Inférence statistique dans les modèles linéaires

Abstract

We study methods for forecasting long-memory processes. We assume that the processes are weakly stationary, linear, causal and invertible, but only a Önitesubset of the past observations is available. We Örst present two approaches when the stochastic structure of the process is known : one is the truncation of the Wiener-Kolmogorov predictor, and the other is the projection of the forecast value on the observations,i.e. the leastsquares predictor. We show that both predictors converge to the WienerKolmogorov predictor. When the stochastic structure is not known, we have to estimate the coe¢ cients of the predictors deÖned in the Örst part. For the truncated Wiener-Kolomogorov, we use a arametric approach and we plug in the forecast coe¢ cients from the Whittle estimator, which is computed on an independent realisation of the series. For the least-squares predictor, we plug the empirical autocovariances (computed on the same realisation or on an independent realisation) into the Yule-Walker equations. For the two predictors, we estimate the mean-squared error and prove the asymptotic normality.