Hausmans Specification Test: OLS Versus 2SLS. This is based on Maddala (1992, page 511). For the simple regression yt = xt + ut t =

Hausman’s Specification Test: OLS Versus 2SLS. This is based on Maddala (1992, page 511). For the simple regression yt = βxt + ut t = 1, 2 . . . , T where β is scalar and ut ∼ IIN(0, σ2). Let wt be an instrumental variable for xt. Run xt on wt and get xt = πwt + νt or xt = ˆxt + νt where ˆxt = πwt.

(a) Show that in the augmented regression yt = βxt + γˆxt + t a test for γ = 0 based on OLS from this regression yields Hausman’s test-statistic. Hint: Show that γOLS = q/(1 − r2 xw)

where r2 xw =

T t=1 xtwt

2

/

T t=1 w2 t

T t=1 x2t

.

Next, show that var(γOLS) = var(βOLS)/r2 xw(1 − r2 xw). Conclude that

γ2 OLS/var(γOLS) = q2r2 xw/[var(βOLS)(1 − r2 xw)]

is the Hausman (1978) test statistic m given in section 11.5.

(b) Show that the same result in part

(a) could have been obtained from the augmented regression yt = βxt + γνt + ηt where νt is the residual from the regression of xt on wt.