVGGδ′β2′+KN1N2(β1δVGG⊗Σ22)+β2δVGG⊗Σ12≡W2,1=W1,2′ cov(vec(UR2R2), vec(UR2G))=2β2δVGG⊗β2δVGGδ′β2′+(IN22+KN2N2)(β2δVGG⊗Σ22)≡W3,1=W1,3′ cov(vec(UR1R2), vec(UR2R2))=2VR2R2⊗VR1R2≡W2,3=W3,2′ To summarize: Tvec(V^R2G−VR2GV^R1R2−VR1R2V^R2R2−VR1R2)=vec(UR2GUR1R2UR2R2)→dN(0,WU),WU=(W1,1W1,2W1,3W2,1W2,2W2,3W3,1W3,2W3,3) This result will be useful, when we derive asymptotic variance of the beta-estimator with mimicking portfolios. Note that WU can be consistently estimated by data, so there exists W^U→pWU. Specifically, W^1,1=V^GG⊗(V^R2R2+B^2V^GGB^2′) W^2,2=V^R2R2⊗(V^R1R1+V^R1R2V^R2R2−1V^R2R1) W^3,3=2V^R2R2⊗V^R2R2 W^2,1=W^1,2′=2B^2V^GG⊗B^1V^GGB^2′+KN1N2(B^1V^GG⊗Σ^22)+B^2V^GG⊗Σ^12 W^3,1=W^1,3′=2B^2V^GG⊗B^2V^GGB^2′+(IN22+KN2N2)(B^2V^GG⊗Σ^22) W^2,3=W^3,2′=2V^R2R2⊗V^R1R2 D.2. Beta Specification with Factors Let Gt be the observed factors, R1t be test assets. Consider the estimand VR1GVGG−1 and the estimator V^R1GV^GG−1 Then: Tvec(V^R1GV^GG−1−VR1GVGG−1)→dN(0,VGG−1⊗(VR1R1−VR1GVGG−1VGR1)) This result allows us to conduct the rank test. That is, the estimator V^R1GV^GG−1 and its estimated variance V^GG−1⊗(V^R1R1−V^R1GV^GG−1V^GR1) are used for the rank test. D.3. Beta Specification with Mimicking Portfolios Let Gt be the observed factors, R1t be test assets. Define β˜1=VR1R2VR2R2−1VR2G(VGR2VR2R2−1VR2G)−1 and the estimator β˜^1=V^R1R2V^R2R2−1V^R2G(V^GR2V^R2R2−1V^R2G)−1 Using that V^R1R2=VR1R2+1TUR1R2, V^R2R2=VR2R2+1TUR2R2, V^R2G=VR2G+1TUR2G: V^R1R2V^R2R2−1V^R2G=[VR1R2+1TUR1R2][VR2R2+1TUR2R2]−1[VR2G+1TUR2G]≈VR1R2VR2R2−1VR2G +1T[UR1R2VR2R2−1VR2G+VR1R2VR2R2−1UR2G−VR1R2VR2R2−1UR2R2VR2R2−1VR2G] and similarly, V^GR2V^R2R2−1V^R2G=[VR2G+1TUR2G]′[VR2R2+1TUR2R2]−1[VR2G+1TUR2G]≈VGR2VR2R2−1VR2G +1T[UR2G′VR2R2−1VR2G+VR2G′VR2R2−1UR2G−VR2G′VR2R2−1UR2R2VR2R2−1VR2G]. Use the expansion of the inverse, β˜^1 is rewritten as: β˜^1≈β˜1 +1T[UR1R2VR2R2−1VR2G+VR1R2VR2R2−1UR2G−VR1R2VR2R2−1UR2R2VR2R2−1VR2G](VGR2VR2R2−1VR2G)−1 −1Tβ˜1[UR2G′VR2R2−1VR2G+VR2G′VR2R2−1UR2G−VR2G′VR2R2−1UR2R2VR2R2−1VR2G](VGR2VR2R2−1VR2G)−1 So Tvec(β˜^1−β˜1)≈[(VGR2VR2R2−1VR2G)−1⊗(VR1R2VR2R2−1−β˜1VGR2VR2R2−1)−(VGR2VR2R2−1VR2G)−1 VGR2VR2R2−1⊗β˜1·KN2K]vec(UR2G) +[(VGR2VR2R2−1VR2G)−1VGR2VR2R2−1⊗IN1]vec(UR1R2) +[(VGR2VR2R2−1VR2G)−1VGR2VR2R2−1⊗(β˜1VGR2VR2R2−1−VR1R2VR2R2−1)]vec(UR2R2) where KN2K is a commutation matrix such that vec(UR2G′)=KN2Kvec(UR2G). Consequently, in order to further derive variance of β˜^1, we use the joint behavior of UR2G, UR1R2, and UR2R2 (which has been derived earlier, see the detail of WU and W^U in the D1 sub-section): vec(UR2GUR1R2UR2R2)→dN(0,WU),WU=(W1,1W1,2W1,3W2,1W2,2W2,3W3,1W3,2W3,3) Combining all these pieces, the asymptotic variance of Tvec(β˜^1−β˜1) reads: vβ˜^1WUv′β˜^1 where vβ˜^1 is defined as ([(VGR2VR2R2−1VR2G)−1⊗(VR1R2VR2R2−1−β˜1VGR2VR2R2−1)−(VGR2VR2R2−1VR2G)−1VGR2VR2R2−1⊗β˜1·KN2K]′[(VGR2VR2R2−1VR2G)−1VGR2VR2R2−1⊗IN1]′[(VGR2VR2R2−1VR2G)−1VGR2VR2R2−1⊗(β˜1VGR2VR2R2−1−VR1R2VR2R2−1)]′)′ with v^β˜^1 equals ([(V^GR2V^R2R2−1V^R2G)−1⊗(V^R1R2V^R2R2−1−β˜^1V^GR2V^R2R2−1)−(V^GR2V^R2R2−1V^R2G)−1V^GR2V^R2R2−1⊗β˜^1·KN2K]′[(V^GR2V^R2R2−1V^R2G)−1V^GR2V^R2R2−1⊗IN1]′[(V^GR2V^R2R2−1V^R2G)−1V^GR2V^R2R2−1⊗(β˜^1V^GR2V^R2R2−1−V^R1R2V^R2R2−1)]′)′ This result allows us to conduct the rank test. That is, the estimator β˜^1 and its estimated variance v^β˜^1W^Uv^β˜^1′ are used for the rank test. D.4. Covariance Specification with Factors Let Gt be the observed factors, R1t be test assets. Consider the estimand VR1G and the estimator V^R1G Then: Tvec(V^R1G−VR1G)→dN(0,VGG⊗(VR1R1+VR1GVGG−1VGR1)). This result allows us to conduct the rank test. That is, the estimator V^R1G and its estimated variance V^GG⊗(V^R1R1+V^R1GV^GG−1V^GR1) are used for the rank test. D.5. Covariance Specification with Mimicking Portfolios Consider the estimator β^1=V^R1R2V^R2R2−1V^R2G Using that V^R1R2=VR1R2+1TUR1R2, V^R2R2=VR2R2+1TUR2R2, V^R2G=VR2G+1TUR2G, we can specify the above estimator as β^1=V^R1R2V^R2R2−1V^R2G=[VR1R2+1TUR1R2][VR2R2+1TUR2R2]−1[VR2G+1TUR2G]=VR1R2VR2R2−1VR2G +1T[UR1R2VR2R2−1VR2G+VR1R2VR2R2−1UR2G−VR1R2VR2R2−1UR2R2VR2R2−1VR2G]+op(T−1/2). Here, we used an expansion of the inverse. Hence, T(β^1−VR1R2VR2R2−1VR2G)≈VR1R2VR2R2−1UR2G+UR1R2VR2R2−1VR2G−VR1R2VR2R2−1UR2R2VR2R2−1VR2G Consequently, in order to further derive variance of β^1, we use the joint behavior of UR2G, UR1R2, and UR2R2 (which has been derived earlier, see the detail of WU and W^U in the D1 sub-section): vec(UR2GUR1R2UR2R2)→dN(0,WU),WU=(W1,1W1,2W1,3W2,1W2,2W2,3W3,1W3,2W3,3) Combining all these pieces, the asymptotic variance of Tvec(β^1−VR1R2VR2R2−1VR2G) reads: vβ^1WUv′β^1, where vβ^1 is defined as (Im⊗VR1R2VR2R2−1 VGR2VR2R2−1⊗IN1 −VGR2VR2R2−1⊗VR1R2VR2R2−1) with v^β^1 equals (Im⊗V^R1R2V^R2R2−1 V^GR2V^R2R2−1⊗IN1 −V^GR2V^R2R2−1⊗V^R1R2V^R2R2−1) This result allows us to conduct the rank test. That is, the estimator β^1 and its estimated variance v^β^1W^Uv^β^1′ are used for the rank test. E. Sensitivity to the Strength of the Factor Structure Instead of calibrating Ω to the estimated Ω^ as described in the main text, we also consider two alternatives in our simulation experiments: Ω=0.04Ω^ and Ω=25Ω^. The strength of the factor structure alters when the magnitude of Ω changes with Ω=0.04Ω^ having a very strong factor structure and Ω=25Ω^ a weak factor structure. The other settings remain unchanged in the simulation. Our purpose is to re-examine the result reported in Table 1, as the strength of the factor structure changes. Tables A.1 and A.2 present the updated results, for Ω=0.04Ω^ and Ω=25Ω^, respectively. It is found that the strength of the factor structure does not alter the two main findings conveyed in Table 1: (i) for rank testing, factors seem to perform better than mimicking portfolios; (ii) improperly treating mimicking portfolios as factors may yield to spurious rank test outcomes that favor poor factors. Table A.1. Rejection frequencies of the rank test by Monte Carlo ( Ω=0.04Ω^) Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.063 0.110 0.066 0.116 1.000 1.000 0.062 0.110 0.064 0.116 1.000 1.000 2. 0.053 0.103 0.016 0.035 0.956 0.960 0.051 0.102 0.016 0.034 0.955 0.960 3. 0.055 0.111 0.005 0.014 0.915 0.931 0.051 0.103 0.005 0.015 0.914 0.931 4. 0.052 0.100 0.006 0.009 0.911 0.921 0.047 0.095 0.005 0.009 0.908 0.920 5. 0.045 0.091 0.001 0.004 0.884 0.902 0.042 0.086 0.001 0.004 0.881 0.900 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.063 0.110 0.066 0.116 1.000 1.000 0.062 0.110 0.064 0.116 1.000 1.000 2. 0.053 0.103 0.016 0.035 0.956 0.960 0.051 0.102 0.016 0.034 0.955 0.960 3. 0.055 0.111 0.005 0.014 0.915 0.931 0.051 0.103 0.005 0.015 0.914 0.931 4. 0.052 0.100 0.006 0.009 0.911 0.921 0.047 0.095 0.005 0.009 0.908 0.920 5. 0.045 0.091 0.001 0.004 0.884 0.902 0.042 0.086 0.001 0.004 0.881 0.900 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1, Ω=0.04Ω^. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. Table A.1. Rejection frequencies of the rank test by Monte Carlo ( Ω=0.04Ω^) Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.063 0.110 0.066 0.116 1.000 1.000 0.062 0.110 0.064 0.116 1.000 1.000 2. 0.053 0.103 0.016 0.035 0.956 0.960 0.051 0.102 0.016 0.034 0.955 0.960 3. 0.055 0.111 0.005 0.014 0.915 0.931 0.051 0.103 0.005 0.015 0.914 0.931 4. 0.052 0.100 0.006 0.009 0.911 0.921 0.047 0.095 0.005 0.009 0.908 0.920 5. 0.045 0.091 0.001 0.004 0.884 0.902 0.042 0.086 0.001 0.004 0.881 0.900 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.063 0.110 0.066 0.116 1.000 1.000 0.062 0.110 0.064 0.116 1.000 1.000 2. 0.053 0.103 0.016 0.035 0.956 0.960 0.051 0.102 0.016 0.034 0.955 0.960 3. 0.055 0.111 0.005 0.014 0.915 0.931 0.051 0.103 0.005 0.015 0.914 0.931 4. 0.052 0.100 0.006 0.009 0.911 0.921 0.047 0.095 0.005 0.009 0.908 0.920 5. 0.045 0.091 0.001 0.004 0.884 0.902 0.042 0.086 0.001 0.004 0.881 0.900 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1, Ω=0.04Ω^. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. Table A.2. Rejection frequencies of the rank test by Monte Carlo ( Ω=25Ω^) Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 0.465 0.583 0.429 0.556 1.000 1.000 0.319 0.486 0.426 0.554 2. 1.000 1.000 0.663 0.760 0.636 0.742 1.000 1.000 0.489 0.648 0.629 0.742 3. 1.000 1.000 0.688 0.782 0.666 0.761 1.000 1.000 0.495 0.650 0.656 0.758 4. 1.000 1.000 0.878 0.930 0.856 0.914 1.000 1.000 0.642 0.793 0.848 0.911 5. 1.000 1.000 0.922 0.953 0.902 0.936 1.000 1.000 0.635 0.805 0.896 0.935 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 0.465 0.583 0.429 0.556 1.000 1.000 0.319 0.486 0.426 0.554 2. 1.000 1.000 0.663 0.760 0.636 0.742 1.000 1.000 0.489 0.648 0.629 0.742 3. 1.000 1.000 0.688 0.782 0.666 0.761 1.000 1.000 0.495 0.650 0.656 0.758 4. 1.000 1.000 0.878 0.930 0.856 0.914 1.000 1.000 0.642 0.793 0.848 0.911 5. 1.000 1.000 0.922 0.953 0.902 0.936 1.000 1.000 0.635 0.805 0.896 0.935 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.055 0.105 0.003 0.014 0.429 0.556 0.054 0.105 0.004 0.015 0.426 0.554 2. 0.055 0.100 0.001 0.004 0.335 0.443 0.051 0.099 0.000 0.003 0.330 0.439 3. 0.057 0.107 0.000 0.000 0.252 0.344 0.055 0.103 0.000 0.000 0.247 0.337 4. 0.055 0.098 0.001 0.001 0.275 0.367 0.050 0.092 0.001 0.001 0.266 0.358 5. 0.046 0.094 0.000 0.000 0.237 0.328 0.044 0.086 0.000 0.000 0.224 0.320 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.055 0.105 0.003 0.014 0.429 0.556 0.054 0.105 0.004 0.015 0.426 0.554 2. 0.055 0.100 0.001 0.004 0.335 0.443 0.051 0.099 0.000 0.003 0.330 0.439 3. 0.057 0.107 0.000 0.000 0.252 0.344 0.055 0.103 0.000 0.000 0.247 0.337 4. 0.055 0.098 0.001 0.001 0.275 0.367 0.050 0.092 0.001 0.001 0.266 0.358 5. 0.046 0.094 0.000 0.000 0.237 0.328 0.044 0.086 0.000 0.000 0.224 0.320 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1, Ω=25Ω^. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. Table A.2. Rejection frequencies of the rank test by Monte Carlo ( Ω=25Ω^) Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 0.465 0.583 0.429 0.556 1.000 1.000 0.319 0.486 0.426 0.554 2. 1.000 1.000 0.663 0.760 0.636 0.742 1.000 1.000 0.489 0.648 0.629 0.742 3. 1.000 1.000 0.688 0.782 0.666 0.761 1.000 1.000 0.495 0.650 0.656 0.758 4. 1.000 1.000 0.878 0.930 0.856 0.914 1.000 1.000 0.642 0.793 0.848 0.911 5. 1.000 1.000 0.922 0.953 0.902 0.936 1.000 1.000 0.635 0.805 0.896 0.935 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 0.465 0.583 0.429 0.556 1.000 1.000 0.319 0.486 0.426 0.554 2. 1.000 1.000 0.663 0.760 0.636 0.742 1.000 1.000 0.489 0.648 0.629 0.742 3. 1.000 1.000 0.688 0.782 0.666 0.761 1.000 1.000 0.495 0.650 0.656 0.758 4. 1.000 1.000 0.878 0.930 0.856 0.914 1.000 1.000 0.642 0.793 0.848 0.911 5. 1.000 1.000 0.922 0.953 0.902 0.936 1.000 1.000 0.635 0.805 0.896 0.935 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.055 0.105 0.003 0.014 0.429 0.556 0.054 0.105 0.004 0.015 0.426 0.554 2. 0.055 0.100 0.001 0.004 0.335 0.443 0.051 0.099 0.000 0.003 0.330 0.439 3. 0.057 0.107 0.000 0.000 0.252 0.344 0.055 0.103 0.000 0.000 0.247 0.337 4. 0.055 0.098 0.001 0.001 0.275 0.367 0.050 0.092 0.001 0.001 0.266 0.358 5. 0.046 0.094 0.000 0.000 0.237 0.328 0.044 0.086 0.000 0.000 0.224 0.320 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.055 0.105 0.003 0.014 0.429 0.556 0.054 0.105 0.004 0.015 0.426 0.554 2. 0.055 0.100 0.001 0.004 0.335 0.443 0.051 0.099 0.000 0.003 0.330 0.439 3. 0.057 0.107 0.000 0.000 0.252 0.344 0.055 0.103 0.000 0.000 0.247 0.337 4. 0.055 0.098 0.001 0.001 0.275 0.367 0.050 0.092 0.001 0.001 0.266 0.358 5. 0.046 0.094 0.000 0.000 0.237 0.328 0.044 0.086 0.000 0.000 0.224 0.320 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1, Ω=25Ω^. