Engle (1984, pp. 787-788). Consider the linear regression model y = X + u = X11 + X22 + u given in (7.8), where u

Engle (1984, pp. 787-788). Consider the linear regression model y = Xβ + u = X1β1 + X2β2 + u given in (7.8), where u ∼ N(0, σ2IT ).

(a) Write down the log-likelihood function, find the MLE of β and σ2.

(b) Write down the score S(β) and show that the information matrix is block-diagonal between

β and σ2.

(c) Derive the W, LR and LM test statistics in order to test Ho; β1 = βo 1, versus HA; β1

= βo 1, where β1 is say the first k1 elements of β. Show that if X = [X1,X2], then W = (βo 1

− β1)[X



1

¯ PX2X1](βo 1

− β1)/σ2 LM = u



X1[X



1

¯ PX2X1]−1X



1u/σ2 LR = T log(u

u/u

u)

where u = y − Xβ, u = y − Xβ and σ2 = uu/T , σ2 = uu/T . β is the unrestricted MLE, whereas β is the restricted MLE.

(d) Using the above results, show that W = T (u

u − u

u)/u

u LM = T (u

u − u

u)/u

u Also, that LR = T log[1 + (W/T )]; LM = W/[1 + (W/T )]; and (T − k)W/Tk1 ∼ Fk1,T−k under Ho. As in problem 16, we use the inequality x ≥ log(1 + x) ≥ x/(1 + x) to conclude that W ≥ LR ≥ LM. Hint: Use x = W/T . However, it is important to note that all the test statistics are monotonic functions of the F-statistic and exact tests for each would produce identical critical regions.

(e) For the cigarette consumption data given in Table 3.2, run the following regression:

logC = α + βlogP + γlogY + u compute the W, LR, LM given in part

(c) for the null hypothesis Ho; β = −1.

(f) Compute the Wald statistics for HA o ; β = −1, HB o ; β5 = −1 and HC o ; β

−5 = −1. How do these statistics compare?