Testing the Equality of Regression Coefficients (β2 = β3) without New Regression

To test the hypothesis H0: β2 = β3 versus H1: β2 ≠ β3, we can use the t-test to compare the estimated coefficients β2 and β3.

The null hypothesis H0 states that the coefficient β2 is equal to the coefficient β3. This means that the two variables represented by these coefficients have the same impact on the dependent variable. The alternative hypothesis H1, on the other hand, suggests that the coefficients β2 and β3 are not equal, implying that the two variables have different impacts on the dependent variable.

To construct the test, we need to calculate the test statistic and compare it to the critical value at the 5% significance level. The test statistic is given by:

t = (β2 - β3) / sqrt(Var(β2 - β3))

where Var(β2 - β3) is the variance of the difference between the two coefficients, which can be obtained from the covariance matrix of the regression coefficients.

The critical value at the 5% significance level can be found from the t-distribution table or calculated using software. We compare the absolute value of the test statistic with the critical value. If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis in favor of the alternative hypothesis.

It's important to note that without running a new regression, we need to have the estimated coefficients β2 and β3 and their covariance matrix from a previous regression analysis.