Το υπόδειγμα τυχαίου περιπάτου με αυτοπαλίνδρομα σφάλματα

In this study we show that a random walk model with drift and first order autocorrelated errors, AR(1), behaves like an ARIMA(1,1,0). The last one is extracted from the unrestricted model of the Augmented Dickey Fuller test using as an explanatory variable a lag of order one difference of the series under consideration when H0 is true. Through Monte Carlo simulations we show that when the population model is a random walk with moderate AR(1) autocorrelation in the errors we have a high type II error either in small or large samples. Thus we are accepting as a population model the random walk with unfortunately uncorrelated errors. This causes problems at the stage of making predictions when constructing prediction intervals of the series we use 2 standard deviations of the forecast error above and below the predicted value. More specifically, the actual probability the prediction interval to include the real future value is really smaller than the nominal one of 95.44% even if the number of forecasting periods ahead is relatively small compared to the sample size we are using.