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Abstract We propose an ARMA-based quasi-maximum likelihood estimator for log-generalized autoregressive conditional heteroscedasticity (GARCH) models that is efficient when the conditional error is normal, and prove its consistency and asymptotic normality under mild assumptions. A study of efficiency shows the estimator can provide major improvements, both asymptotically and in finite samples. Next, two empirical applications illustrate the usefulness of our estimator. The first shows how it can be used to obtain volatility estimates in the presence of zeros, that is, inliers, since ARMA-based log-GARCH estimators enable a practical and straightforward solution to the inlier problem—even when the zero-generating process is non-stationary. Our study shows volatility estimates can be substantially underestimated if zeros are not handled appropriately. In our second empirical application, we show how our estimator can readily be used to model high-order volatility dynamics where one or more squared error autocorrelations are negative, a characteristic that is not compatible with ordinary (i.e., non-exponential) GARCH models. Autoregressive conditional heteroscedasticity (ARCH) models have been applied in the modeling of a wide range of phenomena, for example, the uncertainty of inflation (Engle, 1982), the uncertainty of electricity prices (Koopman, Ooms, and Carnero, 2007), temperature variability (Franses, Neele, and Van Dijk, 2001), and—most commonly—the variability of financial returns. In the majority of the applications, however, attention has been restricted to specifications with low-order dynamics, for example, the first-order generalised ARCH (GARCH) of Bollerslev (1986), the GARCH(1,1), and the first-order exponential GARCH (EGARCH) of Nelson (1991), the EGARCH(1,1). One reason is that first-order specifications often provide parsimonious representations of the process they intend to model. However, another reason is that higher-order GARCH and EGARCH specifications are limited. In the GARCH case, for example, restrictive non-negativity constraints on the parameters are required in higher-order specifications in order to ensure positive variance, see for example, Tsai and Chan (2008). This may be difficult to achieve in practice. Moreover, the GARCH is not compatible with data where the squared errors are negatively autocorrelated (this is not unlikely in monthly and quarterly data). In addition, if conditioning variables are added, as in Han and Kristensen (2014) and Francq and Thieu (2015), then additional non-negativity restrictions are needed on both parameters and the conditioning variables. In the EGARCH case, the theory is simply not available for higher-order dynamics. In fact, only recently was consistency and asymptotic normality (CAN) of the Gaussian quasi-maximum likelihood estimator (QMLE) proved, and this was for the first order (i.e., EGARCH(1,1)) only, see Wintenberger (2013) and Kyriakopoulou (2015). The log-GARCH model, by contrast, does not impose positivity constraints on any of the parameters, its statistical properties are much more tractable, and it is capable of accommodating a much wider range of volatility dynamics than GARCH and EGARCH models. Indeed, Asai (1998) showed that the standard stochastic volatility (SV) model in fact has a log-GARCH representation that can be used for estimation, and recently Francq, Wintenberger, and Zakoïan (2017) showed that also the EGARCH model has a log-GARCH representation (but not vice-versa). Finally, recent contributions (Escribano and Sucarrat, 2016; Francq and Sucarrat, 2017) show how the plain log-GARCH model can straightforwardly be extended by additional conditioning variables, for example, realized volatility (RV), volume, and periodicity terms, in the log-volatility equation, both univariately and multivariately. The (plain) univariate log-GARCH(p, q) model can be written as {εt=σtηt, ηt∼iid(0,1),lnσt2=ω0+∑i=1qα0ilnεt−i2+∑j=1pβ0j lnσt−j2, (1) where εt is typically interpreted as a financial return (possibly mean-corrected) or the error in a regression, and where σt>0 is the conditional standard deviation or volatility. Alternatively, the model can be interpreted as a logarithmic version of a model of a positively valued variable, for example, duration (as in Bauwens and Giot, 2000), RV, or volume, so that it becomes a multiplicative error model (MEM), see Brownlees, Cipollini, and Gallo (2012) for a survey of MEMs and their relation to GARCH models. Notable and attractive features of the log-GARCH model include: (1) No non-negativity constraints on the parameters, (2) the possibility of persistence of both high and low levels of volatility, (3) no lower non-zero bound on the volatility σt, (4) invariance to scale and power transformations, (5) the possibility of negative autocorrelations of εt2 , (6) the shape of the autocorrelation function of εt2 depends on the density of ηt, (7) the model admits a strong ARMA representation, and (8) leverage and other covariates can readily be added without imposing restrictive assumptions on the parameters and covariates, see Francq, Wintenberger, and Zakoïan (2013); Sucarrat, Grønneberg, and Escribano (2016); and Francq and Sucarrat (2017). Ordinary GARCH models do not fulfill any of these points. Engle and Bollerslev (1986b) argued against the log-GARCH because of the possibility of applying the log on zero-values of εt . Although this occurs with probability zero according to the model, it may nevertheless occur in practice. In finance, for example, such zeros may occur due to non-trading, discrete pricing (i.e., rounding error), missing values, and other data issues. This led Nelson (1991) to propose the EGARCH model, in which lnεt−i2 is replaced by |εt−i/σt−i| . While this provides a practical (but theoretically unsatisfactory) solution to the inlier problem, it does make a proof of CAN for a QMLE very difficult, see for example, Straumann and Mikosch (2006). Indeed, the currently most general result is by Wintenberger (2013) under the complicated and restrictive condition of continuous invertibility, and this is for the first-order version only, that is, the EGARCH(1,1). Creal, Koopmans, and Lucas (2013) and Harvey (2013) propose exponential GARCH models driven by the score of the log-likelihood. However, these models are by their very nature not amenable to QML estimation, and—to the best of our knowledge—a complete proof of CAN under mild assumptions is yet to be provided. By contrast, a complete proof of CAN for a QMLE (henceforth, the standard QMLE) of the log-GARCH(p, q) model under mild assumptions was provided by Francq, Wintenberger, and Zakoïan (2013), whereas Sucarrat, Grønneberg, and Escribano (2016) propose general estimation and inference methods based on the ARMA representation [henceforth, the Gaussian ARMA–QMLE, although their procedures can be applied in conjunction with other ARMA estimators as well, e.g., least squares(LS)]. The advantage of the former (i.e., the standard QMLE) is that maximum efficiency is achieved in the theoretically and empirically important case where ηt is Gaussian. This is not the case for the latter, that is, the Gaussian ARMA–QMLE. The main advantage of the latter, however, is that zeros and missing values on εt are readily handled with the algorithm proposed in Sucarrat and Escribano (2017). The standard QMLE does not permit this kind of treatment of zeros. This paper proposes a new QMLE that combines the strengths of the standard QMLE with the strengths of the Gaussian ARMA–QMLE: (a) Maximum efficiency is achieved when ηt is Gaussian and (b) zeros or inliers (if any) on εt can easily be handled satisfactorily, since estimation is via the ARMA representation. When ηt is Gaussian, then the distribution of the ARMA-error ut=lnηt2−E lnηt2 is a centered exponential Chi-squared (Cex- χ2 ); hence, we label the new estimator the Cex- χ2 QMLE. Our contributions are three. First, we prove the strong CAN of the Cex- χ2 QMLE under mild conditions for the log-GARCH(p, q) model, and derive expressions for the asymptotic covariance matrix. Second, we study the asymptotic and finite sample efficiency of the first-order log-GARCH specification. In particular, we show that our estimator has exactly the same asymptotic variances as that of the standard QMLE in the first-order case, and that our estimator is generally more efficient than the Gaussian ARMA–QMLE, both asymptotically and in finite samples. Third, we illustrate the usefulness of the estimator in two empirical applications. The first shows that volatility is generally underestimated if zeros are not adjusted for, and that the discrepancies can be large in relative terms. Moreover, the more zeros, the larger the bias. The second illustrates how the Cex- χ2 QMLE can be used to model unrestricted high-order dynamics of monthly inflation uncertainty, where one or more of the squared error autocorrelations are negative (ordinary GARCH models do not permit negative squared error autocorrelations, see Proposition 2.2 in Francq and Zakoïan, 2010). The rest of the paper is organized as follows. The next section, Section 1, contains the details of our estimator, and the main theoretical results, that is, strong CAN Theorems (their proofs are contained in the Appendix). Section 2 compares the asymptotic and finite sample efficiency of our estimator with the Gaussian ARMA–QMLE, and with the standard QMLE. Section 3 illustrates the usefulness of our results in two empirical applications. Finally, Section 4 concludes. 