Goodness of fit of prediction models and two step prediction

Given a second order stationary time series it can be shown that there exists an optimum linear predictor of Xk, say X*k which is constructed from {Xt ,t=O,-l,-2 …} the mean square error of prediction being given by ek = E [|Xk- X*k|2]. In some cases however a series can be considered to have started at a point in the past and an attempt is made to see how well the optimum linear form of the predictor behaves in this case. Using the fundamental result due to Kolmogorov relating the prediction error e1 to the power spectrum f(w) e1 = exp. {1/2 pi Log from – pi to p log 2 pi f(w) dw} estimates of e1 are constructed using the estimated periodogram and power spectrum estimates. As is argued in some detail the quantity e1 is a natural one to look at when considering prediction and estimation problems and the estimates obtained are non-parametric. The characteristic functions of these estimates are obtained and it is shown that asymptotically they have distributions which are approximately normal. The rate of convergence to normality is also investigated. A previous author has used a similar estimate as the basis of a test of white noise and the published results are extended and in the light of the simulation results obtained some modifications are suggested. To increase the value of the estimates e1 their small sample distribution is approximated and extensive tables of percentage points are provided. Using these approximations one can construct a more powerful and versatile test for white noise and simulation results confirm that the theoretical results work well. The same approximation technique is used to derive the small sample distribution of some new estimates of the coefficients in the model generating {Xt}. These estimates are also based on the power spectrum. While it is shown small sample theory is limited in this situation the asymptotic results are very interesting and useful. Several suggestions are made as to further fields of investigation in both the univariate and multivariate cases