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. F. Proof of Theorem 4 Proof. For R¯1, by the central limit theorem, we have: T(R¯1−Γβ2δλG,cov)→dψ1∼N(0,VR1R1) B^2 results from linear regression of R2t on Gt, so: Tvec(B^2−β2δ)→dvec(ψ2)∼N(0,VGG−1⊗Σ22) where Σ22 is the covariance matrix of residuals. Similarly for Γ^ that results from linear regression of R1t on R2t: Tvec(Γ^−Γ)→dvec(ψ3)∼N(0,VR2R2−1⊗(VR1R1−VR1R2VR2R2−1VR2R1)). Note that ψ1 and ψ2 are independent (see Kleibergen, 2009). By the same logic, ψ1 and ψ3 are also independent. So what remains to derive is cov(vec(ψ3),vec(ψ2)). For ψ3, using that V^R1R2=VR1R2+1TUR1R2, V^R2R2=VR2R2+1TUR2R2, Γ^=V^R1R2V^R2R2−1=(VR1R2+1TUR1R2)(VR2R2+1TUR2R2)−1=VR1R2VR2R2−1+1T[UR1R2VR2R2−1−VR1R2VR2R2−1UR2R2VR2R2−1]+op(1T) so, with Γ=VR1R2VR2R2−1, Tvec(Γ^−Γ)→d(VR2R2−1⊗IN1−1)vec(UR1R2)−(VR2R2−1⊗VR1R2VR2R2−1)vec(UR2R2)=vec(ψ3). where as in Appendix D,21 UR1R2=β1δψGGδ′β2′+β1δψe2G′+ψe1Gδ′β2′+ψe1e2UR2R2=β2δψGGδ′β2′+β2δψe2G′+ψe2Gδ′β2′+ψe2e2 For ψ2: ψ2=ψe2GVGG−1. So, cov(vec(UR1R2),vec(ψ2))=K(N1−1)N2(β1δ⊗Σ22)+β2δ⊗Σ12cov(vec(UR2R2),vec(ψ2))=(IN22+KN2N2)(β2δ⊗Σ22) which implies that cov(vec(ψ3),vec(ψ2)) equals K(N1−1)N2(β1δ⊗VR2R2−1Σ22)+VR2R2−1β2δ⊗Σ12−(VR2R2−1⊗VR1R2VR2R2−1)(IN22+KN2N2)(β2δ⊗Σ22)≡C ▪ G. Proof of Corollary 2 Proof. Rewrite R¯1−Γ^B^2λG,cov as follows: R¯1−Γ^B^2λG,cov=R¯1−Γβ2δλG,cov−(Γ^B^2−Γβ2δ)λG,cov=R¯1−Γβ2δλG,cov−(Γ^−Γ)β2δλG,cov−Γ^(B^2−β2δ)λG,cov=R¯1−Γβ2δλG,cov−(Γ^−Γ)β2δλG,cov−Γ(B^2−β2δ)λG,cov−(Γ^−Γ)(B^2−β2δ)λG,cov Applying Theorem 4, we get T(R¯1−Γ^B^2λG,cov)→dψ1−Γψ2λG,cov−ψ3β2δλG,cov. Since Γ has a full-rank value, the last element of the above expression converges to zero for every value of δ since T(Γ^−Γ)→dψ3. For the analytical expression of the resulting variance, we use the elements in W from Theorem 4. Specifically, for the three terms in ψ1−Γψ2λG,cov−ψ3β2δλG,cov, their variances read: VR1R1, λ′G,covVGG−1λG,cov⊗ΓΣ22Γ′ and (β2δλG,cov)′VR2R2−1(β2δλG,cov)⊗(VR1R1−VR1R2VR2R2−1VR2R1). In addition, since ψ2 and ψ3 are not independent, we also consider the resulting covariance: cov(vec(ψ3β2δλG,cov), vec(Γψ2λG,cov))=[(β2δλG,cov)′⊗IN1−1]C(λG,cov⊗Γ′) So the asymptotic variance of T(R¯1−Γ^B^2λG,cov) results from combining the variance and covariance terms above: VR1R1+λ′G,covVGG−1λG,cov⊗ΓΣ22Γ′+(β2δλG,cov)′VR2R2−1(β2δλG,cov)⊗(VR1R1−VR1R2VR2R2−1VR2R1)−{[(β2δλG,cov)′⊗IN1−1]C(λG,cov⊗Γ′)}−{[(β2δλG,cov)′⊗IN1−1]C(λG,cov⊗Γ′)}′ ▪ H. p-Value of FAR and MPAR tests for Table 3 and Figures A.1–A.5 Figure A.1. View largeDownload slide p-Value of FAR and MPAR tests for Table 3, I. Notes: This figure presents the p-values (solid) of the FAR (left) and MPAR (right) tests for testing risk premium equals the corresponding value on the x-axis, using test assets in I. The 5% line is also plotted for benchmark (dotted). Figure A.1. View largeDownload slide p-Value of FAR and MPAR tests for Table 3, I. Notes: This figure presents the p-values (solid) of the FAR (left) and MPAR (right) tests for testing risk premium equals the corresponding value on the x-axis, using test assets in I. The 5% line is also plotted for benchmark (dotted). Figure A.2. View largeDownload slide p-Value of FAR and MPAR tests for Table 3, II. Notes: This figure presents the p-values (solid) of the FAR (left) and MPAR (right) tests for testing risk premium equals the corresponding value on the x-axis, using test assets in II. The 5% line is also plotted for benchmark (dotted). Figure A.2. View largeDownload slide p-Value of FAR and MPAR tests for Table 3, II. Notes: This figure presents the p-values (solid) of the FAR (left) and MPAR (right) tests for testing risk premium equals the corresponding value on the x-axis, using test assets in II. The 5% line is also plotted for benchmark (dotted). Figure A.3. View largeDownload slide p-Value of FAR and MPAR tests for Table 3, III. Notes: This figure presents the p-values (solid) of the FAR (left) and MPAR (right) tests for testing risk premium equals the corresponding value on the x-axis, using test assets in III. The 5% line is also plotted for benchmark (dotted). Figure A.3. View largeDownload slide p-Value of FAR and MPAR tests for Table 3, III. Notes: This figure presents the p-values (solid) of the FAR (left) and MPAR (right) tests for testing risk premium equals the corresponding value on the x-axis, using test assets in III. The 5% line is also plotted for benchmark (dotted). Figure A.4. View largeDownload slide p-Value of FAR and MPAR tests for Table 3, IV. Notes: This figure presents the p-values (solid) of the FAR (left) and MPAR (right) tests for testing risk premium equals the corresponding value on the x-axis, using test assets in IV. The 5% line is also plotted for benchmark (dotted). Figure A.4. View largeDownload slide p-Value of FAR and MPAR tests for Table 3, IV. Notes: This figure presents the p-values (solid) of the FAR (left) and MPAR (right) tests for testing risk premium equals the corresponding value on the x-axis, using test assets in IV. The 5% line is also plotted for benchmark (dotted). Figure A.5. View largeDownload slide p-Value of FAR and MPAR tests for Table 3, V. Notes: This figure presents the p-values (solid) of the FAR (left) and MPAR (right) tests for testing risk premium equals the corresponding value on the x-axis, using test assets in V. The 5% line is also plotted for benchmark (dotted). Figure A.5. View largeDownload slide p-Value of FAR and MPAR tests for Table 3, V. Notes: This figure presents the p-values (solid) of the FAR (left) and MPAR (right) tests for testing risk premium equals the corresponding value on the x-axis, using test assets in V. The 5% line is also plotted for benchmark (dotted). I. Risk Premium of the Leverage Factor and Its Mimicking Portfolio: Alternative Test Assets Table A.3 updates Table 3 in the main text. The difference lies in the involved test assets. In particular, Groups I′–V′ in Table A.3 are constructed by BE/ME (ratio, low-to-high), while Groups I–V for Table 3 are constructed by ME (market cap, small-to-big), using 25 FF portfolios. Neither the FAR test nor the MPAR test rejects zero risk premium in I′–V′ in Table A.3. Therefore, we find little information on the risk premium from the leverage factor. Table A.3. Risk premium of the leverage factor and its mimicking portfolio: alternative test assets LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I′ −45.58 −1.32 −0.53 (− ∞,∞) −3.70 −1.12 −0.58 (− ∞,∞) II′ 5.73 0.69 0.50 (− ∞,∞) 1.76 0.88 0.45 [−42.80,∞) III′ 19.46 2.14 1.34 (− ∞,−17.94]∪[−5.02,∞) 3.96 2.12 1.12 [−13.92,∞) IV′ 32.65 2.29 1.13 (− ∞,∞) 5.48 2.60 1.27 (− ∞,∞) V′ 14.83 1.91 1.51 (− ∞,∞) 4.53 2.20 1.00 (− ∞,−7.53]∪[−1.87,∞) LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I′ −45.58 −1.32 −0.53 (− ∞,∞) −3.70 −1.12 −0.58 (− ∞,∞) II′ 5.73 0.69 0.50 (− ∞,∞) 1.76 0.88 0.45 [−42.80,∞) III′ 19.46 2.14 1.34 (− ∞,−17.94]∪[−5.02,∞) 3.96 2.12 1.12 [−13.92,∞) IV′ 32.65 2.29 1.13 (− ∞,∞) 5.48 2.60 1.27 (− ∞,∞) V′ 14.83 1.91 1.51 (− ∞,∞) 4.53 2.20 1.00 (− ∞,−7.53]∪[−1.87,∞) Notes: LevFac stands for the leverage factor suggested in Adrian, Etula, and Muir (2014), while LevMP stands for the constructed mimicking portfolio of the leverage factor. Test assets are from 25 Fama–French portfolios on size and book-to-market (25 FF) in the sample period of 1968Q1–2009Q4, and we divide them into five groups, I′–V′ (by BE/ME, low to high), so each group contains five portfolios. Three tests on risk premia are employed, namely, the conventional t-test for the FM methodology, the FAR test of Kleibergen (2009), and the proposed MPAR test in this paper. The KRS and JKZ corrections are also considered for the t-test, LevFac, and LevMP, respectively. Table A.3. Risk premium of the leverage factor and its mimicking portfolio: alternative test assets LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I′ −45.58 −1.32 −0.53 (− ∞,∞) −3.70 −1.12 −0.58 (− ∞,∞) II′ 5.73 0.69 0.50 (− ∞,∞) 1.76 0.88 0.45 [−42.80,∞) III′ 19.46 2.14 1.34 (− ∞,−17.94]∪[−5.02,∞) 3.96 2.12 1.12 [−13.92,∞) IV′ 32.65 2.29 1.13 (− ∞,∞) 5.48 2.60 1.27 (− ∞,∞) V′ 14.83 1.91 1.51 (− ∞,∞) 4.53 2.20 1.00 (− ∞,−7.53]∪[−1.87,∞) LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I′ −45.58 −1.32 −0.53 (− ∞,∞) −3.70 −1.12 −0.58 (− ∞,∞) II′ 5.73 0.69 0.50 (− ∞,∞) 1.76 0.88 0.45 [−42.80,∞) III′ 19.46 2.14 1.34 (− ∞,−17.94]∪[−5.02,∞) 3.96 2.12 1.12 [−13.92,∞) IV′ 32.65 2.29 1.13 (− ∞,∞) 5.48 2.60 1.27 (− ∞,∞) V′ 14.83 1.91 1.51 (− ∞,∞) 4.53 2.20 1.00 (− ∞,−7.53]∪[−1.87,∞) Notes: LevFac stands for the leverage factor suggested in Adrian, Etula, and Muir (2014), while LevMP stands for the constructed mimicking portfolio of the leverage factor. Test assets are from 25 Fama–French portfolios on size and book-to-market (25 FF) in the sample period of 1968Q1–2009Q4, and we divide them into five groups, I′–V′ (by BE/ME, low to high), so each group contains five portfolios. Three tests on risk premia are employed, namely, the conventional t-test for the FM methodology, the FAR test of Kleibergen (2009), and the proposed MPAR test in this paper. The KRS and JKZ corrections are also considered for the t-test, LevFac, and LevMP, respectively. Footnotes 1 Such as extending the available data sets (Ang et al., 2006), providing extra restrictions for testing purposes (Huberman, Kandel, and Stambaugh, 1987). 2 We thank the Associate Editor for indicating that the estimation errors associated with the weights of mimicking portfolios also invalidate the conventional GRS test for testing asset pricing models when using mimicking portfolios. 3 We acknowledge that in practice test assets and base assets need not be in the same frequency or in the same time period. For convenience, we focus on the same frequency and same time period setting, although the message of this paper can be similarly extended to more generalized settings. 4 Note that λG is not always well-defined, if βδ does not span the column space of β. This occurs when Gt is useless so δ = 0 or when δ is drifting to zero. We focus on both of these cases. Similarly, λG may not be well-defined when k > m so misspecification of factor pricing occurs. The statistic that we propose lateron indicates if such misspecification is present alongside deviations of the risk premia from the hypothesized one, see also Kleibergen and Zhan (2017). 5 Ω12=0 is necessary to make the beta of the mimicking portfolios a nonsingular transformation of the beta of the factors. When test assets R1t are base assets R2t, Assumption 2 is dropped. 6 With every sample element of β˜^1 converging to its population counterpart, it results that β˜^1→pβ1VFFβ2′VR2R2−1VR2G(VGR2VR2R2−1VR2G)−1+Ω12VR2R2−1VR2G(VGR2VR2R2−1VR2G)−1 so it only provides a consistent estimator for the column space of β1 when Ω12=0. 7 The sample period used for calibration is January 1927–December 2015. The data are downloaded from French’s online data library, “http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html (accessed 28 February 2018).” 8 There are two reasons that may help explain the conservative performance. First, the beta-estimator of the mimicking portfolios (such as β˜^1 in Theorem 1) may have a non-standard distribution, so first-order asymptotics do not approximate its variance well, which further jeopardizes the rank test. Second, as N1 and N2 increase, larger sample sizes are generally needed to estimate the covariance matrix of asset returns, a component that appears when betas are computed using mimicking portfolios. 9 See Appendix E for additional simulation results, where we alter the strength of the factor structure. 10 If VGG is normalized to Im, then λG,cov is the risk premium of the mimicking portfolios in the covariance specification. Furthermore, λG,cov could also be viewed as the scaled risk premium of mimicking portfolios in the beta specification: since E(R1t)=ιN1λ0+VR1R2VR2R2−1VR2G(VGR2VR2R2−1VR2G)−1(VGR2VR2R2−1VR2G)VGG−1λG,cov, (VGR2VR2R2−1VR2G)VGG−1λG,cov is the risk premium in the beta specification. If the projection error during the construction of mimicking portfolios is negligible, that is, VGR2VR2R2−1VR2G≈VGG, then λG,cov approximately equals the risk premium of mimicking portfolios in the beta specification. Finally, if Equation (18) does not apply since the hypothesized value of λG,cov differs from the true one or when there is misspecification, our proposed test will detect this. 11 Our results are invariant with respect to the asset subtracted. 12 Kleibergen (2009) also proposes several other tests (e.g., a score test). Our paper focuses on extending the FAR statistic to the MPAR statistic, because extending other tests appears, because of the structure of the covariance matrix in Theorem 4, difficult. 13 When we use a sample size of 1,000,000, the MPAR test is size correct in all instances. 14 Note that β1λF=VR1R2VR2R2−1VR2GVGG−1λG,cov, so λF=(β1′β1)−1β1′VR1R2VR2R2−1VR2GVGG−1λG,cov is used for d.g.p., and we use 20,000 Monte-Carlo replications for the power plots. 15 We thank the authors for making the data publicly available. 16 Monthly data for these assets are available at French’s online data library, which is then compounded to construct the quarterly data. Excess returns result from the difference of raw returns and risk-free returns. 17 Unlike Adrian, Etula, and Muir (2014), we do not normalize these coefficients to make their sum equal to 1. 18 Groups I–V are built in the order of 25 FF, that is, the first five portfolios of 25 FF make Group I, … , the last five portfolios of 25 FF make Group V. Therefore, I–V differ in market cap. 19 In accordance with the FAR and MPAR tests, we similarly remove the last asset and take all other asset returns in deviation from the return on the last asset when implementing the rank test. This is one way of conducting a rank test on (ιN ⋮ β) which is the regressor matrix in the second pass of the FM procedure. 20 Note that the factor risk premia defined in Equations (2) and (4) and the mimicking portfolio risk premia defined in Equation (18) are not identical in this application, since base assets and test assets differ. 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For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Financial Econometrics Oxford University Press http://www.deepdyve.com/lp/oxford-university-press/identification-robust-inference-on-risk-premia-of-mimicking-portfolios-N277jDPcsN

