Estimation and inference under non-stationarity

This dissertation consists of three essays on estimation and inference with non-stationary time series data. Motivated by the large empirical literature on the existence of structural breaks in the mean or variance of economic data, we propose time series models with stochastic breaks in the first essay. We systematically study the effect of mean or variance breaks on the ordinary least square (OLS) regression inference. In the case of mean-break, we show that when there are breaks in both the regressor and the error term, spurious regression arises. On the contrary, when only the regressor exhibits mean breaks, standard results apply. In this case, we obtain consistent estimator with standard distribution for the t-statistic. In the case of variance-break, problem arises only when the regressor is very persistent. We show that in the case where the regressor is nearly integrated, breaks in the variance sufficiently lead to a non-standard asymptotic distribution for the t-statistic. The second essay focuses on the analysis of non-stationarity in the second moment. We investigate the behavior of variance-covariance estimators under general form of heteroskedasticity and auto correlation. In particular, we allow for non-stationarity in the covariance dynamic, such as unconditional heteroskedasticity or persistent variation in the volatility, and show that the inconsistent estimators proposed by Kiefer, Vogelsang and Bunzel (2000) no longer converge to the pivotal distributions as claimed. Hence they can not be trusted to perform valid inference when the data exhibit second moment nonstationarity. We also suggest estimators that are robust to certain form of non-stationarity. For example, t-statistic normalized by the consistent kernel estimator of Andrews (1991) provides valid inference under our unconditional heteroskedasticity model and the conservative t-test of Ibragimov and Muller (2006) is valid under both heteroskedasticity and persistent stochastic volatility models. The third essay detects the presence of structural breaks, in the form of mean shifts, in the implied and realized volatilities of S&P 500 returns. When studying the information content of option implied volatility, predictive regressions of future realized volatility using implied volatility are often performed. Since regression inference is strongly affected by presence of mean shifts, as indicated in the first essay, standard regressions should not be used here. We perform robust regressions that can accommodate the breaks in the series. While standard predictive regressions support the unbiasedness and efficiency of the implied volatility, as measured by the VIX index here, as a forecast of future realized volatility, the robust regressions lead to different conclusions. It is shown that the VIX was once a biased forecast, but its performance improves as time goes on. The whole sample results are unreliable due to structural breaks that bias up the OLS estimate. The improvement of VIX's forecasting ability over time may be a result of the market's adaption to the better use of options and the improved efficiency and liquidity of the index options market.