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ESTIMATION OF MONOTONIC REGRESSION MODELS UNDER QUANTILE RESTRICTIONS
Summary This paper provides a set of sufficient conditions that point identify a quantile regression model with fixed effects. It also proposes a simple transformation of the data that gets rid of the fixed effects under the assumption that these effects are location shifters. The new estimator is consistent and asymptotically normal as both n and T grow. 1. Introduction Panel data models and quantile regression models are both widely used in applied econometrics and popular topics of research in theoretical papers. Quantile regression models allow the researcher to account for unobserved heterogeneity and heterogeneous covariates effects, while the availability of panel data potentially allows the researcher to include fixed effects to control for some unobserved covariates. There has been little but growing work at the intersection of these two methodologies (e.g. Koenker, 2004, Geraci and Bottai, 2007, Abrevaya and Dahl, 2008, Galvao, 2008, Rosen, 2009, and Lamarche, 2010). This initial lack of attention is possibly due to a fundamental issue associated with conditional quantiles. This is, as it is the case with non‐linear panel data models, standard demeaning (or differencing) techniques do not result in feasible approaches. These techniques rely on the fact that expectations are linear operators, which is not the case for conditional quantiles. This paper provides sufficient conditions under which the parameter of interest is identified for fixed T and shows that there is a simple transformation of the data that eliminates the fixed effects as T →∞, when the fixed effects are viewed as location shift variables (i.e. variables that affect all quantiles in the same way). The resulting two‐step estimator is consistent and asymptotically normal when both n and T go to infinity. Also, the new estimator is extremely simple to compute and can be implemented in standard econometrics packages. The paper is organized as follows. Section 2 presents the model. Section 3 provides an identification result based on deconvolution arguments. Section 4 introduces a two‐step estimator for panel data quantile regression models. Asymptotic properties of the new estimator are presented in the same section. Section 5 includes a small Monte Carlo experiment to study the finite sample properties of the two‐step estimator. Finally, Section 6 concludes. Appendix A provides proofs of results. An estimator of the covariance kernel and the bootstrap method are given in Appendix B. 2. The Model Consider the following model (2.1) where are observable variables and are unobservable. Throughout the paper the vector Xit is assumed to include a constant term, i.e. with . The function τ↦X′θ(τ) is assumed to be strictly increasing in τ∈ (0, 1) and the parameter of interest is assumed to be θ(τ). If αi were observable it would follow that (2.2) under the assumption that Uit ∼ U[0, 1] conditional on and αi. This type of representation has been extensively used in the literature (e.g. Chernozhukov and Hansen, 2006, 2008). The difference with the model in equation (2.1) and the standard quantile regression model introduced by Koenker and Bassett (1978) lies in the presence of the unobserved αi. This random variable could be arbitrarily related to the rest of the random variables in equation (2.1) (i.e. αi=αi(Uit, Xi, ηi) for some i.i.d. sequence ηi) rendering condition (2.2) as not particularly useful in terms of identification. The question is under what additional conditions on (Uit, αi) the parameter θ(τ) can be identified and consistently estimated from the data. Rosen (2009) recently showed that conditional on covariates quantile restriction alone does not identify θ(τ). That is, let QZ(τ | A) denote the τ‐quantile of a random variable Z conditional on another random variable A, let , and write the model in equation (2.1) as (2.3) Then, the conditional quantile restriction does not have sufficient identification power.1Rosen (2009) then provides different assumptions, i.e. support conditions and some form of conditional independence of eit(τ) across time, that (point and partially) identify θ(τ). Abrevaya and Dahl (2008) use the correlated random‐effects model of Chamberlain (1982, 1984) as a way to get an estimator of θ(τ). This model views the unobservable αi as a linear projection onto the observables plus a disturbance, i.e. The authors view the model as an approximation to the true conditional quantile and proceed to get estimates of θ(τ) and ΛT(τ) by running a quantile regression of Yit on Xit and Xi. In cases where there is no disturbance ηi, such a regression identifies θ(τ). However, it is immediate to see that a quantile restriction alone does not identify θ(τ) whenever ηi is present non‐trivially since the conditional behaviour of depends on the joint distribution of the unobservables Uit and ηi. This is problematic since not even a correctly specified function for αi(τ, Xi, ηi) helps in identifying θ(τ), meaning that the correlated random‐effects model might work poorly in many contexts. The simulations of Section 5 illustrate this point. Koenker (2004) takes a different approach and treats as parameters to be jointly estimated with θ(τ) for q different quantiles. He proposes the penalized estimator (2.4) where ρτ(u) = u[τ− I(u < 0)], I(·) denotes the indicator function, and λ≥ 0 is a penalization parameter that shrinks the s towards a common value. Solving equation (2.4) can be computationally demanding when n is large (even for λ= 0) and has the additional complication involved in the choice of λ.2 Finally, there is a related literature on non‐separable panel data models. These type of models are flexible enough to provide quantile treatment effects (see, e.g. Chernozhukov et al., 2010, and Graham and Powell, 2010). For example, Chernozhukov et al. (2010) show that the quantile treatment effect of interest is partially identified (for fixed T) and provide bounds for those effects in the model (2.5) where Xit is assumed discrete. They also derive rates of shrinkage of the identified set to a point as T goes to infinity. The model in Chernozhukov et al. (2010) is more general than the one in equation (2.1) as it is non‐separable in αi and it involves weaker assumptions on the unobservable Uit. However, it leads to less powerful identification results and more complicated estimators. In this context this paper contributes to the literature in two ways. The next section shows that when the model in equation (2.1) is viewed as a deconvolution model, a result from Neumann (2007) can be applied to show that θ(τ) is identified when there are at least two time periods available and αi has a pure location shift effect.3 This identification result could be potentially used to construct estimators of θ(τ) based on non‐parametric estimators of conditional distribution functions. Such non‐parametric estimators would rarely satisfy the end‐goal of this paper, that is, to provide an easy‐to‐use estimator that can be implemented in standard econometric packages, would suffer from the common curse of dimensionality, and would typically involve a delicate choice of tuning parameters for their implementation.4 Thus, when moving from identification to estimation, this paper takes a different approach and shows that there exists a simple transformation of the data that eliminates the fixed effects αi as T →∞. The transformation leads to an extremely simple asymptotically normal estimator for θ(τ) that can be easily computed even for very large values of n. Standard errors for this new estimator can be computed from the asymptotically normal representation. 