1 Model, Estimator, and Main Assumptions We will estimate separately the intercept ω0∈R and the parameter θ0=(α01,…,α0q,β01,…,β0p) , which belongs to a parameter space Θ⊂Rp+q . Before giving the definition of our estimator, we first give the strong ARMA representation of the log-GARCH model. 1.1 The ARMA Representation of the Log-GARCH Model The process (lnεt2) satisfies an ARMA-type equation of the form Aθ0(L) lnεt2=ω0+Bθ0(L)vt, (2) where L denotes the lag operator, vt=lnηt2 , and for all θ=(α1,…,αq,β1,…,βp)∈Θ, the AR and MA polynomials are, respectively, defined by Aθ(z)=1−∑i=1r(αi+βi)zi, Bθ(z)=1−∑i=1pβizi, r=max{p,q}, αi=0 for i > q and βi=0 for i > p. Assuming E ln+|lnη12|<∞ , where ln+x=max{lnx,0} , it is well-known that both Equations (2) and (1) admit a strictly stationary solution if the AR polynomial satisfies Aθ0(z)≠0 when |z|≤1. (3) This condition is also necessary for the existence of a stationary and non-anticipative solution to Equation (2) (and/or Equation (1)), under the additional condition that Pr(η12=1)≠1 or ω0≠0 (otherwise there is the trivial solution lnεt2=0 , regardless of the value of θ0). Under the moment condition E(lnη12)2<∞ , Equation (2) is a standard ARMA(r, p) equation of the form Aθ0(L)(lnεt2−ν0)=Bθ0(L)ut, (4) with ν0=E lnε12 and the white noise ut=vt−μ0 , where μ0=E lnη12 . The squares of an ordinary GARCH model also satisfy ARMA representations, but these ARMA representations are rarely used (in particular for inference), since the innovations ut=(ηt2−1)σt2 are not independent in the ordinary GARCH case (recall Point 7) in the introduction. 1.2 An Estimator Based on the ARMA Representation It is well known that GARCH models can be consistently estimated by QMLEs based on the instrumental N(0,1) density for ηt [see e.g., Gourieroux, Monfort, and Trognon (1984) for a general reference on QMLE, and Berkes, Horvath, and Kokoszka (2003); and Francq and Zakoïan (2004) for applications to GARCH models].1 If ηt∼N(0,1) , then xt=lnηt2 follows the exponential Chi-squared distribution with density χ0(x) . The centered exponential Chi-squared distribution (Cex- χ2 ) of x=lnηt2−μ0 is given by χμ0(x) with μ0=E lnηt2 , where χμ(x)=12πex+μ2−ex+μ2. The Gaussian ARMA–QMLE uses the normal as instrumental density for ut. So the question is whether a QMLE based on the ARMA equation (4) is consistent when using χμ(x) as instrumental density for ut. The answer is negative if one tries to estimate θ0, ν0, and μ0 simultaneously. This is due to lack of identification, since both μ0 and the ARMA intercept serve as constants in the same equation, and since μ0 appears in the instrumental density. As we will show, however, the answer is positive if we work with the mean-corrected ARMA representation, where the mean ν0 is estimated in a first step by its sample mean νn=n−1∑t=1n lnεt2 . Under the invertibility condition ∀θ∈Θ, Bθ(z)≠0 when |z|≤1, (5) the innovations of the mean-corrected ARMA representation (4) are defined by ut(θ)=Bθ−1(L)Aθ(L)(lnεt2−ν0):=∑i=0∞ψi(θ)(lnεt−i2−ν0). These innovations can be approximated by u˜t,n(θ)=∑i=0t−1ψi(θ)(lnεt−i2−νn), νn=1n∑t=1nlnεt2. In practice, u˜t,n(θ) can be obtained by taking the initial values u˜0,n(θ)=⋯=u˜1−q,n(θ)=0 and lnε02=⋯=lnε1−r2=νn , and by computing recursively u˜t,n(θ)=Aθ(L)(lnεt2−νn)+∑i=1qβiu˜t−i,n(θ), t=1,…,n. (6) Now consider the estimator defined by ϑ^n=(θ^n,μ^n)=argmaxϑ∈ΞQ˜n(ϑ), Q˜n(ϑ)=1n∑t=1nℓ˜t,n(ϑ), (7) where ϑ=(θ,μ) , Ξ is a compact set of the form Θ×[a,b] , and ℓ˜t,n(ϑ)=lnχμ{u˜t,n(θ)}+12 ln(2π)=12(u˜t,n(θ)+μ−eu˜t,n(θ)+μ). The intercept can then be estimated by ω^n=Aθ^n(1)νn−Bθ^n(1)μ^n. (8) Note that, for the log-GARCH model given by Equation (1), the estimator of the parameters of interest is (ω^n,θ^n) . This is a multi-step estimator in the spirit of the variance targeting estimator (see e.g., Francq, Horvath, and Zakoian, 2011), since it involves the estimation of a parameter by an empirical mean in a first step, and QML estimation of the remaining parameters in a second step. Note also that ∂Q˜n(ϑ)∂μ=0 ⇔ 1−1n∑t=1neu˜t,n(θ)+μ=0. It follows that μ^n=−ln1n∑t=1neu˜t,n(θ^n). The estimator ϑ^n can thus be obtained by setting μ=−ln1n∑t=1neu˜t,n(θ) and by optimizing on θ∈Θ . 1.3 Asymptotic Behavior of the Cex- χ2 QMLE We now provide CAN results for the Cex- χ2 QMLE defined by Equations (7) and (8). Theorem 1 (Strong Consistency). Let ϑ^n and ω^n be sequences of estimators satisfying (7) and (8), where the εt’s follow the log-GARCH model (1). Assume that θ0∈Θ and Θ is compact, that the stationarity condition (3) and the invertibility condition (5) hold, that the distribution of lnη12 is not degenerate with E|lnη12|<∞, that Aθ0(z) and Bθ0(z) have no common roots, and that p+q>0 with α0r+β0r≠0 or β0p≠0 (with the convention α00=β00=1). Then, almost surely, ϑ^n→ϑ0=(θ0,μ0) and ω^n→ω0 as n→∞. Proof See Appendix A.1. Francq, Wintenberger, and Zakoïan (2013) showed the consistency of the Standard QMLE under the same assumptions as Theorem 1, whereas the consistency of the Gaussian ARMA–QMLE holds under similar assumptions, see Sucarrat, Grønneberg, and Escribano (2016). Theorem 2 (Asymptotic Normality). Let the assumptions of Theorem 1 hold, and suppose in addition that ϑ0 belongs to the interior of Ξ, Eη14<∞, E(lnη12)2<∞ and E|1/η1|s0<∞ for some s0>0. Then, as n→∞, n(ω^n−ω0θ^n−θ0)→dN{0,(Eη14−1)(Bθ02(1)+γ′Σu−1γγ′Σu−1Σu−1γΣu−1)},where γ=(−ν01q′,(μ0−ν0)1p′)′, 1s denoting the all-ones vector of dimension s, and Σu=E∂ut∂θ∂ut∂θ′(θ0). Proof See Appendix A.2. Remark 1 The moment conditions of Theorem 2 are obviously satisfied when η1∼N(0,1). More generally, E|1/η1|s0<∞ for some s0>0 when η1 has a density f such that |x|1−ɛf(x) is bounded in a neighborhood of 0 for some ɛ>0. To show the asymptotic normality of the standard QMLE, Francq, Wintenberger, and Zakoïan (2013) assume the existence of an exponential moment for |lnη12| , which is virtually equivalent to assuming the existence of E|1/η1|s0 (compare Remark 1 and their Proposition 3.2). For the asymptotic normality of the Gaussian ARMA–QMLE, Sucarrat, Grønneberg, and Escribano (2016) do not assume E|1/η1|s0<∞ , but they also need Eη14<∞ and E(lnη12)2<∞ . Nevertheless, even if the CAN of the standard QMLE, Gaussian ARMA–QMLE, and Cex- χ2 QMLE are obtained under similar assumptions, their techniques of proof are quite different, and the ARMA-based estimators are simpler because they inherit the usual techniques developed for linear time-series analysis. Moreover, as we show in Section 2, both the asymptotic and finite sample variances can differ substantially. 