Chapters 1 and 2 introduced the concept of hypothesis testing in regression analysis. We looked at the “Global” F test, which tested the hypothesis of model significance. We also discussed the t tests for the individual coefficients in the model. We will now extend these to testing the joint hypothesis of the coefficients and also to hypothesis tests involving linear combinations of the coefficients. This chapter will conclude with a discussion on testing data for structural breaks and for stability over time.
3.1.1 The General Linear Hypothesis
Hypothesis testing on regression parameters, subsets of parameters, or a linear combination of the parameters can be done by considering a set of linear restrictions on the model y = Xβ + ε. These restrictions are of the form Cβ = d, where C is a j x k matrix of j restrictions on the k parameters (j ≤ k), β is the k x 1 vector of coefficients, and d is a j x 1 vector of constants. Note that here k is used to denote the number of parameters in the regression model. The ith restriction equation can be written as (Greene, 2003, p. 94; Meyers, 1990, p. 103)
To see the general form of C, consider the following hypothetical model:
A linear restriction of the form β2 – β3 = 0 can be written as
The C matrix ...