Test of Hypothesis with a Deterministic Time Trend Model. This is based on Hamilton (1994). In problem 6, we showed that OLS and OLS converged

Test of Hypothesis with a Deterministic Time Trend Model. This is based on Hamilton (1994). In problem 6, we showed that αOLS and βOLS converged at different rates,

√

T and T

√

T respectively.

Despite this fact, the usual least squares t and F-statistics are asymptotically valid even when the ut’s are not Normally distributed.

(a) Show that s2 =

T t=1(yt − αOLS − βOLSt)2/(T − 2) has plim s2 = σ2.

(b) In order to test Ho; α = αo, the usual least squares package computes tα = (αOLS − αo)/[s2(1, 0)(X



X)−1(1, 0)]1/2 where (XX) is given in problem 6. Multiply the numerator and denominator by

√

T and use the results of part

(c) of problem 6 to show that this t-statistic has the same asymptotic distribution as t∗

α =

√

T(αOLS −αo)/σ



q11 where q11 is the (1, 1) element of Q−1 defined in problem 6. t∗

α has an asymptotic N(0, 1) distribution using the results of part

(e) in problem 6.

(c) Similarly, to test Ho; β = βo, the usual least squares package computes tβ = (βOLS

− β)/[s2(0, 1)(X



X)−1(0, 1)]1/2.

Multiply the numerator and denominator by T

√

T and use the results of part

(c) of problem 6 to show that this t-statistic has the same asymptotic distribution as t∗

β = T

√

T(βOLS

−

β)/σ



q22 where q22 is the (2, 2) element of Q−1 defined in problem 6. t∗

β has an asymptotic N(0, 1) distribution using the results of part

(e) in problem 6.