A Time Series Model of Multiple Structural Changes in Level, Trend and Variance

Since the publication of the influential papers by Rappoport and Reichlin (1989) and Perron (1989), which provide evidence that many macroeconomic time series might best be modeled as stationary around a broken trend, the detection of structural change in the trend function of a time series has captured the attention of econometricians and applied researchers. Much of the subsequent research has focused on testing the unit root hypothesis in the presence of one time structural change where the date of structural change may or may not be known. Contributions in this area include Christiano (1992), Banerjee, Lumsdaine and Stock (1992), Zivot and Andrews (1992), Perron (1997) and Perron and Vogelsang (1998). Perron (1994) and Maddala and Kim (1996a) provide useful summaries. In related work, Vogelsang (1997) develops tests for a change in trend that are robust to whether the data are I(0) or I(1) thereby extending the results of Andrews (1993) to some models with trending data. Empirically, the unit root hypothesis has been rejected in favor of a broken trend model with one change for numerous series. Most notably, using various techniques and tests, the unit root hypothesis has been rejected for many international output series by Banerjee et al. (1992), Raj (1992), Perron (1992), De Haan and Zelhorst (1993), Zelhorst and De Haan (1995), Cheung and Chinn (1996), Ben-David and Papell (1995) and Perron (1997). Unit roots have been rejected in favor of a single trend break model for several inflation series by Evans and Lewis (1995) and Culver and Papell (1997), for real exchange rates by Edison and Fisher (1991), Perron and Vogelsang (1992) and Culver and Papell (1995), and for real interest rates by Perron (1990). Overall, there is a large body of evidence to suggest that the trend function of many macroeconomic time series can be modeled as deterministic with at least one structural change. A natural extension of the literature on testing for unit roots in the presence of structural change involves allowing for more than one possible break date under the alternative broken trend stationary model. Indeed, for many macroeconomic time series for which the possibility of structural change is entertained the assumption of at most one break date is unrealistic and restrictive. For example, trend breaks are often motivated by “big events” like wars, oil price shocks, financial crisis or changes in political or institutional regimes and most long time series contain several such events. To this end, Lumsdaine and Papell (1997) extend the Zivot-Andrews (1992) testing procedure to allow for up to two possible endogenous breaks and they find more evidence against the unit root hypothesis than Zivot and Andrews, but less than Perron (1997). Ben-David, Lumsdaine and Papell (1997) find further evidence for at least two structural breaks for three quarters of the per capita real GDP series collected by Maddison (1995). In addition, Papell (1998) finds evidence of multiple breaks in numerous European real exchange rates, Kanas (1998) finds evidence for up to six breaks in ERM exchange rates and Garcia and Perron (1996) find evidence for two breaks in U.S. real interest rates. Indeed, there is a growing body of results that support trend stationary models with multiple breaks for many macroeconomic and financial time series. In addition to changes in level and trend, changes in variance are often found in economic and financial data. For example, Schwert (1990) finds that the stock market volatility is higher during and after the 1987 crash, compared with other periods. Inclán (1993), Inclán and Tiao (1994) and Chen and Gupta (1997) detect multiple changes in variance for various series of stock returns. Lamoureux and Lastrapes (1990) suggest that the empirical persistence of volatility captured by GARCH models might be caused by structural changes in variance and this view has been supported by Wilson, Aggarwal and Inclán (1996) and Fong (1997). Engel and Hakkio (1996) find that EMS exchange rates have higher volatility dur-