Mean Square Forecast Error Research Articles

The exact multi-period mean-square forecast error for the first-order autoregressive model with an intercept

We study the exact finite-sample behaviour of the mean-square forecast error ( MSFE) of multi-period least-squares forecasts in the normal autoregressive model y t = α + β y t - 1 + u t . We obtain necessary and sufficient conditions for the existence of the MSFE and give an exact expression which we use to obtain numerical results for both the stationary and the fixed start-up model. We conclude, inter alia, that the behaviour of the MSFE in the model with intercept can be very different from that in the model without intercept, especially when β is close to unity.

Exact and Approximate Multi-Period Mean-Square Forecast Errors for Dynamic Econometric Models

Both future disturbances and estimated coefficients contribute to the uncertainty in model-based ex ante forecasts, but only the first source is usually taken into account when calculating confidence intervals for practical applications. Schmidt (1974) and Baillie (1979) provide an easily computable second-order approximation to the mean-square forecast error (MSFE) for linear dynamic systems which recognizes both sources of uncertainty. To assess the accuracy of their approximation, and thus its usefulness, we compare it with three sets of estimates of the exact MSFE for the univariate AR(l) process: Monte Carlo estimates for OLS, analytically based values for OLS, and Monte Carlo estimates for maximum likelihood. We find that the Schmidt-Baillie formula is a good approximation to the exact MSFE, and that it helps explain why the exact MSFE can decrease as the forecast horizon increases. In fact, for dynamics typical to econometric models, the MSFE often has a maximum at a forecast horizon of one to twelve periods, i.e., at horizons that are of principal concern to forecasters and policy makers.

The exact multi-period mean-square forecast error for the first-order autoregressive model

The finite-sample behaviour of the multi-period least-squares forecast is considered in the simple normal autoregressive model y t = βy t –1 + u t where ‖ β‖ < 1. Necessary and sufficient conditions are established for the existence of the forecast bias and the mean-square forecast error (MSFE) and an exact expression for the MSFE is given. Exact numerical results are obtained for both the stationary and the fixed start-up case. Our main conclusions are that for small values of β the MSFE is a decreasing function of the number of forecast periods, and that the behaviour of the MSFE in the stationary and the fixed start-up case is very similar, except for values of ‖ β‖ close to 1.

Composite forecasting using ridge regression

Recent studies have shown that composite Forecasting produces superior forecasts when compared to individual forecasts. This paper extends the existing literature by employing ridge regression techniques in composite model building. Security analysts forecasts may be improved when combined with time series forecasts for a diversified sample of 261 firms with a 1980-1982 post-sample estimation period. The mean square error of analyst forecasts may be reduced by combining analyst and univariate time series model forecasts in an ordinary least squares regression model. This reduction is very interesting when one finds that the univatiate time series model forecasts do not substatially deviate from those produced by ARIMA (0,1,1) processes. Multicollinearity exists between analyst and time series model forecasts and ridge regression techniques are used to estimate composite earnings models. Moreover, the estimated mean square ridge regression errors are not statistically different by standardizing different b...

Linear constraints, robust‐weighting and efficient composite modeling

AbstractRecent studies have shown that composite forecasting produces superior forecasts when compared to individual forecasts. This paper extends the existing literature by employing linear constraints and robust regression techniques in composite model building. Security analysts forecasts may be improved when combined with time series forecasts for a diversified sample of 261 firms with a 1980‐1982 post‐sample estimation period. The mean square error of analyst forecasts may be reduced by combining analyst and univariate time series model forecasts in constrained and unconstrained ordinary least squares regression models. These reductions are very interesting when one finds that the univariate time series model forecasts do not substantially deviate from those produced by ARIMA (0,1,1) processes. Moreover, security analysts' forecast errors may be significantly reduced when constrained and unconstrained robust regression analyses are employed.

An Analysis~of Forecast Error in the NMC Hemispheric Primitive Equation Model

Abstract Errors in 72 h forecasts of 500 and 1000mb height and 1000-500mb thickness from the NMC hemispheric primitive equation model during six recent winter seasons are examined in order to determine the geographical distribution of the systematic and non-systematic components of the forecast error. The systematic errors are in the sense as to make the equator-to-pole temperature gradient, the time-averaged jet streams,and the standing waves weaker in the forecasts than in the observed climatology. The non-systematic forecast errors are closely related to the observed high-frequency temporal variability, with largest values in the storm tracks. The systematic error accounts for up to about one-third of the mean-square forecast error in middle latitudes.

How to Improve Accuracy by Combining Independent Forecasts

Abstract It is shown that the ensemble mean-square error of forecasts constructed from a particular linear combination of independent and imperfectly correlated predictions is less than that of any of the individual predictions. The weights to be attached to each prediction are determined by the Gaussian method of least squares and depend on the covariances between independent predictions and between prediction and verification. At the present stage of development of numerical prediction methods, it appears possible to reduce the error variance by about 20%, simply by the optimum combination of two independent predictions.

The Optimality of General-Order Exponential Smoothing

This paper derives the class of nonstationary time-series representations for which exponential smoothing of arbitrary order minimizes mean-square forecast error. It points out that these representations are included in the class of integrated moving averages developed by Box and Jenkins, permitting various procedures to be applied to estimating the smoothing constant and determining the appropriate order of smoothing. These results further permit the principle of parsimony in parameterization to be applied to any choice between exponential smoothing and alternative forecasting procedures.