Autocorrelation is a common phenomenon in time series data. In the presence of autocorrelation, the ordinary least squares (OLS) estimator of regression parameters is still consistent in general. However, the usual formulae for standard errors are invalid. Many practical methods in econometrics make use of heteroskedasticity and autocorrelation consistent standard errors in order to obtain robust inferences. These standard errors are now widely implemented in statistical packages. The practical problem is that the practitioner is required to choose the so-called bandwidth, a tuning parameter in constructing these standard errors. The tuning parameter is important because different tuning parameters may lead to qualitatively different conclusions. Existing methods for bandwidth selection are all based on minimizing the mean square error (MSE) criterion of the relevant nonparametric quantity, which in this context is the long-run variance estimator. Such a choice of the bandwidth is designed to be optimal in the MSE sense for point estimation of the long run variance, but is not necessarily best suited for hypothesis testing and confidence interval estimation. This project proposes to choose the bandwidth to minimize a criterion function or loss function that is directed at hypothesis testing and interval estimation. For hypothesis testing, the loss function is taken to be a weighted average of the type I error (the probability of false rejection) and the type II error (the probability of false acceptance). For interval estimation, the loss function is taken to be the error in coverage probabilities, i.e. the difference between the true coverage probability and the nominal coverage probability. The newly proposed bandwidth choice rules thus address the central concerns of interest in hypothesis testing and interval estimation.
Optimal bandwidth selection for semiparametric testing and interval estimation is a long-standing problem in time series regressions. Developing an optimal selection procedure is not straightforward and involves some conceptual as well as technical challenges. The present project confronts this challenge by proposing an approach that is theoretically sounded and empirically relevant. This project significantly advances the frontiers of current time series research. Preliminary calculations show that, in order to optimize the new criteria, one would choose the bandwidth to balance the asymptotic bias and variance of the long run variance estimator. This is in sharp contrast with the conventional MSE criterion that balances the squared asymptotic bias with variance. Some limited simulations show that the proposed approach is promising.
Broad Impacts: This project will have significant and far-reaching impacts on both theoretical and practical analyses of time series data. While the theoretical framework is developed specifically for heteroskedasticity and autocorrelation robust test and interval estimation, the idea and approach can be used to optimally select the tuning parameter in general nonparametric and semiparametric models. The tuning parameter can be the bandwidth in kernel smoothing or the number of terms in sieve approximation. From a practical perspective, the new bandwidth selection rule has the potential for developing a standard of practice for the computation of autocorrelation robust standard errors. Researchers in social sciences and natural sciences may use the newly developed bandwidth choice rule to perform more precise and reliable inferences. This will improve empirical studies in related fields.
