Question

Hypothesis testing--F test for two variables

Suppose that you wants to test whether the returns on a company stock (y) show no sensitivity to two factors (factor x2 and factor x3) among three considered. That is, the null hypothesis is: β2=0, β3=0. The regression is carried out on 144 monthly observations, which means the sample size is 144. The regression is:

y = β1 + β2x2 + β3x3 + β4x4 + u

(1) What are the restricted and unrestricted regressions?

(2) If the two F-test statistic value is 36.8, and if the critical value is 4.79(1% significance level), would the null hypothesis be rejected at 1% significance levels?

Answer 1

1)

Restricted model: We do not include the variables whose coefficients are assumed to be zero in the NULL hypothesis

Unrestricted: We write the whole model, without any restriction imposed

Restricted equation

Since β2 and  β3 are assumed to be zero as per null

y = β1 + β4x4 + u

Unrestricted equation

y = β1 + β2x2 + β3x3 + β4x4 + u

2)

F-test statistic = 36.8

F- critical = 4.791

Since the F-test statistic > critical value, we reject the null hypothesis