3. Identification In this section, I prove that the parameter of interest θ(τ) is identified for T ≥ 2 under independence restrictions and existence of moments. The intuition behind the result is quite simple. Letting (the dependence on i is omitted for convenience here), it follows from equation (2.1) that Yt = St +α is a convolution of St and α conditional on X, provided α and Ut are independent conditional on X. It then follows that the conditional distributions of St and α can be identified from the conditional distribution of Yt by using a deconvolution argument similar to that in Neumann (2007). This in turn results in identification of θ(τ) after exploiting the fact that Ut is conditionally U[0, 1] together with some regularity conditions. For ease of exposition let T = 2 and consider the following assumption where the lower case x = (x1, x2) denotes a realization of the random variable X = (X1, X2). Assumption 3.1. Denote byand ϕα|xthe conditional onX = xcharacteristic functions of the distributionsandPα|x, respectively. Then(a)conditional onX = xthe random variablesS1, S2and α are independent for all, wheredenotes the support ofX = (X1, X2); (b)the setis dense infor all. Assumption 3.1 follows Neumann (2007), but it does not impose for all , which does not hold in equation (2.1). Assumption 3.1(a) implies that αi does not change across quantiles as αi is independent of (U1, U2). Assumption 3.1(b) excludes characteristic functions that vanish on non‐empty open subsets of but allows the characteristic function to have countably many zeros. This includes cases ruled out in the deconvolution analysis of Li and Vuong (1998) and Evdokimov (2010), among others. More generally, any deconvolution analysis based on Kotlarski’s Lemma (which has been widely applied to identify and estimate a variety of models in economics) will fail to include such cases.5 For example, if α∼ U[ − 1, 1], or if α is discrete uniform, ϕα has countably many zeros and so Kotlarski’s Lemma would not apply while Assumption 3.1(b) would be satisfied. Having this extra dimension of generality is important here as the distribution of αi is left unspecified in this paper. Assumption 3.1 implies that Yt is a convolution of St and α conditional on X = x and so (3.1) (3.2) Lemma 3.1. Suppose thatandPα|xare distributions with characteristic functionsand ϕα|xsatisfying Assumption 3.1. Letandbe further distributions with respective characteristic functionsand. If now (3.3)and all, then there exist constantssuch that (3.4) That is, andfort ∈{1, 2} as well asandPα|xare equal up to a location shift. Remark 3.1. Lemma 3.1 extends immediately to models that are slightly more general than the one in equation (2.1). For example, consider the case where Yit = q(Uit, Xit) +αi and q(τ, x) is strictly increasing in τ for all . This model reduces to equation (2.1) if . Lemma 3.1 holds in this case by letting St = q(Ut, Xt). Another example would be a random coefficients model where and . This model can be written as and so the additional generality relative to equation (2.1) comes from varying across i and t. Lemma 3.1 also applies to this model conditional on the event X1= X2. Remark 3.2. It is worth noting that the identification result in Lemma 3.1 also applies to the correlated random effects model in Abrevaya and Dahl (2008) where (3.5) Here Assumption 3.1 must hold for and ϕη|x where St = (θ(Ut) +Λt(Ut))Xt +Λ−t(Ut)X−t. Lemma 3.1 identifies the distributions up to location. Typically, to be able to identify the entire distribution one would need to add a location assumption. In the case of quantile regression the standard cross‐section specification assumes that U ∼ U[0, 1] independent of X. The extension of this assumption to the panel case together with some additional regularity conditions allows identification of the location of St conditional on X = x as well as the parameter θ(τ). This is the role of Assumption 3.2. Assumption 3.2. (a)Uit⊥(Xi, αi) andUit ∼ U[0, 1]; (b) ΩUU≡ E[(θ(Uit) −θμ)(θ(Uit) −θμ)′], where θμ≡ E[θ(Uit)], is non‐singular with finite norm; (c) lettingfort = 1, 2, there exists nosuch thatAhas probability 1 under the distribution ofandAis a proper linear subspace of; (d) (Yt, Xt) have finite first moments fort ={1, 2}. Assumption 3.2(a) is standard in quantile regression models except that here Uit is also assumed independent of αi. Assumption 3.2(b) implies that exists and this implies that the location of St is well defined. The restriction on ΩUU is not used in Lemma 3.2 but it is important for the derivation of the asymptotic variance of the two‐step estimator of the next section. Assumption 3.2(c) is a standard rank‐type condition on the subvector of regressors that excludes the constant term. Assumption 3.2(d) is implied by Assumption 4.1 in the next section. It is immediate to see that Assumption 3.2(a) (b) implies and , where . Assumption 3.2(c) (d) then implies that θμ is identified.6 Since , the location of St conditional on X = x is identified. Also, note that Assumption 3.2 implies (3.6) The following Lemma then follows immediately. Lemma 3.2. Under Assumptions 3.1 and 3.2 the location θμand the function θ(τ) for τ∈ (0, 1) are identified. Remark 3.3. The extension to the case T > 2 is straightforward under the same assumptions. The exception is Assumption 3.1(ii), which should be replaced by is dense in for all ; see Neumann (2007). The results of this section show that the parameter of interest θ(τ) is point identified from the distribution of the observed data and Assumptions 3.1 and 3.2. The next step then is to derive a consistent estimator for this parameter and study its asymptotic properties. Section 4 presents a two‐step estimator that follows a different intuition relative to the one behind the identification result, but has the virtue of being extremely simple to compute and has an asymptotically normal distribution as both n and T go to infinity. 