On the relationship between the sample size and the number of variables in a linear regression model

The problem of determining the number of variables to be included in the linear regression model is considered under the assumption that the dependent and independent variables have a joint normal distribution. It is shown that for a given sample size n there exists an optimal number k₀ (0 ≤ k₀ < n-2) of variables among all independent variables in the model, such that the expectation of the mean squared error corresponding to the prediction equation with k₀ variables is minimal.Application of this result to ustepwise procedures is discussed.