## Se fa fa

follows the t distribution with (n - 4) df because (8.6.1) is a four-variable model or, more generally, with (n - k) df, where k is the total number of parameters estimated, including the constant term. The se (fa - fa) is obtained from the following well-known formula (see Appendix A for details):

If we substitute the null hypothesis and the expression for the se (fa — fa) into (8.6.3), our test statistic becomes fa 3 — fa 4

Now the testing procedure involves the following steps:

1. Estimate fa3 and fa. Any standard computer package can do that.

2. Most standard computer packages routinely compute the variances and covariances of the estimated parameters.11 From these estimates the standard error in the denominator of (8.6.5) can be easily obtained.

3. Obtain the t ratio from (8.6.5). Note the null hypothesis in the present case is (fa — fa) = 0.

4. If the t variable computed from (8.6.5) exceeds the critical t value at the designated level of significance for given df, then you can reject the null hypothesis; otherwise, you do not reject it. Alternatively, if the p value of the

"The algebraic expression for the covariance formula is rather involved. Appendix C provides a compact expression for it, however, using matrix notation.

266 PART ONE: SINGLE-EQUATION REGRESSION MODELS

t statistic from (8.6.5) is reasonably low, one can reject the null hypothesis. Note that the lower the p value, the greater the evidence against the null hypothesis. Therefore, when we say that a p value is low or reasonably low, we mean that it is less than the significance level, such as 10, 5, or 1 percent. Some personal judgment is involved in this decision.