4. Two‐Step Estimator The two‐step estimator that I introduce in this section exploits two direct consequences of Assumption 3.2 and the fact that αi is a location shift (Assumption 3.1(a)). The first implication is in equation (2.3), where only θ(τ) and eit(τ) depend on τ. The second implication arises by letting and writing a conditional mean equation for Yit as follows7 (4.1)Equation (4.1) implies that αi is also present in the conditional mean of Yit. Therefore, from equation (4.1) I can compute a ‐consistent estimator of αi given a ‐consistent estimator of θμ. This includes for example the standard within estimator of θμ given in equation (A.17) in Appendix A. Then, using equation (2.3) I estimate θ(τ) by a quantile regression of the random variable on Xit. To be precise, the two steps are described below where I use the notation and . Step 1. Let be a ‐consistent estimator of θμ. Define . (4.2) Step 2. Let and define the two‐step estimator as: (4.2) Intuitively, the two‐step estimator in equation (4.2) works because as T →∞, where denotes weak convergence. This is so because , where (4.3) Then, the random variable converges in probability, as T →∞, to the variable which implies weak convergence, . The next proposition shows that the two‐step estimator defined in equation (4.2) is consistent and asymptotically normal under the following assumptions. Assumption 4.1. Let ϕτ(u) =τ− I(u < 0), I(·) denote the indicator function, W = (Y*, X), andgτ(W, θ, r) ≡ϕτ(Y*− X′θ+ r)X. (a)are i.i.d. defined on the probability space, take values in a compact set, and E(αi) = 0; (b)for all, θ(τ) ∈ int Θ, where Θ is compact and convex andis a closed subinterval of (0, 1); (c)has bounded conditional onXdensity a.s., and Π(θ, τ, r) ≡ E[gτ(W, θ, r)] has Jacobian matrixthat is continuous and has full rank uniformly over, andis uniformly continuous over. Assumption 4.1 imposes similar conditions to those in Chernozhukov and Hansen (2006). Note that 4.1(a) imposes i.i.d. on the unobserved variable and not on Yit. Condition 4.1(c) is used for asymptotic normality. Finally, in order to derive the expression for the covariance kernel of the limiting Gaussian process provided in equation (4.7), I use the following assumption on the preliminary estimator . Assumption 4.2. The preliminary estimatoradmits the expansion (4.4)where ψitis an i.i.d. sequence of random variables withE[ψit] = 0 and finite. Theorem 4.1. Letn/Ts → 0 for somes ∈ (1, ∞). Under Assumptions 3.2, 4.1 and 4.2 and (4.5) (4.6)where, , , μX≡ E[Xit], J1(τ) ≡ J1(θ(τ), τ, 0), J2(τ) ≡ J2(θ(τ), τ, 0), is a mean‐zero Gaussian process with covariance function, Ψ(τ, τ′) is defined in equation (4.7) below, andin the set of all uniformly bounded functions on. The matrix Ψ(τ, τ′) is given by (4.7)whereS(τ, τ′) ≡ (min{τ, τ′}−ττ′)E(XX′), Ωgξ(τ) ≡ E[gτ(W, θ(τ))ξ], and Ωξξ≡ E[ξ2]. The asymptotic expansion for presented in Theorem 4.1 has two terms. The first term is the standard term in quantile regressions while the second term captures the fact αi is being estimated by . Assumption 4.2 is not necessary for convergence to a limit Gaussian process but it is used to derive the expression of Ψ(τ, τ′) in equation (4.7). This is a mild assumption as, for example, the usual within estimator of θμ satisfies Assumption 4.2 under Assumptions 3.2 and 4.1 (see Lemma A.4 in Appendix A). Appendix B contains expressions for estimating each element entering the covariance function J1(τ)−1Ψ(τ, τ′)[J1(τ′)−1]′ that can be used to make inference on . Alternatively, I conjecture that the standard i.i.d. bootstrap on (Yi, Xi) would work in this context and therefore provide a different approach to inference ( Appendix B). The proof of this conjecture is however beyond the scope of this paper and the supporting evidence is limited to the simulations of the Section 5. 5. Simulations To illustrate the performance of the two‐step estimator I conduct a small simulation study. Tables 1 and 2 summarize the results by reporting percentage bias (%Bias) and mean squared error (MSE) for each estimator considered. The simulated model is (5.1) where Xit ∼ Beta(1, 1), ηi∼ N(0, 1) and the distribution of εit changes with the model specification. In Model 1, εit∼ N(2, 1); in Model 2, εit∼ exp(1) + 2; while in Model 3, εit∼ BitN(1, .1) + (1 − Bit)N(3, .1) with Bit ∼ Bernoulli(p = 0.3). The conditional τth quantiles are given by θ[0](τ) +θ[1](τ)X +α. In Model 1, for example, θ[0](τ) =Φ−1(τ) + 1 and θ[1](τ) =Φ−1(τ) + 2 for Φ−1(τ) the inverse of the normal CDF evaluated at τ. For all models I set n ={100, 5000}, T ={5, 10, 20}, τ={0.25, 0.90}, γ= 2 and consider four estimators. These are a standard quantile regression estimator of Yit on Xit (QR), a correlated random effect estimator (CRE) from Abrevaya and Dahl (2008), an infeasible estimator of on Xit (INFE) and the two‐step estimator (2‐STEP) from Section 4.8 Note that QR and CRE are inconsistent estimators. Finally, I set the number of Monte Carlo simulations to 1000. Table 1. Bias and MSE for Model 5.1: γ= 2 and 0.25. . Estimator . n = 100 . n = 5000 . θ(·) ∼ N(2, 1), θ[1](τ) = 1.3255 . θ(·) ∼ N(2, 1), θ[1](τ) = 1.3255 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 1.7569 0.2157 0.0068 0.1494 1.7444 0.2074 0.0004 0.1496 MSE 5.8220 0.2794 0.1032 0.1473 5.3543 0.0798 0.0020 0.0414 T = 10 %Bias 1.7660 0.2127 0.0025 0.0793 1.7687 0.2063 −0.0020 0.0757 MSE 5.7557 0.1639 0.0494 0.0605 5.5021 0.0765 0.0010 0.0111 T = 20 %Bias 1.7750 0.1903 0.0032 0.0377 1.8084 0.2070 −0.0001 0.0397 MSE 5.7678 0.1044 0.0228 0.0264 5.7506 0.0762 0.0005 0.0032 . Estimator . n = 100 . n = 5000 . θ(·) ∼ N(2, 1), θ[1](τ) = 1.3255 . θ(·) ∼ N(2, 1), θ[1](τ) = 1.3255 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 1.7569 0.2157 0.0068 0.1494 1.7444 0.2074 0.0004 0.1496 MSE 5.8220 0.2794 0.1032 0.1473 5.3543 0.0798 0.0020 0.0414 T = 10 %Bias 1.7660 0.2127 0.0025 0.0793 1.7687 0.2063 −0.0020 0.0757 MSE 5.7557 0.1639 0.0494 0.0605 5.5021 0.0765 0.0010 0.0111 T = 20 %Bias 1.7750 0.1903 0.0032 0.0377 1.8084 0.2070 −0.0001 0.0397 MSE 5.7678 0.1044 0.0228 0.0264 5.7506 0.0762 0.0005 0.0032 . Estimator . θ(·) ∼ exp(1) + 2, θ[1](τ) = 2.2877 . θ(·) ∼ exp(1) + 2, θ[1](τ) = 2.2877 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 1.0202 0.1291 −0.0003 0.0976 1.0285 0.1367 −0.0000 0.0961 MSE 5.8106 0.2066 0.0174 0.1003 5.5434 0.1002 0.0003 0.0493 T = 10 %Bias 1.0310 0.1184 −0.0002 0.0487 1.0393 0.1368 −0.0001 0.0489 MSE 5.8306 0.1198 0.0087 0.0286 5.6593 0.0990 0.0002 0.0128 T = 20 %Bias 1.0524 0.0919 0.0008 0.0215 1.0593 0.1361 −0.0001 0.0209 MSE 6.0432 0.0641 0.0045 0.0087 5.8777 0.0975 0.0001 0.0024 . Estimator . θ(·) ∼ exp(1) + 2, θ[1](τ) = 2.2877 . θ(·) ∼ exp(1) + 2, θ[1](τ) = 2.2877 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 1.0202 0.1291 −0.0003 0.0976 1.0285 0.1367 −0.0000 0.0961 MSE 5.8106 0.2066 0.0174 0.1003 5.5434 0.1002 0.0003 0.0493 T = 10 %Bias 1.0310 0.1184 −0.0002 0.0487 1.0393 0.1368 −0.0001 0.0489 MSE 5.8306 0.1198 0.0087 0.0286 5.6593 0.0990 0.0002 0.0128 T = 20 %Bias 1.0524 0.0919 0.0008 0.0215 1.0593 0.1361 −0.0001 0.0209 MSE 6.0432 0.0641 0.0045 0.0087 5.8777 0.0975 0.0001 0.0024 . Estimator . θ(·) ∼ Mixture, θ[1](τ) = 1.3097 . θ(·) ∼ Mixture, θ[1](τ) = 1.3097 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 2.0565 0.5369 0.0190 0.2561 2.1061 0.5370 0.0012 0.1897 MSE 7.6837 0.7855 0.5115 0.4963 7.5625 0.4965 0.0036 0.0682 T = 10 %Bias 2.1221 0.5683 0.0026 0.0424 