Maximum likelihood unit rooting test in the presence GARCH: A new test with increased power Steve Cook To cite this version: Steve Cook. Maximum likelihood unit rooting test in the presence GARCH: A new test with increased power. Communications in Statistics - Simulation and Computation, Taylor & Francis, 2008, 37 (04), pp.756-765. 10.1080/03610910701779726. hal-00514322 HAL Id: hal-00514322 https://hal.archives-ouvertes.fr/hal-00514322 Submitted on 2 Sep 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Communications in Statistics - Simulation and Computation For Peer Review Only Maximum likelihood unit rooting test in the presence GARCH: A new test with increased power Journal: Communications in Statistics - Simulation and Computation Manuscript ID: LSSP-2007-0036.R1 Manuscript Type: Original Paper Date Submitted by the 16-Oct-2007 Author: Complete List of Authors: Cook, Steve; Swansea University, Economics Keywords: Unit roots, GARCH, Monte Carlo, Maximum likelihood The literature on testing the unit root hypothesis in the presence of GARCH errors is extended. A new test based upon the combination of local-to-unity detrending and joint maximum likelihood estimation of the autoregressive parameter and GARCH process is presented. The finite sample distribution of the test is derived under Abstract: alternative decisions regarding the deterministic terms employed. Using Monte Carlo simulation, the newly proposed ML t-test is shown to exhibit increased power of relative to rival tests. Finally, the empirical relevance of the simulation results is illustrated via an application to real GDP for the UK Note: The following files were submitted by the author for peer review, but cannot be converted to PDF. You must view these files (e.g. movies) online. GLS_GARCH_2.tex URL: http://mc.manuscriptcentral.com/lssp E-mail: [email protected] Page 1 of 15 Communications in Statistics - Simulation and Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 For Peer Review Only 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 URL: http://mc.manuscriptcentral.com/lssp E-mail: [email protected] Communications in Statistics - Simulation and Computation Page 2 of 15 1 2 3 4 5 6 7 8 Maximum likelihood unit rooting test in the presence 9 10 GARCH: A new test with increased power 11 12 13 Steven Cook∗ 14 15 16 For PeerOctober Review 16, 2007 Only 17 18 19 20 ABSTRACT 21 22 23 The literature on testing the unit root hypothesis in the presence of GARCH errors is extended. 24 A new test based upon the combination of local-to-unity detrending and joint maximum likelihood 25 26 estimation of the autoregressive parameter and GARCH process is presented. The finite sample 27 distribution of the test is derived under alternative decisions regarding the deterministic terms 28 employed. Using Monte Carlo simulation, the newly proposed ML t-test is shown to exhibit in- 29 creased power of relative to rival tests. Finally, the empirical relevance of the simulation results is 30 illustrated via an application to real GDP for the UK. 31 32 Keywords: Unit roots; GARCH; Monte Carlo simulation; Maximum likelihood estimation; GDP. 33 34 JEL codes: C12; C15. 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ∗ Professor Steven Cook, School of Business and Economics, Swansea University, Singleton Park, Swansea, SA2 8PP. Tel: (01792) 602106. E-mail: [email protected]. 1 URL: http://mc.manuscriptcentral.com/lssp E-mail: [email protected] Page 3 of 15 Communications in Statistics - Simulation and Computation 1 2 3 4 1Introduction 5 6 7 It has long been recognised that financial time series exhibit volatility clustering. In response, 8 9 Engle (1982) introduced the notion of autoregressive conditional heteroskedasticity (ARCH) as a 10 11 means capturing this phenomenon via modelling of the temporal dependency of the conditional 12 13 variances of time series processes. This research was subsequently extended in the independent 14 15 studies of Bollerslev (1986) and Taylor (1986) where the generalised autoregressive conditional het- 16 For Peer Review Only 17 eroskedasticity (GARCH) model was proposed. The GARCH model and its various modifications 18 19 now occupy as central position in empirical and theoretical research, exhibiting attractive properties 20 21 in widespread application to financial data (see Anderson and Bollerslev 1998). In recent years 22 23 the noted