Estimation and Testing for Smooth Structural Changes in GARCH Models

Detecting and modelling structural changes in GARCH processes have attracted a great amount of attention in time series econometrics over the past few years. In this paper, we …rst generalize Dahlhaus and Subba Rao (2006 2008)’s time-varying ARCH processes to time-varying GARCH processes and derive the consistency and asymptotic normality of the weighted quasi maximum likelihood estimator of the smooth time-varying coe¢ cients. A class of generalized likelihood ratio tests are proposed to check smooth structural changes as well as abrupt structural breaks with possibly 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. The test has a 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 macroeconomic and …nancial time series. However, underlying all these models is the key assumption of stationarity. Given the changing pace of the underlying economic mechanism, modeling economic variables over a long time horizon under the stationarity assumption may not be suitable. It is quite plausible that structural changes may occurr, causing the time series to deviate from stationarity. Indeed, various economic factors may lead to structural changes in economic 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. The prevalence of structural instability in …nancial time series has been documented 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 sensible 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 volatility 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 in the literature. For example, Chu (1995) generalizes Andrews’(1993) supremum Lagrange multiplier (LM) test to GARCH models. However, the test just considers one-time shift as the alternative so it does not have good power against multiple breaks. Berkes, Gombay, Horvath and Kokoszka (2004) develop a sequential likelihood-ratio (LR) based test for evaluating the stability of GARCH parameters. Their test can be used to check which parameter of a GARCH model has a change point and