2 Asymptotic and Finite Sample Efficiency Comparison A nice feature of the log-GARCH model is that, contrary to the ordinary GARCH model, the asymptotic coefficient covariance matrix of its QMLE is available in closed form. This enables direct comparison of the asymptotic variances of the Cex- χ2 QMLE, the Gaussian ARMA–QMLE, and the standard QMLE. For brevity, we do the comparison for the log-GARCH(1,1) specification only. The asymptotic variance of the Cex- χ2 QMLE of (α0,β0) is (see the Supplementary Appendix for the derivation) E(ηt4−1)·Σu−1, (9) where Σu−1=1Var(lnη12)·(1−β02(α0+β0)2−(α0+β0)(1−β02)(1−β0(α0+β0))α0−(α0+β0)(1−β02)(1−β0(α0+β0))α0(1−β02)(1−β0(α0+β0))2α02). The variance of β0 explodes when α0→0 , which is not surprising since the existence of common roots in the AR and MA polynomials is excluded for the consistency. The asymptotic variance of ω^ for the log-GARCH(1,1) is given by E(ηt2−1)(Bθ02(1)+γ′Σu−1γ) with Bθ02(1)=(1−β0)2 and γ=(ν0,μ0−ν0)′ , where μ0=E(lnηt2) and ν0=(ω0+(1−β0)μ0)/(1−α0−β0) . The full asymptotic covariance matrix of the Cex- χ2 QMLE is thus (Eη14−1)(Bθ02(1)+γ′Σu−1γγ′Σu−1Σu−1γΣu−1), (10) where γ=(−ν01q′,(μ0−ν0)1p′)′ and 1s denote the all-ones vector of dimension s, see Theorem 2. The asymptotic variance of the μ0 estimate is Var(ηt2−lnηt2) , see Equation (27). In comparing Equation (10) with the asymptotic covariance of the standard QMLE, it can be shown that their diagonal is the same. This is somewhat surprising, since one might have expected that the two-step nature of the Cex- χ2 QMLE would make it less efficient asymptotically. In comparing Equation (10) with the asymptotic covariance of the Gaussian ARMA–QMLE, only the variances of the estimates of α0, β0, and μ0 can be compared, since the asymptotic variance of the ω0-estimate is (currently) not available. The variance of μ0 is exactly the same, see Sucarrat, Grønneberg, and Escribano (2016), whereas the covariance of α0 and β0 is Var(lnη12)·Σu−1 . In other words, the asymptotic variances are higher than those of the Cex- χ2 when E(ηt4−1)/Var(lnηt2)=Var(η2)/Var(lnηt2)<1 . In most situations this will indeed be the case, for example, when ηt is N(0,1) , which yields a fraction of about 0.41, see the upper part of Table 1. For some very fat-tailed and/or very skewed densities,2 however, for example, ηt∼t(5) and ηt∼t(5,0.7) , then the fraction is approximately 1.5 and 2.1, respectively. In fact, a detailed inspection reveals that when ηt is symmetric t, then the Cex- χ2 QMLE is more efficient for degrees of freedom greater than 5.79. Table 1 Relative efficiency comparison (see Section 2) Cex- χ2 vs. Gaussian ARMA Cex- χ2 vs. Standard QMLE Density ω0 α0 β0 μ0 ω0 α0 β0 N(0,1) : – 0.405 0.405 1.000 1.000 1.000 1.000 t(7): – 0.760 0.760 1.000 1.000 1.000 1.000 t(7,0.8) : – 0.840 0.840 1.000 1.000 1.000 1.000 t(5): – 1.475 1.475 1.000 1.000 1.000 1.000 t(5,0.7) : – 2.074 2.074 1.000 1.000 1.000 1.000 Finite sample (n = 1000): Cex- χ2 vs. Gaussian ARMA Cex- χ2 vs. Standard QMLE Density DGP(ω0,α0,β0) ω0 α0 β0 μ0 ω0 α0 β0 N(0,1) : 0,0.1,0.8 0.106 0.367 0.146 0.800 0.982 0.966 0.954 0,0.05,0.94 0.006 0.221 0.015 0.294 0.787 1.000 0.927 t(5): 0,0.1,0.8 0.415 0.902 0.494 0.753 1.136 0.911 0.924 0,0.05,0.94 0.081 0.701 0.158 0.327 1.485 1.277 1.658 t(5,0.7) : 0,0.1,0.8 0.878 1.696 0.917 0.965 1.406 1.216 1.174 0,0.05,0.94 0.064 1.155 0.156 0.350 0.348 0.769 0.804 Cex- χ2 vs. Gaussian ARMA Cex- χ2 vs. Standard QMLE Density ω0 α0 β0 μ0 ω0 α0 β0 N(0,1) : – 0.405 0.405 1.000 1.000 1.000 1.000 t(7): – 0.760 0.760 1.000 1.000 1.000 1.000 t(7,0.8) : – 0.840 0.840 1.000 1.000 1.000 1.000 t(5): – 1.475 1.475 1.000 1.000 1.000 1.000 t(5,0.7) : – 2.074 2.074 1.000 1.000 1.000 1.000 Finite sample (n = 1000): Cex- χ2 vs. Gaussian ARMA Cex- χ2 vs. Standard QMLE Density DGP(ω0,α0,β0) ω0 α0 β0 μ0 ω0 α0 β0 N(0,1) : 0,0.1,0.8 0.106 0.367 0.146 0.800 0.982 0.966 0.954 0,0.05,0.94 0.006 0.221 0.015 0.294 0.787 1.000 0.927 t(5): 0,0.1,0.8 0.415 0.902 0.494 0.753 1.136 0.911 0.924 0,0.05,0.94 0.081 0.701 0.158 0.327 1.485 1.277 1.658 t(5,0.7) : 0,0.1,0.8 0.878 1.696 0.917 0.965 1.406 1.216 1.174 0,0.05,0.94 0.064 1.155 0.156 0.350 0.348 0.769 0.804 Notes: Cex- χ2 vs. Gaussian ARMA, ratio of variances between the Cex- χ2 QMLE and the Gaussian ARMA-QMLE. Cex- χ2 vs. Standard QMLE, ratio of variances between the Cex- χ2 QMLE and the Standard QMLE. N(0,1) , ηt is standard normal. t(df), ηt is standardized t with df degrees of freedom. t(df, skew), ηt is standardized skew t with df degrees of freedom and skew > 0. Symmetry obtains when skew = 1, and left-skewness (right-skewness) obtains when skew < 1 (skew > 1). The skewing method is that of Fernández and Steel (1998). All computations in R, see R Core Team (2014). Table 1 Relative efficiency comparison (see Section 2) Cex- χ2 vs. Gaussian ARMA Cex- χ2 vs. Standard QMLE Density ω0 α0 β0 μ0 ω0 α0 β0 N(0,1) : – 0.405 0.405 1.000 1.000 1.000 1.000 t(7): – 0.760 0.760 1.000 1.000 1.000 1.000 t(7,0.8) : – 0.840 0.840 1.000 1.000 1.000 1.000 t(5): – 1.475 1.475 1.000 1.000 1.000 1.000 t(5,0.7) : – 2.074 2.074 1.000 1.000 1.000 1.000 Finite sample (n = 1000): Cex- χ2 vs. Gaussian ARMA Cex- χ2 vs. Standard QMLE Density DGP(ω0,α0,β0) ω0 α0 β0 μ0 ω0 α0 β0 N(0,1) : 0,0.1,0.8 0.106 0.367 0.146 0.800 0.982 0.966 0.954 0,0.05,0.94 0.006 0.221 0.015 0.294 0.787 1.000 0.927 t(5): 0,0.1,0.8 0.415 0.902 0.494 0.753 1.136 0.911 0.924 0,0.05,0.94 0.081 0.701 0.158 0.327 1.485 1.277 1.658 t(5,0.7) : 0,0.1,0.8 0.878 1.696 0.917 0.965 1.406 1.216 1.174 0,0.05,0.94 0.064 1.155 0.156 0.350 0.348 0.769 0.804 Cex- χ2 vs. Gaussian ARMA Cex- χ2 vs. Standard QMLE Density ω0 α0 β0 μ0 ω0 α0 β0 N(0,1) : – 0.405 0.405 1.000 1.000 1.000 1.000 t(7): – 0.760 0.760 1.000 1.000 1.000 1.000 t(7,0.8) : – 0.840 0.840 1.000 1.000 1.000 1.000 t(5): – 1.475 1.475 1.000 1.000 1.000 1.000 t(5,0.7) : – 2.074 2.074 1.000 1.000 1.000 1.000 Finite sample (n = 1000): Cex- χ2 vs. Gaussian ARMA Cex- χ2 vs. Standard QMLE Density DGP(ω0,α0,β0) ω0 α0 β0 μ0 ω0 α0 β0 N(0,1) : 0,0.1,0.8 0.106 0.367 0.146 0.800 0.982 0.966 0.954 0,0.05,0.94 0.006 0.221 0.015 0.294 0.787 1.000 0.927 t(5): 0,0.1,0.8 0.415 0.902 0.494 0.753 1.136 0.911 0.924 0,0.05,0.94 0.081 0.701 0.158 0.327 1.485 1.277 1.658 t(5,0.7) : 0,0.1,0.8 0.878 1.696 0.917 0.965 1.406 1.216 1.174 0,0.05,0.94 0.064 1.155 0.156 0.350 0.348 0.769 0.804 Notes: Cex- χ2 vs. Gaussian ARMA, ratio of variances between the Cex- χ2 QMLE and the Gaussian ARMA-QMLE. Cex- χ2 vs. Standard QMLE, ratio of variances between the Cex- χ2 QMLE and the Standard QMLE. N(0,1) , ηt is standard normal. t(df), ηt is standardized t with df degrees of freedom. t(df, skew), ηt is standardized skew t with df degrees of freedom and skew > 0. Symmetry obtains when skew = 1, and left-skewness (right-skewness) obtains when skew < 1 (skew > 1). The skewing method is that of Fernández and Steel (1998). All computations in R, see R Core Team (2014). An extensive set of Monte-Carlo simulations were undertaken in order to compare the finite sample properties of the Cex- χ2 QMLE, the Gaussian ARMA–QMLE, and the standard QMLE. To recall, if the ARMA representation of a log-GARCH(1,1) is lnεt2=ω0*+φ0 lnεt−12+θ0ut−1+ut , then the Gaussian ARMA–QMLE proceeds in two steps. First, estimate ω*, φ0 , and θ0 by using the Gaussian as instrumental density for ut. Second, estimate E lnηt2 by means of the negative of the log of the smearing estimator of Duan (1983). Under mild assumptions, this provides consistent estimates of ω0, α0, and β0, see Sucarrat, Grønneberg, and Escribano (2016). The lower part of Table 1 contains the results for n = 1000 only (additional simulations are contained in the Supplementary Appendix). Unsurprisingly, the Cex- χ2 is substantially more efficient than the Gaussian ARMA–QMLE when ηt∼N(0,1) . Interestingly, however, the Cex- χ2 is in fact also more efficient in general when ηt is t or skewed t with five degrees of freedom, that is, situations where it is less efficient asymptotically. This is particularly the case for ω0, β0, and μ0. Compared with the standard QMLE, the Cex- χ2 is more efficient about half of the times, but there is no clear pattern. An interesting exception, perhaps, is when ηt∼N(0,1) . In this case, the Cex- χ2 is slightly more efficient for all but one estimate. Another finding of interest is that skewness can matter a lot. For example, the standard QMLE is more efficient when ηt∼t(5) (i.e., symmetry) and volatility is at its most persistent (i.e., α0=0.05 and β0=0.94 ). By contrast, in the skewed case (i.e., ηt∼t(5,0.7), α0=0.05 , and β0=0.94 ) then the Cex- χ2 is (substantially) more efficient. 