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Publisher: Oxford University Press
ISSN: 1479-8409
eISSN: 1479-8417
DOI: 10.1093/jjfinec/nby005
Publisher site: See Article on Publisher Site

Abstract

Abstract Mimicking portfolios of economic (non-traded) factors are commonly constructed by projecting the factors on a set of base assets. When these factors are associated with small betas, the beta-estimator using their mimicking portfolios has non-standard limit behavior. This jeopardizes inference on risk premia in the commonly used Fama and MacBeth (1973) two-pass procedure. Using sorting or the average excess returns on the mimicking portfolios to estimate the risk premia leads to similar non-standard behavior. We therefore propose a novel test for the risk premia on mimicking portfolios. Its validity does not depend on the magnitude of the betas. Simulation evidence suggests that it performs well in terms of size and power. We use it to analyze the risk premium on the leverage factor of Adrian, Etula, and Muir (2014). Our results indicate that the leverage factor is a weak factor which leads to substantially different results for its risk premium. An intersection of economics and finance is that a large group of economic factors are found useful for financial asset pricing. These factors include, for example, consumption growth in Breeden, Gibbons, and Litzenberger (1989), labor income growth in Jagannathan and Wang (1996), consumption–wealth ratio in Lettau and Ludvigson (2001), GDP growth in Vassalou (2003), investment growth in Li, Vassalou, and Xing (2006), among many others. The prevalence of these factors has recently led to a sizeable and growing literature that scrutinizes their usefulness. One major concern for economic factors is that their minor correlation with asset returns (see, e.g., Bai and Ng, 2006) invalidates the conventional inference methods used in asset pricing studies, such as the t-test on risk premia in the Fama and MacBeth (1973) (FM) two-pass methodology. Consequently, empirical support for economic factors based on conventional inference methods is up to careful scrutiny. An early contribution along this line is Kan and Zhang (1999), who show that the t-test in the FM methodology can spuriously support useless factors that are independent of asset returns. More recently, Kleibergen (2009) further warns that when factors are only weakly correlated with asset returns, so their betas are small, traditional inference on the risk premia based on the FM two-pass procedure is also spurious. Instead of using economic factors themselves, their mimicking portfolios are also widely used to replace these non-traded factors in asset pricing studies. The theoretical support for such practice is provided by Breeden (1979) and Huberman, Kandel, and Stambaugh (1987), who establish that factors can be replaced by their mimicking portfolios for asset pricing tests. In terms of economic factors, the norm for constructing their mimicking portfolios is to project these factors on a set of base assets that span the asset space. This projection is implemented by regressing factors on the returns of base assets in a time-series regression. The resulting mimicking portfolios after projection are also called maximum correlation portfolios. See, for example, Breeden, Gibbons, and Litzenberger (1989); Lamont (2001); Vassalou (2003); Avramov and Chordia (2006); and Adrian, Etula, and Muir (2014). Alternatively, mimicking portfolios are also commonly constructed after sorting the assets using their betas or other characteristics of the assets. On the one hand, besides other potential advantages,1 using mimicking portfolios instead of non-traded factors appears able to bypass the weak statistical correlation issue studied in Kleibergen (2009), etc. For instance, by projecting factors on base assets, it is natural that the resulting mimicking portfolios exhibit improved correlation with asset returns so their betas are amplified. In existing studies (e.g., Vassalou, 2003; Adrian, Etula, and Muir, 2014), this projection is commonly interpreted as a way to remove the noise in factors while keeping only their relevant information for asset pricing. Because of this interpretation, inference on risk premia using mimicking portfolios is believed to be more informative compared with that using the original non-traded factors. On the other hand, using mimicking portfolios instead of factors has its own costs which are rarely discussed in the existing literature. In this paper, we reveal the consequences of using mimicking portfolios in the FM two-pass procedure when their background factors are only minorly correlated with asset returns. We show that although these mimicking portfolios may appear relevant for asset pricing, the risk premia estimator in the FM two-pass procedure has a non-standard distribution which jeopardizes the t-test on the risk premia. This results since when the betas of the non-traded factors are small, the beta-estimator of their mimicking portfolios has non-standard limiting behavior which makes the t-test unreliable. Similarly, we find that the excess returns of such mimicking portfolios also fail to appropriately reflect the risk premia when the quality of the corresponding factors is poor. Our findings therefore suggest that using mimicking portfolios instead of economic factors for asset pricing does not necessarily imply improved inference on risk premia or more generally, improved performance of asset pricing tests. Ironically, since economic factors are almost indistinguishable from useless factors in terms of their minor correlation with asset returns in finite samples, empirical findings based on mimicking portfolios could be driven by the noise occurred in the construction of such portfolios rather than by the relevant pricing information contained in economic factors and preserved in the construction of their mimicking portfolios.2 Since our result puts the validity of the conventional t-test on risk premia of mimicking portfolios resulting from economic factors under doubt, we propose a novel test. Unlike the t-test, the asymptotic size of our test does not depend on the magnitude of the betas so it remains trustworthy when economic factors have small betas. The suggested test is in line with the robust tests proposed by Kleibergen (2009) for the risk premia on factors but the extension to a mimicking portfolios setting is new. The performance of the test is examined using Monte-Carlo simulation. For illustrative purposes, we apply the proposed test to the leverage factor model from Adrian, Etula, and Muir (2014). Following Adrian, Etula, and Muir (2014), we construct mimicking portfolios by regressing the leverage factor on seven base assets: excess returns of the six Fama–French portfolios on size and book-to-market, plus the momentum factor. By inverting the proposed test, we obtain confidence intervals for the risk premium of the constructed mimicking portfolios, with the usage of various test assets. We show that these confidence intervals differ substantially from those that result from inverting the conventional t-test, which is consistent with the fact that the leverage factor is associated with a small beta. Related to our paper, there exists a sizeable literature that examines identification and inference issues in asset pricing. Besides those related articles we cite above, see, for example, Burnside (2016) and Balduzzi and Robotti (2008). Unlike this paper, Burnside (2016) does not focus on mimicking portfolios but shows that when factors are weak, lack of identification exists in linear stochastic discount factor models. The rank test of Kleibergen and Paap (2006) is consequently suggested in Burnside (2016) as a model diagnostic. Balduzzi and Robotti (2008) do not focus on the FM two-pass procedure but argue that the average excess return on the mimicking portfolios provides an estimator for the risk premium. We show that this argument no longer applies when the correlation between factors and asset returns is small. The paper proceeds as follows. In Section 1, we describe a linear factor model with observed factors and present the limiting behavior of the beta-estimator using mimicking portfolios. In Section 2, we propose the robust test on the risk premia of mimicking portfolios and provide simulation evidence for its size and power. In Section 3, we re-evaluate the pricing performance of the leverage factor using the mimicking portfolios as in Adrian, Etula, and Muir (2014). Section 4 concludes the paper. Proofs, technical details, and additional results are provided in the Appendix. Throughout the paper, we use the following notation: μ is used for mean, while V is for covariance; μ^ and V^ are the sample analogs of μ and V, respectively; ιN is the N×1 dimensional vector of ones, vec(A) stands for the column vectorization of a matrix A, PA=A(A′A)−1A′, MA=I−PA,I is the identity matrix; “ →p”and “ →d” stand for convergence in probability and convergence in distribution, respectively. 1 Weak Identification with Mimicking Portfolios 1.1 Linear Factor Model and Observed Factors There is ample empirical evidence that asset returns are to a large extent explained by a small number of factors, see, for example, Merton (1973), Ross (1976), Roll and Ross (1980), Chamberlain and Rothschild (1983), and Connor and Korajczyk (1988, 1989). We therefore use a linear factor model for asset returns Rt (N×1) with k potentially unobserved factors Ft (k×1) to reflect the factor structure: Rt=ιNλ0+β(F¯t+λF)+vt, (1) where F¯t=Ft−μF, vt is the mean zero error term, t=1, … , T, β is the N × k full-rank matrix. Equation (1) implies the moment condition for the risk premia, that is, E(Rt)=ιNλ0+βλF, (2) with λ0: the zero-β return, λF: the k×1 vector of factor risk premia. Moreover, there exists a partition of Rt=(R1t′,R2t′)′, with R1t: N1×1, R2t: N2×1, N1+N2=N. From now on, we consider R1t as the returns on test assets, and R2t as the returns on base assets that are used for constructing the mimicking portfolios.3 Correspondingly, β=(β1′,β2′)′ with β1: N1×k, β2: N2×k, and vt=(v1t′,v2t′)′ with v1t: N1×1, v2t: N2×1. To accommodate that the factors are potentially unobserved, we use an m×1 vector of observed proxy factors Gt. We relate Ft to Gt by F¯t=δG¯t+ut, (3) where G¯t=Gt−μG and ut is the error term, see Kleibergen and Zhan (2015). The quality of the approximation of the true factors by the proxy factors is reflected by the k × m matrix δ. Ideally, Ft is observed so Gt coincides with it and δ=Ik,ut = 0. On the other hand, if δ is approximately equal to zero then Ft is poorly approximated by Gt. Plugging Equation (3) into Equation (1), and if the vector of risk premia of Gt, denoted by λG, is defined by the moment condition E(Rt)=ιNλ0+βδλG, then λF and λG are also related by δ since λF=δλG.4 In addition, Equation (1) is re-written as, Rt=ιNλ0+βδ(G¯t+λG)+et, (4) where et=(e1t′,e2t′)′=βut+vt and Σ=βΛβ′+Ω with Σ=var(et), Λ=var(ut), Ω=var(vt). The beta corresponding to Gt is thus βδ, which is small if δ is close to zero. We make the following assumptions for the model described above. Assumption 1. 