Consider the linear regression model yt = ????′xt + ut, t = 1,…, T, where yt and xt (k × 1) are observed and ut is a random disturbance.The following assumptions are made about this model:

(i) E(ut) = 0 for each t.

(ii) E(u2t

) = ????2 (constant) for each t.

(iii) E(utus) = 0 for each t and s ≠ t.

(iv) u1,…, uT are jointly normally distributed.

(v) x1,…, xT are fixed in repeated samples.

Let V = s2 (
ΣT t=1 xtx′t )−1 (k × k) where s2 = 1 T − k ΣT t=1 (yt − ????̂
′
xt)2 and ????̂ is the least squares estimator. Let ????j denote the jth element of ????, ????̂j the corresponding element of ????̂ , and ????jj the corresponding diagonal element of V.

(a) Derive from ????̂j a statistic having the Student’s t distribution.

(b) Explain the roles of each of assumptions (i)–(v) in validating the result of part (a). Should the assumption of random sampling have been included in this list?

(c) Does the result of part

(a) depend on having xkt = 1 for each t (inclusion of an intercept)? Explain your answer.

(d) Explain how the hypothesis ????j = 0 is tested using the result of part (a).