Rank-Based Estimation for Garch Processes

Abstract

We consider a rank-based technique for estimating GARCH model parameters, some of which are scale transformations of conventional GARCH parameters. The estimators are obtained by minimizing a rank-based residual dispersion function similar to the one given in Jaeckel (1972).They are useful for GARCH order selection and preliminary estimation. We give a limiting distribution for the rank estimators which holds when the true parameter vector is in the interior of its parameter space, and when some GARCH parameters are zero. The limiting theory is used to show that the rank estimators are robust, can have the same asymptotic efficiency as maximum likelihood estimators, and are relatively efficient compared to traditional Gaussian and Laplace quasi-maximum likelihood estimators. The behavior of the estimators for finite samples is studied via simulation, and we use rank estimation to fit a GARCH model to exchange rate log-returns.

Comment

“…In Andrews (2012), the limiting distribution for R-estimators is given not only when the true parameter vector is in the interior of its parameter space and the estimators are asymptotically 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 (Andrews 2012, sec.…”

“…Root mean squared errors for R-estimates and QMLEs of GARCH model parameters when the noise distribution is N(0,1) and standardized t 3 (3) is 1.041 when the noise {ε t } are N(0,1), and ARE is 1.052 when the {ε t } are standardized t 3 (Andrews 2012). Since the RMSEs in Table 1 for R-estimation are mostly smaller than the corresponding values for QMLE, it appears the asymptotic relative efficiencies for R-estimation with respect to non-Gaussian QMLE can be indicative of finite sample behavior for sample size 250 ≤ T ≤ 1000.…”

A New Pearson-Type QMLE for Conditionally Heteroscedastic Models

“…Note that the value of (ν, m) can be anywhere in Γ, and a different value of (ν, m) will imply a different stationarity region of y t . To see this, Figure 4 plots the strict stationarity region of the GARCH(1, 1) model: y t = ε t σ t and σ 2 (1,4), and (2, 4). As a comparison, the stationarity regions for the cases that ε t ∼ N (0, 1) in Nelson (1990) and ε t ∼ Laplace(0, 1) in Zhu and Ling (2011) are also re-plotted in Figure 4.…”

“…Second, one can retain the identification condition Eε 2 t = 1 for the NGQMLE by re-parameterizing models (1)- (2). This method has been used for the semi-parametric estimator in Drost and Klaassen (1997), the rank-based estimator in Andrews (2012), and the generalized NGQMLE (GNGQMLE) in Fan, Qi, and Xiu (2014). By introducing a scale adjustment parameter, the GNGQMLE is consistent and asymptotical normal when ε t has a finite second moment, while the semi-parametric and rank-based estimators can only estimate the heteroscedastic parameters α i and β j under the same re-parameterized GARCH(p, q) model.…”

Spline Estimation of a Semiparametric Garch Model

“…Engle and Ng (1993), Glosten, Jaganathan, and Runkle (1993), Hentschel (1995), Duan (1997), Hafner and Herwartz (2006), and Hafner (2008) examined various useful extensions of model (1), mostly providing empirical evidence without establishing asymptotic results. For related theoretical works on GARCH model, see Peng and Yao (2003), Sun and Stengos (2006), Chan, Deng, Peng, and Xia (2007), Giraitis, Leipus, and Surgailis (2010), Meitz and Saikkonen (2011), and Andrews (2012). Linton, Pan, and Wang (2010) and Zhang and Ling (2014) established asymptotic results for heavy-tailed noises.…”