1T∑t=1T((1Ft)⊗(Rt−ιNλ0−β(F¯t+λF)))→d(ϕRϕF), where (ϕRϕF)∼N(0,QF⊗Ω), QF=(1μF′μFVFF+μFμF′), Ω=var((v1t′,v2t′)′)=(Ω11Ω12Ω21Ω22). Assumption 2. All the covariance of R1t and R2t is captured by the factors so Ω12=Ω21′=0 and cov (R1t,R2t)=β1VFFβ2′. Assumption 3. 1T∑t=1T((1Gt)⊗(Rt−ιNλ0−β(F¯t+λF)))→d(ϕRϕG), where (ϕRϕG)∼N(0,QG⊗Ω), QG=(1μG′μGVGG+μGμG′). Assumption 1 is a central limit theorem that holds under rather mild conditions, like, for example, Assumptions 1 and 2 from Shanken (1992). Because of the independence of the disturbances over time and their finite variance, Assumption 1 holds under the linear factor model (1) with a constant covariance matrix as well. Assumption 2 is a necessary condition for the beta of the mimicking portfolios to be a nonsingular transformation of β1 which we further show below in Footnotes 5 and 6.5 Assumption 3 is a corollary of Assumption 1, with the observed Gt replacing the unknown Ft. 1.2 Beta of Mimicking Portfolios In accordance with common practice, we first construct mimicking portfolios by projecting Gt on base assets R2t. We also briefly discuss mimicking portfolios constructed by sorting on betas later on. The feasible mimicking portfolios that result from projection are then specified by: V^GR2V^R2R2−1R2t, (5) where V^GR2V^R2R2−1 is the sample counterpart of the infeasible VGR2VR2R2−1, and results from regressing Gt on R2t, t=1,…,T. With R1t as test assets, the beta-estimator for the mimicking portfolios in Equation (5) reads6: β˜^1=V^R1R2V^R2R2−1V^R2G(V^GR2V^R2R2−1V^R2G)−1. (6) Theorem 1. When Assumptions 1–3 hold, the limiting behavior of β˜^1 is described by: When δ is fixed and the number of elements of Gt equals the number of elements of Ft, so δ is a square invertible matrix: β˜^1→pβ1VFFδ−1′VGG−1,so when Gt=Ft, δ=Ik, and β˜^1→pβ1. If the proxy factors Gt are weak factors so δ=d/T, with d a fixed full-rank matrix, then regardless of k = m: T−12β˜^1→dβ1VFFβ2′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G) [(β2(dVGG+ψuG)+ψv2G)′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G)]−1,where 1T∑t=1T(G¯t⊗ut)→dvec (ψuG), 1T∑t=1T(G¯t⊗v2t)→dvec (ψv2G). Proof. See Appendix A. ▪ The first part of Theorem 1 shows that the beta-estimator of the mimicking portfolios converges to a non-singular transformation of the beta of the factors when the mimicking portfolios are constructed from good proxies of the underlying unobserved factors. The proof of Theorem 1 in Appendix A also shows that the covariance of test assets and base assets must be fully captured by the factors to render the beta-estimator consistent as implied by Assumption 2. Put differently, if Assumption 2 is not satisfied, then the estimated beta of the mimicking portfolios does not converge to the transformed beta of the factors, even when the true underlying factors are used to construct the mimicking portfolios. The motivation of this paper comes from the second part of Theorem 1, that is, when mimicking portfolios result from observed factors that are only poor (weak) proxies for underlying factors. It is well known that economic factors commonly exhibit minor correlation with asset returns, so they are likely to be poor proxies for the underlying factors, which we reflect by the weak factor specification: δ=d/T. (7) This drifting specification for δ, where it decreases with the sample size, is taken from the weak-instrument assumption made in econometrics (see, e.g., Staiger and Stock, 1997), in order to reflect that Gt is a poor proxy for Ft. A similar treatment can be found in Kleibergen (2009) and Kleibergen and Zhan (2015). The weak factor specification should not be taken literally but captures the often observed setting where an F-test of δ in Equation (3) equal to zero is rather small. When δ is fixed this F-test should be proportional to the sample size. The weak factor specification in Equation (7) allows for such small values of the F-test, see Kleibergen and Zhan (2015). For the weak factor setting, Theorem 1 shows that the beta-estimator of the mimicking portfolios is increasing with the sample size and has a non-standard limiting distribution. In other words, when the betas of the economic factors are small, the mimicking portfolios of such factors can be associated with betas that are spuriously large. Theorem 1 does not articulate a mixed setting where some of the unobserved factors are well approximated by the observed factors and some are not. In this case, δ would have both elements that drift to zero according to Equation (7) and elements that remain fixed. We left out this mixed case because it results in a combination of the limiting behaviors discussed under 1 and 2 in Theorem 1 which makes it notationally burdensome without altering the main conclusion from Theorem 1 that the magnitude of δ affects the limiting behavior of the beta-estimator. It is also worth noting that if k > m so βδ does not span the column space of β, then misspecification of factor pricing occurs. Our statistic that we propose lateron in the paper also indicates if such misspecification is present alongside deviations of the risk premia from the hypothesized one. In the commonly used FM two-pass procedure, the risk premia of the mimicking portfolios are obtained by regressing the sample average of the test assets μ^R1=1T∑t=1TR1t on β˜^1 and an intercept, that is, (λ˜^0λ˜^G)=[(ιN1⋮β˜^1)′(ιN1⋮β˜^1)]−1(ιN1⋮β˜^1)′μ^R1, (8) where λ˜^G (m×1) denotes the estimated risk premia for the mimicking portfolios that are constructed from Gt. If β˜^1 has a non-standard distribution, it is natural to expect that the behavior of λ˜^G is also non-standard as stated in the corollary below. Corollary 1. When Assumptions 1–3 hold, the limiting behavior of λ˜^G can be described by: When δ is fixed and the number of elements of Gt equals the number of elements of Ft, so δ is a square invertible matrix: λ˜^G→pVGGδ′VFF−1λF, so when Gt=Ft, δ=Ik, and λ˜^G→pλF. When δ=d/T: Tλ˜^G→d(Ψ′β1MιN1Ψβ1)−1Ψ′β1MιN1μR1+1T(Ψ′β1MιN1Ψβ1)−1Ψ′β1MιN1ψR1,where ψR1 is from Assumption 1 with ψR=(ψ′R1,ψ′R2)′, MιN1 is a projection matrix with MιN1=IN1−ιN1(ι′N1ιN1)−1ι′N1, Ψβ1≡β1VFFβ2′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G) [(β2(dVGG+ψuG)+ψv2G)′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G)]−1,see Theorem 1. Proof. See Appendix B. ▪ In line with Theorem 1, Corollary 1 also contains two cases. In the ideal first case, where accurate proxies of the underlying factors are used for constructing mimicking portfolios, λ˜^G is a consistent risk premia estimator. On the other hand in the second case where weak factors are observed, as reflected by δ=d/T, because of the non-standard behavior of β˜^1, λ˜^G also has a non-standard limiting distribution. Due to the non-standard behavior of λ˜^G in Corollary 1 under mimicking portfolios of weak factors, the conventional FM t-statistic for testing risk premia does not have an asymptotic standard normal distribution. Consequently, Corollary 1 implies that the conventional FM t-test on risk premia is under doubt when it is used for mimicking portfolios of economic factors. 1.3 Covariance Specification Instead of the beta specification above, the so-called covariance specification, see, for example, Kan, Robotti, and Shanken (2013) (KRS) suggests that for the mimicking portfolios in Equation (5), we can also use as an estimator for β1: β^1=V^R1R2V^R2R2−1V^R2G. Different from β˜^1, the inverted matrix (V^GR2V^R2R2−1V^R2G)−1 from β˜^1 is omitted in β^1. Theorem 2. When Assumptions 1–3 hold, the limiting behavior of β^1 can be described by: When δ is fixed and the number of elements of Gt equals the number of elements of Ft, so δ is a square invertible matrix: β^1→pβ1VFFβ2′(β2VFFβ2′+Ω22)−1β2δVGG. When δ=d/T: T12β^1→dβ1VFFβ2′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G).The specifications of ψuG and ψv2G are stated in Theorem 1. Proof. See Appendix C. ▪ Unlike Theorem 1, the second part of Theorem 2 shows that the large sample behavior of the weak factor mimicking portfolio’s beta-estimator is now comparable to that of the pure weak factor’s beta-estimator, see Kleibergen (2009) and Kleibergen and Zhan (2015). Put differently, under the covariance specification, the beta-estimator of the mimicking portfolios (denoted by β^1) does not suffer from the exaggeration problem of its counterpart (denoted by β˜^1) which exists under the beta specification, as stated in Theorem 1. However, it is known that small betas jeopardize risk premia estimation in the FM methodology, see for example, Kleibergen (2009). Consequently, a malfunction of the FM two-pass procedure also exists in the covariance specification, albeit for reasons opposite to those of the beta specification: if mimicking portfolios are constructed from weak factors, the beta-estimator is large in magnitude in the beta specification, and small in the covariance specification, both could induce failure of risk premia estimation in the FM methodology. In order to resolve this issue, we develop a new inference method for risk premia of mimicking portfolios, which is presented later on. 1.4 Simulation Study To further illustrate the results in Theorems 1 and 2, we conduct a simple simulation experiment. Asset returns are generated from the factor model (1), with T=1000 and k = 1. Specifically, Ft∼NID (0,VFF), vt∼NID (0,Ω), where VFF is calibrated from the returns on the market portfolio in Fama–French (1993), Ω is calibrated from a regression of the returns on N1 industry portfolios, and N2 size- and book-to-market-sorted portfolios on the returns on the market portfolio. We consider N1=N2 for convenience in the data generation process (d.g.p.) and set them equal to 1, 2, 3, 4, and 5, as reported in the first column of Table 1. The values of the parameters λ0, λF, and β used in the d.g.p. result from the FM two-pass procedure using the described portfolios.7 Table 1. Rejection frequencies of the rank test by Monte Carlo Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.051 0.107 0.060 0.112 1.000 1.000 0.050 0.106 0.059 0.112 1.000 1.000 2. 0.053 0.105 0.014 0.031 0.936 0.949 0.051 0.102 0.013 0.030 0.935 0.949 3. 0.050 0.104 0.005 0.012 0.898 0.916 0.048 0.101 0.005 0.012 0.895 0.914 4. 0.052 0.102 0.003 0.011 0.879 0.899 0.050 0.097 0.002 0.009 0.878 0.897 5. 0.050 0.090 0.001 0.002 0.841 0.864 0.046 0.085 0.000 0.002 0.837 0.860 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.051 0.107 0.060 0.112 1.000 1.000 0.050 0.106 0.059 0.112 1.000 1.000 2. 0.053 0.105 0.014 0.031 0.936 0.949 0.051 0.102 0.013 0.030 0.935 0.949 3. 0.050 0.104 0.005 0.012 0.898 0.916 0.048 0.101 0.005 0.012 0.895 0.914 4. 0.052 0.102 0.003 0.011 0.879 0.899 0.050 0.097 0.002 0.009 0.878 0.897 5. 0.050 0.090 0.001 0.002 0.841 0.864 0.046 0.085 0.000 0.002 0.837 0.860 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. Table 1. Rejection frequencies of the rank test by Monte Carlo Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.051 0.107 0.060 0.112 1.000 1.000 0.050 0.106 0.059 0.112 1.000 1.000 2. 0.053 0.105 0.014 0.031 0.936 0.949 0.051 0.102 0.013 0.030 0.935 0.949 3. 0.050 0.104 0.005 0.012 0.898 0.916 0.048 0.101 0.005 0.012 0.895 0.914 4. 0.052 0.102 0.003 0.011 0.879 0.899 0.050 0.097 0.002 0.009 0.878 0.897 5. 0.050 0.090 0.001 0.002 0.841 0.864 0.046 0.085 0.000 0.002 0.837 0.860 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.051 0.107 0.060 0.112 1.000 1.000 0.050 0.106 0.059 0.112 1.000 1.000 2. 0.053 0.105 0.014 0.031 0.936 0.949 0.051 0.102 0.013 0.030 0.935 0.949 3. 0.050 0.104 0.005 0.012 0.898 0.916 0.048 0.101 0.005 0.012 0.895 0.914 4. 0.052 0.102 0.003 0.011 0.879 0.899 0.050 0.097 0.002 0.009 0.878 0.897 5. 