TABLE OF CONTENTS If a regressor in a linear regression model (5.6) is equal to a linear combination of one or more of the other regressors, then all the involved regressors are said to be linearly dependent. In that case, the parameter estimation routine cannot assign specific values to the parameters for those terms, because many different weighted combinations of the linearly dependent regressors could be used equally well to model the same variation in the dependent variable. This makes the parameter estimation problem ill-conditioned. In practice, the regressors might be almost linearly dependent, but not exactly so. But the fundamental difficulty remains the same, with its severity becoming worse as regressors get closer to being perfectly correlated, i.e., as r jk, , j, k ? (1, 2, , n), from Eq. (5.77) approaches 1. Any situation where regressors are correlated at a high enough level to cause problems in the parameter estimation is called data collinearity. Operationally, when data collinearity is present, the parameter estimation routines will produce inaccurate parameter estimates with large variances, or in severe cases the parameter estimation routine may fail.
This section discusses methods for assessing data collinearity and some approaches for getting good parameter estimation results when data collinearity exists. For the following discussion and analysis, it is convenient to use standardized regressors introduced in Sec. 5.1.5.
As shown in Sec. 5.1.5, the matrix X * T X * is the n n matrix of pair-wise...