2.1433 0.5413 0.0023 0.0280 MSE 7.9763 0.6679 0.1450 0.2088 7.8295 0.5016 0.0018 0.0046 T = 20 %Bias 2.1770 0.6010 −0.0039 −0.0008 2.1750 0.5412 −0.0013 −0.0032 MSE 8.3200 0.6744 0.0536 0.0763 8.0616 0.5001 0.0009 0.0013 . Estimator . θ(·) ∼ Mixture, θ[1](τ) = 1.3097 . θ(·) ∼ Mixture, θ[1](τ) = 1.3097 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 2.0565 0.5369 0.0190 0.2561 2.1061 0.5370 0.0012 0.1897 MSE 7.6837 0.7855 0.5115 0.4963 7.5625 0.4965 0.0036 0.0682 T = 10 %Bias 2.1221 0.5683 0.0026 0.0424 2.1433 0.5413 0.0023 0.0280 MSE 7.9763 0.6679 0.1450 0.2088 7.8295 0.5016 0.0018 0.0046 T = 20 %Bias 2.1770 0.6010 −0.0039 −0.0008 2.1750 0.5412 −0.0013 −0.0032 MSE 8.3200 0.6744 0.0536 0.0763 8.0616 0.5001 0.0009 0.0013 Note 1000 MC replications. QR: standard quantile regression estimator; CRE: correlated random effect estimator; INFE: infeasible estimator, 2‐STEP: two‐step estimator from 4.2. All regressions include an intercept term. Open in new tab Table 1. Bias and MSE for Model 5.1: γ= 2 and 0.25. . Estimator . n = 100 . n = 5000 . θ(·) ∼ N(2, 1), θ[1](τ) = 1.3255 . θ(·) ∼ N(2, 1), θ[1](τ) = 1.3255 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 1.7569 0.2157 0.0068 0.1494 1.7444 0.2074 0.0004 0.1496 MSE 5.8220 0.2794 0.1032 0.1473 5.3543 0.0798 0.0020 0.0414 T = 10 %Bias 1.7660 0.2127 0.0025 0.0793 1.7687 0.2063 −0.0020 0.0757 MSE 5.7557 0.1639 0.0494 0.0605 5.5021 0.0765 0.0010 0.0111 T = 20 %Bias 1.7750 0.1903 0.0032 0.0377 1.8084 0.2070 −0.0001 0.0397 MSE 5.7678 0.1044 0.0228 0.0264 5.7506 0.0762 0.0005 0.0032 . Estimator . n = 100 . n = 5000 . θ(·) ∼ N(2, 1), θ[1](τ) = 1.3255 . θ(·) ∼ N(2, 1), θ[1](τ) = 1.3255 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 1.7569 0.2157 0.0068 0.1494 1.7444 0.2074 0.0004 0.1496 MSE 5.8220 0.2794 0.1032 0.1473 5.3543 0.0798 0.0020 0.0414 T = 10 %Bias 1.7660 0.2127 0.0025 0.0793 1.7687 0.2063 −0.0020 0.0757 MSE 5.7557 0.1639 0.0494 0.0605 5.5021 0.0765 0.0010 0.0111 T = 20 %Bias 1.7750 0.1903 0.0032 0.0377 1.8084 0.2070 −0.0001 0.0397 MSE 5.7678 0.1044 0.0228 0.0264 5.7506 0.0762 0.0005 0.0032 . Estimator . θ(·) ∼ exp(1) + 2, θ[1](τ) = 2.2877 . θ(·) ∼ exp(1) + 2, θ[1](τ) = 2.2877 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 1.0202 0.1291 −0.0003 0.0976 1.0285 0.1367 −0.0000 0.0961 MSE 5.8106 0.2066 0.0174 0.1003 5.5434 0.1002 0.0003 0.0493 T = 10 %Bias 1.0310 0.1184 −0.0002 0.0487 1.0393 0.1368 −0.0001 0.0489 MSE 5.8306 0.1198 0.0087 0.0286 5.6593 0.0990 0.0002 0.0128 T = 20 %Bias 1.0524 0.0919 0.0008 0.0215 1.0593 0.1361 −0.0001 0.0209 MSE 6.0432 0.0641 0.0045 0.0087 5.8777 0.0975 0.0001 0.0024 . Estimator . θ(·) ∼ exp(1) + 2, θ[1](τ) = 2.2877 . θ(·) ∼ exp(1) + 2, θ[1](τ) = 2.2877 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 1.0202 0.1291 −0.0003 0.0976 1.0285 0.1367 −0.0000 0.0961 MSE 5.8106 0.2066 0.0174 0.1003 5.5434 0.1002 0.0003 0.0493 T = 10 %Bias 1.0310 0.1184 −0.0002 0.0487 1.0393 0.1368 −0.0001 0.0489 MSE 5.8306 0.1198 0.0087 0.0286 5.6593 0.0990 0.0002 0.0128 T = 20 %Bias 1.0524 0.0919 0.0008 0.0215 1.0593 0.1361 −0.0001 0.0209 MSE 6.0432 0.0641 0.0045 0.0087 5.8777 0.0975 0.0001 0.0024 . Estimator . θ(·) ∼ Mixture, θ[1](τ) = 1.3097 . θ(·) ∼ Mixture, θ[1](τ) = 1.3097 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 2.0565 0.5369 0.0190 0.2561 2.1061 0.5370 0.0012 0.1897 MSE 7.6837 0.7855 0.5115 0.4963 7.5625 0.4965 0.0036 0.0682 T = 10 %Bias 2.1221 0.5683 0.0026 0.0424 2.1433 0.5413 0.0023 0.0280 MSE 7.9763 0.6679 0.1450 0.2088 7.8295 0.5016 0.0018 0.0046 T = 20 %Bias 2.1770 0.6010 −0.0039 −0.0008 2.1750 0.5412 −0.0013 −0.0032 MSE 8.3200 0.6744 0.0536 0.0763 8.0616 0.5001 0.0009 0.0013 . Estimator . θ(·) ∼ Mixture, θ[1](τ) = 1.3097 . θ(·) ∼ Mixture, θ[1](τ) = 1.3097 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 2.0565 0.5369 0.0190 0.2561 2.1061 0.5370 0.0012 0.1897 MSE 7.6837 0.7855 0.5115 0.4963 7.5625 0.4965 0.0036 0.0682 T = 10 %Bias 2.1221 0.5683 0.0026 0.0424 2.1433 0.5413 0.0023 0.0280 MSE 7.9763 0.6679 0.1450 0.2088 7.8295 0.5016 0.0018 0.0046 T = 20 %Bias 2.1770 0.6010 −0.0039 −0.0008 2.1750 0.5412 −0.0013 −0.0032 MSE 8.3200 0.6744 0.0536 0.0763 8.0616 0.5001 0.0009 0.0013 Note 1000 MC replications. QR: standard quantile regression estimator; CRE: correlated random effect estimator; INFE: infeasible estimator, 2‐STEP: two‐step estimator from 4.2. All regressions include an intercept term. Open in new tab Table 2. Bias and MSE for Model 5.1: γ= 2 and 0.90. . Estimator . n = 100 . n = 5000 . θ(·) ∼ N(2, 1), θ[1](τ) = 3.2815 . θ(·) ∼ N(2, 1), θ[1](τ) = 3.2815 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.4069 −0.1697 −0.0065 −0.1223 0.4321 −0.1575 −0.0002 −0.1144 MSE 2.4048 0.6455 0.1509 0.3162 2.0220 0.2743 0.0032 0.1439 T = 10 %Bias 0.4039 −0.1547 −0.0069 −0.0645 0.4079 −0.1594 −0.0006 −0.0603 MSE 2.1894 0.3995 0.0763 0.1228 1.8001 0.2768 0.0015 0.0406 T = 20 %Bias 0.3777 −0.1307 0.0013 −0.0280 0.3833 −0.1579 0.0001 −0.0301 MSE 1.8410 0.2466 0.0393 0.0479 1.5884 0.2700 0.0008 0.0105 . Estimator . n = 100 . n = 5000 . θ(·) ∼ N(2, 1), θ[1](τ) = 3.2815 . θ(·) ∼ N(2, 1), θ[1](τ) = 3.2815 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.4069 −0.1697 −0.0065 −0.1223 0.4321 −0.1575 −0.0002 −0.1144 MSE 2.4048 0.6455 0.1509 0.3162 2.0220 0.2743 0.0032 0.1439 T = 10 %Bias 0.4039 −0.1547 −0.0069 −0.0645 0.4079 −0.1594 −0.0006 −0.0603 MSE 2.1894 0.3995 0.0763 0.1228 1.8001 0.2768 0.0015 0.0406 T = 20 %Bias 0.3777 −0.1307 0.0013 −0.0280 0.3833 −0.1579 0.0001 −0.0301 MSE 1.8410 0.2466 0.0393 0.0479 1.5884 0.2700 0.0008 0.0105 . Estimator . θ(·) ∼ exp(1) + 2, θ[1](τ) = 4.3026 . θ(·) ∼ exp(1) + 2, θ[1](τ) = 4.3026 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.3120 −0.1517 −0.0015 −0.0814 0.3134 −0.1366 −0.0002 −0.0804 MSE 2.5781 0.9598 0.4730 0.4583 1.8333 0.3566 0.0095 0.1264 T = 10 %Bias 0.2911 −0.1442 0.0012 −0.0407 0.2984 −0.1361 −0.0005 −0.0417 MSE 2.0493 0.6100 0.2337 0.2444 1.6585 0.3479 0.0049 0.0363 T = 20 %Bias 0.2748 −0.1261 0.0008 −0.0224 0.2823 −0.1371 −0.0003 −0.0216 MSE 1.8136 0.4023 0.1166 0.1181 1.4834 0.3501 0.0023 0.0107 . Estimator . θ(·) ∼ exp(1) + 2, θ[1](τ) = 4.3026 . θ(·) ∼ exp(1) + 2, θ[1](τ) = 4.3026 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.3120 −0.1517 −0.0015 −0.0814 0.3134 −0.1366 −0.0002 −0.0804 MSE 2.5781 0.9598 0.4730 0.4583 1.8333 0.3566 0.0095 0.1264 T = 10 %Bias 0.2911 −0.1442 0.0012 −0.0407 0.2984 −0.1361 −0.0005 −0.0417 MSE 2.0493 0.6100 0.2337 0.2444 1.6585 0.3479 0.0049 0.0363 T = 20 %Bias 0.2748 −0.1261 0.0008 −0.0224 0.2823 −0.1371 −0.0003 −0.0216 MSE 1.8136 0.4023 0.1166 0.1181 1.4834 0.3501 0.0023 0.0107 . Estimator . θ(·) ∼ Mixture, θ[1](τ) = 3.335 . θ(·) ∼ Mixture, θ[1](τ) = 3.335 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.4921 −0.0843 −0.0040 −0.0984 0.4922 −0.0948 0.0004 −0.0958 MSE 3.2573 0.2499 0.0196 0.1751 2.7050 0.1047 0.0004 0.1034 T = 10 %Bias 0.4779 −0.0808 −0.0005 −0.0584 0.4784 −0.0949 0.0009 −0.0568 MSE 2.9631 0.1275 0.0101 0.0590 2.5542 0.1021 0.0002 0.0363 T = 20 %Bias 0.4556 −0.0732 −0.0002 −0.0319 0.4585 −0.0953 0.0006 −0.0325 MSE 2.6537 