0.050 0.090 0.001 0.002 0.841 0.864 0.046 0.085 0.000 0.002 0.837 0.860 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. In the simulation exercise, we consider two different types of observed factors which are encountered in empirical studies. The first case is the one of observed factors that are good/strong proxies for the underlying factors Ft. Here, we use Gt=Ft. The second case is the one of observed factors that are poor/weak proxies of the underlying factors. We therefore use observed factors Gt that are independently generated as N(0,VFF) distributed random variables so the observed factor is completely useless for asset returns. In Panel A of Table 1, we present the outcome of the simulation study for the strong factor case, while the useless factor case is in Panel B. Both the beta specification and the covariance specification for constructing the beta-estimator are considered in our simulation study. In addition, for each specification, we consider three different manners of using the simulated factors. Under “Fac” in Table 1, the simulated factor is directly used for estimating β1 in the time-series regression using the simulated asset returns R1t. Under “MP” in Table 1, we construct mimicking portfolios of simulated factors as in Equation (5); with the constructed mimicking portfolios, we proceed to compute their beta-estimator β˜^1 (beta specification) or β^1 (covariance specification); furthermore, the asymptotic variances of such estimators are derived by first-order asymptotics taking into account the estimation error contained in the mimicking portfolios (see Appendix D for details). Under “MP as Fac” in Table 1, we treat the constructed mimicking portfolios in the same manner as the observed factors for constructing the variance of the beta-estimator (hence, the estimation error contained in the mimicking portfolios is ignored when computing the variance of their beta-estimator), which is commonly done in existing empirical studies. We apply the rank test of Kleibergen and Paap (2006) for the estimator of β1 in the various scenarios described above. In particular, we test whether the estimand has reduced rank k − 1 and document the rejection frequency of the null at the nominal 5% and 10% level using standard χ2 critical values. Since β1 has full rank under strong factors, we expect that the rank test will strongly reject the null. This is in line with the rejection frequencies reported in Panel A of Table 1. In fact, we show that the rejection frequencies based on 2000 Monte-Carlo replications are all equal to 1, so the outcome of the rank test supports the strong factor, as expected. On the contrary, under useless factors, we expect the rejection frequency of the rank test to be close to the nominal size, since useless factors are associated with zero beta’s so the null hypothesis holds. This is in line with the rejection frequencies reported under “Fac” in Panel B of Table 1. Under “MP,” the rank test appears conservative, since the rejection frequencies quickly decrease from nominal sizes as N1 and N2 increase.8 What is astonishing in Panel B of Table 1 lies in the columns of “MP as Fac”, that is, mimicking portfolios are naively treated as alternatives of factors and used for rank tests. Panel B shows that the rejection frequencies under “MP as Fac” are very large (albeit decrease as N1 and N2 increase). Consequently, mimicking portfolios of useless factors may signal strong factor pricing in rank tests, if they are treated in the same manner as factors. Theorems 1 and 2 show that such treatment is improper, since the beta-estimator under mimicking portfolios does not have the same limiting distribution as when using observed factors. Overall, Table 1 suggests that the rank test can serve as a diagnostic tool for the quality of the factors, if implemented properly so correcting for the estimation error that results from using mimicking portfolios. When factors are strong (weak), the null is rejected (accepted with the probability close to the nominal size), if these factors are used for beta estimation. When mimicking portfolios are used for estimating the betas, Table 1 suggests that the rank test outcome needs to be taken with caution. In particular, if mimicking portfolios are improperly treated as observed factors, so ignoring their estimation error, then the rank test may spuriously favor strong factors.9 1.5 Average Excess Returns and Sorting Risk premia can also be estimated using the average excess returns of the mimicking portfolios, see, for example, Balduzzi and Robotti (2008). For the mimicking portfolios obtained by projection, V^GR2V^R2R2−1R2t, the risk premia estimator then equals the time-series average: V^GR2V^R2R2−1R¯2, (9) with R¯2=1T∑t=1TR2t. The limiting behavior of the risk premia estimator (9) becomes non-standard when the observed factors are weakly correlated with the true underlying factors. Theorem 3. When Assumptions 1–3 hold, the limiting behavior of V^GR2V^R2R2−1R¯2 under excess returns with λ0=0 is described by: When δ is fixed: V^GR2V^R2R2−1R¯2→pVGGδ′β2′VR2R2−1β2λF. When δ=d/T: TV^GR2V^R2R2−1R¯2→d[β2(dVGG+ψuG)+ψv2G]′VR2R2−1β2λF. Proof. It results straightforwardly from the proof of Theorem 1. The specifications of ψuG and ψv2G are stated in Theorem 1. ▪ Theorem 3 shows that when δ is sizeable, the average excess returns of the mimicking portfolios converge to a linear combination of the risk premia λF. The risk premia estimator (9) thus provides an adequate estimator of the risk premia when the observed factors are good proxies of the underlying true factors. In contrast, when the proxy factors only weakly correlate with the true underlying factors, so δ is local to zero, the risk premia estimator V^GR2V^R2R2−1R¯2 is no longer a valid estimator of the risk premia. On top of this, its limiting behavior differs considerably from the one under strong factors, which further complicates inference. We have so far focused on mimicking portfolios constructed by projection. The challenge induced by weak factors, however, also applies to mimicking portfolios constructed by sorting which we briefly discuss next. To facilitate the analysis, consider a single observed factor Gt, so m=1. We construct so-called high-minus-low mimicking portfolios of Gt based on sorting the N2×1 vector V^R2GV^GG−1 of estimated betas. We denote the N2×N2 matrix with the sorting (WLOG in descending order) indicators of V^R2GV^GG−1 by S(V^R2GV^GG−1) so: S(V^R2GV^GG−1)×V^R2GV^GG−1=[max(V^R2GV^GG−1)⋮min(V^R2GV^GG−1)], (10) where max(V^R2GV^GG−1) is the maximal element of V^R2GV^GG−1 and min(V^R2GV^GG−1) the minimal element. Correspondingly, the N2×1 returns of R2t after sorting by V^R2GV^GG−1 read: S(V^R2GV^GG−1)×R2t=[R2t,(1)⋮R2t,(N2)], (11) where the (estimated) beta of R2t,(1) is the largest, the beta of R2t,(N2) is the smallest, and the betas are obtained from V^R2GV^GG−1. The return on the high-minus-low mimicking portfolios is then R2t,(1)−R2t,(N2), if only one asset is taken as “high” and one asset is taken as “low.” In accordance with empirical practice, we consider R2t,(1), … , R2t,(λN2) as the first λN2 asset returns after sorting, where 0<λ<1. Similarly, R2t,((1−λ)N2+1), … , R2t,(N2) are the last λN2 asset returns after sorting. We take the first λN2 assets as “high,” and take the last λN2 assets as “low.” The return on the high-minus-low mimicking portfolios (equally weighted) then reads: 1λN2∑i=1λN2R2t,(i)−1λN2∑i=(1−λ)N2+1N2R2t,(i); in particular, when λ=1/N2, this expression simplifies to R2t,(1)−R2t,(N2). In matrix notation, the return on the high-minus-low portfolios then reads: 1λN2[1,...,1,0,...,0,−1,...,−1]×S(V^R2GV^GG−1)×R2t, (12) where both the number of 1s and −1s above are equal to λN2. If the average returns of the high-minus-low portfolio are used to estimate the risk premia, the risk premia estimator reads: ΔλN2×S(V^R2GV^GG−1)×R¯2, (13) with ΔλN2=1λN2[1,...,1,0,...,0,−1,...,−1] a fixed constant row vector. The sorting operator S(V^R2GV^GG−1) is therefore at the center of our analysis. We distinguish two different settings for its limiting behavior which result from the quality of the approximation of the true underlying factors by the proxy factors. When the observed factors Gt provide good proxies so δ is sizeable: V^R2GV^GG−1→pβ2δ. (14) Hence, in the limit S(V^R2GV^GG−1) effectively sorts β2δ which is a fixed vector. Consequently, as T→∞, the limiting behavior of the risk premia estimator is characterized by ΔλN2×S(V^R2GV^GG−1)×R¯2→pΔλN2×S(β2δ)×β2λF. (15) Since R¯2→pιN2λ0+β2λF. It is worth noting that assets with large elements in β2δs tend to be in “high,” while the assets with small elements in β2δs tend to be in “low,” since sorting is effectively based on β2δ as T gets large. The risk premia estimator is therefore equal to a linear combination of the risk premia λF. Sorting using high-minus-low portfolios thus provides an estimator of the risk premia when the observed factors are good proxies. In contrast, when the observed factors are poor proxies of the true factors so δ=d/T: TV^R2GV^GG−1→dβ2d+(β2ψuG+ψv2G)VGG−1≡[Z1⋮ZN2], (16) where Z1, … , ZN2 are N2 (non-identical) correlated normal random variates. Each of these N2 variates has a non-zero probability of being the largest one in realizations; similarly, each of them also has a non-zero probability of being the smallest one in realizations. Since the scaling by T does not affect the sorting, the limiting behavior of the risk premia estimator is now characterized by ΔλN2×S(V^R2GV^GG−1)×R¯2→dΔλN2×S(β2d+(β2ψuG+ψv2G)VGG−1)×β2λF, (17) which shows that when Gt is a poor proxy, the sorting is conducted on a random vector even when T is large. The risk premia estimator now converges to a random function of the risk premia since the sorting on the beta-estimator is arbitrary as it converges to a random variable. This also shows that identical to the other risk premia estimators, the behavior of the risk premia estimator based on sorting becomes non-standard when the observed factors are poor proxies of the true factors. 2 Robust Inference The previous section discussed issues associated with the conventional approaches for inference on the risk premia when mimicking portfolios result from non-traded factors which are poor proxies of the true factors. To resolve this problem, we propose a robust inference procedure for testing the risk premia of mimicking portfolios. 2.1 Scaled Risk Premia under Mimicking Portfolios Specifically, we suggest to test the risk premium λG,cov on mimicking portfolios using the moment condition: E(R1t)=ιN1λ0+VR1R2VR2R2−1VR2GVGG−1λG,cov (18) where VR1R2VR2R2−1VR2G is the covariance of the test assets R1t with the (infeasible) mimicking portfolios VGR2VR2R2−1R2t, so λG,cov is the scaled risk premium in the covariance specification (scaled by VGG−1).10 If the base assets in R2t span the test assets in R1t, then VR1R2VR2R2−1VR2G reduces to VR1G and λG,cov equals λG in Equation (4). For instance, in the special case that R1t=R2t, so test and base assets coincide, λG,cov reduces to λG. If so, inference on λG,cov reduces to inference on λG, which has been discussed by Kleibergen (2009). Since it is common that R2t does not fully span R1t, λG,cov is consequently not necessarily equal to λG. We thus proceed to consider inference on λG,cov defined in Equation (18). As a starting point, we remove λ0 since our interest lies in λG,cov. This is done by, without loss of generality, removing the return on the N1-th test asset and taking all other test asset returns in deviation from the return on the N1-th asset.11Equation (18) is then re-written as: E(R1t)=VR1R2VR2R2−1VR2GVGG−1λG,cov=Γβ2δλG,cov, (19) where Γ=VR1R2VR2R2−1, R1t=R1t,1:(N1−1)−ιN1−1R1t,N1, with R1t=(R′1t,1:(N1−1),R′1t,N1)′, and VR2GVGG−1=β2δ as in Equation (4). 