Joint maximum likelihood estimation of unit root testing equations and GARCH processes: Some finite-sample issues

https://doi.org/10.1016/j.matcom.2007.02.009Get rights and content

Abstract

In recent research [B. Seo, Distribution theory for unit root tests with conditional heteroskedasticity, J. Econometrics 91 (1999) 113–144] has suggested that the examination of the unit root hypothesis in series exhibiting GARCH behaviour should proceed via joint maximum likelihood (ML) estimation of the unit root testing equation and GARCH process. The results presented show the asymptotic distribution of the resulting ML t-test to be a mixture of the Dickey–Fuller and standard normal distributions. In this paper, the relevance of these asymptotic arguments is considered for the finite samples encountered in empirical research. In particular, the influences of sample size, alternative values of the parameters of the GARCH process and the use of the Bollerslev–Wooldridge covariance matrix estimator upon the finite-sample distribution of the ML t-statistic are explored. It is shown that the resulting critical values for the ML t-statistic are similar to those of the Dickey–Fuller distribution rather than the standard normal, unless a large sample size and empirically unrealistic values of the volatility parameter of the GARCH process are considered. Use of the Bollerslev–Wooldridge standard covariance matrix estimator exaggerates this finding, causing a leftward shift in the finite-sample distribution of the ML t-statistic. The results of the simulation analysis are illustrated via an application to U.S. short term interest rates.

Introduction

Following the seminal research of Engle [14], the concept of autoregressive conditional heteroskedasticity (ARCH) has come to occupy a central position in the empirical finance and financial econometrics literatures. Under this approach, the volatility clustering exhibited by financial time series data is captured via modelling of the temporal dependency of their conditional variances. In subsequent independent research, Bollerslev [2] and Taylor [35] have extended ARCH, proposing the generalised autoregressive conditional heteroskedasticity (GARCH) model. The GARCH model and its various modifications are now the subject of widespread application in financial research, proving to provide accurate forecasts (see [1]). Given the noted prevalence of GARCH in financial time series, it is unsurprising that a number of studies have emerged examining its interaction with unit root testing, itself a cornerstone of empirical research in time series econometrics. The resulting literature can be divided into two broad themes. Considering the first theme, a number of authors have analysed the behaviour of existing unit root tests when applied to series exhibiting GARCH. Examples of studies considering the impact of GARCH upon unit root tests include Kim and Schmidt [21], Haldrup [18], Brooks and Rew [4], Ling et al. [24] and Cook [5], [6], [7].1 While Kim and Schmidt [21] and Haldrup [18] consider the seminal unit root test of Dickey and Fuller [9], Brooks and Rew [4] examine the behaviour of Perron [31] tests which allow for structural change under the alternative hypothesis of stationarity. Adopting a different approach, Ling et al. [24] consider the properties of unit root tests in the presence of GARCH under least squares (LS), maximum likelihood (ML) and mixed LS and ML estimation. The analysis of symmetric, linear unit root tests is extended by Cook [5] where the behaviour of the modified unit root tests of Park and Fuller [30], Shin and So [34], Elliott et al. [11], Leybourne [22] and Granger and Hallman [17] are examined. The general finding of this body of research is that GARCH has little impact upon the size of linear unit root tests. In contrast to this, the results of Cook [6], [7] show that asymmetric and non-linear unit root tests can exhibit severe size distortion in the presence of GARCH. In particular, it was shown that asymmetric unit root tests based upon threshold autoregression (TAR) and momentum–threshold autoregression (MTAR) (see [13], [12], [8]) and the exponential smooth transition autoregressive (ESTAR) based non-linear unit root test of Kapetanios et al. [20] can suffer oversizing when applied to unit root processes with GARCH errors. While the above results all relate to the issue of neglected GARCH, a second theme of research into the interaction between GARCH and unit root testing has suggested a different approach whereby the autoregressive (AR) unit root and parameters of a GARCH model are estimated jointly. The results of Seo [32] show that under joint ML estimation of a Dickey–Fuller testing equation and a GARCH process, the asymptotic distribution of the t-statistic for the unit root hypothesis is a mixture of the non-standard Dickey–Fuller distribution and the standard normal. The weightings attached to these respective distributions are captured by a nuisance parameter (ρ) which represents the strength of the GARCH effect. Seo [32] shows that as the GARCH effect increases and ρ moves from its lower limit of 0 towards its upper limit of 1, the asymptotic distribution of the ML t-statistic moves from the Dickey–Fuller towards the standard normal.