prevalence of GARCH in financial time series has led to the emergence of a 24 25 literature examining its impact upon examination of the unit root hypothesis, which 26 27 following the seminal work of Dickey and Fuller (1979) (DF) has itself become a cor- 28 29 nerstone of empirical research in time series econometrics. In its simplest case, the 30 31 DF test examines the unit root hypothesis (H0 : ρ =1)in the equation below: 32 33 34 yt = ρyt 1 + εt (1) − 35 36 37 where yt denotes a variable of interest and εt is an error term. Following DF, an 38 39 enormous literature has emerged examining and extending the analysis of the unit root 40 41 hypothesis in a variety of ways. The various developments made include examination 42 43 of unit root testing in the presence of structural change (see, inter alia, Perron 1989, 44 45 1997; Leybourne et al. 1998; Leybourne and Newbold 2000; Cook 2002) and the use 46 of alternative estimators to increase test power and/or reduce bias in the estimation 47 48 of the autoregressive parameter ρ (see, inter alia, Pantula et al. 1995; Park and Fuller 49 50 1995; Shin and So 2001). For a full discussion of the literature on the testing of the unit 51 52 root hypothesis, the reader is referred to the excellent survey provided by Maddala 53 54 and Kim (1999). 55 56 Research involving the joint consideration of the unit root hypothesis and GARCH behaviour 57 58 has followed two approaches. Under the first approach, investigators have examined the behaviour of 59 60 a range of unit root tests when GARCH is present but not addressed by the practitioner. Examples 2 URL: http://mc.manuscriptcentral.com/lssp E-mail: [email protected] Communications in Statistics - Simulation and Computation Page 4 of 15 1 2 3 4 of studies considering the impact of neglected GARCH behaviour upon unit root tests include Kim 5 6 and Schmidt (1993), Haldrup (1994), Brooks and Rew (2002) and Cook (2006a, 2006b, 2006c). 7 8 The general conclusion to be drawn from these studies is that while GARCH has little impact 9 10 upon the size of linear unit root tests such as those of Dickey and Fuller (1979), Elliott et al. 11 12 (1996), Granger and Hallman (1991), Leybourne (1995), Park and Fuller (1995), Perron (1997) 13 14 and Shin and So (2001), asymmetric and non-linear unit root tests (see Enders and Granger 15 16 1998, Enders 2001,For Cook andPeer Manning 2003Review and Kapetanios etOnly al. 2003) can suffer severe size 17 18 distortion in the presence of GARCH. In contrast to this work, the second approach to the analysis 19 20 of the interaction between unit root testing and GARCH adopts a different stance whereby the 21 22 presence of GARCH is explicitly addressed by the practitioner. This research is exemplified by 23 24 Seo (1999) where it is suggested that in the presence of GARCH, a higher powered approach to 25 26 testing of the unit root hypothesis is possible via joint maximum likelihood (ML) estimation of the 27 28 autoregressive (AR) unit root and a GARCH process. The results of this analysis show that under 29 30 ML estimation of a DF testing equation and a GARCH process, the asymptotic distribution of the 31 32 t-statistic for the unit root hypothesis is a mixture of the non-standard DF distribution and the 33 34 standard normal. The weightings attached to each of these distributions is dependent upon the 35 36 strength of the GARCH effect which is captured by a nuisance parameter (ρ).AstheGARCH 37 38 effect increases and ρ moves from its lower limit of 0 towards its upper limit of 1, the asymptotic 39 40 distribution of the ML t-statistic moves from the DF distribution towards the standard normal. In 41 42 subsequent research, Cook (2007) has examined the relevance of these asymptotic arguments for 43 44 the types of finite sample encountered in empirical research. Using Monte Carlo simulation, it was 45 46 shown that the proposed movement of the distribution of the resulting ML t-statistic from the DF 47 48 distribution towards the standard normal is dependent upon both a very large sample size and an 49 50 empirically large value for the volatility parameter of the GARCH process. In response to this, 51 52 the finite-sample distribution of the ML t-statistic under joint estimation of a DF testing equation 53 54 and GARCH process, denoted as DF-GARCH, was derived for alternative sample sizes and values 55 of the GARCH parameters.