0.0784 0.0048 0.0191 2.3448 0.1018 0.0001 0.0119 . Estimator . θ(·) ∼ Mixture, θ[1](τ) = 3.335 . θ(·) ∼ Mixture, θ[1](τ) = 3.335 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.4921 −0.0843 −0.0040 −0.0984 0.4922 −0.0948 0.0004 −0.0958 MSE 3.2573 0.2499 0.0196 0.1751 2.7050 0.1047 0.0004 0.1034 T = 10 %Bias 0.4779 −0.0808 −0.0005 −0.0584 0.4784 −0.0949 0.0009 −0.0568 MSE 2.9631 0.1275 0.0101 0.0590 2.5542 0.1021 0.0002 0.0363 T = 20 %Bias 0.4556 −0.0732 −0.0002 −0.0319 0.4585 −0.0953 0.0006 −0.0325 MSE 2.6537 0.0784 0.0048 0.0191 2.3448 0.1018 0.0001 0.0119 Note 1000 MC replications. QR: standard quantile regression estimator; CRE: correlated random effect estimator; INFE: infeasible estimator, 2‐STEP: two‐step estimator from 4.2. All regressions include an intercept term. Open in new tab Table 2. Bias and MSE for Model 5.1: γ= 2 and 0.90. . Estimator . n = 100 . n = 5000 . θ(·) ∼ N(2, 1), θ[1](τ) = 3.2815 . θ(·) ∼ N(2, 1), θ[1](τ) = 3.2815 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.4069 −0.1697 −0.0065 −0.1223 0.4321 −0.1575 −0.0002 −0.1144 MSE 2.4048 0.6455 0.1509 0.3162 2.0220 0.2743 0.0032 0.1439 T = 10 %Bias 0.4039 −0.1547 −0.0069 −0.0645 0.4079 −0.1594 −0.0006 −0.0603 MSE 2.1894 0.3995 0.0763 0.1228 1.8001 0.2768 0.0015 0.0406 T = 20 %Bias 0.3777 −0.1307 0.0013 −0.0280 0.3833 −0.1579 0.0001 −0.0301 MSE 1.8410 0.2466 0.0393 0.0479 1.5884 0.2700 0.0008 0.0105 . Estimator . n = 100 . n = 5000 . θ(·) ∼ N(2, 1), θ[1](τ) = 3.2815 . θ(·) ∼ N(2, 1), θ[1](τ) = 3.2815 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.4069 −0.1697 −0.0065 −0.1223 0.4321 −0.1575 −0.0002 −0.1144 MSE 2.4048 0.6455 0.1509 0.3162 2.0220 0.2743 0.0032 0.1439 T = 10 %Bias 0.4039 −0.1547 −0.0069 −0.0645 0.4079 −0.1594 −0.0006 −0.0603 MSE 2.1894 0.3995 0.0763 0.1228 1.8001 0.2768 0.0015 0.0406 T = 20 %Bias 0.3777 −0.1307 0.0013 −0.0280 0.3833 −0.1579 0.0001 −0.0301 MSE 1.8410 0.2466 0.0393 0.0479 1.5884 0.2700 0.0008 0.0105 . Estimator . θ(·) ∼ exp(1) + 2, θ[1](τ) = 4.3026 . θ(·) ∼ exp(1) + 2, θ[1](τ) = 4.3026 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.3120 −0.1517 −0.0015 −0.0814 0.3134 −0.1366 −0.0002 −0.0804 MSE 2.5781 0.9598 0.4730 0.4583 1.8333 0.3566 0.0095 0.1264 T = 10 %Bias 0.2911 −0.1442 0.0012 −0.0407 0.2984 −0.1361 −0.0005 −0.0417 MSE 2.0493 0.6100 0.2337 0.2444 1.6585 0.3479 0.0049 0.0363 T = 20 %Bias 0.2748 −0.1261 0.0008 −0.0224 0.2823 −0.1371 −0.0003 −0.0216 MSE 1.8136 0.4023 0.1166 0.1181 1.4834 0.3501 0.0023 0.0107 . Estimator . θ(·) ∼ exp(1) + 2, θ[1](τ) = 4.3026 . θ(·) ∼ exp(1) + 2, θ[1](τ) = 4.3026 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.3120 −0.1517 −0.0015 −0.0814 0.3134 −0.1366 −0.0002 −0.0804 MSE 2.5781 0.9598 0.4730 0.4583 1.8333 0.3566 0.0095 0.1264 T = 10 %Bias 0.2911 −0.1442 0.0012 −0.0407 0.2984 −0.1361 −0.0005 −0.0417 MSE 2.0493 0.6100 0.2337 0.2444 1.6585 0.3479 0.0049 0.0363 T = 20 %Bias 0.2748 −0.1261 0.0008 −0.0224 0.2823 −0.1371 −0.0003 −0.0216 MSE 1.8136 0.4023 0.1166 0.1181 1.4834 0.3501 0.0023 0.0107 . Estimator . θ(·) ∼ Mixture, θ[1](τ) = 3.335 . θ(·) ∼ Mixture, θ[1](τ) = 3.335 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.4921 −0.0843 −0.0040 −0.0984 0.4922 −0.0948 0.0004 −0.0958 MSE 3.2573 0.2499 0.0196 0.1751 2.7050 0.1047 0.0004 0.1034 T = 10 %Bias 0.4779 −0.0808 −0.0005 −0.0584 0.4784 −0.0949 0.0009 −0.0568 MSE 2.9631 0.1275 0.0101 0.0590 2.5542 0.1021 0.0002 0.0363 T = 20 %Bias 0.4556 −0.0732 −0.0002 −0.0319 0.4585 −0.0953 0.0006 −0.0325 MSE 2.6537 0.0784 0.0048 0.0191 2.3448 0.1018 0.0001 0.0119 . Estimator . θ(·) ∼ Mixture, θ[1](τ) = 3.335 . θ(·) ∼ Mixture, θ[1](τ) = 3.335 . QR . CRE . INFE . 2‐STEP . QR . CRE . INFE . 2‐STEP . T = 5 %Bias 0.4921 −0.0843 −0.0040 −0.0984 0.4922 −0.0948 0.0004 −0.0958 MSE 3.2573 0.2499 0.0196 0.1751 2.7050 0.1047 0.0004 0.1034 T = 10 %Bias 0.4779 −0.0808 −0.0005 −0.0584 0.4784 −0.0949 0.0009 −0.0568 MSE 2.9631 0.1275 0.0101 0.0590 2.5542 0.1021 0.0002 0.0363 T = 20 %Bias 0.4556 −0.0732 −0.0002 −0.0319 0.4585 −0.0953 0.0006 −0.0325 MSE 2.6537 0.0784 0.0048 0.0191 2.3448 0.1018 0.0001 0.0119 Note 1000 MC replications. QR: standard quantile regression estimator; CRE: correlated random effect estimator; INFE: infeasible estimator, 2‐STEP: two‐step estimator from 4.2. All regressions include an intercept term. Open in new tab Table 3. Bias and MSE for Model 5.1: γ= 2 and n = 100. τ= 0.25 . Estimator . Model 1 . Model 2 . Model 3 . %Bias . MSE . Time . %Bias . MSE . Time . %Bias . MSE . Time . T = 5 2‐STEP 0.1494 0.1469 0.009 0.0976 0.1003 0.009 0.2561 0.4963 0.009 KOEN 0.1084 0.1473 0.133 −0.0014 0.0219 0.136 0.5038 0.5565 0.135 T = 10 2‐STEP 0.0722 0.0579 0.017 0.0487 0.0286 0.018 0.0424 0.2088 0.019 KOEN 0.0504 0.0545 0.258 −0.0061 0.0086 0.254 0.3488 0.2841 0.284 T = 20 2‐STEP 0.0377 0.0264 0.016 0.0215 0.0087 0.020 −0.0008 0.0763 0.015 KOEN 0.0223 0.0268 0.240 −0.0024 0.0048 0.241 0.1924 0.1129 0.233 τ= 0.25 . Estimator . Model 1 . Model 2 . Model 3 . %Bias . MSE . Time . %Bias . MSE . Time . %Bias . MSE . Time . T = 5 2‐STEP 0.1494 0.1469 0.009 0.0976 0.1003 0.009 0.2561 0.4963 0.009 KOEN 0.1084 0.1473 0.133 −0.0014 0.0219 0.136 0.5038 0.5565 0.135 T = 10 2‐STEP 0.0722 0.0579 0.017 0.0487 0.0286 0.018 0.0424 0.2088 0.019 KOEN 0.0504 0.0545 0.258 −0.0061 0.0086 0.254 0.3488 0.2841 0.284 T = 20 2‐STEP 0.0377 0.0264 0.016 0.0215 0.0087 0.020 −0.0008 0.0763 0.015 KOEN 0.0223 0.0268 0.240 −0.0024 0.0048 0.241 0.1924 0.1129 0.233 τ= 0.90 . Estimator . %Bias . MSE . Time . %Bias . MSE . Time . %Bias . MSE . Time . T = 5 2‐STEP −0.1223 0.3162 0.009 −0.0814 0.4583 0.013 −0.0984 0.1751 0.011 KOEN −0.1860 0.5377 0.137 −0.2382 1.2574 0.187 −0.0617 0.0691 0.158 T = 10 2‐STEP −0.0645 0.1228 0.019 −0.0407 0.2444 0.018 −0.0584 0.0590 0.022 KOEN −0.0555 0.1371 0.262 −0.0850 0.3708 0.239 −0.0177 0.0180 0.270 T = 20 2‐STEP −0.0280 0.0479 0.021 −0.0224 0.1181 0.024 −0.0319 0.0191 0.026 KOEN −0.0280 0.0543 0.241 −0.0484 0.1594 0.253 −0.0081 0.0068 0.275 τ= 0.90 . Estimator . %Bias . MSE . Time . %Bias . MSE . Time . %Bias . MSE . Time . T = 5 2‐STEP −0.1223 0.3162 0.009 −0.0814 0.4583 0.013 −0.0984 0.1751 0.011 KOEN −0.1860 0.5377 0.137 −0.2382 1.2574 0.187 −0.0617 0.0691 0.158 T = 10 2‐STEP −0.0645 0.1228 0.019 −0.0407 0.2444 0.018 −0.0584 0.0590 0.022 KOEN −0.0555 0.1371 0.262 −0.0850 0.3708 0.239 −0.0177 0.0180 0.270 T = 20 2‐STEP −0.0280 0.0479 0.021 −0.0224 0.1181 0.024 −0.0319 0.0191 0.026 KOEN −0.0280 0.0543 0.241 −0.0484 0.1594 0.253 −0.0081 0.0068 0.275 Note 1000 MC replications. KOEN: Koenker’s estimator for panel data quantile regression, 2‐STEP: two‐step estimator from 4.2. All regressions include an intercept term. Time is reported in seconds. Open in new tab Table 3. Bias and MSE for Model 5.1: γ= 2 and n = 100. τ= 0.25 . Estimator . Model 1 . Model 2 . Model 3 . %Bias . MSE . Time . %Bias . MSE . Time . %Bias . MSE . Time . T = 5 2‐STEP 0.1494 0.1469 0.009 0.0976 0.1003 0.009 0.2561 0.4963 0.009 KOEN 0.1084 0.1473 0.133 −0.0014 0.0219 0.136 0.5038 0.5565 0.135 T = 10 2‐STEP 0.0722 0.0579 0.017 0.0487 0.0286 0.018 0.0424 0.2088 0.019 KOEN 