It often occurs in multiple regression (see eq. 2.2.1) that the predictor variables (Fi(t)) exhibit very large mutual correlations. This is referred to as multicollinearity, in which the sampling distributions of the estimated regression coefficients can become very broad. The practical consequence is that a forecasting equation may perform very badly when implemented on future data independent of the training sample. The multicollinearity problem can be rectified by first subjecting the predictor variables to an empirical orthogonal function (EOF) analysis, then using the first M principal components as predictors in place of the original variables . Let ej(i) denote the i-th element of the j-th EOF. Then, using the training data set again, we compute the j-th principal component (PC),
Here, the PCs are orthogonal, if Similar to, but instead of equation (2.2.1), we write our forecast as
where bj is the j-th principal component weight. Since the PCs are uncorrelated, they do not exhibit the multicollinearity problem, and their regression coefficients will be estimated with much better precision.