3 Empirical Applications 3.1 The Effect of Zero Returns on Financial Volatility Estimates A critique that has been directed toward the log-GARCH model is that its log-volatility specification does not exist in the presence of zero returns, see for example, Engle and Bollerslev (1986a). Sucarrat and Escribano (2017) propose a solution that distinguishes between true and observed returns: zeros constitute unobserved or missing values of the true return, and estimation via the ARMA representation is combined with a missing data procedure. Since the Cex- χ2 QMLE undertakes estimation via the ARMA representation, the algorithm can straightforwardly be combined with the estimator (see Section 3 in the Supplementary Appendix for the details). The algorithm is not available for the standard log-GARCH QMLE, but it is available for the standard GARCH (i.e., non-exponential) QMLE. This is of interest, since the standard GARCH QMLE is inconsistent if the zero-probability is time-varying (ηt is not iid if the zero probability is assumed to be zero. For comparison, we thus adapt the algorithm of Sucarrat and Escribano (2017) to the standard GARCH QMLE (see Section 3.4 in the Supplementary Appendix for the details). Sucarrat and Escribano (2017) show that volatility estimates can differ substantially if zeros are not accommodated correctly. Here, we revisit three of their return series to undertake a similar illustration. Specifically, we show that the standard log-GARCH QMLE systematically underestimates volatility in the presence of zeros (the more zeros the greater the bias) compared with the zero-adjusted Cex- χ2 QMLE. Then, we compare the log-GARCH results with those of the standard GARCH (i.e., non-exponential) QMLE and its zero-adjusted counterpart. As we will see, the comparison reveals that also the standard GARCH QMLE systematically underestimates volatility if not properly adjusted for zeros, and that the size of the bias is higher the greater the zero-probability. The three financial series in question are the daily Standard and Poor’s 500 stock market index (SP500) return, the daily Apple stock price return, and the daily Ekornes stock price return. The first two series are well-known, whereas the latter is a leading Nordic furniture manufacturer listed on the Oslo Stock Exchange. Ekornes is a medium-sized company in international terms, its market value being approximately 300 million euros toward the end of the series. Our interest in Ekornes is due to its relatively large proportion of zero returns over the sample (about 19%).3 However, in comparison with intra-day data its zero proportion may actually be small. The source of the data is Yahoo Finance (http://finance.yahoo.com). The sample dates and the descriptive statistics are contained in Table 2, whereas Figure 1 (left) contains graphs of the returns. The table and figure confirm that the returns exhibit the usual properties of excess kurtosis compared with the normal, and ARCH as measured by first-order serial correlation in the squared return. The number of zeros varies from only two observations (about 0.1% of the sample) for SP500 to 667 observations (about 19% of the sample) for Ekornes. Table 2 Descriptive statistics of financial returns (see Figure 1) s2 κ^ ARCH1[p-val] n 0s π^0 SP500 (January 4, 1999–August 23, 2013) 1.73 10.30 143.07[0.00] 3684 2 0.001 Apple (September 10, 1984–August 23, 2013) 9.25 55.03 7.11[0.01] 7303 294 0.040 Ekornes (January 4, 2000–August 26, 2013) 5.70 10.32 54.00[0.00] 3546 667 0.188 s2 κ^ ARCH1[p-val] n 0s π^0 SP500 (January 4, 1999–August 23, 2013) 1.73 10.30 143.07[0.00] 3684 2 0.001 Apple (September 10, 1984–August 23, 2013) 9.25 55.03 7.11[0.01] 7303 294 0.040 Ekornes (January 4, 2000–August 26, 2013) 5.70 10.32 54.00[0.00] 3546 667 0.188 Notes: s2, sample variance. κ^ , sample kurtosis. ARCH1, Ljung and Box (1979) test statistic of first-order serial correlation in the squared return. n, number of returns. 0s, number of zero returns in the sample. π^0 , proportion of zero returns in the sample. All computations in R, see R Core Team (2014). Table 2 Descriptive statistics of financial returns (see Figure 1) s2 κ^ ARCH1[p-val] n 0s π^0 SP500 (January 4, 1999–August 23, 2013) 1.73 10.30 143.07[0.00] 3684 2 0.001 Apple (September 10, 1984–August 23, 2013) 9.25 55.03 7.11[0.01] 7303 294 0.040 Ekornes (January 4, 2000–August 26, 2013) 5.70 10.32 54.00[0.00] 3546 667 0.188 s2 κ^ ARCH1[p-val] n 0s π^0 SP500 (January 4, 1999–August 23, 2013) 1.73 10.30 143.07[0.00] 3684 2 0.001 Apple (September 10, 1984–August 23, 2013) 9.25 55.03 7.11[0.01] 7303 294 0.040 Ekornes (January 4, 2000–August 26, 2013) 5.70 10.32 54.00[0.00] 3546 667 0.188 Notes: s2, sample variance. κ^ , sample kurtosis. ARCH1, Ljung and Box (1979) test statistic of first-order serial correlation in the squared return. n, number of returns. 0s, number of zero returns in the sample. π^0 , proportion of zero returns in the sample. All computations in R, see R Core Team (2014). Figure 1 Open in new tabDownload slide Daily financial log-returns in percent (left, descriptive statistics in Table 2) and ratios of fitted conditional standard deviations (right, see Table 4). Figure 1 Open in new tabDownload slide Daily financial log-returns in percent (left, descriptive statistics in Table 2) and ratios of fitted conditional standard deviations (right, see Table 4). Table 3 contains the estimates of the log-GARCH(1,1) specification lnσt2=ω0+α0 lnεt−12+β0 lnσt−12 , where εt is the log-return in percent (i.e., the log-difference of price multiplied by 100). For SP500, in which there are only two zeros, the estimates of ω0, α0, and β0 are almost identical. Unsurprisingly, therefore, the ratios of their fitted volatilities are also almost identical, apart from slight differences around the two zero return days, see Figure 1 (right). The maximum difference between the fitted volatilities is 3% in relative terms, see Table 4. For Apple, in which there are 294 zeros (i.e., about 4% of the sample), the parameter estimates differ more, and Table 4 and Figure 1 show that these discrepancies can lead to substantial differences in the volatility estimates. In particular, on single days the Cex- χ2 QMLE can produce fitted volatilities that are up to 12% larger. In finance, this discrepancy can be huge in economic terms. Another interesting characteristic of the Apple graph in Figure 1 is that the fitted values are closer to each other on average in the second part of the sample. This reflects that there are fewer zeros in the second part of the sample (the zero probability trends toward over the sample). Finally, for Ekornes, in which there are 667 zeros (i.e., about 19% of the sample), the parameter estimates do not differ more than for Apple. However, from Table 4 and Figure 1, it is very clear that the standard log-GARCH QMLE systematically estimates volatility to be substantially lower compared with the Cex- χ2 QMLE. This is most clearly exhibited in the bottom graph (right). On average, the fitted volatility of Cex- χ2 is about 12% higher than that of the standard log-GARCH QMLE, and the maximum difference is 21% higher. Table 3 Estimates of log-GARCH(1,1) and GARCH(1,1) specifications for different estimators (see Section 3.1) Estimator ω^ α^ se(α^) β^ se(β^) μ^ se(μ^) SP500: Log-GARCH Standard 0.077 0.049 0.005 0.937 0.007 – – Gaussian 0.071 0.046 0.006 0.946 0.007 −1.532 0.032 Cex- χ2 0.077 0.049 0.005 0.937 0.007 −1.528 0.023 GARCH Standard 0.015 0.083 0.008 0.908 0.009 – – 0-adjusted 0.015 0.083 0.008 0.908 0.009 – – Apple: Log-GARCH Standard 0.113 0.054 0.009 0.927 0.014 – – Gaussian 0.048 0.029 0.003 0.967 0.004 −1.399 0.042 Cex- χ2 0.114 0.053 0.009 0.928 0.014 −1.395 0.017 GARCH Standard 0.168 0.087 0.008 0.901 0.010 – – 0-Adjusted 0.165 0.091 0.010 0.898 0.011 – – Ekornes: Log-GARCH Standard 0.024 0.023 0.008 0.977 0.008 – – Gaussian 0.075 0.047 0.007 0.943 0.009 −1.179 0.050 Cex- χ2 0.037 0.024 0.008 0.971 0.010 −1.170 0.026 GARCH Standard 0.036 0.019 0.002 0.974 0.004 – – 0-Adjusted 0.049 0.025 0.004 0.967 0.005 – – Estimator ω^ α^ se(α^) β^ se(β^) μ^ se(μ^) SP500: Log-GARCH Standard 0.077 0.049 0.005 0.937 0.007 – – Gaussian 0.071 0.046 0.006 0.946 0.007 −1.532 0.032 Cex- χ2 0.077 0.049 0.005 0.937 0.007 −1.528 0.023 GARCH Standard 0.015 0.083 0.008 0.908 0.009 – – 0-adjusted 0.015 0.083 0.008 0.908 0.009 – – Apple: Log-GARCH Standard 0.113 0.054 0.009 0.927 0.014 – – Gaussian 0.048 0.029 0.003 0.967 0.004 −1.399 0.042 Cex- χ2 0.114 0.053 0.009 0.928 0.014 −1.395 0.017 GARCH Standard 0.168 0.087 0.008 0.901 0.010 – – 0-Adjusted 0.165 0.091 0.010 0.898 0.011 – – Ekornes: Log-GARCH Standard 0.024 0.023 0.008 0.977 0.008 – – Gaussian 0.075 0.047 0.007 0.943 0.009 −1.179 0.050 Cex- χ2 0.037 0.024 0.008 0.971 0.010 −1.170 0.026 GARCH Standard 0.036 0.019 0.002 0.974 0.004 – – 0-Adjusted 0.049 0.025 0.004 0.967 0.005 – – Notes: The estimated log-GARCH(1,1) specification is εt=σtηt, lnσt2=ω0+α0lnεt−12+β0lnσt−12, μ=E(lnηt2) . Standard, Standard log-GARCH QMLE. Gaussian, 0-adjusted Gaussian ARMA-QMLE. Cex- χ2 , 0-adjusted Cex- χ2 QMLE. The estimated GARCH(1,1) specification is εt=σtηt, σt2=ω0+α0εt−12+β0σt−12 . Standard, the standard (non-exponential) GARCH QMLE. 0-Adjusted, 0-adjusted standard (non-exponential) GARCH QMLE. se(·) , standard error of estimate. All computations in R, see R Core Team (2014). Table 3 Estimates of log-GARCH(1,1) and GARCH(1,1) specifications for different estimators (see Section 3.1) Estimator ω^ α^ se(α^) β^ se(β^) μ^ se(μ^) SP500: Log-GARCH Standard 0.077 0.049 0.005 0.937 0.007 – – Gaussian 0.071 0.046 0.006 0.946 0.007 −1.532 0.032 Cex- χ2 0.077 0.049 0.005 0.937 0.007 −1.528 0.023 GARCH Standard 0.015 0.083 0.008 0.908 0.009 – – 0-adjusted 0.015 0.083 0.008 0.908 0.009 – – Apple: Log-GARCH Standard 0.113 0.054 0.009 0.927 0.014 – – Gaussian 0.048 0.029 0.003 0.967 0.004 −1.399 0.042 Cex- χ2 0.114 0.053 0.009 0.928 0.014 −1.395 0.017 GARCH Standard 0.168 0.087 0.008 0.901 0.010 – – 0-Adjusted 0.165 0.091 0.010 0.898 0.011 – – Ekornes: Log-GARCH Standard 0.024 0.023 0.008 0.977 0.008 – – Gaussian 0.075 0.047 0.007 0.943 0.009 −1.179 0.050 Cex- χ2 0.037 0.024 0.008 0.971 0.010 −1.170 0.026 GARCH Standard 0.036 0.019 0.002 0.974 0.004 – – 0-Adjusted 0.049 0.025 0.004 0.967 0.005 – – Estimator ω^ α^ se(α^) β^ se(β^) μ^ se(μ^) SP500: Log-GARCH Standard 0.077 0.049 0.005 0.937 0.007 – – Gaussian 0.071 0.046 0.006 0.946 0.007 −1.532 0.032 Cex- χ2 0.077 0.049 0.005 0.937 0.007 −1.528 0.023 GARCH Standard 0.015 0.083 0.008 0.908 0.009 – – 0-adjusted 0.015 0.083 0.008 0.908 0.009 – – Apple: Log-GARCH Standard 0.113 0.054 0.009 0.927 0.014 – – Gaussian 0.048 0.029 0.003 0.967 0.004 −1.399 0.042 Cex- χ2 0.114 0.053 0.009 0.928 0.014 −1.395 0.017 GARCH Standard 0.168 0.087 0.008 0.901 0.010 – – 0-Adjusted 0.165 0.091 0.010 0.898 0.011 – – Ekornes: Log-GARCH Standard 0.024 0.023 0.008 0.977 0.008 – – Gaussian 0.075 0.047 0.007 0.943 0.009 −1.179 0.050 Cex- χ2 0.037 0.024 0.008 0.971 0.010 −1.170 0.026 GARCH Standard 0.036 0.019 0.002 0.974 0.004 – – 0-Adjusted 0.049 0.025 0.004 0.967 0.005 – – Notes: The estimated log-GARCH(1,1) specification is εt=σtηt, lnσt2=ω0+α0lnεt−12+β0lnσt−12, μ=E(lnηt2) . Standard, Standard log-GARCH QMLE. Gaussian, 0-adjusted Gaussian ARMA-QMLE. Cex- χ2 , 0-adjusted Cex- χ2 QMLE. The estimated GARCH(1,1) specification is εt=σtηt, σt2=ω0+α0εt−12+β0σt−12 . Standard, the standard (non-exponential) GARCH QMLE. 0-Adjusted, 0-adjusted standard (non-exponential) GARCH QMLE. se(·) , standard error of estimate. All computations in R, see R Core Team (2014). Table 4 Descriptive statistics of fitted conditional standard deviations, and their ratios (see Section 3.1 and Figure 1) Estimator Mean s2 Max. Min. SP500: Log-GARCH Standard (S) 1.182 0.287 4.470 0.438 Gaussian (G) 1.196 0.336 4.779 0.433 Cex- χ2 (X) 1.182 0.287 4.476 0.438 Ratio X/S 1.000 0.000 1.002 0.976 GARCH Standard (S) 1.168 0.359 5.210 0.532 0-Adjusted (A) 1.169 0.359 5.210 0.532 Ratio A/S 1.000 0.000 1.045 1.000 Apple: Log-GARCH Standard (S) 2.894 0.657 6.581 1.013 Gaussian (G) 2.992 0.799 6.190 1.100 Cex- χ2 (X) 2.960 0.691 6.635 1.029 Ratio X/S 1.022 0.000 1.119 1.004 GARCH Standard (S) 2.915 1.364 22.123 1.431 0-Adjusted (A) 2.976 1.431 22.622 1.399 Ratio A/S 1.020 0.001 1.181 0.978 Ekornes: Log-GARCH Standard (S) 2.299 0.281 4.667 1.371 Gaussian (G) 2.580 0.392 5.480 1.353 Cex- χ2 (X) 2.561 0.334 4.891 1.509 Ratio X/S 1.115 0.001 1.214 1.025 GARCH Standard (S) 2.287 0.363 5.364 1.520 0-Adjusted (A) 2.539 0.439 5.948 1.676 Ratio A/S 1.111 0.001 1.225 1.038 Estimator Mean s2 Max. Min. SP500: Log-GARCH Standard (S) 1.182 0.287 4.470 0.438 Gaussian (G) 1.196 0.336 4.779 0.433 Cex- χ2 (X) 1.182 0.287 4.476 0.438 Ratio X/S 1.000 0.000 1.002 0.976 GARCH Standard (S) 1.168 0.359 5.210 0.532 0-Adjusted (A) 1.169 0.359 5.210 0.532 Ratio A/S 1.000 0.000 1.045 1.000 Apple: Log-GARCH Standard (S) 2.894 0.657 6.581 1.013 Gaussian (G) 2.992 0.799 6.190 1.100 Cex- χ2 (X) 2.960 0.691 6.635 1.029 Ratio X/S 1.022 0.000 1.119 1.004 GARCH Standard (S) 2.915 1.364 22.123 1.431 0-Adjusted (A) 2.976 1.431 22.622 1.399 Ratio A/S 1.020 0.001 1.181 0.978 Ekornes: Log-GARCH Standard (S) 2.299 0.281 4.667 1.371 Gaussian (G) 2.580 0.392 5.480 1.353 Cex- χ2 (X) 2.561 0.334 4.891 1.509 Ratio X/S 1.115 0.001 1.214 1.025 GARCH Standard (S) 2.287 0.363 5.364 1.520 0-Adjusted (A) 2.539 0.439 5.948 1.676 Ratio A/S 1.111 0.001 1.225 1.038 Notes: Standard, standard log-GARCH QMLE. Gaussian, 0-adjusted Gaussian ARMA-QMLE. Cex- χ2 , 0-adjusted Cex- χ2 QMLE. Ordinary, the ordinary (non-exponential) GARCH(1,1) specification. 0-Adjusted, 0-adjusted ordinary GARCH(1,1). Mean, sample average. s2, sample variance. Max., maximum value. Min., minimum value. Ratio, the ratio between fitted conditional standard deviations. All computations in R, see R Core Team (2014). Table 4 Descriptive statistics of fitted conditional standard deviations, and their ratios (see Section 3.1 and Figure 1) Estimator Mean s2 Max. Min. SP500: Log-GARCH Standard (S) 1.182 0.287 4.470 0.438 Gaussian (G) 1.196 0.336 4.779 0.433 Cex- χ2 (X) 1.182 0.287 4.476 0.438 Ratio X/S 1.000 0.000 1.002 0.976 GARCH Standard (S) 1.168 0.359 5.210 0.532 0-Adjusted (A) 1.169 0.359 5.210 0.532 Ratio A/S 1.000 0.000 1.045 1.000 Apple: Log-GARCH Standard (S) 2.894 0.657 6.581 1.013 Gaussian (G) 2.992 0.799 6.190 1.100 Cex- χ2 (X) 2.960 0.691 6.635 1.029 Ratio X/S 1.022 0.000 1.119 1.004 GARCH Standard (S) 2.915 1.364 22.123 1.431 0-Adjusted (A) 2.976 1.431 22.622 1.399 Ratio A/S 1.020 0.001 1.181 0.978 Ekornes: Log-GARCH Standard (S) 2.299 0.281 4.667 1.371 Gaussian (G) 2.580 0.392 5.480 1.353 Cex- χ2 (X) 2.561 0.334 4.891 1.509 Ratio X/S 1.115 0.001 1.214 1.025 GARCH Standard (S) 2.287 0.363 5.364 1.520 0-Adjusted (A) 2.539 0.439 5.948 1.676 Ratio A/S 1.111 0.001 1.225 1.038 Estimator Mean s2 Max. Min. SP500: Log-GARCH Standard (S) 1.182 0.287 4.470 0.438 Gaussian (G) 1.196 0.336 4.779 0.433 Cex- χ2 (X) 1.182 0.287 4.476 0.438 Ratio X/S 1.000 0.000 1.002 0.976 GARCH Standard (S) 1.168 0.359 5.210 0.532 0-Adjusted (A) 1.169 0.359 5.210 0.532 Ratio A/S 1.000 0.000 1.045 1.000 Apple: Log-GARCH Standard (S) 2.894 0.657 6.581 1.013 Gaussian (G) 2.992 0.799 6.190 1.100 Cex- χ2 (X) 2.960 0.691 6.635 1.029 Ratio X/S 1.022 0.000 1.119 1.004 GARCH Standard (S) 2.915 1.364 22.123 1.431 0-Adjusted (A) 2.976 1.431 22.622 1.399 Ratio A/S 1.020 0.001 1.181 0.978 Ekornes: Log-GARCH Standard (S) 2.299 0.281 4.667 1.371 Gaussian (G) 2.580 0.392 5.480 1.353 Cex- χ2 (X) 2.561 0.334 4.891 1.509 Ratio X/S 1.115 0.001 1.214 1.025 GARCH Standard (S) 2.287 0.363 5.364 1.520 0-Adjusted (A) 2.539 0.439 5.948 1.676 Ratio A/S 1.111 0.001 1.225 1.038 Notes: Standard, standard log-GARCH QMLE. Gaussian, 0-adjusted Gaussian ARMA-QMLE. Cex- χ2 , 0-adjusted Cex- χ2 QMLE. Ordinary, the ordinary (non-exponential) GARCH(1,1) specification. 0-Adjusted, 0-adjusted ordinary GARCH(1,1). Mean, sample average. s2, sample variance. Max., maximum value. Min., minimum value. Ratio, the ratio between fitted conditional standard deviations. All computations in R, see R Core Team (2014). The results for the GARCH(1,1) models, which are also contained in Table 3, are very similar: Volatilities are systematically underestimated if zeros are not treated appropriately, and the downward bias is larger the more zeros. Nominally, the parameter estimates of the 0-adjusted standard GARCH QMLE are not very different to those of the standard GARCH QMLE, but the discrepancy between fitted volatilities can be large in relative terms. Moreover, both the ranges, that is, the differences between the maximum- and minimum-fitted standard deviations, and the nominal sizes of the downward biases, are very similar to those of the X/S ratios, that is, the comparison between the Cex- χ2 ARMA–QMLE and the standard log-GARCH QMLE. 3.2 The Uncertainty of Monthly Euro-inflation Conditional forecasts of inflation play an important part when the policy interest rate is set, so the uncertainty associated with those forecasts are central to financial market participants. When Engle (1982) proposed his ARCH model, he used forecasts of the uncertainty of quarterly UK inflation to illustrate the usefulness of the model. However, the ARCH(4) specification he used in his illustration was severely restricted in order to ensure the positivity of the variance estimates (see Engle, 1982, p. 1002): Instead of freely estimating the ARCH parameters, he imposed a linearly declining relationship. Moreover, the ordinary GARCH model cannot accommodate negative autocorrelations in the squared errors (see Proposition 2.2 in Francq and Zakoïan, 2010), a characteristic which is not unlikely in monthly, quarterly, and yearly data. Here, we illustrate the versatility and usefulness of the log-GARCH model by fitting specifications of up to 12 orders—without any parameter restrictions—to the error of a dynamic model of monthly Euro-area inflation. The underlying series is the Harmonized Index of Consumer Prices (HICP) from January 2001 to June 2013 (n = 150 observations), and the source of the data is the European Central Bank (http://www.ecb.int/). We denote