0.0504 0.0545 0.258 −0.0061 0.0086 0.254 0.3488 0.2841 0.284 T = 20 2‐STEP 0.0377 0.0264 0.016 0.0215 0.0087 0.020 −0.0008 0.0763 0.015 KOEN 0.0223 0.0268 0.240 −0.0024 0.0048 0.241 0.1924 0.1129 0.233 τ= 0.25 . Estimator . Model 1 . Model 2 . Model 3 . %Bias . MSE . Time . %Bias . MSE . Time . %Bias . MSE . Time . T = 5 2‐STEP 0.1494 0.1469 0.009 0.0976 0.1003 0.009 0.2561 0.4963 0.009 KOEN 0.1084 0.1473 0.133 −0.0014 0.0219 0.136 0.5038 0.5565 0.135 T = 10 2‐STEP 0.0722 0.0579 0.017 0.0487 0.0286 0.018 0.0424 0.2088 0.019 KOEN 0.0504 0.0545 0.258 −0.0061 0.0086 0.254 0.3488 0.2841 0.284 T = 20 2‐STEP 0.0377 0.0264 0.016 0.0215 0.0087 0.020 −0.0008 0.0763 0.015 KOEN 0.0223 0.0268 0.240 −0.0024 0.0048 0.241 0.1924 0.1129 0.233 τ= 0.90 . Estimator . %Bias . MSE . Time . %Bias . MSE . Time . %Bias . MSE . Time . T = 5 2‐STEP −0.1223 0.3162 0.009 −0.0814 0.4583 0.013 −0.0984 0.1751 0.011 KOEN −0.1860 0.5377 0.137 −0.2382 1.2574 0.187 −0.0617 0.0691 0.158 T = 10 2‐STEP −0.0645 0.1228 0.019 −0.0407 0.2444 0.018 −0.0584 0.0590 0.022 KOEN −0.0555 0.1371 0.262 −0.0850 0.3708 0.239 −0.0177 0.0180 0.270 T = 20 2‐STEP −0.0280 0.0479 0.021 −0.0224 0.1181 0.024 −0.0319 0.0191 0.026 KOEN −0.0280 0.0543 0.241 −0.0484 0.1594 0.253 −0.0081 0.0068 0.275 τ= 0.90 . Estimator . %Bias . MSE . Time . %Bias . MSE . Time . %Bias . MSE . Time . T = 5 2‐STEP −0.1223 0.3162 0.009 −0.0814 0.4583 0.013 −0.0984 0.1751 0.011 KOEN −0.1860 0.5377 0.137 −0.2382 1.2574 0.187 −0.0617 0.0691 0.158 T = 10 2‐STEP −0.0645 0.1228 0.019 −0.0407 0.2444 0.018 −0.0584 0.0590 0.022 KOEN −0.0555 0.1371 0.262 −0.0850 0.3708 0.239 −0.0177 0.0180 0.270 T = 20 2‐STEP −0.0280 0.0479 0.021 −0.0224 0.1181 0.024 −0.0319 0.0191 0.026 KOEN −0.0280 0.0543 0.241 −0.0484 0.1594 0.253 −0.0081 0.0068 0.275 Note 1000 MC replications. KOEN: Koenker’s estimator for panel data quantile regression, 2‐STEP: two‐step estimator from 4.2. All regressions include an intercept term. Time is reported in seconds. Open in new tab Tables 1 and 2 show equivalent patterns. CRE has a larger bias than 2‐STEP and its bias does not improve as T grows.9 2‐STEP does show a bias that decreases as T grows. This is consistent with the analysis presented in Sections 2 and 4. Also, it is worth noticing that the bias of 2‐STEP is not affected as N increases and T remains fixed. Comparisons based on MSE arrive to similar conclusions. Finally, additional simulations not reported show that these results also hold for other quantiles, and that the two‐step estimator performs similarly when αi is specified as a non‐linear function of Xi and ηi. Table 3 reports bias, MSE and computational time for 2‐STEP and KOEN, the estimator proposed by Koenker (2004), see equation (2.4). Only the case n = 100 is shown since KOEN had problems handling the big matrices for the case n = 5000. The results are quite mixed. 2‐STEP performs better than KOEN in about half of the cases. However, the worst performance of 2‐STEP (bias: 25%, MSE: 0.50) is better than that of KOEN (bias: 50%, MSE: 1.26). Note also that 2‐STEP is about 15 times faster than KOEN. Finally, Table 4 reports standard errors and 95% confidence intervals computed using the formulas provided in Appendix B. The coverage is very close to the nominal level when T = 20 and below the nominal level for T = 5. It is worth noting that the coverage is expected to deteriorate in two circumstances. Given a value of n, a smaller T implies a larger finite sample bias and so a finite sample distribution centred further away from the truth. In addition, given a value of T, a larger value of n keeps the finite sample bias unaffected but implies a finite sample distribution that is more concentrated about the wrong place. However, even for a case with 12% of bias (model 1, T = 5), the actual coverage levels are about 85%, which look very decent for such small values of T and bias above 10%. Table 4. Standard Errors and 95% Confidence Intervals for Model 5.1: γ= 2 and n = 100. τ= 0.25 . . τ= 0.90 . T = 5 . T = 10 . T = 20 . T = 5 . T = 10 . T = 20 . Estimator 1.5329 1.4194 1.3719 Estimator 2.9057 3.0995 3.1822 Asy SE 0.2934 0.2162 0.1555 Asy SE 0.3503 0.2620 0.1901 Model 1 Boot SE 0.2868 0.2131 0.1537 Boot SE 0.3689 0.2713 0.1944 Asy Cov 0.8540 0.9230 0.9420 Asy Cov 0.7600 0.8920 0.9240 Boot Cov 0.8320 0.9030 0.9340 Boot Cov 0.7930 0.8870 0.9240 Estimator 2.5028 2.4046 2.3369 Estimator 3.9575 4.1257 4.2009 Asy SE 0.1961 0.1231 0.0773 Asy SE 0.5336 0.4290 0.3254 Model 2 Boot SE 0.1904 0.1181 0.0738 Boot SE 0.5697 0.4447 0.3335 Asy Cov 0.7760 0.8240 0.8910 Asy Cov 0.8620 0.9090 0.9220 Boot Cov 0.7260 0.7980 0.8910 Boot Cov 0.8720 0.9090 0.9220 Estimator 1.6305 1.3778 1.3110 Estimator 3.0058 3.1488 3.2298 Asy SE 0.4751 0.3943 0.2630 Asy SE 0.2218 0.1320 0.0822 Model 3 Boot SE 0.5364 0.4494 0.2918 Boot SE 0.2333 0.1368 0.0842 Asy Cov 0.8630 0.9290 0.9460 Asy Cov 0.6520 0.7050 0.7400 Boot Cov 0.8720 0.9310 0.9460 Boot Cov 0.6800 0.7050 0.7400 τ= 0.25 . . τ= 0.90 . T = 5 . T = 10 . T = 20 . T = 5 . T = 10 . T = 20 . Estimator 1.5329 1.4194 1.3719 Estimator 2.9057 3.0995 3.1822 Asy SE 0.2934 0.2162 0.1555 Asy SE 0.3503 0.2620 0.1901 Model 1 Boot SE 0.2868 0.2131 0.1537 Boot SE 0.3689 0.2713 0.1944 Asy Cov 0.8540 0.9230 0.9420 Asy Cov 0.7600 0.8920 0.9240 Boot Cov 0.8320 0.9030 0.9340 Boot Cov 0.7930 0.8870 0.9240 Estimator 2.5028 2.4046 2.3369 Estimator 3.9575 4.1257 4.2009 Asy SE 0.1961 0.1231 0.0773 Asy SE 0.5336 0.4290 0.3254 Model 2 Boot SE 0.1904 0.1181 0.0738 Boot SE 0.5697 0.4447 0.3335 Asy Cov 0.7760 0.8240 0.8910 Asy Cov 0.8620 0.9090 0.9220 Boot Cov 0.7260 0.7980 0.8910 Boot Cov 0.8720 0.9090 0.9220 Estimator 1.6305 1.3778 1.3110 Estimator 3.0058 3.1488 3.2298 Asy SE 0.4751 0.3943 0.2630 Asy SE 0.2218 0.1320 0.0822 Model 3 Boot SE 0.5364 0.4494 0.2918 Boot SE 0.2333 0.1368 0.0842 Asy Cov 0.8630 0.9290 0.9460 Asy Cov 0.6520 0.7050 0.7400 Boot Cov 0.8720 0.9310 0.9460 Boot Cov 0.6800 0.7050 0.7400 Note 1000 MC replications. Asy SE: asymptotic standard errors. Boot SE: bootstrap standard errors. Asy Cov: Coverage of the asymptotic confidence interval. Boot Cov: Coverage of the Bootstrap percentile interval. Open in new tab Table 4. Standard Errors and 95% Confidence Intervals for Model 5.1: γ= 2 and n = 100. τ= 0.25 . . τ= 0.90 . T = 5 . T = 10 . T = 20 . T = 5 . T = 10 . T = 20 . Estimator 1.5329 1.4194 1.3719 Estimator 2.9057 3.0995 3.1822 Asy SE 0.2934 0.2162 0.1555 Asy SE 0.3503 0.2620 0.1901 Model 1 Boot SE 0.2868 0.2131 0.1537 Boot SE 0.3689 0.2713 0.1944 Asy Cov 0.8540 0.9230 0.9420 Asy Cov 0.7600 0.8920 0.9240 Boot Cov 0.8320 0.9030 0.9340 Boot Cov 0.7930 0.8870 0.9240 Estimator 2.5028 2.4046 2.3369 Estimator 3.9575 4.1257 4.2009 Asy SE 0.1961 0.1231 0.0773 Asy SE 0.5336 0.4290 0.3254 Model 2 Boot SE 0.1904 0.1181 0.0738 Boot SE 0.5697 0.4447 0.3335 Asy Cov 0.7760 0.8240 0.8910 Asy Cov 0.8620 0.9090 0.9220 Boot Cov 0.7260 0.7980 0.8910 Boot Cov 0.8720 0.9090 0.9220 Estimator 1.6305 1.3778 1.3110 Estimator 3.0058 3.1488 3.2298 Asy SE 0.4751 0.3943 0.2630 Asy SE 0.2218 0.1320 0.0822 Model 3 Boot