Finite sample criteria for autoregressive order selection

The quality of selected AR models depends on the true process in the finite sample practice, on the number of observations, on the estimation algorithm, and on the order selection criterion. Samples are considered to be finite if the maximum candidate model order for selection is greater than 10, where denotes the number of observations. Finite sample formulae give empirical approximations for the statistical average of the residual energy and of the squared error of prediction for several autoregressive estimation algorithms. This leads to finite sample criteria for order selection that depend on the estimation method. The special finite sample information criterion (FSIC) and combined information criterion (CIC) are necessary because of the increase of the variance of the residual energy for higher model orders that has not been accounted for in other criteria. Only the expectation of the logarithm of the residual energy, as a function of the model order, has been the basis for the previous classes of asymptotical and finite sample criteria. However, the behavior of the variance causes an undesirable tendency to select very high model orders without the special precautions of FSIC or CIC.