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Journal of Business & Economic Statistics
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Comment
Beth Andrews
To cite this article: Beth Andrews (2014) Comment, Journal of Business & Economic Statistics, 32:2, 191-193, DOI: 10.1080/07350015.2013.875921
To link to this article: http://dx.doi.org/10.1080/07350015.2013.875921
Published online: 16 May 2014.
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Andrews: Comment 191
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Comment
Beth A
NDREWSDepartment of Statistics, Northwestern University, Evanston, IL 60208 ([email protected]; [email protected])
In their article, Fan, Qi, and Xiu develop non-Gaussian quasi-maximum likelihood estimators (QMLEs) for the parameters
θ=(σ,γ′)′=(σ, a
1, . . . , ap, b1, . . . , bq)′of a generalized au-toregressive conditional heteroscedasticity (GARCH) process {xt}, where
and the noise {εt}are assumed to be independent and iden-tically distributed with mean zero and variance one. The QMLEs ofγare shown to be√T-consistent (Trepresents sam-ple size) and asymptotically Normal under general conditions. When E{ε4
t}<∞, the QMLEs of the scale parameterσare also √
T-consistent and asymptotically Normal, but, as is the case for Gaussian QMLEs of GARCH model parameters, the estimator ofσ has a slower rate of convergence otherwise (Hall and Yao 2003). As Fan, Qi, and Xiu mention, a rank-based technique for estimatingθwas presented in Andrews (2012). These rank (R
)-estimators are also consistent under general conditions, with the same rates of convergence as the non-Gaussian QMLEs. Hence, theR-estimators have robustness properties similar to the QMLEs. In this comment, I make some methodological and efficiency comparisons between the two techniques, and suggest R-estimation be used prior to QMLE for preliminary GARCH estimation. Once anR-estimate has been found, corresponding model residuals can be used to identify one or more suitable noise distributions and QMLE/MLE can then be used. As Fan, Qi, and Xiu suggest in Section 6, one can optimize over a pool of appropriate likelihoods in an effort to improve efficiency. Ad-ditionally, MLEs of all elements ofθ are consistent with rate √
T under general conditions (Berkes and Horv´ath2004).
© 2014American Statistical Association Journal of Business & Economic Statistics April 2014, Vol. 32, No. 2 DOI:10.1080/07350015.2013.875921
192 Journal of Business & Economic Statistics, April 2014 constant and nondecreasing weight function from (0,1) to IR. In practice, when the noise distribution for the GARCH process is unknown, I recommend using the weight func-tion λt7(x)=[7{F− 5], whereFt7 represents the distribution function for
standard-izedt7 noise, or a similar weight function (Andrews2012, Re-mark 7). When compared with other techniques,R-estimation with weight function λt7 performs well for light-, medium-,
and heavier-tailed noise distributions. Note that, if λ=(T − p)−1T negative, continuous function (Andrews2012). Because it tends to be near zero when the elements of{ξt(γ)}are similar and gets larger as the values of{|ξ(t)(γ)−ξ(γ)|}increase,DT can be thought of as a measure of the dispersion of the residuals {ξt(γ)}. This rank-based estimation technique is similar to the one introduced in Jaeckel (1972) for estimating linear regression parameters. In Remark 9 of Andrews (2012), the corresponding R-estimator ofσ2(which, following the notation of Bollerslev 1986, I denote asα0) is given by ˆσR2=n−1tT=p+1xt2/v2t( ˆγR). obtained via a two-step procedure: (a) minimizeDT(γ) in Equa-tion (1) to find ˆγR and (b) obtain ˆσR via Equation (2). In con-trast, the non-Gaussian QMLEs ˆθT proposed by Fan, Qi, and Xiu are obtained via a three-step procedure, where optimization (i.e., maximization) is required in all three steps. From this per-spective,R-estimation is a simpler method than non-Gaussian QMLE.
2. RELATIVE EFFICIENCY
Let ˜f and ˜F represent the density and distribution functions for ln(ε2
t). In Andrews (2012), I showed that, when the distribu-tion for the noise{εt}is symmetric about zero and a weight func-tionλ(x)∝ −f˜′( ˜F−1(x))/f˜( ˜F−1(x)) is used,R-estimators ofγ
have the same asymptotic efficiency as MLEs. Whenεt has a standardizedt7distribution,λt7(x)∝ −f˜
′( ˜F−1
(x))/f˜( ˜F−1(x)). Furthermore, when the weight functionλused forR-estimation and the densityf used for QMLE correspond to the same noise distribution,R-estimation tends to be asymptotically as efficient or more efficient than QMLE. This was discussed in Andrews
(2012, Remarks 6 and 7) and also observed in Fan, Qi, and Xiu (Section 7.2, Table 3) whenλandf correspond to the standard-izedt7distribution. Note that the asymptotic relative efficiency (ARE) for R-estimators of γ with respect to non-Gaussian QMLEs is given by in Fan, Qi, and Xiu and depend onf, the density being used for QMLE.
whereM−1[1,1] represents the element in row one, column one of matrixM−1;Mis defined in the statement of Theorem 2. Via matrix algebra, it can be shown that
σ2E
so theR-estimator ofσis asymptotically more efficient than the QMLE ofσ when the ARE in Equation (3) for estimators ofγ
is larger than one. As demonstrated in Table 3 of Fan, Qi, and Xiu, this is often the case (see also Andrews2012, Remark 7).