SE 0.5364 0.4494 0.2918 Boot SE 0.2333 0.1368 0.0842 Asy Cov 0.8630 0.9290 0.9460 Asy Cov 0.6520 0.7050 0.7400 Boot Cov 0.8720 0.9310 0.9460 Boot Cov 0.6800 0.7050 0.7400 τ= 0.25 . . τ= 0.90 . T = 5 . T = 10 . T = 20 . T = 5 . T = 10 . T = 20 . Estimator 1.5329 1.4194 1.3719 Estimator 2.9057 3.0995 3.1822 Asy SE 0.2934 0.2162 0.1555 Asy SE 0.3503 0.2620 0.1901 Model 1 Boot SE 0.2868 0.2131 0.1537 Boot SE 0.3689 0.2713 0.1944 Asy Cov 0.8540 0.9230 0.9420 Asy Cov 0.7600 0.8920 0.9240 Boot Cov 0.8320 0.9030 0.9340 Boot Cov 0.7930 0.8870 0.9240 Estimator 2.5028 2.4046 2.3369 Estimator 3.9575 4.1257 4.2009 Asy SE 0.1961 0.1231 0.0773 Asy SE 0.5336 0.4290 0.3254 Model 2 Boot SE 0.1904 0.1181 0.0738 Boot SE 0.5697 0.4447 0.3335 Asy Cov 0.7760 0.8240 0.8910 Asy Cov 0.8620 0.9090 0.9220 Boot Cov 0.7260 0.7980 0.8910 Boot Cov 0.8720 0.9090 0.9220 Estimator 1.6305 1.3778 1.3110 Estimator 3.0058 3.1488 3.2298 Asy SE 0.4751 0.3943 0.2630 Asy SE 0.2218 0.1320 0.0822 Model 3 Boot SE 0.5364 0.4494 0.2918 Boot SE 0.2333 0.1368 0.0842 Asy Cov 0.8630 0.9290 0.9460 Asy Cov 0.6520 0.7050 0.7400 Boot Cov 0.8720 0.9310 0.9460 Boot Cov 0.6800 0.7050 0.7400 Note 1000 MC replications. Asy SE: asymptotic standard errors. Boot SE: bootstrap standard errors. Asy Cov: Coverage of the asymptotic confidence interval. Boot Cov: Coverage of the Bootstrap percentile interval. Open in new tab 6. Discussion This paper provided an identification result for quantile regression in panel data models and introduced a two‐step estimator that is attractive for its computational simplicity. There are many issues that remain to be investigated. First, several panels available have a short time span and therefore approximations taking T to infinity might result in poor approximations for those cases. However, a computationally simple estimator that works for fixed T and large N is extremely challenging since, even under the assumption that αi is independent of the rest of the variables of the model, we would still have to face similar problems to those discussed in Sections 2 and 3. Second, the assumption that αi does not depend on the quantiles restricts the type of unobserved heterogeneity that the model can handle. Improvements in any of these directions are important for future research. Footnotes 1 " Note that the distribution of eit(τ) need not be identical across t even when Uit is i.i.d., but that eit(τ) has the same τ‐quantile for all t. 2 " Lamarche (2010) proposes a method to chose λ under the additional assumption that αi and Xi are independent. Galvao (2008) further extends this idea to dynamic panels. 3 " This means that if αi captures unobserved covariates that enter the model and are constant over time, such variables must have coefficients that are constant across τ, that is, β=β(τ) for all τ. 4 " The advantage of such non‐parametric estimators would be their consistency for asymptotics in which n →∞ and T remains fixed. 5 " An extension of the result by Kotlarski for the case of characteristic functions with zeros was recently proposed by Evdokimov and White (2010) by using conditions on the derivatives of the characteristic functions. 6 " This follows from being a proper linear subspace of if . In addition, if Xt does not include a constant then Assumption 3.2(c) must hold for X2− X1. 7 " Note that unless θμ=θ(τ). Also, , depending on whether θ(Uit)⋚θμ. 8 " I use the within estimator of equation (A.17) as first step estimator. 9 " The performance of CRE depends on the parameter γ, so as γ grows its performance deteriorates. Acknowledgments This paper was previously circulated under the title ‘A Note on Quantile Regression for Panel Data Models.’ I am deeply grateful to Jack Porter and Bruce Hansen for thoughtful discussions. I would also like to thank the Editor and two anonymous referees whose comments have led to an improved version of this paper. References Abrevaya , J. and C. M. Dahl ( 2008 ). The effects of birth inputs on birthweight: Evidence from quantile estimation on panel data . Journal of Business and Economic Statistics 26 , 379 – 97 . Google Scholar Crossref Search ADS WorldCat Chamberlain , G. ( 1982 ). Multivariate regression models for panel data . Journal of Econometrics 18 , 5 – 46 . Google Scholar Crossref Search ADS WorldCat Chamberlain , G. ( 1984 ). Panel data. In Z. Griliches and M. D. Intriligator (Eds.), Handbook of Econometrics, Volume 2 , 1247 – 318 . New York : Academic Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Chernozhukov , V. , I. Fernández‐Val, J. Hahn, and W. Newey ( 2010 ). Average and quantile effects in nonseparable panel models . Working paper, M.I.T. , Cambridge . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Chernozhukov , V. and C. Hansen ( 2006 ). Instrumental quantile regression inference for structural and treatment effect models . Journal of Econometrics 123 , 491 – 525 . Google Scholar Crossref Search ADS WorldCat Chernozhukov , V. and C. Hansen ( 2008 ). Instrumental variable quantile regression: a robust inference approach . Journal of Econometrics 142 , 379 – 98 . Google Scholar Crossref Search ADS WorldCat Evdokimov , K. ( 2010 ). Identification and estimation of a nonparametric panel data model with unobserved heterogeneity . 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Cambridge : Cambridge University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Rosen , A. ( 2009 ). Set identification via quantile restrictions in short panels . Working paper, University College , London . van der Vaart , A. W. ( 1998 ). Asymptotic Statistics . Cambridge : Cambridge University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC van der Vaart , A. W. and J. A. Wellner ( 1996 ). Weak Convergence and Empirical Processes . New York : Springer‐Verlag . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Appendix A: Proof of the Lemmas and Theorems Throughout the Appendix I use the following notation. For W = (Y*, X), In addition, denotes weak convergence, CMT refers to the Continuous Mapping Theorem and LLN refers to the Law of Large Numbers. The symbols o, O, op and Op denote the usual order of magnitudes for non‐random and random sequences. Proof of Lemma 3.1: The proof is a simple extension of the result Lemma 2.1 from Neumann (2007). I write it here for completeness. Since and ϕα|x are characteristic functions there exists an ω0 > 0 such that for t ∈{1, 2} and ϕα|x(2ω) ≠ 0 if |ω| ≤ω0 (In this case, ω0 might depend on x but we omit this dependence for simplicity). For |ω| ≤ 2ω0 define, a continuous complex function which equals 1 at 0. It follows from equation (3.3), for ω1, ω2∈ [ −ω0, ω0], that (A.1) which implies The unique solution to this equation satisfying gα|x(0) = 1 and (a Hermitian function) is gα|x(ω) = eicω, from some real c. Therefore we conclude that, (A.2) Furthermore, equation (3.3) yields that for ω2= 0, so that, (A.3) Setting ω1= 0 it follows that equation (A.3) holds for S2 as well. Now it remains to extend these results to the whole real line. Let ω∈Γ be arbitrary. We obtain, analogously to equation (A.1), that after iterating. Using this equation with a k large enough such that |ω2−k| ≤ 2ω0 we conclude from equation (A.2) that Since Γ is dense in and is a closed set, we conclude that , , this is, . This implies, again by equation (3.3) that , , which yields for t ∈{1, 2}. □ Lemma A.1. Under Assumptions 3.2 and 4.1, the following statements are true.