the HICP index-value at t by pt, and define the 12-month inflation or %-change as yt=100·(pt−pt−12)/pt−12 . The estimation results of the dynamic model of inflation—an AR(12) specification—is contained in Table 5. The diagnostics tests suggest there is little or no autocorrelation in the residuals ε^t , since the p-values of the tests of no autocorrelation up to the 12th and 13th lags, respectively, are 0.12 and 0.11. However, the diagnostic tests suggest that there is significant ARCH, since the p-values are equal to 0.00 in the two tests of no ARCH up to the 12th and 13th lags, respectively. Moreover, several of the autocorrelations of ε^t2 are negative. Next, the diagnostic tests of the log-GARCH(1,12) model of volatility show that the model successfully removes the ARCH, since the two p-values associated with the tests of no ARCH increase to 0.75 and 0.77, respectively. The tests for autocorrelation also improve, since the p-values increase to 0.52 and 0.51, respectively. The GARCH term is estimated to be negative and significant at usual significance levels, and several of the ARCH-lags are estimated to be negative. But only the fifth and sixth ARCH-lag—which are both positive—are significant at 5%. Sequential backward elimination of regressors with t-ratios smaller than 2 in absolute value leads to the second log-GARCH specification in the same table.4 There, the first-order GARCH term together with ARCH terms 5, 6, and 11 are significant, and the latter is negative. The value of the GARCH(1) parameter β^1=−0.644 is negative and substantially smaller that the value of the ARCH(1) parameter α^1=0.046 , so the dynamics is contrarian (i.e., exhibit negative autocorrelations at some lags). The log-moment μ0 is estimated to –1.228, which is very close to normality (i.e., –1.27), and the test-statistic (−1.228+1.27)/0.133 is equal to –0.316. So the null of normality is not rejected at the usual significance levels. For comparison, we also include an ordinary GARCH(2,2) specification. It does not pass the diagnostics tests in the sense that η^t2 is autocorrelated at lags 12 and 13, which suggests a more flexible specification is needed. We also tried a GARCH(1,1)—see the Supplementary Appendix, which also failed the diagnostics. Table 5 Empirical monthly Eurozone inflation estimates (see Section 3.2) AR(12) model of inflation (yt): y^t = 0.372(0.095)+1.169(0.094)yt−1−0.193(0.139)yt−2−0.177(0.141)yt−3+0.362(0.127)yt−4−0.493(0.133)yt−5+0.356(0.139)yt−6 +0.014(0.150)yt−7−0.337(0.150)yt−8+0.323(0.130)yt−9−0.239(0.142)yt−10+0.201(0.136)yt−11−0.171(0.086)yt−12 AR12(ε^t)[p-val.]:17.76[0.12] AR13(ε^t)[p-val.]:19.55[0.11] ARCH12(ε^t)[p-val.]:30.87[0.00] ARCH13(ε^t)[p-val.]:32.51[0.00] rε^t2,ε^t−i2 : 0.07i=1:,−0.03i=2:,−0.11i=3:,0.00i=4:,0.12i=5:,0.35i=6:,−0.05i=7:,0.01i=8:,0.02i=9:,−0.05i=10:,−0.04i=11:,0.25i=12:, Log-GARCH(1,12) model of the variance: lnσ^t2 = −4.292+0.076(0.067)lnε^t−12+0.030(0.056)lnε^t−22−0.029(0.070)lnε^t−32+0.034(0.075)lnε^t−42+0.263(0.065)lnε^t−52 +0.205(0.083)lnε^t−62−0.087(0.083)lnε^t−72+0.044(0.070)lnε^t−82−0.046(0.066)lnε^t−92−0.100(0.090)lnε^t−102 −0.094(0.082)lnε^t−112+0.050(0.067)lnε^t−122−0.522(0.230)lnσ^t−12 , μ^0=−1.205(0.133) AR12(η^t)[p-val.]:11.08[0.52] AR13(η^t)[p-val.]:12.17[0.51], ARCH12(η^t)[p-val.]:8.49[0.75] ARCH13(η^t)[p-val.]:9.01[0.77] Simplified log-GARCH model of the variance: lnσ^t2 = −4.754+0.046(0.061)lnε^t−12+0.226(0.059)lnε^t−52+0.251(0.057)lnε^t−62−0.128(0.052)lnε^t−112−0.644(0.137)lnσ^t−12 , μ^0=−1.228(0.133) AR12(η^t)[p-val.]:11.95[0.53] AR13(η^t)[p-val.]:12.47[0.49], ARCH12(η^t)[p-val.]:10.47[0.57] ARCH13(η^t)[p-val.]:10.51[0.65] Ordinary GARCH(2,2) model of the variance: σ^t2 = 0.049(0.405)+0.063(0.174)ε^t−12+3e-15(0.476)ε^t−22+0.049(7.566)σ^t−12+0.038(2.623)σ^t−22 , AR12(η^t)[p-val.]:15.19[0.23] AR13(η^t)[p-val.]:16.61[0.22], ARCH12(η^t)[p-val.]:35.70[0.00] ARCH13(η^t)[p-val.]:37.72[0.00] AR(12) model of inflation (yt): y^t = 0.372(0.095)+1.169(0.094)yt−1−0.193(0.139)yt−2−0.177(0.141)yt−3+0.362(0.127)yt−4−0.493(0.133)yt−5+0.356(0.139)yt−6 +0.014(0.150)yt−7−0.337(0.150)yt−8+0.323(0.130)yt−9−0.239(0.142)yt−10+0.201(0.136)yt−11−0.171(0.086)yt−12 AR12(ε^t)[p-val.]:17.76[0.12] AR13(ε^t)[p-val.]:19.55[0.11] ARCH12(ε^t)[p-val.]:30.87[0.00] ARCH13(ε^t)[p-val.]:32.51[0.00] rε^t2,ε^t−i2 : 0.07i=1:,−0.03i=2:,−0.11i=3:,0.00i=4:,0.12i=5:,0.35i=6:,−0.05i=7:,0.01i=8:,0.02i=9:,−0.05i=10:,−0.04i=11:,0.25i=12:, Log-GARCH(1,12) model of the variance: lnσ^t2 = −4.292+0.076(0.067)lnε^t−12+0.030(0.056)lnε^t−22−0.029(0.070)lnε^t−32+0.034(0.075)lnε^t−42+0.263(0.065)lnε^t−52 +0.205(0.083)lnε^t−62−0.087(0.083)lnε^t−72+0.044(0.070)lnε^t−82−0.046(0.066)lnε^t−92−0.100(0.090)lnε^t−102 −0.094(0.082)lnε^t−112+0.050(0.067)lnε^t−122−0.522(0.230)lnσ^t−12 , μ^0=−1.205(0.133) AR12(η^t)[p-val.]:11.08[0.52] AR13(η^t)[p-val.]:12.17[0.51], ARCH12(η^t)[p-val.]:8.49[0.75] ARCH13(η^t)[p-val.]:9.01[0.77] Simplified log-GARCH model of the variance: lnσ^t2 = −4.754+0.046(0.061)lnε^t−12+0.226(0.059)lnε^t−52+0.251(0.057)lnε^t−62−0.128(0.052)lnε^t−112−0.644(0.137)lnσ^t−12 , μ^0=−1.228(0.133) AR12(η^t)[p-val.]:11.95[0.53] AR13(η^t)[p-val.]:12.47[0.49], ARCH12(η^t)[p-val.]:10.47[0.57] ARCH13(η^t)[p-val.]:10.51[0.65] Ordinary GARCH(2,2) model of the variance: σ^t2 = 0.049(0.405)+0.063(0.174)ε^t−12+3e-15(0.476)ε^t−22+0.049(7.566)σ^t−12+0.038(2.623)σ^t−22 , AR12(η^t)[p-val.]:15.19[0.23] AR13(η^t)[p-val.]:16.61[0.22], ARCH12(η^t)[p-val.]:35.70[0.00] ARCH13(η^t)[p-val.]:37.72[0.00] Notes: (·) , standard error of estimate. AR12 and AR13, Ljung and Box (1979) tests for 12th and 13th order autocorrelation in either ε^t or η^t . ARCH12 and ARCH13, Ljung and Box (1979) tests for 12th and 13th order autocorrelation in either ε^t2 or η^t2 . p-val., p-value. rε^t2,ε^t−i2 , the sample correlation between εt2 and εt−i2 . The log-GARCH models are estimated with the lgarch package version 0.6-2 (Sucarrat, 2015), whereas the GARCH model is estimated with the garch function from the tseries package version 0.10.35 (Trapletti and Hornik, 2013). All computations in R, see R Core Team (2014). Table 5 Empirical monthly Eurozone inflation estimates (see Section 3.2) AR(12) model of inflation (yt): y^t = 0.372(0.095)+1.169(0.094)yt−1−0.193(0.139)yt−2−0.177(0.141)yt−3+0.362(0.127)yt−4−0.493(0.133)yt−5+0.356(0.139)yt−6 +0.014(0.150)yt−7−0.337(0.150)yt−8+0.323(0.130)yt−9−0.239(0.142)yt−10+0.201(0.136)yt−11−0.171(0.086)yt−12 AR12(ε^t)[p-val.]:17.76[0.12] AR13(ε^t)[p-val.]:19.55[0.11] ARCH12(ε^t)[p-val.]:30.87[0.00] ARCH13(ε^t)[p-val.]:32.51[0.00] rε^t2,ε^t−i2 : 0.07i=1:,−0.03i=2:,−0.11i=3:,0.00i=4:,0.12i=5:,0.35i=6:,−0.05i=7:,0.01i=8:,0.02i=9:,−0.05i=10:,−0.04i=11:,0.25i=12:, Log-GARCH(1,12) model of the variance: lnσ^t2 = −4.292+0.076(0.067)lnε^t−12+0.030(0.056)lnε^t−22−0.029(0.070)lnε^t−32+0.034(0.075)lnε^t−42+0.263(0.065)lnε^t−52 +0.205(0.083)lnε^t−62−0.087(0.083)lnε^t−72+0.044(0.070)lnε^t−82−0.046(0.066)lnε^t−92−0.100(0.090)lnε^t−102 −0.094(0.082)lnε^t−112+0.050(0.067)lnε^t−122−0.522(0.230)lnσ^t−12 , μ^0=−1.205(0.133) AR12(η^t)[p-val.]:11.08[0.52] AR13(η^t)[p-val.]:12.17[0.51], ARCH12(η^t)[p-val.]:8.49[0.75] ARCH13(η^t)[p-val.]:9.01[0.77] Simplified log-GARCH model of the variance: lnσ^t2 = −4.754+0.046(0.061)lnε^t−12+0.226(0.059)lnε^t−52+0.251(0.057)lnε^t−62−0.128(0.052)lnε^t−112−0.644(0.137)lnσ^t−12 , μ^0=−1.228(0.133) AR12(η^t)[p-val.]:11.95[0.53] AR13(η^t)[p-val.]:12.47[0.49], ARCH12(η^t)[p-val.]:10.47[0.57] ARCH13(η^t)[p-val.]:10.51[0.65] Ordinary GARCH(2,2) model of the variance: σ^t2 = 0.049(0.405)+0.063(0.174)ε^t−12+3e-15(0.476)ε^t−22+0.049(7.566)σ^t−12+0.038(2.623)σ^t−22 , AR12(η^t)[p-val.]:15.19[0.23] AR13(η^t)[p-val.]:16.61[0.22], ARCH12(η^t)[p-val.]:35.70[0.00] ARCH13(η^t)[p-val.]:37.72[0.00] AR(12) model of inflation (yt): y^t = 0.372(0.095)+1.169(0.094)yt−1−0.193(0.139)yt−2−0.177(0.141)yt−3+0.362(0.127)yt−4−0.493(0.133)yt−5+0.356(0.139)yt−6 +0.014(0.150)yt−7−0.337(0.150)yt−8+0.323(0.130)yt−9−0.239(0.142)yt−10+0.201(0.136)yt−11−0.171(0.086)yt−12 AR12(ε^t)[p-val.]:17.76[0.12] AR13(ε^t)[p-val.]:19.55[0.11] ARCH12(ε^t)[p-val.]:30.87[0.00] ARCH13(ε^t)[p-val.]:32.51[0.00] rε^t2,ε^t−i2 : 0.07i=1:,−0.03i=2:,−0.11i=3:,0.00i=4:,0.12i=5:,0.35i=6:,−0.05i=7:,0.01i=8:,0.02i=9:,−0.05i=10:,−0.04i=11:,0.25i=12:, Log-GARCH(1,12) model of the variance: lnσ^t2 = −4.292+0.076(0.067)lnε^t−12+0.030(0.056)lnε^t−22−0.029(0.070)lnε^t−32+0.034(0.075)lnε^t−42+0.263(0.065)lnε^t−52 +0.205(0.083)lnε^t−62−0.087(0.083)lnε^t−72+0.044(0.070)lnε^t−82−0.046(0.066)lnε^t−92−0.100(0.090)lnε^t−102 −0.094(0.082)lnε^t−112+0.050(0.067)lnε^t−122−0.522(0.230)lnσ^t−12 , μ^0=−1.205(0.133) AR12(η^t)[p-val.]:11.08[0.52] AR13(η^t)[p-val.]:12.17[0.51], ARCH12(η^t)[p-val.]:8.49[0.75] ARCH13(η^t)[p-val.]:9.01[0.77] Simplified log-GARCH model of the variance: lnσ^t2 = −4.754+0.046(0.061)lnε^t−12+0.226(0.059)lnε^t−52+0.251(0.057)lnε^t−62−0.128(0.052)lnε^t−112−0.644(0.137)lnσ^t−12 , μ^0=−1.228(0.133) AR12(η^t)[p-val.]:11.95[0.53] AR13(η^t)[p-val.]:12.47[0.49], ARCH12(η^t)[p-val.]:10.47[0.57] ARCH13(η^t)[p-val.]:10.51[0.65] Ordinary GARCH(2,2) model of the variance: σ^t2 = 0.049(0.405)+0.063(0.174)ε^t−12+3e-15(0.476)ε^t−22+0.049(7.566)σ^t−12+0.038(2.623)σ^t−22 , AR12(η^t)[p-val.]:15.19[0.23] AR13(η^t)[p-val.]:16.61[0.22], ARCH12(η^t)[p-val.]:35.70[0.00] ARCH13(η^t)[p-val.]:37.72[0.00] Notes: (·) , standard error of estimate. AR12 and AR13, Ljung and Box (1979) tests for 12th and 13th order autocorrelation in either ε^t or η^t . ARCH12 and ARCH13, Ljung and Box (1979) tests for 12th and 13th order autocorrelation in either ε^t2 or η^t2 . p-val., p-value. rε^t2,ε^t−i2 , the sample correlation between εt2 and εt−i2 . The log-GARCH models are estimated with the lgarch package version 0.6-2 (Sucarrat, 2015), whereas the GARCH model is estimated with the garch function from the tseries package version 0.10.35 (Trapletti and Hornik, 2013). All computations in R, see R Core Team (2014). 