Empirical Limits for Time Series Econometric Models

This paper characterizes empirically achievable limits for time series econometric modeling
and forecasting. 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 (the DGP) in models that are
of econometric interest. The approach utilizes joint probability measures over the combined
space of parameters and observables and the results apply for models with stationary,
integrated, and cointegrated data. A theorem due to Rissanen is extended 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. For practical purposes, the
proximity bound has the asymptotic form (K/2) logn, where K is a new dimensionality
factor that depends on the nature of the data as well as the number of parameters in the
model. When 'good' model selection principles are employed in modeling time series data,
we are able to show that our proximity bound quantifies empirical limits even in situations
where the models may be incorrectly specified.
One of the main implications of the new result is that time trends are more costly
than stochastic trends, which are more costly in turn than stationary regressors in achieving
proximity to the true density. Thus, in a very real sense and quantifiable manner, the
DGP is more elusive when there is nonstationarity in the data. The implications for prediction
are explored and a second proximity theorem is given, which provides a bound
that measures how close feasible predictors can come to the optimal predictor. Again, the
bound has the asymptotic form (K/2) log n, showing that forecasting trends is fundamentally
more difficult than forecasting stationary time series, even when the correct form of
the model for the trends is known.