Testing for Smooth Structural Changes in GARCH Models

Modelling and detecting structural changes in GARCH processes have attracted a great amount of attention in econometrics over the past few years. We generalize Dahlhaus and Rao (2006)’s time varying ARCH processes to time varying GARCH processes and show the consistency of the weighted quasi maximum likelihood estimator. A class of generalized likelihood ratio tests are proposed to check smooth structural changes as well as abrupt structural breaks with known or unknown change points in GARCH models. The idea is to compare the log likelihood of the unrestricted nonparametric time-varying GARCH model and the restricted constant parameter GARCH model, which can be viewed as a generalization of likelihood ratio tests from the parametric framework to the nonparametric framework. The tests have a convenient asymptotic N(0,1) distribution and do not require any prior information about the alternatives. A simulation study highlights the merits of the proposed tests. JEL Classi…cations: C1, C4, E0. Key words: GARCH, Kernel, Model stability, Parameter constancy, QMLE, Smooth structural change 1. INTRODUCTION Since the seminal works by Engle (1982) and Bollerslev (1986), Generalized Autoregressive conditional heteroskedasticity (GARCH) type models have been commonly used to capture volatility dynamics of …nancial time series. However, underlying all these models is the assumption of stationarity. Given the changing pace of the underlying economic mechanism, modeling …nancial variables over a long time horizon may not be suitable. It is quite plausible that structural changes have occurred, causing the time series to deviate from stationarity. Indeed, various economic factors may lead to structural changes detected in …nancial time series. For example, one driving force for structural changes are“shocks” induced by institutional changes, such as changes of exchange rate systems from the …xed exchange rate mechanism to the ‡oating exchange rate mechanism, or the introduction of Euro. As Lucas (1976) points out, “any change in policy will systematically alter the structure of econometric models”, given that the structure of an econometric model depends crucially on agents’expectations, which in turn vary systematically with changes in the structure of time series relevant to decision makers. The prevalence of structural instability in …nancial time series has been con…rmed by numerous empirical studies. For example, Andreou and Ghysels (2002) examine the change-point hypothesis in volatility dynamics of international stock market indices and foreign exchange returns and …nd multiple breaks associated with the Asian and Russian …nancial crisis; Mikosch and Starica (2004) apply their goodness-of-…t test to the S&P500 returns and detect structural changes related to shifts of unconditional variance. Model stability is crucial for statistical inference, out-of-sample forecasts, and any policy implications drawn from the model. In particular, ignoring structural changes in …nancial time series can easily lead to spurious persistence in the conditional volatility parameters. Diebold (1986), Hendry (1986) and Lamoureux and Lastrapes (1990) are among the …rst to suggest that structural changes unaccounted for can yield Integrated GARCH or long memory e¤ects. More recently, Mikosch and Starica (2004) and Hillebrand (2005) provide some theoretical explanation for this phenomenon. The spurious IGARCH e¤ects imply that shocks have a permanent impact on volatility so current information remains relevant when forecasting the conditional variance for all horizons while for the short memory process, shocks to variance do decay over time. Moreover, model instability may a¤ect asset allocation or lead to large errors in pricing, hedging and managing risk. Pettenuzzo and Timmerman (2005) show that the possibility of future breaks has its largest e¤ect at long investment horizons, but historical breaks can signi…cantly change investment decisions even at short horizons through its e¤ect on current parameter estimates. Some tests have been proposed to test structural breaks in GARCH models. For example, Chu (1995) generalizes Andrews’(1993) supremum Lagrange multiplier (LM) test to GARCH framework. However, the test just considers one-time shift as the alternative so does not have 1 good power against multiple breaks. Berkes, Gombay, Horvath and Kokoszka (2004) develop a sequential likelihood-ratio (LR) based test for evaluating the stability of the GARCH parameters. Their test can be used to check which parameter of a GARCH model has a change point and hence is more informative than some existing tests. However, it is computationally intensive as it involves the calculation of quasilikelihood scores. Kulperger and Yu (2005) derive the properties of structural break tests based on the partial sums of residuals of GARCH models. Almost all existing change-point tests for GARCH models are to detect abrupt changes. One exception is the test of Amado and Ter :: asvirta (2008), who considers testing for a smooth time varying structure of GARCH models. In fact, smooth changes may be more realistic because volatility usually evolves over time in a continuous manner and volatility jumps are rare. Empirical evidences show that various economic events, such as liberalization of emerging markets, integration of world equity markets, changes in exchange rate or interest rate regimes, may lead to structural changes in volatility models. This paper proposes a class of consistent tests for smooth structural changes as well as abrupt structural breaks in GARCH models with known or unknown change points. The idea is to estimate the smooth time-varying parameters of GARCH model by weighted Quasi maximum likelihood estimation (WQMLE) and compare them with the QMLE parameter estimator. The tests compare the log likelihood of the unrestricted nonparametric time-varying GARCH model and the restricted constant parameter GARCH model, which can be viewed as a generalization of LR tests from the parametric framework to the nonparametric framework. Compared with the existing tests for structural breaks in GARCH models in the literature, the proposed tests have a number of appealing features. First, the proposed tests are consistent against a large class of smooth time-varying parameter alternatives. They are also consistent against multiple sudden structural breaks in GARCH models with unknown break points. Second, no prior information on a structural change GARCH alternative is needed. In particular, we do not need to know whether the structural changes are smooth or abrupt, and in the cases of abrupt structural breaks, we do not need to know the dates or the number of breaks. Third, unlike most tests for structural breaks in GARCH models in the literature, which often have nonstandard asymptotic distributions, the proposed tests have a convenient null asymptotic N(0,1) distribution. The only inputs required are the QMLE andWQMLE parameter estimators. Hence, any standard econometric software can carry out computational implementation easily. Fourth, the nonparametric time-varying parameter estimator is sensitive to the local behavior of time-varying parameters. Because only local information is employed in estimating parameters at each time point, the proposed tests have symmetric power against structural breaks that occur either in the …rst or second half of the sample period. This is di¤erent from some existing tests