On predictability of ultra short AR(1) sequences

Abstract: This paper addresses short term forecast of ultra short AR(1) sequences (4 to 6 terms only) with a single structural break at an unknown time and of unknown sign and magnitude. As prediction of autoregressive processes requires estimated coefficients, the efficiency of which relies on the large sample properties of the estimator, it is a common perception that prediction is practically impossible for such short series with structural break. However, we obtain a heuristic result that some universal predictors represented in the frequency domain allow certain predictability based on these ultra short sequences. The predictors that we use are universal in a sense that they are not oriented on particular types of autoregressions and do not require explicit modelling of structural break. The shorter the sequence, the better the one-step-ahead forecast performance of the smoothed predicting kernel. If the structural break entails a model parameter switch from negative to positive value, the forecast performance of the smoothed predicting kernel is better than that of the linear predictor that utilize AR(1) coefficient estimated from the ultra short sequence without taking the structural break into account regardless whether the innovation terms in the learning sequences are constructed from independent and identically distributed random Gaussian or Gamma variables, scaled pseudo-uniform variables, or first-order auto-correlated Gaussian process.