2.2 Mimicking Portfolio Anderson–Rubin Test To conduct inference on λG,cov, we state the joint behavior of R¯1, B^2, and Γ^ in Theorem 4, with R¯1=1T∑t=1TR1t, B^2=V^R2GV^GG−1, and Γ^=V^R1R2V^R2R2−1. Theorem 4. When Assumptions 1–3 hold: T(R¯1−Γβ2δλG,covvec(B^2−β2δ)vec(Γ^−Γ))→d(ψ1vec(ψ2)vec(ψ3))∼N(0,W), (20)and W=(VR1R1000VGG−1⊗Σ22C′0CVR2R2−1⊗(VR1R1−VR1R2VR2R2−1VR2R1))with C=K(N1−1)N2(VR1FVFF−1δ⊗VR2R2−1Σ22)+(VR2R2−1β2δ⊗Σ12) −(VR2R2−1⊗Γ)(IN22+KN2N2)(β2δ⊗Σ22),K is the commutation matrix and Σ12=cov(e1t,e2t) with e1t=R1t−VR1FVFF−1δ(G¯t+λG). Proof. See Appendix F. ▪ Theorem 4 implies that the limiting distribution of T(R¯1−Γ^B^2λG,cov) is normal. Corollary 2. When Assumptions 1–3 hold and Γ has a full-rank value: T(R¯1−Γ^B^2λG,cov)→dψ1−Γψ2λG,cov−ψ3β2δλG,cov∼N(0,V), (21)with V=VR1R1+(λ′G,covVGG−1λG,cov⊗ΓΣ22Γ′)+((β2δλG,cov)′VR2R2−1(β2δλG,cov)⊗(VR1R1−VR1R2VR2R2−1VR2R1))−{[(β2δλG,cov)′⊗IN1−1]C(λG,cov⊗Γ′)}−{[(β2δλG,cov)′⊗IN1−1]C(λG,cov⊗Γ′)}′. Proof. See Appendix G. ▪ Since the unobserved factor Ft affects both R1 and R2, Γ has a full-rank value. We do not make a full-rank assumption on δ so we allow for weak correlation between observed and unobserved factors. Using Corollary 2, it is straightforward to obtain an asymptotic result to test H0:λG,cov=λG,cov,0. Corollary 3 (MPAR). Under H0:λG,cov=λG,cov,0, MPAR(λG,cov,0)=T(R¯1−Γ^B^2λG,cov,0)′V^−1(R¯1−Γ^B^2λG,cov,0)→dχN1−12 (22)with V^=V^R1R1+(λ′G,cov,0V^GG−1λG,cov,0⊗Γ^Σ^22Γ^′)+((B^2λG,cov,0)′V^R2R2−1(B^2λG,cov,0)⊗(V^R1R1−V^R1R2V^R2R2−1V^R2R1))−[(B^2λG,cov,0)′⊗IN1−1]C^(λG,cov,0⊗Γ^′)−(λ′G,cov,0⊗Γ^)C^′[(B^2λG,cov,0)⊗IN1−1] C^=K(N1−1)N2(B^1⊗V^R2R2−1Σ^22)+(V^R2R2−1B^2⊗Σ^12)−(V^R2R2−1⊗Γ^)(IN22+KN2N2)(B^2⊗Σ^22)and Σ^12=1T∑t=1Te^1te^′2t, Σ^22=1T∑t=1Te^2te^′2t, e^2t=R2t−R¯2−B^2(Gt−μ^G), e^1t=R1t−R¯1−B^1(Gt−μ^G), B^1=V^R1GV^GG−1. Proof. Results from Corollary 2, with V^ a consistent estimator for V. ▪ We refer to this statistic as the Mimicking Portfolio Anderson–Rubin (MPAR) statistic, since it is in line with the Anderson and Rubin (1949) statistic for robust inference and it is proposed using mimicking portfolios. Similar tests in the factor rather than mimicking portfolios setting include the Factor Anderson–Rubin (FAR) test proposed in Kleibergen (2009) and the Hotelling-type statistic proposed in Beaulieu, Dufour, and Khalaf (2013).12 Because of Corollary 3, a test of H0:λG,cov=λG,cov,0 that rejects H0, if MPAR(λG,cov,0) exceeds the 1−α quantile of the χ2 distribution with N1−1 degrees of freedom has an asymptotic size that equals α irrespective of the value of δ. Values of λG,cov,0 that are not rejected by this test thus constitute a 100×(1−α)% confidence set of λG,cov. 2.3 Size of the MPAR Statistic in Simulations To examine the size of the proposed MPAR test, we conduct a simple simulation experiment. The d.g.p. is similar to the one used in Section 1. Specifically, asset returns are generated from Equation (1), with T = 1000 and k = 1. In addition, Ft and vt are generated as independent NID (0,VFF) and NID (0,Ω) random variables, with VFF calibrated from the returns on the market portfolio in Fama–French (1993) and Ω calibrated from the regression of the returns on N1 industry and N2 size- and book-to-market-sorted portfolios on the return on the market portfolio. We consider N1=N2 for convenience and set them equal to 5, 10, 15, 20, and 25, as reported in the first column of Table 2. The values of λ0, λF, and β used in d.g.p. result from the FM two-pass procedure using the portfolios described above. Table 2. Actual sizes of the MPAR test by Monte Carlo, T = 1000 δ=0.01 δ=0.25 δ=0.50 δ=0.75 δ=0.99 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% N1, N2=5 0.050 0.105 0.059 0.105 0.058 0.103 0.057 0.104 0.057 0.104 N1, N2=10 0.020 0.051 0.052 0.108 0.051 0.108 0.050 0.108 0.051 0.108 N1, N2=15 0.018 0.047 0.064 0.115 0.063 0.116 0.063 0.116 0.063 0.116 N1, N2=20 0.002 0.008 0.063 0.127 0.064 0.128 0.063 0.126 0.062 0.126 N1, N2=25 0.001 0.002 0.064 0.124 0.064 0.127 0.063 0.127 0.063 0.126 δ=0.01 δ=0.25 δ=0.50 δ=0.75 δ=0.99 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% N1, N2=5 0.050 0.105 0.059 0.105 0.058 0.103 0.057 0.104 0.057 0.104 N1, N2=10 0.020 0.051 0.052 0.108 0.051 0.108 0.050 0.108 0.051 0.108 N1, N2=15 0.018 0.047 0.064 0.115 0.063 0.116 0.063 0.116 0.063 0.116 N1, N2=20 0.002 0.008 0.063 0.127 0.064 0.128 0.063 0.126 0.062 0.126 N1, N2=25 0.001 0.002 0.064 0.124 0.064 0.127 0.063 0.127 0.063 0.126 Notes: The reported sizes are rejection frequencies of the MPAR test for H0:λG,cov=λG,cov,0 at the nominal 5% and 10%, respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000. Table 2. Actual sizes of the MPAR test by Monte Carlo, T = 1000 δ=0.01 δ=0.25 δ=0.50 δ=0.75 δ=0.99 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% N1, N2=5 0.050 0.105 0.059 0.105 0.058 0.103 0.057 0.104 0.057 0.104 N1, N2=10 0.020 0.051 0.052 0.108 0.051 0.108 0.050 0.108 0.051 0.108 N1, N2=15 0.018 0.047 0.064 0.115 0.063 0.116 0.063 0.116 0.063 0.116 N1, N2=20 0.002 0.008 0.063 0.127 0.064 0.128 0.063 0.126 0.062 0.126 N1, N2=25 0.001 0.002 0.064 0.124 0.064 0.127 0.063 0.127 0.063 0.126 δ=0.01 δ=0.25 δ=0.50 δ=0.75 δ=0.99 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% N1, N2=5 0.050 0.105 0.059 0.105 0.058 0.103 0.057 0.104 0.057 0.104 N1, N2=10 0.020 0.051 0.052 0.108 0.051 0.108 0.050 0.108 0.051 0.108 N1, N2=15 0.018 0.047 0.064 0.115 0.063 0.116 0.063 0.116 0.063 0.116 N1, N2=20 0.002 0.008 0.063 0.127 0.064 0.128 0.063 0.126 0.062 0.126 N1, N2=25 0.001 0.002 0.064 0.124 0.064 0.127 0.063 0.127 0.063 0.126 Notes: The reported sizes are rejection frequencies of the MPAR test for H0:λG,cov=λG,cov,0 at the nominal 5% and 10%, respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000. The observed Gt is simulated as follows: Gt=δ·Ft+1−δ2·NID(0,VFF), so δ coincides with the specification in Equation (3) and reflects the quality of Gt for approximating Ft. When δ is close to zero, Gt is a weak factor which becomes stronger for an increasing value of δ. We consider a sequence of values for δ: δ∈{0.01,0.25,0.50,0.75,0.99}, which covers a wide range of settings of Gt as the observed proxy for Ft. With the simulated asset returns and Gt, we conduct the MPAR test on the risk premia of the mimicking portfolios as described in Corollary 3. The resulting sizes of the test are reported in Table 2. Table 2 shows that the MPAR test does not severely over reject in any of the settings. Its rejection frequency approximately equals the size of the test in many instances. Table 2 shows that the MPAR test is conservative when the factors are weak. It indicates that the covariance matrix estimator V^ is on average too large in this simulation setting. Using the properties of the partitioned inference and that Γ^ does not depend on the strength of the factors, the too large value of V^ can be attributed to the realized values of B^1 and B^2 being away from their expected values (zero). This explains also why the under rejection increases when N1 and N2 get larger. When we increase the sample size, the estimates of B^1 and B^2 become more precise and the under rejection of the MPAR test disappears.13 Table 2 shows that the behavior of the MPAR test accords with Corollary 3 since the under rejection reported in Table 2 results from the estimation of the covariance matrix V. The size of a test is defined as the maximal rejection frequency under the null hypothesis so under rejection does not indicate size distortion. 2.4 Power of the MPAR Test To illustrate the power of the MPAR test, we consider a sequence of values of λG,cov (between −5 and 5) in the d.g.p. described above with T = 1000 and N1=N2=5.14 We then test H0:λG,cov=0 at the 5% level using the generated data. The resulting rejection frequencies of the MPAR test are in Figure 1. Figure 1. View largeDownload slide Power plots of the MPAR test. Notes: This figure presents the power plot of the MPAR test for H0:λG,cov=0 at the 5% level, with δ=0.01 (plus), δ=0.25 (dotted), δ=0.50 (dash-dot), δ=0.75 (dashed), and δ=0.99 (solid line). The d.g.p. is described in the main text, with T=1000 and N1=N2=5. Figure 1. View largeDownload slide Power plots of the MPAR test. Notes: This figure presents the power plot of the MPAR test for H0:λG,cov=0 at the 5% level, with δ=0.01 (plus), δ=0.25 (dotted), δ=0.50 (dash-dot), δ=0.75 (dashed), and δ=0.99 (solid line). The d.g.p. is described in the main text, with T=1000 and N1=N2=5. We use a range of values of δ to reflect the different qualities of the observed factors. Figure 1 has five power plots of the MPAR test, corresponding to δ=0.01 (plus), δ=0.25 (dotted), δ=0.50 (dash-dot), δ=0.75 (dashed), and δ=0.99 (solid line), respectively. Figure 1 shows that the MPAR test has good power when δ gets large. This is as expected since a larger value of δ implies more informative factors. Consequently, when confidence sets of risk premia are to be constructed by inverting the MPAR test, wider confidence sets indicate that the corresponding factors are less informative, while narrower sets signal that the factors are stronger for asset pricing. Note that when δ=0.01 so the factor is close to being useless, Figure 1 shows that the corresponding power plot is close to the nominal 5%. We further compare the MPAR test with the FAR test in Kleibergen (2009). As stated above, in the special case that test assets and base assets coincide so R1t=R2t, the mimicking portfolio risk premia λG,cov coincides with the factor risk premia λG. In this scenario, both MPAR and FAR tests are applicable, and we plot their power curves in Figure 2. As shown in Figure 2, the power of the MPAR test is almost identical to that of the FAR test. Figure 2. View largeDownload slide Power plots of the MPAR test and the FAR test. Notes: This figure presents the power plots of the MPAR test (solid) and the FAR test (dashed) for H0:λG,cov=0 at the 5% level, with δ=0.01 in (a), δ=0.25 in (b), δ=0.50 in (c), δ=0.75 in (d), and δ=0.99 in (e). The d.g.p. is described in the main text, with T = 1000 and N1=N2=5. Figure 2. View largeDownload slide Power plots of the MPAR test and the FAR test. Notes: This figure presents the power plots of the MPAR test (solid) and the FAR test (dashed) for H0:λG,cov=0 at the 5% level, with δ=0.01 in (a), δ=0.25 in (b), δ=0.50 in (c), δ=0.75 in (d), and δ=0.99 in (e). The d.g.p. is described in the main text, with T = 1000 and N1=N2=5. 3 Re-evaluating the Pricing Performance of the Leverage Factor We illustrate the practical usage of the MPAR test by employing it to the leverage factor model proposed by Adrian, Etula, and Muir (2014). Specifically, Adrian, Etula, and Muir (2014) define the leverage level as the ratio of total assets over the difference between total assets and total liabilities of security broker–dealers. The resulting log change of the leverage level makes up their leverage factor. Adrian, Etula, and Muir (2014) find that the leverage factor prices the cross-section of various test portfolios remarkably well, outperforming the standard benchmarks in the literature. Their empirical study uses data for the leverage factor “LevFac” from 1968Q1–2009Q4, so T = 168 and k = 1. For illustrative purposes, we adopt the same data set.15 For base assets, we follow Adrian, Etula, and Muir (2014) who consider seven assets, so N2=7: the excess returns on the six Fama–French portfolios on size and book-to-market (“BL”, “BM”, “BH”, “SL”, “SM”, “SH”), plus the momentum factor “Mom.” These seven assets are widely acknowledged for their ability to span the asset space.16 To construct the mimicking portfolios, we project the leverage factor on these seven base assets. We obtained estimates for the coefficients (denoted by V^GR2V^R2R2−1 in Section 1) by projecting the leverage factor “LevFac” on the base assets (“BL”, “BM”, “BH”, “SL”, “SM”, “SH”, “Mom”) equal to: (−0.22,−0.10,0.56,−0.57,1.24,−0.43,0.43). These coefficients (after normalizing their sum to one) are almost identical to those in Adrian, Etula, and Muir (2014) for the reported weights of the mimicking portfolio.17 Adrian, Etula, and Muir (2014) show that the mimicking portfolio of the leverage factor performs well in various asset pricing tests, for example, it is associated with large R-squareds and low intercepts in cross-sectional regressions. Adrian, Etula, and Muir (2014) do not study the risk premium associated with the mimicking portfolio in cross-sectional regressions while we focus on it. Specifically, we are interested in the risk premium of the mimicking portfolio “LevMP” that results from projecting “LevFac” on the seven base assets. We use as test assets the commonly used 25 Fama–French portfolios on size and book-to-market (25 FF). Since both test assets and base assets are based on size and book-to-market, they are likely driven by the same underlying factors, as in the model setup in Equation (1). In addition, since T = 168 is relatively small, while our simulation study suggests that the size of the MPAR test is more reliable under small N1 and N2 for small T, we divide the 25 FF portfolios into five equal-sized groups (denoted by I–V in the first column of Table 3, by market capitalization), and use each group as test assets with N1=5.18 Table 3. Risk premium of the leverage factor (LevFac) and its mimicking portfolio (LevMP) LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I 