Peter Hall, Australian National University
Qiwei Yao, London School of Economics
Abstract
We develop a general methodology for tilting time series data. Attention is focused on a large class of regression problems, where errors are expressed through autoregressive processes. The class has a range of important applications, and in the context of our work may be used to illustrate the application of tilting methods to interval estimation in regression, robust statistical inference, and estimation subject to constraints. The interval estimation example includes empirical likelihood, where earlier applications to time series have involved either the multivariate ``dual likelihood'' approach introduced by Per Mykland, or a ``Whittle likelihood'' method suggested by Anna Monti. One advantage of our form of empirical likelihood is that it admits a wide range of distance, or more correctly divergence, functions. (We favour a non-traditional form of Kullback-Leibler divergence, because of its robustness properties.) Another is its simplicity; it is based directly on computed residuals, which are of course very familiar to time series analysts, and it does not involve the complexities of dual likelihood. A third advantage is its flexibility; it is readily applied to constructing confidence intervals or confidence bands in general regression problems. And a fourth is its context as a particular example of a very general methodology; our empirical likelihood approach is no more than a special case of a very broad class of tilting-based techniques for inference in time series problems.