4 Conclusions We propose a QMLE for log-GARCH models via the ARMA representation with the centred exponential Chi-squared (Cex- χ2 ) distribution as instrumental density. We prove the strong CAN of the Cex- χ2 QMLE under mild conditions, and compare its efficiency both asymptotically and in finite samples. Both asymptotically and in finite samples the Cex- χ2 QMLE performs better—in general—than the Gaussian ARMA–QMLE. Finally, two empirical applications illustrate the usefulness and versatility of the estimator. In the first we show how volatility estimates can be obtained by treating zeros (i.e., inliers) as missing values, since estimation via the ARMA representation enables the solution proposed by Sucarrat and Escribano (2017). The results show that volatility is systematically underestimated if zeros are not handled appropriately, and that the bias can be substantial. Moreover, the more zeros, the greater the bias. In our second empirical application, we show how the estimator can be used to model high-order volatility dynamics when one or more squared error autocorrelations are negative. The results in this paper are likely to be extendable in several ways. The most straightforward concerns the addition of leverage or asymmetry terms, and of additional conditional variables (“X”). This is because the relationships between the log-GARCH and ARMA parameters are not affected by the linear addition of terms. We conjecture that CAN results should not be too difficult to establish in these instances. More generally, since estimation is via the ARMA representation, a large body of results are likely to be applicable as straightforward extensions. Appendix A: Proofs of Theorems A.1 Proof of Theorem 1: Strong Consistency Let the random variables On(ϑ)=1n∑t=1nℓt(ϑ), Qn(ϑ)=1n∑t=1nℓt,n(ϑ), where up to the unimportant constant ln(2π)/2 , ℓt(ϑ)=lnχμ{ut(θ)}, ℓt,n(ϑ)=lnχμ{ut,n(θ)} , and ut,n(θ)=∑i=0∞ψi(θ)(lnεt−i2−νn). We also need to introduce subsets Λ of Ξ satisfying Esupϑ∈Λeu1(θ)<∞. (11) Note that, without loss of generality, we can assume that θ0∈Λ because Eeu1(θ0)=Eeu1=Eelnη12−μ0=e−μ0 . The proof of the consistency of ϑ^n is split into the following steps. For any Λ satisfying Equation (11),limn→∞supϑ∈Λ|On(ϑ)−Qn(ϑ)|=0 a.s.; limn→∞supϑ∈Ξ|Qn(ϑ)−Q˜n(ϑ)|=0 a.s.; if u1(θ)+μ=u1(θ0)+μ0 a.s. then θ=θ0 and μ=μ0; if ϑ≠ϑ0 then Eℓ1(ϑ)<Eℓ1(ϑ0); any ϑ≠ϑ0 has a neighborhood V(ϑ) such that limsupn→∞supϑ*∈V(ϑ)∩ΞQ˜n(ϑ*)<Eℓ1(ϑ0)=limn→∞Q˜n(ϑ0) a.s. (12) In the rest of the paper, the letters K≥0 and ρ∈[0,1) denote generic constants, or measurable functions of {εu, u≤0} , that do not vary with n. We first show (i). Note that Equation (3) and E|lnη12|<∞ entail E|lnε12|<∞ , so that ν0 is well defined. By the invertibility condition (5), we immediately have ut(θ)−ut,n(θ)=Kθ(νn−ν0), where Kθ=Bθ−1(1)Aθ(1)=∑i=0∞ψi(θ). The compactness of Θ then entails that supθ∈Θ|ut(θ)−ut,n(θ)|≤K|νn−ν0|, a.s. (13) We now need to show a similar bound for |eut(θ)−eut,n(θ)| when ϑ∈Λ . We have eut(θ)−eut,n(θ)=Xt(θ){e−ν0Kθ−e−νnKθ}, where Xt(θ)=e∑i=0∞ψi(θ) lnεt−i2 . By a Taylor expansion, we then obtain |eut(θ)−eut,n(θ)|=Kθ|νn−ν0|e−ν*KθXt(θ), where ν* lies between ν0 and νn. By the ergodic theorem, we have strong convergence of νn to ν0 as n→∞ . We thus obtain |eut(θ)−eut,n(θ)|≤K|νn−ν0|Xt(θ), (14) where Esupϑ∈ΛXt(θ)<∞ . Since |ℓt(ϑ)−ℓt,n(ϑ)|≤K(|ut(θ)−ut,n(θ)|+|eut(θ)−eut,n(θ)|), we obtain (i) from Equations (13) and (14), together with the ergodic theorem. We now show (ii). The compactness of Θ and the invertibility condition (5) entail that Bθ−1(L)=∑i=0∞ϕi(θ)Li, where supθ∈Θ|ϕi(θ)|≤Kρi. (15) Note that Equation (6) still holds true for any t when u˜t,n(θ) is replaced by ut,n(θ) . For all t > r we thus have Bθ(L){ut,n(θ)−u˜t,n(θ)}=0. Iterating this relation, we obtain in the log-GARCH (1,q) case ut,n(θ)−u˜t,n(θ)=βt−r{ur,n(θ)−u˜r,n(θ)}, with the simplified notation β=β1 . For the general log-GARCH(p, q) model, by Equation (15) we also obtain supθ∈Θ|ut,n(θ)−u˜t,n(θ)|≤Kρt, a.s. (16) We now study the difference eut,n(θ)−eu˜t,n(θ) . For simplicity, we focus on the log-GARCH(1,1) case, but the same arguments apply to the general model (however, with more complex notation). We then have, for t≥2 , ut,n(θ)=dt,n(θ)+βt−1u1,n(θ), dt,n(θ)=∑i=0t−2βiAθ(L){lnεt−i2−νn}. The same expression holds true for u˜t,n(θ) when u1,n(θ) is replaced by u˜1,n(θ) . Doing a Taylor expansion, it follows that eut,n(θ)−eu˜t,n(θ)={eβt−1u1,n(θ)−eβt−1u˜1,n(θ)}edt,n(θ)=βt−1{u1,n(θ)−u˜1,n(θ)}eβt−1u*edt,n(θ), where u* is between u1,n(θ) and u˜1,n(θ) . It follows that 1t ln|eut,n(θ)−eu˜t,n(θ)|≤Kt+dt,n(θ)t+ln|β|. (17) Note that supθ∈Θ|dt,n(θ)|≤dt+K|νn|, dt=K∑i=0∞ρi|lnεt−i2|. Since suptEdt<∞ , a.s. limt→∞dt/t=0 (see Lemma 7.1 in Francq, Wintenberger, and Zakoïan, 2013). Noting also that supn≥1|νn|<∞ with probability 1 (because the sequence (νn) converges a.s.), the first two terms of the right-hand side of Equation (17) tend a.s. to 0 as t→∞ , uniformly in n and θ. Since |β|<1 on the compact Θ, limsupt→∞supn≥1supθ∈Θ1t ln|eut,n(θ)−eu˜t,n(θ)|∈[−∞,0). We thus obtain supn≥1supθ∈Θ|eut,n(θ)−eu˜t,n(θ)|≤Kρt a.s. (18) and the conclusion follows from Equations (16) and (18). Let us turn to (iii). If ut(θ)+μ=ut(θ0)+μ0 then ∑i=0∞{ψi(θ)−ψi(θ0)}(lnεt−i2−ν0)=μ0−μ. Since the left-hand side of the equality is a centered random variable, we must have μ=μ0 . If ψ1(θ)−ψ1(θ0)≠0 then lnεt−12 is a linear combination of its past values. This is impossible because the linear innovations ut are not a.s. equal to zero. By recursion, ψi(θ)−ψi(θ0)=0 for all i, which entails θ=θ0 under the conditions on the AR and MA polynomials. To show (iv), note that Eℓ1(ϑ)=−∞ when Eeu1(θ)=∞ . It is thus sufficient to consider the case Eeu1(θ)<∞ . Then, we have 2{Eℓ1(ϑ)−Eℓ1(ϑ0)}=E{u1(θ)+μ−u1(θ0)−μ0+1−eu1(θ)+μ}=1+μ−μ0−Eeu1(θ)−u1(θ0)+μ−μ0+u1(θ0)+μ0=1+μ−μ0−Eeu1(θ)−u1(θ0)+μ−μ0≤0, with equality iff u1(θ)+μ=u1(θ0)+μ0 with probability 1. For the first equality, we used the fact that eu1(θ0)+μ0=eu1+μ0=η12, (19) for the second equality we note that Eu1(θ)=0 for all θ, for the third equality we use the independence between u1(θ)−u1(θ0) and u1(θ0)=lnηt2−μ0 , and for the inequality we argue that ex≥x+1 with equality iff x = 0. The conclusion comes from (iii). It remains to show (v). For any real function f, let f+(x)=max{f(x),0} and f−(x)=max{−f(x),0} . First note that, by the ergodic theorem, Equations (3), (5), and the compactness of Ξ, limsupn→∞supϑ∈ΞQn+(ϑ)≤K0:=supϑ∈Ξ{2∑i=0∞|ψi(θ)|E|lnε12|+|μ|}<∞. Moreover, applying the ergodic theorem for stationary and ergodic processes having an expectation in [0,∞] [see Billingsley, (1995, pp. 284 and 495)], we have liminfn→∞{supϑ∈Vk(ϑ*)∩ΞQn(ϑ)}−≥limn→∞1n∑t=1ninfϑ∈Vk(ϑ*)∩Ξℓt,n−(θ)≥−K0+infϑ∈Ξe−Kθν0+μEinfϑ∈Vk(ϑ*)∩ΞX1(θ). By Beppo Levi’s theorem Einfϑ∈Vk(ϑ*)∩ΞX1(θ) increases to EX1(θ*) as k→∞ . Using (ii), it thus follows that Equation (12) hold true when EX1(θ)=∞ . Since EX1(θ)=∞ if and only if Eeu1(θ)=∞ , we can now assume that ϑ∈Λ . The function θ↦Eeu1(θ) being continuous, we can also assume k large enough so that Vk(ϑ)∩Ξ⊂Λ . Using successively (i) and (ii), the ergodic process, the dominated convergence theorem, and (iv), we then obtain almost surely limsupn→∞supϑ*∈Vk(ϑ)∩ΞQ˜n(ϑ*)≤limsupn→∞supϑ*∈Vk(ϑ)∩ΞQn(ϑ*)+limsupn→∞supϑ∈Λ1|Q˜n(ϑ)−Qn(ϑ)|≤limsupn→∞n−1∑t=1nsupϑ*∈Vk(ϑ)∩Ξℓt(ϑ*)=Esupϑ*∈Vk(ϑ)∩Ξℓ1(ϑ*)<Eℓ1(ϑ0). for k large enough, when