THREE ESSAYS ON STRUCTURAL BREAKS IN FINANCIAL TIME SERIES

Abstract

In the first essay, Forecast Combinations for Realized Volatility in Presence of Structural Breaks, the problem of instability due to changes in the parameters of some Realized Volatil ity models has been addressed. The analysis is based on 5-minute RV of four U.S. stock market indices. Three different representations of the log Realized Volatility have been con sidered and rewritten as linear regression models. In order to test the presence of structural breaks in Realized Volatility, we propose the use of standard tests for parameter instability, usually implemented in a regression context. In par ticular the attention has been focused on the recursive estimates test proposed by Ploberger et al. (1989). This choice is based on the fact that this test does not assume a particular pattern of deviation from the null hypothesis and it does not require the specification of the locations of the break points. Moreover, it is suitable for high dimensional data ensuring, at the same time, not high computational costs. In order to account for potential structural breaks when generating forecasts, we have pro posed the use of several forecast combinations. These tools, originally defined in a regression context, have been successfully adapted to the considered Realized Volatility models. All of the proposed forecast combinations are based on different estimation windows, with alterna tive weighting schemes, They do not take into account explicitly estimated break dates and, again, they are appropriate for high dimensional data. In order to evaluated the advantages of the considered forecast combinations, for each Real ized Volatility model, an out-of-sample forecasting exercise has been performed. The fore casting performances of the proposed approaches has been compared, in terms of two differ ent loss functions, by using the Model Confidence Set procedure (Hansen et al., 2011). Our analysis has highlighted the importance of taking into account structural breaks in the Realized Volatility models. moreover it gives empirical evidence of the effectiveness of the proposed forecast combinations in this context. In the second essay, A Bootstrap Bias Correction of long run fourth order moment estima tion in the CUSUM of squares test, we have focused on the CUSUM of squares test proposed by Sansó et al. (2004) for the detection of structural breaks in financial data. It makes adjust ments to the original proposal by Inclan and Tiao (1994) that allow the time series to obey a wide class of dependent processes, including GARCH and log-normal stochastic volatility processes, under the null. The test statistic is based on a consistent estimation of the the long run fourth order moment which can be obtained by using a kernel Heteroskedasticity Au tocorrelation Consistent (HAC) estimator. Unfortunately, in the case of strong dependence and in relative small samples, the HAC estimation of the long run fourth order moment could be inefficient and, as a consequence, significant biases may arise leading to significant size distortion of the test (Rodriguez and Rubia, 2007). In this essay we have proposed a bias correction of the estimation of the long run fourth or der moment based on the stationary bootstrap proposed by Politis and Romano (1994). The choice of this resampling technique is justified by the stationarity and weakly dependence of the time series under the assumptions which ensure the existence of the limiting distribution of the test statistic, under the null hypothesis. In order to evaluate the effect of the proposed bias correction we have implemented an ex tensive Monte Carlo experiment considering two alternative data generating processes, the GARCH(1,1) and the log-normal stochastic volatility. For both the specifications, we have considered a parametric space which includes empirical values typically observed in practice and different sample sizes. The results give evidence that the bootstrap approach is better able to correctly identify the presence of structural breaks in the data. More specifically, in the GARCH(1,1) data gen erating process without breaks, the false rejection rates could be considered non statistically different from the nominal values when the persistence is not high and for large sample size even in the case of high persistence. These results are confirmed in the case of the log-normal stochastic volatility model. When the data are generated assuming the presence of a single break in the middle of the sample, the bias corrected estimation is able to correctly identify the break and its location, for all the sample sizes and in both the considered models. The proposed procedure has been applied to analyse the presence of structural breaks in two real time series, IPC-Mexico and CNX Nifty-India. In both cases it seems to work quite well resulting more robust with respect to extreme observations. In the last essay, Forecasting with GARCH models under structural breaks: an approach based on combinations across estimation windows, we have addressed the problem of fore casting in presence of structural breaks in the unconditional variance of a GARCH (1,1) model. The existence of structural breaks in the volatility can be the cause of estimation problems and forecast failures. If they are present in the data generating process but they are not considered in the specification of the model, the analysis could be biased toward a spu rious persistence. The problem is not new in the econometric literature and generally it has been solved by selecting an arbitrary single estimation window in which only a fraction of the most recent observations is used to estimate the parameters and to generate the forecasts. In order to solve the problems arising with the choice of a single estimation window, it can be useful to consider forecast combinations generated by the same model but over different estimation windows. In this essay, we have proposed some weighting schemes to average forecasts across differ ent estimation windows to account for structural changes in the unconditional variance of a GARCH (1,1) model. Each combination is obtained by averaging forecasts generated by re cursively increasing an initial estimation window of a fixed number of observations v. Three different choices of the combination weights have been proposed. In the first scheme, the forecast combination is obtained by using equal weights to average the individual forecasts; the second weighting method simply assigns heavier weights to forecasts that use more recent information; the third is a trimmed version of the forecast combination with equal weights where a fixed fraction of forecasts with the worst performance are discarded. In order to evaluate the effects of the choice of the tuning parameter v and the effectiveness of the proposed procedures in accounting for structural breaks, an extensive Monte Carlo ex periment has been implemented. The results give evidence that forecast combinations with high values of v seem to be effective in accounting for structural breaks. Moreover, they are also able to perform better with respect to alternative schemes proposed in the literature by Rapach and Strauss (2008) and Rapach et al. (2008). The proposed procedures have been evaluated on real data by considering the same data set used by Rapach and Strauss (2008). This empirical application has confirmed the results obtained in the simulations.