In Section 7.2, Fan, Qi, and Xiu give simulation results for the GARCH(1,1) model with parameters (σ, a1, b1)= (0.5,0.6,0.3) when sample size T =250, 500, and 1000. In these simulations, R-estimation with weight function λt7
is essentially as efficient as QMLE. For comparison, I con-sidered GARCH(1,1) models with parameters (σ, a1, b1)= (0.1,50.0,0.4) and (σ, a1, b1)=(0.1,10.0,0.8), and the same three values ofT. (The model (σ, a1, b1)=(0.1,50.0,0.4) is considered in Andrews2012, and (σ, a1, b1)=(0.1,10.0,0.8) was selected because, for many observed series, a value ofb1 near one appears appropriate.) In each case, I simulated 1000 GARCH processes with N(0,1) and standardizedt3 noise, and found the correspondingR-estimates and QMLEs, also using the weight functionλt7forR-estimation and the standardizedt7
density for QMLE. Root mean squared errors for the estimates are listed inTable 1. For these estimation methods, the value of
Fiorentini and Sentana: Comment 193
Table 1. Root mean squared errors forR-estimates and QMLEs of GARCH model parameters when the noise distribution is N(0,1) and
standardizedt3
RMSEs
R-estimates QMLEs
T Model parameters (N(0,1), t3) (N(0,1), t3)
250 σ =0.1 0.020, 0.026 0.029, 0.031
a1=50.0 22.004, 27.196 20.135, 27.213
b1=0.4 0.119, 0.184 0.129, 0.192
500 σ =0.1 0.014, 0.020 0.021, 0.026
a1=50.0 14.700, 18.399 14.312, 18.544
b1=0.4 0.080, 0.125 0.083, 0.129
1000 σ =0.1 0.009, 0.016 0.016, 0.023
a1=50.0 9.465, 12.982 9.496, 13.085
b1=0.4 0.052, 0.084 0.054, 0.086
250 σ =0.1 0.090, 0.067 0.113, 0.085
a1=10.0 6.877, 7.716 6.812, 6.932
b1=0.8 0.350, 0.330 0.442, 0.408
500 σ =0.1 0.059, 0.046 0.079, 0.056
a1=10.0 5.111, 5.182 5.365, 5.143
b1=0.8 0.222, 0.215 0.299, 0.265
1000 σ =0.1 0.033, 0.028 0.047, 0.038
a1=10.0 3.680, 3.564 3.943, 3.745
b1=0.8 0.111, 0.109 0.164, 0.166
ARE in Equation (3) is 1.041 when the noise {εt}are N(0,1), and ARE is 1.052 when the{εt}are standardizedt3 (Andrews 2012). Since the RMSEs inTable 1forR-estimation are mostly smaller than the corresponding values for QMLE, it appears the asymptotic relative efficiencies forR-estimation with respect to
non-Gaussian QMLE can be indicative of finite sample behavior for sample size 250≤T ≤1000.
3. CONCLUDING REMARKS
In Andrews (2012), the limiting distribution forR-estimators is given not only when the true parameter vector is in the in-terior of its parameter space and the estimators are asymptot-ically Normal, but also when some GARCH parameters are zero and the limiting distribution is non-Normal. The results are used to develop hypothesis tests for GARCH order selection (Andrews2012, sec. 3.2). SinceR-estimates are straightforward to compute and tend to be relatively efficient, I recommend R-estimation be used not only for preliminary GARCH estima-tion, but also for order selection when the noise distribution is unknown. If further model accuracy is desired, residuals from R-estimation can be used to identify one or more suitable noise distributions, and then the GARCH model can be estimated via QMLE/MLE.
REFERENCES
Andrews, B. (2012), “Rank-Based Estimation for GARCH Processes,” Econo-metric Theory, 28, 1037–1064. [191,192,193]
Berkes, I., and Horv´ath, L. (2004), “The Efficiency of the Estimators of the Parameters in GARCH Processes,”The Annals of Statistics, 32, 633–655. [191]
Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroskedas-ticity,”Journal of Econometrics, 31, 307–327. [192]
Hall, P., and Yao, Q. (2003), “Inference in ARCH and GARCH Models With Heavy-Tailed Errors,”Econometrica, 71, 285–317. [191]
Jaeckel, L. A. (1972), “Estimating Regression Coefficients by Minimizing the Dispersion of the Residuals,”Annals of Mathematical Statistics, 43, 1449– 1458. [192]
Comment
Gabriele F
IORENTINIUniversit `a di Firenze and RCEA, I-50134 Firenze, Italy ([email protected])
Enrique S
ENTANACEMFI, E-28014 Madrid, Spain ([email protected])
The Gaussian pseudo-maximum likelihood (PML) estimators advocated by Bollerslev and Wooldridge (1992) among many others remain root-Tconsistent for the conditional variance pa-rameters of univariate generalized autoregressive conditional heteroscedasticity (GARCH) models with a zero conditional mean irrespective of the degree of asymmetry and kurtosis of the conditional distribution of the observed variables, so long as the first two conditional moments are correctly specified and the fourth conditional moments are bounded. Nevertheless, many empirical researchers prefer to specify a non-Gaussian paramet-ric distribution for the standardized innovations, which they use to estimate the conditional variance parameters by maximum likelihood (ML). The most important commercially available econometric packages have responded to this demand by offer-ing ML procedures that either jointly estimate the parameters
characterizing the shape of the assumed distribution or allow the user to fix them to some prespecified values (see, e.g., the uni-variate ARCH sections of IHS Global Inc.2013and StataCorp LP2013).
However, while such ML estimators will often yield more ef-ficient estimators than Gaussian PML if the assumed conditional distribution is correct, they may end up sacrificing consistency when it is not, as shown by Newey and Steigerwald (1997). For that reason, Fan, Qi, and Xiu (2014) must be congratulated
© 2014American Statistical Association Journal of Business & Economic Statistics April 2014, Vol. 32, No. 2 DOI:10.1080/07350015.2013.878661 Color versions of one or more of the figures in the article can be found online atwww.tandfonline.com/r/jbes.