22.35 4.02 2.30 (− ∞,−126.02]∪[13.96,∞) 4.11 4.11 1.74 (− ∞,−104.78]∪[16.82,∞) II 11.78 2.72 2.23 [−1.23,98.81] 2.73 2.72 1.62 (− ∞,−25.57]∪[−2.43,∞) III 16.59 2.57 1.79 (− ∞,−43.11]∪[1.67,∞) 2.32 2.38 1.52 (− ∞,−191.49]∪[3.08,∞) IV 8.32 1.31 1.25 (− ∞,∞) 1.01 1.21 1.03 (− ∞,∞) V 4.85 0.92 0.86 (− ∞,∞) 0.80 0.86 0.79 (− ∞,∞) LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I 22.35 4.02 2.30 (− ∞,−126.02]∪[13.96,∞) 4.11 4.11 1.74 (− ∞,−104.78]∪[16.82,∞) II 11.78 2.72 2.23 [−1.23,98.81] 2.73 2.72 1.62 (− ∞,−25.57]∪[−2.43,∞) III 16.59 2.57 1.79 (− ∞,−43.11]∪[1.67,∞) 2.32 2.38 1.52 (− ∞,−191.49]∪[3.08,∞) IV 8.32 1.31 1.25 (− ∞,∞) 1.01 1.21 1.03 (− ∞,∞) V 4.85 0.92 0.86 (− ∞,∞) 0.80 0.86 0.79 (− ∞,∞) Notes: LevFac stands for the leverage factor suggested in Adrian, Etula, and Muir (2014), while LevMP stands for the constructed mimicking portfolio of the leverage factor. Test assets are from the 25 Fama–French portfolios on size and book-to-market (25 FF) in the sample period of 1968Q1–2009Q4, and we divide them into five groups of test assets, I–V, so each group contains five portfolios. Three tests on risk premia are employed, namely, the conventional t-test for the FM methodology, the FAR test of Kleibergen (2009), and the proposed MPAR test in this paper. The KRS and JKZ corrections are also considered for the t-test, LevFac, and LevMP, respectively. Table 3. Risk premium of the leverage factor (LevFac) and its mimicking portfolio (LevMP) LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I 22.35 4.02 2.30 (− ∞,−126.02]∪[13.96,∞) 4.11 4.11 1.74 (− ∞,−104.78]∪[16.82,∞) II 11.78 2.72 2.23 [−1.23,98.81] 2.73 2.72 1.62 (− ∞,−25.57]∪[−2.43,∞) III 16.59 2.57 1.79 (− ∞,−43.11]∪[1.67,∞) 2.32 2.38 1.52 (− ∞,−191.49]∪[3.08,∞) IV 8.32 1.31 1.25 (− ∞,∞) 1.01 1.21 1.03 (− ∞,∞) V 4.85 0.92 0.86 (− ∞,∞) 0.80 0.86 0.79 (− ∞,∞) LevFac LevMP t-test FAR test t-test MPAR test Coef. FM t KRS t 95% CI Coef. FM t JKZ t 95% CI I 22.35 4.02 2.30 (− ∞,−126.02]∪[13.96,∞) 4.11 4.11 1.74 (− ∞,−104.78]∪[16.82,∞) II 11.78 2.72 2.23 [−1.23,98.81] 2.73 2.72 1.62 (− ∞,−25.57]∪[−2.43,∞) III 16.59 2.57 1.79 (− ∞,−43.11]∪[1.67,∞) 2.32 2.38 1.52 (− ∞,−191.49]∪[3.08,∞) IV 8.32 1.31 1.25 (− ∞,∞) 1.01 1.21 1.03 (− ∞,∞) V 4.85 0.92 0.86 (− ∞,∞) 0.80 0.86 0.79 (− ∞,∞) Notes: LevFac stands for the leverage factor suggested in Adrian, Etula, and Muir (2014), while LevMP stands for the constructed mimicking portfolio of the leverage factor. Test assets are from the 25 Fama–French portfolios on size and book-to-market (25 FF) in the sample period of 1968Q1–2009Q4, and we divide them into five groups of test assets, I–V, so each group contains five portfolios. Three tests on risk premia are employed, namely, the conventional t-test for the FM methodology, the FAR test of Kleibergen (2009), and the proposed MPAR test in this paper. The KRS and JKZ corrections are also considered for the t-test, LevFac, and LevMP, respectively. To gauge the statistical quality of the leverage factor, we conduct the Kleibergen and Paap (2006) rank test, which is commonly used as a diagnostic tool in the asset pricing literature. The null of the rank test is that the leverage factor beta has reduced rank, and the resulting p-values are found to be 0.09, 0.01, 0.26, 0.27, and 0.08 for I–V, respectively.19 Since most p-values exceed 5%, the leverage factor appears to be weakly correlated with asset returns and its beta is small. Consequently, the conventional t-test on the risk premium of the leverage factor is under doubt. Table 3 starts out with the FM two-pass methodology. When the leverage factor LevFac is tested using the FM procedure (in this scenario, its risk premium corresponds to that of mimicking portfolios with base assets equal test assets), Table 3 reports that the estimated risk premium is positive and the conventional FM t-statistics are significant in I, II, and III at the 5% significance level. Similarly, when the mimicking portfolio LevMP is tested in the FM procedure, Table 3 shows positive risk premium, associated with FM t-statistics comparable to the factor counterparts. These results thus appear to support the leverage factor for asset pricing. If the KRS and Jiang, Kan, and Zhan (2015) (JKZ) corrections are adopted to account for estimation uncertainty and model misspecification, however, t-statistics only remain significant in I and II under LevFac. It is now known that the FM methodology and its associated t-test are doubtful when the factor is associated with a small beta. As an alternative to the t-test for the factor risk premia, the FAR test is proposed by Kleibergen (2009). Unlike the FM t-test, the FAR test is size-correct since its limiting distribution does not depend on the quality of the observed factors. Table 3 presents the 95% confidence sets of the risk premium that results from inverting the FAR test. These sets are found to be substantially different from those obtained by inverting the FM t-test. Such differences put the quality of the leverage factor under doubt, see also Kleibergen (2009) for a further comparison of the FM t-test and the FAR test. Furthermore, in Appendix H, we present the p-value plots of the FAR test which show how we obtained the 95% FAR confidence sets reported in Table 3. The last column of Table 3 shows the 95% confidence sets that result from the MPAR test. In line with the conventional FM t-test, it shows rejection of the null hypothesis of a zero risk premium for LevMP for I and III; however, unlike the FM t-test, it does not reject a zero risk premium in II. In IV and V, identical to the FAR test but unlike the FM t-test, we find that no information about the risk premium is contained in the leverage factor or its mimicking portfolio. In Appendix H, we also present the p-value plots of the MPAR test, which helps to explain the 95% MPAR confidence sets reported in Table 3. Overall, Table 3 indicates that inference on the risk premium of the leverage factor may substantially change, when robust tests (FAR or MPAR) are employed.20 This is consistent with the fact that the leverage factor is only weakly correlated with asset returns and thus likely to be a weak proxy for the underlying factor(s). Similar findings are presented in Appendix I, where we adopt alternative test assets. 4 Conclusions We document the threats involved in using mimicking portfolios of non-traded factors in the FM two-pass procedure. When these factors have small betas, we show that their mimicking portfolios have betas that are spurious. These spurious betas induce non-standard behavior of the risk premia estimator so conventional t-tests on risk premia become unreliable. A rank test on beta is used in the literature to serve as a diagnostic tool for the quality of the factors. We, however, find that the outcome of the rank test needs to be taken with caution when using mimicking portfolios. This results from the estimation error in the mimicking portfolio which we have to account for in the rank test. It implies a more challenging expression for the covariance matrix estimator employed in the rank test. When we do not account for this estimation error, the rank test performs poorly. When we account for it, the rank test still has some issues when the covariance matrix becomes of large dimension but generally works well. Instead of gauging the quality of factors or mimicking portfolios, inference methods are available for analyzing risk premia that are reliable irrespective of the quality of the factors. These methods are robust in the sense that their limiting distributions do not depend on the quality of factors as reflected by the magnitude of the betas. To the best of our knowledge, the method we propose here is the first one which deals with mimicking portfolios. Robust methods do exist for tests on risk premia in the standard factor pricing setting, see Kleibergen (2009) and Beaulieu, Dufour, and Khalaf (2013). This clearly indicates the need for our developed methods for which the empirical relevance is further emphasized by our application to the risk premium on the leverage factor from Adrian, Etula, and Muir (2014). APPENDIX A. Proof of Theorem 1 Proof. Let us start with V^R1R2, V^R2R2, and V^R2G. Assumptions 1 and 2 imply that: V^R1R2→pβ1VFFβ2′V^R2R2→pβ2VFFβ2′+Ω22, where we used that V^RR→pβVFFβ′+Ω and Ω12=0. The convergence of V^R2G is a bit more tricky. Note that: Rt=ιNλ0+β[(δG¯t+ut)+λF]+vt=ιNλ0+βλF+βδG¯t+βut+vt. When δ is fixed and the number of elements of G equals the number of elements of F, so δ is a square invertible matrix: V^R2G→pβ2δVGG, so β˜^1→pβ1VFFβ2′(β2VFFβ2′+Ω22)−1β2δVGG[VGGδ′β2′(β2VFFβ2′+Ω22)−1β2δVGG]−1=β1VFFδ−1′VGG−1, and when Gt=Ft, δ=Ik, β˜^1→pβ1. When δ=d/T: TV^R2G→dβ2(dVGG+ψuG)+ψv2G, where 1T∑t=1T(G¯t⊗ut)→dvec (ψuG), 1T∑t=1T(G¯t⊗v2t)→dvec (ψv2G), so T−12β˜^1→dβ1VFFβ2′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G) [(β2(dVGG+ψuG)+ψv2G)′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G)]−1. ▪ B. Proof of Corollary 1 Proof. (λ˜^0λ˜^G)=[(ιN1⋮β˜^1)′(ιN1⋮β˜^1)]−1(ιN1⋮β˜^1)′μ^R1=((ι′N1Mβ˜^1ιN1)−1ι′N1Mβ˜^1μ^R1(β˜^1′MιN1β˜^1)−1β˜^1′MιN1μ^R1), where from Assumption 1: μ^R1=μR1+ψR1T+op(1/T), while the limiting behavior of β˜^1 is provided by Theorem 1. The strong factor case: β˜^1→pβ1VFFδ−1′VGG−1, μ^R1→pιN1λ0+β1λF, so λ˜^G=(β˜^1′MιN1β˜^1)−1β˜^1′MιN1μ^R1→p((β1VFFδ−1′VGG−1)′MιN1(β1VFFδ−1′VGG−1))−1(β1VFFδ−1′VGG−1)′MιN1(ιN1λ0+β1λF)=((β1VFFδ−1′VGG−1)′MιN1(β1VFFδ−1′VGG−1))−1(β1VFFδ−1′VGG−1)′MιN1β1λF=VGGδ′VFF−1λF. The weak factor case with δ=d/T: for convenience, we write T−12β˜^1→dΨβ1, where Ψβ1 is the non-standard distribution in Theorem 1, so λ˜^G=(β˜^1′MιN1β˜^1)−1β˜^1′MιN1μ^R1≃ 1T(Ψβ1′MιN1Ψβ1)−1Ψβ1′MιN1(μR1+ψR1T) which implies Tλ˜^G−T(Ψ′β1MιN1Ψβ1)−1Ψ′β1MιN1μR1→d(Ψ′β1MιN1Ψβ1)−1Ψ′β1MιN1ψR1. ▪ C. Proof of Theorem 2 Proof. To derive the limiting behavior of β^1, we first study V^R1R2, V^R2R2, and V^R2G: V^R1R2→pβ1VFFβ2′V^R2R2→pβ2VFFβ2′+Ω22. When δ is fixed and the number of elements of G equals the number of elements of F, so δ is a square invertible matrix: V^R2G→pβ2δVGG so β^1→pβ1VFFβ2′(β2VFFβ2′+Ω22)−1β2δVGG. When δ=d/T: TV^R2G→dβ2(dVGG+ψuG)+ψv2G so T12β^1→dβ1VFFβ2′(β2VFFβ2′+Ω22)−1(β2(dVGG+ψuG)+ψv2G) ▪ D. Asymptotic Variance of Beta Estimators for Rank Testing D.1. Joint Behavior of V^R2G, V^R1R2, and V^R2R2 Here, we present the joint behavior of V^R2G, V^R1R2, and V^R2R2, since they make the beta-estimator of mimicking portfolios. We use the following notation: 1T∑t=1T(G¯t⊗e1tG¯t⊗e2te1t⊗e2t−vec(Σ12)e2t⊗e2t−vec(Σ22)G¯t⊗G¯t−vec(VGG))→d(vec(ψe1G)vec(ψe2G)vec(ψe1e2)vec(ψe2e2)vec(ψGG)), V^R2G, V^R1R2, and V^R2R2 can be rewritten as follows: V^R2G=1T∑t=1TR¯2tG¯t′=1T∑t=1T(β2δG¯t+e2t)G¯t′=β2δVGG+β2δ(1T∑t=1TG¯tGt′−VGG)+1T∑t=1Te2tG¯t′ V^R1R2=1T∑t=1TR¯1tR¯2t′=1T∑t=1T(β1δG¯t+e1t)(β2δG¯t+e2t)′=β1δVGGδ′β2′+β1δ(1T∑t=1TG¯tG¯t′−VGG)δ′β2′+β1δ(1T∑t=1TG¯te2t′)+(1T∑t=1Te1tG¯t′)δ′β2′+1T∑t=1Te1te2t′−Σ12+Σ12 V^R2R2=1T∑t=1TR¯2tR¯2t′=1T∑t=1T(β2δG¯t+e2t)(β2δG¯t+e2t)′=β2δVGGδ′β2′+β2δ(1T∑t=1TG¯tG¯t′−VGG)δ′β2′+β2δ(1T∑t=1TG¯te2t′)+(1T∑t=1Te2tG¯t′)δ′β2′+1T∑t=1Te2te2t′−Σ22+Σ22 The convergence of V^R2G, V^R2R1, and V^R2R2 is thus characterized by: T(V^R2G−β2δVGG)→dβ2δψGG+ψe2Gdenoted by UR2GT(V^R1R2−β1δVGGδ′β2′−Σ12)→dβ1δψGGδ′β2′+β1δψe2G′+ψe1Gδ′β2′+ψe1e2denoted by UR1R2T(V^R2R2−β2δVGGδ′β2′−Σ22)→dβ2δψGGδ′β2′+β2δψe2G′+ψe2Gδ′β2′+ψe2e2denoted by UR2R2 and in our simulation setup, their variances read: vec(UR2G)∼N(0,VGG⊗(VR2R2+β2δVGGδ′β2′)≡W1,1) vec(UR1R2)∼N(0,VR2R2⊗(VR1R1+VR1R2VR2R2−1VR2R1)≡W2,2) vec(UR2R2)∼N(0,2VR2R2⊗VR2R2≡W3,3) In addition, their covariances read: cov(vec(UR1R2), vec(UR2G))=2β2δVGG⊗β1δVGGδ′β2′+KN1N2(β1δVGG⊗Σ22)+β2δVGG⊗Σ12≡W2,1=W1,2′ cov(vec(UR2R2), vec(UR2G))=2β2δVGG⊗β2δVGGδ′β2′+(IN22+KN2N2)(β2δVGG⊗Σ22)≡W3,1=W1,3′ cov(vec(UR1R2), vec(UR2R2))=2VR2R2⊗VR1R2≡W2,3=W3,2′ To summarize: Tvec(V^R2G−VR2GV^R1R2−VR1R2V^R2R2−VR1R2)=vec(UR2GUR1R2UR2R2)→dN(0,WU),WU=(W1,1W1,2W1,3W2,1W2,2W2,3W3,1W3,2W3,3) This result will be useful, when we derive asymptotic variance of the beta-estimator with mimicking portfolios. Note that WU can be consistently estimated by data, so there exists W^U→pWU. Specifically, W^1,1=V^GG⊗(V^R2R2+B^2V^GGB^2′) W^2,2=V^R2R2⊗(V^R1R1+V^R1R2V^R2R2−1V^R2R1) W^3,3=2V^R2R2⊗V^R2R2 W^2,1=W^1,2′=2B^2V^GG⊗B^1V^GGB^2′+KN1N2(B^1V^GG⊗Σ^22)+B^2V^GG⊗Σ^12 W^3,1=W^1,3′=2B^2V^GG⊗B^2V^GGB^2′+(IN22+KN2N2)(B^2V^GG⊗Σ^22) W^2,3=W^3,2′=2V^R2R2⊗V^R1R2 D.2. Beta