(a)inwhereis a Gaussian process with covariance function. (b)If and then Proof: The proof follows by similar arguments to those in Lemma B.2 of Chernozhukov and Hansen (2006) after noticing that the class of functions, is Donsker (it is formed by taking products and sums of bounded Donsker classes) by Theorem 2.10.6 in van der Vaart and Wellner (1996). □ Lemma A.2. Under Assumptions 3.2 and 4.1, providedfor somes ∈ (1, ∞). Proof: Let and note that by the triangle inequality, Xit and Yit have compact support so that maxi ≤ n|Xit| ≤ Cx < ∞ and maxi ≤ n|uit| ≤ Cz < ∞. It is immediate then that Since E(uit) = 0 and for all s ∈ (1, ∞), it follows from the Markov inequality that for any η > 0, and then □ Proof of Theorem 4.1: Consistency. Define the following two criterion functions, The first step shows that QnT(θ, τ) converge uniformly to Q(θ, τ). To this end, note that since , it follows from van der Vaart (1998, Lemma 2.2) that since ρτ(·) is a bounded Lipschitz function (it is bounded due to 4.1(a) and 4.1(b)). Due to the compactness of and the continuity of and implied by 4.1(c), the above convergence is also uniform, (A.4) Next note that functions in the class are bounded, uniformly Lipschitz over and form a Donsker class. This also means that is Glivenko‐Cantelli so that, (A.5) It follows from equations (A.4) and (A.5) that QnT(θ, τ) converges uniformly to Q(θ, τ) as both n and T go to infinity. Under Assumption 4.1(c) θ(τ) uniquely solves and Q(θ, τ) is continuous over so that after invoking Chernozhukov and Hansen (2006, Lemma B.1). Asymptotic Normality. From the properties of standard quantile regression it follows that is op(1), and then the following expansion is valid, Here follows from Lemma A.1. Now, expand . where θ* is on the line connecting and θ(τ) for each τ and is on the line connecting 0 and . The second equality follows from (A.6) which in turn follows from the uniform continuity assumption, and the fact that by Lemma A.2. Solving for , where is a gaussian process with covariance kernel J1(τ)−1Ψ(τ, τ′)[J1(τ′)−1]′, where Ψ(τ, τ′) is defined in equation (A.10). This follows from the first term converging to a Gaussian process by Lemma A.1, and the second term being asymptotically normal due to , and both being asymptotically normal. Covariance Kernel. We first need to derive the expression for Ψ(τ, τ′). Under Assumption 4.2 we can write the expansion as (A.7) where , μX≡ E(X), and . In addition, where fɛ(τ)(0|X) denotes the conditional on X density of ɛ(τ) ≡ Y*− X′θ(τ) at 0 and Under Assumptions 3.2 and 4.1 it follows that (A.8) where is a zero‐mean gaussian process with covariance kernel (A.9) and S(τ, τ′) ≡ (min{τ, τ′}−ττ′)E(XX′), Ωgξ(τ) ≡ E[gτ(W, θ(τ))ξ], and Ωξξ≡ E[ξ2]. The above result implies that (A.10) We can conclude from equation (A.7) that (A.11) where is a gaussian process with covariance kernel (A.12) □ Lemma A.3. If ||vi, T− v|| → 0 a.s. uniformly iniasT →∞, and there exists a functionqi ≥ 0 such that ||vi, T|| ≤ qifor alliandTwith, then, asT →∞. Proof: Let hi, T= ||vi, T− v|| and note that . By Fatou’s Lemma (A.13) (A.14) meaning that . Then, the result directly follows from (A.15) □ Lemma A.4. Assumeis non‐singular with finite norm, for somea ∈ (0, ∞) and let Assumptions 3.2 and 4.1 hold. The within estimator of θμsatisfies Assumption 4.2 with the influence function (A.16)where, , μY≡ E(Yit), uitis i.i.d. withE[uit | Xi] = 0 and, and ΩUUnon‐singular with finite norm. Proof: Use the partition , where and . Then (A.17) where and . By equations (4.1) and (A.17) it follows that (A.18) where . By Assumption 3.2, E[uit] = 0, , and ΩUU is non‐singular and has finite norm. By Assumptions 3.2 and 4.1 (A.19) for ΩXX non‐singular and therefore (A.20) Next note that (A.21) meaning that the result for the slope coefficients would follow provided the second term in equation (A.21) is op(1). To show this, write this term as (A.22) where ςi, T is i.i.d. across i for all T and satisfies E[ςi, T] = 0 and (A.23) (A.24) since as T →∞ by Lemma A.4 (note that by Assumption 4.1, the function qi is just the upper bound of the support ) and the fact that uniformly over i as T →∞ by similar arguments to those in Lemma A.2 provided for some a ∈ (0, ∞). Finally, from equation (A.17), , the expansion in equation (A.21), and a few algebraic manipulations, it follows that (A.25) Letting , the result follows. □ Appendix B: Estimator of the Covariance Kernel and the Bootstrap The components of the asymptotic variance in equation (A.10) can be estimated using sample analogs. The expressions below correspond to the case where is the within estimator and so ψit is given by equation (A.16). They can be naturally extended to cover any other preliminary estimator satisfying Assumption 4.2. The matrix S(τ, τ′) can be estimated by its sample counterpart (B.1) For the matrices J1(τ) and J2(τ), I follow Powell (1991) and propose where , , and hn is an appropriately chosen bandwidth such that hn → 0 and . Following Koenker (2005, pp. 81 and 140), one possible choice is (B.2) where κ is a robust estimate of scale, zα=Φ−1(1 −α/2) and α denotes the desired size of the test. For the terms involving ξit and gτ(W, θ(τ)) define (B.3) where , , , and . This way, we can define , where . Finally, letting we have the following sample counterparts for the remaining terms (B.4) The estimator of the covariance matrix would then be , where is the matrix in equation (A.10) where all matrices have been replaced by their respective sample analogs. In the simulations of Section 5 I use the following bootstrap algorithm to compute standard errors and confidence intervals for . Let , j = 1, … , B, denote the jth i.i.d. sample of size n distributed according to , the empirical measure of , where Yi = (Yi1, … , YiT) and Xi = (Xi1, … , XiT). For each j = 1, … , B compute the two step estimator as described in Section 4 and denote this estimator by . This involves computing preliminary estimators and fixed effects for each bootstrap sample j = 1, … , B. The bootstrap estimate of the variance covariance matrix for is given by (B.5) where . In the simulations of Section 5 I also report the coverage of the 1 −α percentile interval (B.6) where is the α‐quantile of the empirical distribution of . This confidence interval is translation invariant, which is a good property when working with quantile regressions. Symmetric and equally‐tailed intervals can be alternatively computed using the same algorithm. © 2011 The Author(s). The Econometrics Journal © 2011 Royal Economic Society.
Econometrics Journal – Oxford University Press
Published: Oct 1, 2011
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