ϑ≠ϑ0 , which completes the proof of (v). The proof of the consistency ϑ^n then follows from a standard compactness argument, as in Wald (1949). Taking the expectation on both sides of Equation (2), we obtain Aθ0(1)ν0=ω0+Bθ0(1)μ0. The consistency of ω^n follows from that of ϑ^n and νn. □ A.2 Proof of Theorem 2: Asymptotic Normality The following preliminary result shows that the conditions of Theorem 2 entail the existence of a moment of order 4 for eut(θ) in a neighborhood of θ0. Lemma 1 Assume that E|η1|2r<∞ and E|1/η1|s0<∞ for some s0>0. Then there exists a compact set V¯(θ0) which contains a neighborhood V(θ0) of θ0, and which is such that Esupθ∈V̲(θ0)|eut(θ)|r<∞. Proof of Lemma 1 First, define the sequence {πi(θ)}i by ut(θ)=Bθ−1(L)Aθ(L)Aθ0−1(L)Bθ0(L)ut:=∑i=0∞πi(θ)ut−i. Note that in the previous MA (∞) representation, we have π0(θ)=1 and, for V¯(θ0) small enough, supθ∈V̲(θ0)2r|πi(θ)|<s<s0 for all i≥1 . We then have supθ∈V̲(θ0)|ηt|2rπi(θ)≤1+|ηt|s+|ηt|−s. Moreover, a Taylor expansion yields |ηt|2rπi(θ)=1+2rπi(θ)|ηt|a ln(|ηt|) for some a between 0 and 2rπi(θ) . Therefore, Esupθ∈V̲(θ0)|ηt|2rπi(θ)≤1+2rcsupθ∈V̲(θ0)|πi(θ)|, with c=E{(1+|ηt|s+|ηt|−s)|ln|ηt||}<∞, assuming s<2r . It follows that ∑i=0∞ lnEsupθ∈V̲(θ0)|ηt|2rπi(θ)≤2rc∑i=1∞supθ∈V̲(θ0)|πi(θ)|<∞, since the MA coefficients tend to zero exponentially fast and uniformly in θ when i→∞ . It follows that ∏i=0∞Esupθ∈V̲(θ0)|ηt|2rπi(θ)<∞. Since eut=ηt2e−μ0 , we then have Esupθ∈V̲(θ0)|eut(θ)|r≤supθ∈V̲(θ0)e−μ0r∑i=0∞πi(θ)Esupθ∈V̲(θ0)∏i=0∞|ηt−i|2rπi(θ)<∞. We now decompose the proof of the asymptotic normality into several steps. (a) Asymptotic distribution of νn Note that our estimation procedure shares some similarities with the variance targeting estimator (see Francq, Horvath, and Zakoian, 2011) because they are both two-step estimators, involving an empirical moment estimation in the first step. Taking the average of both sides of Equation (4), for t varying from 1 to n, we obtain Aθ0(1)(νn−ν0)=Bθ0(1)1n∑t=1nut+OP(1n), which is the analog of Equation (A.15) in Francq, Horvath, and Zakoian (2011). The central limit theorem then entails n(νn−ν0)=Bθ0(1)Aθ0(1)1n∑t=1nut+oP(1)→dN(0,σν2), σν2=Bθ02(1)Aθ02(1)Var ln η12. (b) Negligible impact of the initial values Similarly to (ii) in the proof of Theorem 1, we now show that limn→∞nsupϑ∈Ξ∩V̲(ϑ0)|∂Qn(ϑ)∂ϑ−∂Q˜n(ϑ)∂ϑ|=0 a.s. (20) The components of the derivatives ∂ut,n(θ)/∂θ and ∂u˜t,n(θ)/∂θ have ARMA representations similar to those of ut,n(θ) and u˜t,n(θ) . By the arguments used to show Equation (16), we thus obtain supθ∈Θ‖∂ut,n(θ)∂θ−∂u˜t,n(θ)∂θ‖≤Kρt, a.s. (21) Using this inequality, Equation (18) and ∂eut,n(θ)+μ∂ϑ′=eut,n(θ)+μ(∂ut,n(θ)∂θ′,1), we obtain supϑ∈Ξ∩V̲(ϑ0)‖∂eut,n(θ)∂ϑ−∂eu˜t,n(θ)∂ϑ‖≤Kρtsupϑ∈V̲(ϑ0)(eut,n(θ)+‖∂ut,n(θ)∂θ‖). In view of Remark 1, it follows that Esupϑ∈Ξ∩V̲(ϑ0)‖∂eut,n(θ)∂ϑ−∂eu˜t,n(θ)∂ϑ‖≤Kρt. (22) We easily obtain Equation (20) from Equations (21) and (22). (c) A Taylor expansion for the derivative of the criterion Note that Qn(ϑ) and On(ϑ) are values of the same function at the points (ϑ,νn) and (ϑ,ν0) , respectively. More precisely, with some abuse of notation, we have ut,n(θ)=ut(θ,νn) and ut(θ)=ut(θ,ν0), (23) ℓt,n(ϑ^n)=ℓt(ϑ^n,νn) and ℓt(ϑ0)=ℓt(ϑ0,ν0). (24) Using Equation (20), the notation a=cb when a=b+c , the consistency of ϑ^n , and Taylor expansions, we then obtain, for n large enough, 0d=1n∑t=1n∂ℓ˜t,n(ϑ^n)∂ϑ=oP(1)1n∑t=1n∂ℓt,n(ϑ^n)∂ϑ=1n∑t=1n∂ℓt(ϑ0)∂ϑ+Jnn(ϑ^n−ϑ0)+Knn(νn−ν0), (25) where d=p+q+1 , and the elements of the d × d matrix Jn and the d×1 vector Kn are defined by Jn(i,j)=1n∑t=1n∂2ℓt(ϑi*,νi*)∂ϑi∂ϑj and Kn(i)=1n∑t=1n∂2ℓt(ϑi*,νi*)∂ϑi∂ν, for some ϑi* between ϑ^n and ϑ0 and some νi* between νn and ν0. (d) A CLT for stationary martingale increments Noting that ∂lnχμ(x)∂x=∂lnχμ(x)∂μ=12(1−ex+μ) and using Equation (19), we obtain 1n∑t=1n∂ℓt(ϑ0)∂ϑ=12n∑t=1n(1−ηt2)(∂ut(θ0)∂θ1). Because E(lnηt2)2<∞ and ∂ut(θ0)/∂θi is an absolutely summable linear combination of the lnηu2’s, for u≤t−1 , we have E||∂ut(θ0)/∂θ||2<∞ . The central limit theorem for martingale difference thus implies (1n∑t=1n∂ℓt(ϑ0)∂ϑn(νn−ν0))=oP(1)(12n∑t=1n(1−ηt2)(∂ut(θ0)∂θ1)Bθ0(1)Aθ0(1)1n∑t=1nut),→dN{0,(τηE∂ut∂θ∂ut∂θ′(θ0)0d−10d−10d−1′τηξ0d−1′ξσν2)} where τη=Eη14−14 and ξ=Bθ0(1)2Aθ0(1)E(1−η12) ln η12 . (e) Existence and invertibility of some information matrices We have ∂2ℓt(ϑ0)∂ϑ∂ϑ′=1−ηt22(∂2ut(θ0)∂θ∂θ′0d−10d−1′0)−ηt22(∂ut(θ0)∂θ∂ut(θ0)∂θ′∂ut(θ0)∂θ∂ut(θ0)∂θ′1). Noting that ∂ut(θ0)/∂θ and ηt are independent, that E||∂2ut(θ0)/∂θ∂θ′||<∞ and E||∂ut(θ0)/∂θ||2<∞ , one can set J:=E∂2ℓt(ϑ0)∂ϑ∂ϑ′=−12(E∂ut(θ0)∂θ∂ut(θ0)∂θ′0d−10d−1′1). Arguing by contradiction, we assume that J is singular. Then there exists λ∈Rp+q such that λ′∂ut(θ0)∂θ=0 a.s. Taking the derivative of both sides of Equation (4), we obtain Bθ0(L)∂ut(θ0)∂θ−(0⋮0ut−1(θ0)⋮ut−p(θ0))=−(lnεt−12−ν0⋮lnεt−q2−ν0lnεt−12−ν0⋮lnεt−p2−ν0). Multiplying the two sides of this equation by λ′=(λ1,…,λp+q) , it can be seen that λ1=0 (otherwise the linear innovation of (lnεt2) would be degenerated) and ∑i=2qλi(lnεt−i2−ν0)+∑j=1pλj+q{lnεt−j2−ν0−ut−j(θ0)}. The process (lnεt2−ν0) thus satisfies an ARMA (r−1,p−1) . This is impossible under the identifiability conditions of Theorem 1. Therefore, J is invertible. Introducing the notation K˙θ:=∂2ut(θ)∂θ∂ν=−∂∂θAθ(1)Bθ(1), we also have ∂2ℓt(ϑ0)∂ϑ∂ν=1−ηt22(K˙θ0)+ηt22Aθ(1)Bθ(1)(∂ut(θ0)∂θ1). Thus, K:=E∂2ℓt(ϑ0)∂ϑ∂ν=(0d−1Aθ(1)2Bθ(1)). (f) Convergence of (Jn, Kn) to (J, K) With the notation (23), for i,j∈{1,p+q}2 , we have ∂2ℓt(ϑ,ν)∂θi∂θj=1−eut(θ,ν)+μ2∂2ut(θ,ν)∂θi∂θj−eut(θ,ν)+μ2∂ut(θ,ν)∂θi∂ut(θ,ν)∂θj. Write 2∂2ℓt(ϑ,ν)∂θi∂θj−2∂2ℓt(ϑ0,ν0)∂θi∂θj=c1+c2+c3, where for m = 1, 2, 3, the cm=cmt(θ,ν) are defined by c1=∂2ut(θ,ν)∂θi∂θj−∂2ut(θ0,ν0)∂θi∂θj,c2={eut(θ,ν)−eut(θ0,ν0)}{∂2ut(θ,ν)∂θi∂θj+∂ut(θ,ν)∂θi∂ut(θ,ν)∂θj},c3=eut(θ0,ν0){c1+∂ut(θ,ν)∂θi∂ut(θ,ν)∂θj−∂ut(θ0,ν0)∂θi∂ut(θ0,ν0)∂θj}. Introducing the notation ψki,j(θ)=∂2ψk(θ)/∂θi∂θj , we have c1=∑ℓ=0∞{ψℓi,j(θ)−ψℓi,j(θ0)}(lnεt−ℓ2−ν)−∑ℓ=0∞ψℓi,j(θ0)(ν−ν0). Recall that Vk(ϑ0) denotes the ball of center ϑ0 and radius 1/k . Since there is no risk of confusion, we also denote by Vk(θ0) the ball of center θ0 and radius 1/k . Noting that ψℓi,j(θ)≤Kρℓ, E|lnεt2|<∞ , and θ↦ψℓi,j(θ) are continuous, the dominated convergence theorem entails limk→∞Esup(θ,ν)∈Vk(θ0)×[ν0−1k,ν0+1k]|cmt(θ,ν)|=0, (26) for m=1. We now consider the term c2. By already given arguments, it is easy to show that Esupθ∈Θ,ν∈[a,b]{∂2ut(θ,ν)∂θi∂θj+∂ut(θ,ν)∂θi∂ut(θ,ν)∂θj}2<∞. In view of Lemma 1, by the dominated convergence theorem limk→∞Esup(θ,ν)∈Vk(θ0)×[ν0−1k,ν0+1k](eut(θ,ν)−eut(θ0,ν0))2=0. We then obtain Equation (26) for m = 2 by the Cauchy–Schwarz inequality. Similar arguments show Equation (26) for m = 3. Doing the same derivations when ∂θi and/or ∂θj are replaced by ∂μ in the second-order derivatives, we finally obtain that ∀ɛ>0 , there exists an integer kɛ such that Esup(ϑ,ν)∈Vk(ϑ0)×[ν0−1k,ν0+1k]‖∂2ℓt(ϑ,ν)∂ϑ∂ϑ′−∂2ℓt(ϑ0,ν0)∂ϑ∂ϑ′‖<ɛ for all k≥kɛ . By the ergodic theorem, with probability 1 we have limn→∞1n∑t=1n∂2ℓt(ϑ0,ν0)∂ϑ∂ϑ′=J. Because (ϑ^n,νn)∈Vkɛ(ϑ0)×[ν0−1kɛ,ν0+1kɛ] for n large enough, we have limn→∞||Jn−J||<ɛ a.s. Since ɛ is an arbitrary positive number, the limit is actually zero. By exactly the same arguments, it can be show that Kn→K a.s. (g) Joint asymptotic distribution of ϑ^n and νn By Equation (25), (d), (e), and (f), we have n(ϑ^n−ϑ0νn−ν0)=oP(1)(−J−1−J−1K0d′1)(1n∑t=1n∂ℓt(ϑ0)∂ϑn(νn−ν0))→dN{0,(Σθ0d−10d−10d−1′σμ2σμν0d−1′σμνσν2)}, where Σθ=4τηΣu−1, Σu=E∂ut∂θ∂ut∂θ′(θ0), σμ2=Var(η12−ln ηt2) (27) and σμν=2ξ+σν2Aθ0(1)/Bθ0(1) . (h) Joint asymptotic distribution of (ω^n,ϑ^n′,νn). By Equation (8), we have ω^n−ω0=−Bθ0(1)(μ^n−μ0)−{Bθ^n(1)−Bθ0(1)}μ^n+Aθ0(1)(νn−ν0)+{Aθ^n(1)−Aθ0(1)}νn=−Bθ0(1)(μ^n−μ0)+Aθ0(1)(νn−ν0)+∑j=1p(β^j−β0j)μ^n−∑i=1r(α^i+β^i−α0i−β0i)νn. Recalling the notation γ=(−ν01q′,(μ0−ν0)1p′)′ , we have n(ω^n−ω0ϑ^n−ϑ0νn−ν0)=oP(1)(γ′−Bθ0(1)Aθ0(1)Id+1)n(ϑ^n−ϑ0νn−ν0)→dN{0,(σω2γ′Σθσωμσωνγ′ΣθΣθ0d−10d−1σωμ0d−1′σμ2σμνσων0d−1′σμνσν2)}, where σωμ=−Bθ0(1)σμ2+Aθ0(1)σμν=−Bθ0(1){Var(η12)−Cov(η12,lnη12)},σων=−Bθ0(1)σμν+Aθ0(1)σν2=−Bθ02(1)Aθ0(1)Cov(η12,ln η12),σω2=γ′Σθγ+Bθ02(1)σμ2−2Bθ0(1)Aθ0(1)σμν+Aθ02(1)σν2=γ′Σθγ+Bθ02(1)Var(η12). The conclusion follows by direct computation. Supplementary Data Supplementary data are available at Journal of Financial Econometrics online. Footnotes * We are grateful to the Editor, Associate Editor, two anonymous reviewers, and conference participants at the CFE 2013 (London) and CEF 2014 (Oslo) for useful comments and suggestions. 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Journal of Financial Econometrics – Oxford University Press
Published: Jan 1, 2018
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