Maddala (1992, pp. 120-127). Consider the simple linear regression yi = + Xi + ui i = 1, 2, . . ., n. where

Maddala (1992, pp. 120-127). Consider the simple linear regression yi = α + βXi + ui i = 1, 2, . . ., n.

where α and β are scalars and ui ∼ IIN(0, σ2). For Ho; β = 0,

(a) Derive the Likelihood Ratio (LR) statistic and show that it can be written as nlog[1/(1−r2)]

where r2 is the square of the correlation coefficient between X and y.

(b) Derive the Wald (W) statistic for testing Ho; β = 0. Show that it can be written as nr2/(1−

r2). This is the square of the usual t-statistic on β with σ2 MLE =

n i=1e2i

/n used instead of s2 in estimating σ2. β is the unrestricted MLE which is OLS in this case, and the ei’s are the usual least squares residuals.

(c) Derive the Lagrange Multiplier (LM) statistic for testing Ho; β = 0. Show that it can be written as nr2. This is the square of the usual t-statistic on with σ2 RMLE =

n i=1(yi − ¯y)2/n used instead of s2 in estimating σ2. The σ2 RMLE is restricted MLE of σ2 (i.e., imposing Ho and maximizing the likelihood with respect to σ2).

(d) Show that LM/n = (W/n)/[1 + (W/n)], and LR/n = log[1 + (W/n)]. Using the following inequality x ≥ log(1 + x) ≥ x/(1 + x), conclude that W ≥ LR ≥ LM. Hint: Use x = W/n.

(e) For the cigarette consumption data given in Table 3.2, compute the W, LR, LM for the simple regression of logC on logP and demonstrate the above inequality given in part (d)
for testing that the price elasticity is zero?