In this paper, the analysis of Seo [32] is extended in three ways via the use of Monte Carlo simulation. First, the relevance of the Seo’s asymptotic results is examined for the types of finite samples encountered in empirical research. Second, the contribution of the individual parameters of the GARCH model to the movement from the Dickey–Fuller distribution towards the standard normal distribution is analysed. It is argued that for GARCH parameter values and sample sizes often encountered in empirical research, the resulting finite-sample critical values of the ML t-statistic for the unit root hypothesis are much closer to the Dickey–Fuller distribution than the standard normal. Indeed, for some empirically plausible combinations of the GARCH parameters and sample size, the resulting critical values are larger in absolute terms (more negative) than the relevant Dickey–Fuller critical values. Therefore, while the asymptotic findings of Seo [32] may lead practitioners to assume that a much less stringent criterion (less negative critical value) is required to reject the unit root hypothesis under the joint ML approach, the finite sample findings for explicit parameter values of the GARCH process show that the relevant critical values to employ are often very similar to, or may even exceed, those of the Dickey–Fuller distribution. Third, the impact of Bollerslev and Wooldridge [3] robust standard errors upon the finite-sample distribution of the ML t-statistic is examined. Given widespread application of the Bollerslev–Wooldridge covariance matrix in the empirical analysis of financial time series, analysis of its impact upon the distribution of the resulting ML t-statistic for a unit root test is clearly warranted. The results obtained from this analysis show the application of Bollerslev–Wooldridge standard errors to reduce the movement of the distribution of the ML t-statistic towards the standard normal. That is, the finite-sample critical values of the ML t-statistic are found to be greater in absolute terms when the robust Bollerslev–Wooldridge covariance matrix rather than the standard ML covariance matrix is employed. Finally, it should be noted that unit root testing equations in the following analysis are considered in their empirically relevant forms with either an intercept, or an intercept and trend included in the underlying testing equation.

The analysis presented in this paper is therefore deliberately specific. Attention is paid solely to the impact of GARCH(1,1) errors upon the finite sample distribution of the Dickey–Fuller unit root test under joint ML estimation of the AR parameter and the GARCH process. The analysis also considers two specific sets of values for the parameters of the GARCH process. The first set contains a number of values corresponding to those employed in Seo [32]. The second set contains values which are more typical of those observed in empirical research.2 It is found that the distinction between the two sets of values highlights the central finding of this paper that convergence towards the standard normal distribution is dependent upon unrealistically large values of the volatility parameter of the GARCH process. As a consequence of the specific focus of this paper, a range of further issues is left for subsequent research. For example, the extension of the present study to consider, inter alia, cointegration analysis, the impact of misspecification of the conditional variance process and stochastic volatility are all potential avenues of future research.3 Readers interested in more general results concerning the impact of GARCH in time series models and the issue of stochastic volatility are referred to the excellent coverage provided by Li et al. [23], McAleer [25] and Shephard [33].

This paper proceeds as follows. Section 2 presents the simulation framework employed to examine the finite-sample distribution of the ML t-statistic for the AR unit root under joint estimation of a GARCH process. The results of this analysis are provided in Section 3. In Section 4, an empirical analysis of the U.S. Federal Funds interest rate is provided to illustrate the simulation results. Section 5 concludes.

Section snippets

Joint ML estimation of an AR unit root testing equation and GARCH process

Consider the familiar Dickey–Fuller testing equation below for a time series {yt}t=0T:Δyt=dt+βyt1+ɛtwhere the deterministic components are denoted as dt. Typically in empirical research, these deterministic components are specified as either an intercept (dt=α0) or an intercept and linear trend term (dt=α0+α1t). The Dickey–Fuller test for a unit root is based upon the t-ratio of β in (1). Following Kim and Schmidt [21] and Haldrup [18] it has been noted that when error ɛt in (1) is a GARCH

Monte Carlo simulation results

The results obtained from the above Monte Carlo simulation are presented in Table 1, Table 2, Table 3, Table 4 . From inspection of the results in Table 1 for the intercept model (dt=α0) using the first set of GARCH parameter values, a number of features are immediately apparent. With regard to Seo’s [32] argument that the asymptotic distribution of the ML t-statistic tβ is a mixture of the Dickey–Fuller and the standard normal, it can be seen that this is broadly supported in finite samples.

Testing the order of integration of U.S. short-term interest rates

To illustrate the above arguments, the unit root hypothesis is examined for short term U.S. interest rates as measured by the Federal Funds interest rate. The data used are monthly observations covering the period from January 1962 to December 2005, thus giving a sample size of 528 observations. Ignoring the above comments concerning the incorporation of GARCH effects into the unit root testing procedure, the augmented Dickey–Fuller (ADF) test and the higher powered GLS-based Dickey–Fuller test

Conclusion

In this paper, the finite-sample distribution of the t-test of the unit root hypothesis has been examined under joint maximum likelihood estimation of a Dickey–Fuller unit root testing equation and a GARCH process. In particular, the finite-sample distribution has been examined under alternative sample sizes, values of the parameters of the GARCH process and the covariance matrix estimators. This extends the analysis of Seo [32] which has shown the asymptotic distribution of the ML t-test to be

Acknowledgement

I am grateful to an anonymous referee for numerous comments which have helped improve the content and presentation of this paper.

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