Question

Suppose OLS assumptions A1-A4 hold and we want to test the joint null hypothesis that β 1 = β 2 = 0. We obtain the t-statistics t1 and t2 for the single null hypothesis that β 1 = β 2 = 0 respectively. Further denote C alpha/2 as the critical value that satisfies P(|N(0;1)|> C alpha/2) = Alpha/2, where N(0,1) is the standard normal. Our rule is if one of the two t-statistics is greater than C alpha/2, we will reject the joint null hypothesis. Show that based on this rule, the probability that we reject the joint null while it holds (i.e., β 1= 0 and β 2= 0 but we mistakenly reject it) is less than or equal to alpha asymptotically. Hint: let A and B be two events, then the union bound says P(AUB) is less than or equal to P(A) +P(B)

Answer 1

Thank You.