Empirical Limits for Time Series Econometric Models

This paper seeks to characterize empirically achievable limits for time series econometric modeling. The approach involves the concept of minimal information loss in time series regression and the paper shows how to derive bounds that delimit the proximity of empirical measures to the true probability measure in models that are of econometric interest. The approach utilizes generally valid asymptotic expressions for Bayesian data densities and works from joint measures over the sample space and parameter space. A theorem due to Rissanen is modified so that it applies directly to probabilities about the relative likelihood (rather than averages), a new way of proving results of the Rissanen type is demonstrated, and the Rissanen theory is extended to nonstationary time series with unit roots, near unit roots and cointegration of unknown order. The corresponding bound for the minimal information loss in empirical work is shown not to be a constant, in general, but to be proportional to the logarithm of the determinant of the (possibility stochastic) Fisher-information matrix. In fact, the bound that determines proximity to the DGP is generally path dependent, and it depends specifically on the type as well as the number of regressors. Time trends are more costly than stochastic trends, which, in turn, are more costly than stationary regressors in achieving proximity to the true density. The conclusion is that, in a very real sense, the 'true' DGP is more elusive when there is nonstationarity in the data. Some implications of these results for prediction and for the achieving proximity to the optimal predictor are explored.