Applied Bayesian Modelling - Free

MODELLING STRUCTURAL SHIFTS 2153.53Volatility ofExchange Rate Difference2.521.510.500 100 200 300 400 500Day600 700 800 900 1000Figure 5.13Volatility of exchange rate series5.7 MODELLING STRUCTURAL SHIFTSState space models are designed to accommodate gradual or smooth shifts in time seriesparameters. Often, however, there are temporary or permanent shifts in time seriesparameters that occur more abruptly, and a more appropriate model allows for changesin regression regimes and other shifts in structure. Section 5.2 considered innovationand additive outliers. Here we consider models that allow for repeated switchingbetween distinct regimes according to a latent Markov series, and models for shifts inboth the mean and variance of autoregressive series. A further category includesswitching regression models (Maddala and Kim, 1996).5.7.1 Binary indicators for mean and variance shiftsMcCulloch and Tsay (1994) consider autoregressive models allowing for shifts in meanand/or variance. By allowing for variance shifts as well as changes in level, nonstationarytrends that might otherwise have been attributed to changes in level may moreappropriately be seen as due to heteroscedasticity. McCulloch and Tsay choose to focusexplicitly on autoregressive models, rather than introduce moving average effects becausegiven a sufficiently large AR model, one may achieve similar results to a stipulatedARMA model. Thus lety t ˆ m twhere a change in level is accommodated by lettinge tm t ˆ m t1 d 1t n t (5:47)The d 1t are binary variables for each time point which equal 1 if a shift in mean occursand n t models the shift that occurs, conditional on d 1t ˆ 1. The n t are usually modelledas normal with mean zero and low precision t n . The autoregressive component of theseries is the p-lag model for e t , namely