Specification with Factors Let Gt be the observed factors, R1t be test assets. Consider the estimand VR1GVGG−1 and the estimator V^R1GV^GG−1 Then: Tvec(V^R1GV^GG−1−VR1GVGG−1)→dN(0,VGG−1⊗(VR1R1−VR1GVGG−1VGR1)) This result allows us to conduct the rank test. That is, the estimator V^R1GV^GG−1 and its estimated variance V^GG−1⊗(V^R1R1−V^R1GV^GG−1V^GR1) are used for the rank test. D.3. Beta Specification with Mimicking Portfolios Let Gt be the observed factors, R1t be test assets. Define β˜1=VR1R2VR2R2−1VR2G(VGR2VR2R2−1VR2G)−1 and the estimator β˜^1=V^R1R2V^R2R2−1V^R2G(V^GR2V^R2R2−1V^R2G)−1 Using that V^R1R2=VR1R2+1TUR1R2, V^R2R2=VR2R2+1TUR2R2, V^R2G=VR2G+1TUR2G: V^R1R2V^R2R2−1V^R2G=[VR1R2+1TUR1R2][VR2R2+1TUR2R2]−1[VR2G+1TUR2G]≈VR1R2VR2R2−1VR2G +1T[UR1R2VR2R2−1VR2G+VR1R2VR2R2−1UR2G−VR1R2VR2R2−1UR2R2VR2R2−1VR2G] and similarly, V^GR2V^R2R2−1V^R2G=[VR2G+1TUR2G]′[VR2R2+1TUR2R2]−1[VR2G+1TUR2G]≈VGR2VR2R2−1VR2G +1T[UR2G′VR2R2−1VR2G+VR2G′VR2R2−1UR2G−VR2G′VR2R2−1UR2R2VR2R2−1VR2G]. Use the expansion of the inverse, β˜^1 is rewritten as: β˜^1≈β˜1 +1T[UR1R2VR2R2−1VR2G+VR1R2VR2R2−1UR2G−VR1R2VR2R2−1UR2R2VR2R2−1VR2G](VGR2VR2R2−1VR2G)−1 −1Tβ˜1[UR2G′VR2R2−1VR2G+VR2G′VR2R2−1UR2G−VR2G′VR2R2−1UR2R2VR2R2−1VR2G](VGR2VR2R2−1VR2G)−1 So Tvec(β˜^1−β˜1)≈[(VGR2VR2R2−1VR2G)−1⊗(VR1R2VR2R2−1−β˜1VGR2VR2R2−1)−(VGR2VR2R2−1VR2G)−1 VGR2VR2R2−1⊗β˜1·KN2K]vec(UR2G) +[(VGR2VR2R2−1VR2G)−1VGR2VR2R2−1⊗IN1]vec(UR1R2) +[(VGR2VR2R2−1VR2G)−1VGR2VR2R2−1⊗(β˜1VGR2VR2R2−1−VR1R2VR2R2−1)]vec(UR2R2) where KN2K is a commutation matrix such that vec(UR2G′)=KN2Kvec(UR2G). Consequently, in order to further derive variance of β˜^1, we use the joint behavior of UR2G, UR1R2, and UR2R2 (which has been derived earlier, see the detail of WU and W^U in the D1 sub-section): vec(UR2GUR1R2UR2R2)→dN(0,WU),WU=(W1,1W1,2W1,3W2,1W2,2W2,3W3,1W3,2W3,3) Combining all these pieces, the asymptotic variance of Tvec(β˜^1−β˜1) reads: vβ˜^1WUv′β˜^1 where vβ˜^1 is defined as ([(VGR2VR2R2−1VR2G)−1⊗(VR1R2VR2R2−1−β˜1VGR2VR2R2−1)−(VGR2VR2R2−1VR2G)−1VGR2VR2R2−1⊗β˜1·KN2K]′[(VGR2VR2R2−1VR2G)−1VGR2VR2R2−1⊗IN1]′[(VGR2VR2R2−1VR2G)−1VGR2VR2R2−1⊗(β˜1VGR2VR2R2−1−VR1R2VR2R2−1)]′)′ with v^β˜^1 equals ([(V^GR2V^R2R2−1V^R2G)−1⊗(V^R1R2V^R2R2−1−β˜^1V^GR2V^R2R2−1)−(V^GR2V^R2R2−1V^R2G)−1V^GR2V^R2R2−1⊗β˜^1·KN2K]′[(V^GR2V^R2R2−1V^R2G)−1V^GR2V^R2R2−1⊗IN1]′[(V^GR2V^R2R2−1V^R2G)−1V^GR2V^R2R2−1⊗(β˜^1V^GR2V^R2R2−1−V^R1R2V^R2R2−1)]′)′ This result allows us to conduct the rank test. That is, the estimator β˜^1 and its estimated variance v^β˜^1W^Uv^β˜^1′ are used for the rank test. D.4. Covariance Specification with Factors Let Gt be the observed factors, R1t be test assets. Consider the estimand VR1G and the estimator V^R1G Then: Tvec(V^R1G−VR1G)→dN(0,VGG⊗(VR1R1+VR1GVGG−1VGR1)). This result allows us to conduct the rank test. That is, the estimator V^R1G and its estimated variance V^GG⊗(V^R1R1+V^R1GV^GG−1V^GR1) are used for the rank test. D.5. Covariance Specification with Mimicking Portfolios Consider the estimator β^1=V^R1R2V^R2R2−1V^R2G Using that V^R1R2=VR1R2+1TUR1R2, V^R2R2=VR2R2+1TUR2R2, V^R2G=VR2G+1TUR2G, we can specify the above estimator as β^1=V^R1R2V^R2R2−1V^R2G=[VR1R2+1TUR1R2][VR2R2+1TUR2R2]−1[VR2G+1TUR2G]=VR1R2VR2R2−1VR2G +1T[UR1R2VR2R2−1VR2G+VR1R2VR2R2−1UR2G−VR1R2VR2R2−1UR2R2VR2R2−1VR2G]+op(T−1/2). Here, we used an expansion of the inverse. Hence, T(β^1−VR1R2VR2R2−1VR2G)≈VR1R2VR2R2−1UR2G+UR1R2VR2R2−1VR2G−VR1R2VR2R2−1UR2R2VR2R2−1VR2G Consequently, in order to further derive variance of β^1, we use the joint behavior of UR2G, UR1R2, and UR2R2 (which has been derived earlier, see the detail of WU and W^U in the D1 sub-section): vec(UR2GUR1R2UR2R2)→dN(0,WU),WU=(W1,1W1,2W1,3W2,1W2,2W2,3W3,1W3,2W3,3) Combining all these pieces, the asymptotic variance of Tvec(β^1−VR1R2VR2R2−1VR2G) reads: vβ^1WUv′β^1, where vβ^1 is defined as (Im⊗VR1R2VR2R2−1 VGR2VR2R2−1⊗IN1 −VGR2VR2R2−1⊗VR1R2VR2R2−1) with v^β^1 equals (Im⊗V^R1R2V^R2R2−1 V^GR2V^R2R2−1⊗IN1 −V^GR2V^R2R2−1⊗V^R1R2V^R2R2−1) This result allows us to conduct the rank test. That is, the estimator β^1 and its estimated variance v^β^1W^Uv^β^1′ are used for the rank test. E. Sensitivity to the Strength of the Factor Structure Instead of calibrating Ω to the estimated Ω^ as described in the main text, we also consider two alternatives in our simulation experiments: Ω=0.04Ω^ and Ω=25Ω^. The strength of the factor structure alters when the magnitude of Ω changes with Ω=0.04Ω^ having a very strong factor structure and Ω=25Ω^ a weak factor structure. The other settings remain unchanged in the simulation. Our purpose is to re-examine the result reported in Table 1, as the strength of the factor structure changes. Tables A.1 and A.2 present the updated results, for Ω=0.04Ω^ and Ω=25Ω^, respectively. It is found that the strength of the factor structure does not alter the two main findings conveyed in Table 1: (i) for rank testing, factors seem to perform better than mimicking portfolios; (ii) improperly treating mimicking portfolios as factors may yield to spurious rank test outcomes that favor poor factors. Table A.1. Rejection frequencies of the rank test by Monte Carlo ( Ω=0.04Ω^) Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.063 0.110 0.066 0.116 1.000 1.000 0.062 0.110 0.064 0.116 1.000 1.000 2. 0.053 0.103 0.016 0.035 0.956 0.960 0.051 0.102 0.016 0.034 0.955 0.960 3. 0.055 0.111 0.005 0.014 0.915 0.931 0.051 0.103 0.005 0.015 0.914 0.931 4. 0.052 0.100 0.006 0.009 0.911 0.921 0.047 0.095 0.005 0.009 0.908 0.920 5. 0.045 0.091 0.001 0.004 0.884 0.902 0.042 0.086 0.001 0.004 0.881 0.900 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.063 0.110 0.066 0.116 1.000 1.000 0.062 0.110 0.064 0.116 1.000 1.000 2. 0.053 0.103 0.016 0.035 0.956 0.960 0.051 0.102 0.016 0.034 0.955 0.960 3. 0.055 0.111 0.005 0.014 0.915 0.931 0.051 0.103 0.005 0.015 0.914 0.931 4. 0.052 0.100 0.006 0.009 0.911 0.921 0.047 0.095 0.005 0.009 0.908 0.920 5. 0.045 0.091 0.001 0.004 0.884 0.902 0.042 0.086 0.001 0.004 0.881 0.900 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1, Ω=0.04Ω^. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. Table A.1. Rejection frequencies of the rank test by Monte Carlo ( Ω=0.04Ω^) Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.063 0.110 0.066 0.116 1.000 1.000 0.062 0.110 0.064 0.116 1.000 1.000 2. 0.053 0.103 0.016 0.035 0.956 0.960 0.051 0.102 0.016 0.034 0.955 0.960 3. 0.055 0.111 0.005 0.014 0.915 0.931 0.051 0.103 0.005 0.015 0.914 0.931 4. 0.052 0.100 0.006 0.009 0.911 0.921 0.047 0.095 0.005 0.009 0.908 0.920 5. 0.045 0.091 0.001 0.004 0.884 0.902 0.042 0.086 0.001 0.004 0.881 0.900 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.063 0.110 0.066 0.116 1.000 1.000 0.062 0.110 0.064 0.116 1.000 1.000 2. 0.053 0.103 0.016 0.035 0.956 0.960 0.051 0.102 0.016 0.034 0.955 0.960 3. 0.055 0.111 0.005 0.014 0.915 0.931 0.051 0.103 0.005 0.015 0.914 0.931 4. 0.052 0.100 0.006 0.009 0.911 0.921 0.047 0.095 0.005 0.009 0.908 0.920 5. 0.045 0.091 0.001 0.004 0.884 0.902 0.042 0.086 0.001 0.004 0.881 0.900 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1, Ω=0.04Ω^. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. Table A.2. Rejection frequencies of the rank test by Monte Carlo ( Ω=25Ω^) Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 0.465 0.583 0.429 0.556 1.000 1.000 0.319 0.486 0.426 0.554 2. 1.000 1.000 0.663 0.760 0.636 0.742 1.000 1.000 0.489 0.648 0.629 0.742 3. 1.000 1.000 0.688 0.782 0.666 0.761 1.000 1.000 0.495 0.650 0.656 0.758 4. 1.000 1.000 0.878 0.930 0.856 0.914 1.000 1.000 0.642 0.793 0.848 0.911 5. 1.000 1.000 0.922 0.953 0.902 0.936 1.000 1.000 0.635 0.805 0.896 0.935 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 0.465 0.583 0.429 0.556 1.000 1.000 0.319 0.486 0.426 0.554 2. 1.000 1.000 0.663 0.760 0.636 0.742 1.000 1.000 0.489 0.648 0.629 0.742 3. 1.000 1.000 0.688 0.782 0.666 0.761 1.000 1.000 0.495 0.650 0.656 0.758 4. 1.000 1.000 0.878 0.930 0.856 0.914 1.000 1.000 0.642 0.793 0.848 0.911 5. 1.000 1.000 0.922 0.953 0.902 0.936 1.000 1.000 0.635 0.805 0.896 0.935 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.055 0.105 0.003 0.014 0.429 0.556 0.054 0.105 0.004 0.015 0.426 0.554 2. 0.055 0.100 0.001 0.004 0.335 0.443 0.051 0.099 0.000 0.003 0.330 0.439 3. 0.057 0.107 0.000 0.000 0.252 0.344 0.055 0.103 0.000 0.000 0.247 0.337 4. 0.055 0.098 0.001 0.001 0.275 0.367 0.050 0.092 0.001 0.001 0.266 0.358 5. 0.046 0.094 0.000 0.000 0.237 0.328 0.044 0.086 0.000 0.000 0.224 0.320 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.055 0.105 0.003 0.014 0.429 0.556 0.054 0.105 0.004 0.015 0.426 0.554 2. 0.055 0.100 0.001 0.004 0.335 0.443 0.051 0.099 0.000 0.003 0.330 0.439 3. 0.057 0.107 0.000 0.000 0.252 0.344 0.055 0.103 0.000 0.000 0.247 0.337 4. 0.055 0.098 0.001 0.001 0.275 0.367 0.050 0.092 0.001 0.001 0.266 0.358 5. 0.046 0.094 0.000 0.000 0.237 0.328 0.044 0.086 0.000 0.000 0.224 0.320 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1, Ω=25Ω^. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. Table A.2. Rejection frequencies of the rank test by Monte Carlo ( Ω=25Ω^) Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 0.465 0.583 0.429 0.556 1.000 1.000 0.319 0.486 0.426 0.554 2. 1.000 1.000 0.663 0.760 0.636 0.742 1.000 1.000 0.489 0.648 0.629 0.742 3. 1.000 1.000 0.688 0.782 0.666 0.761 1.000 1.000 0.495 0.650 0.656 0.758 4. 1.000 1.000 0.878 0.930 0.856 0.914 1.000 1.000 0.642 0.793 0.848 0.911 5. 1.000 1.000 0.922 0.953 0.902 0.936 1.000 1.000 0.635 0.805 0.896 0.935 Panel A: Strong factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 1.000 1.000 0.465 0.583 0.429 0.556 1.000 1.000 0.319 0.486 0.426 0.554 2. 1.000 1.000 0.663 0.760 0.636 0.742 1.000 1.000 0.489 0.648 0.629 0.742 3. 1.000 1.000 0.688 0.782 0.666 0.761 1.000 1.000 0.495 0.650 0.656 0.758 4. 1.000 1.000 0.878 0.930 0.856 0.914 1.000 1.000 0.642 0.793 0.848 0.911 5. 1.000 1.000 0.922 0.953 0.902 0.936 1.000 1.000 0.635 0.805 0.896 0.935 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.055 0.105 0.003 0.014 0.429 0.556 0.054 0.105 0.004 0.015 0.426 0.554 2. 0.055 0.100 0.001 0.004 0.335 0.443 0.051 0.099 0.000 0.003 0.330 0.439 3. 0.057 0.107 0.000 0.000 0.252 0.344 0.055 0.103 0.000 0.000 0.247 0.337 4. 0.055 0.098 0.001 0.001 0.275 0.367 0.050 0.092 0.001 0.001 0.266 0.358 5. 0.046 0.094 0.000 0.000 0.237 0.328 0.044 0.086 0.000 0.000 0.224 0.320 Panel B: Useless factor Beta specification Covariance specification Fac MP MP as Fac Fac MP MP as Fac 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 1. 0.055 0.105 0.003 0.014 0.429 0.556 0.054 0.105 0.004 0.015 0.426 0.554 2. 0.055 0.100 0.001 0.004 0.335 0.443 0.051 0.099 0.000 0.003 0.330 0.439 3. 0.057 0.107 0.000 0.000 0.252 0.344 0.055 0.103 0.000 0.000 0.247 0.337 4. 0.055 0.098 0.001 0.001 0.275 0.367 0.050 0.092 0.001 0.001 0.266 0.358 5. 0.046 0.094 0.000 0.000 0.237 0.328 0.044 0.086 0.000 0.000 0.224 0.320 Notes: This table reports the rejection frequencies of the Kleibergen and Paap (2006) rank test of the null H0:rank(β1)=k−1 at the nominal 5% and 10% respectively, based on the average of 2000 replications. The d.g.p. is described in the main text, with T = 1000 and k=1, Ω=25Ω^. Column 1 lists five choices of N1 = N2: 1, 2, 3, 4, or 5. We consider two types of observed factors: Panel A for a strong factor and Panel B for a useless factor in the single factor model. For “Fac”: the observed factor is used for beta estimation. For “MP”: the mimicking portfolio of the observed factor is used for the beta estimation, and the variance of the beta estimator is derived by first-order asymptotics (see Appendix D). For “MP as Fac”: the mimicking portfolio of the observed factor is used for the beta estimation, as if the mimicking portfolio is an observed factor. F. Proof of Theorem 4 Proof. For R¯1, by the central limit theorem, we have: T(R¯1−Γβ2δλG,cov)→dψ1∼N(0,VR1R1) B^2 results from linear regression of R2t on Gt, so: Tvec(B^2−β2δ)→dvec(ψ2)∼N(0,VGG−1⊗Σ22) where Σ22 is the covariance matrix of residuals. Similarly for Γ^ that results from linear regression of R1t on R2t: Tvec(Γ^−Γ)→dvec(ψ3)∼N(0,VR2R2−1⊗(VR1R1−VR1R2VR2R2−1VR2R1)). Note that &ps

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Identification-Robust Inference on Risk Premia of Mimicking Portfolios of Non-traded Factors

Identification-Robust Inference on Risk Premia of Mimicking Portfolios of Non-traded Factors