For the simple dynamic model with AR(1) disturbances given in (6.18), (a) Verify that plim(OLS ) = (12)/(1+). Hint: From (6.18), Yt1 = Yt2+t1 and

For the simple dynamic model with AR(1) disturbances given in (6.18),

(a) Verify that plim(βOLS

−β) = ρ(1−β2)/(1+ρβ). Hint: From (6.18), Yt−1 = βYt−2+νt−1 and

ρYt−1 = ρβYt−2 + ρνt−1. Subtracting this last equation from (6.18) and re-arranging terms, one gets Yt = (β+ρ)Yt−1−ρβYt−2+t. Multiply both sides by Yt−1 and sum

T t=2 YtYt−1 =

(β + ρ)

T t=2 Y 2 t−1

− ρβ

T t=2 Yt−1Yt−2 +

T t=2 Yt−1t. Now divide by

T t=2 Y 2 t−1 and take probability limits. See Griliches (1961).

(b) For various values of |ρ| < 1 and |β| < 1, tabulate the asymptotic bias computed in part (a).

(c) Verify that plim(ρ − ρ) = −ρ(1 − β2)/(1 + ρβ) = −plim(βOLS

− β).

(d) Using part (c), show that plim d = 2(1− plim ρ) = 2[1 − βρ(β + ρ)

1 + βρ

] where d =

T t=2(νt −

νt−1)2/

T t=1 ν2 t denotes the Durbin-Watson statistic.

(e) Knowing the true disturbances, the Durbin-Watson statistic would be d∗ =

T t=2(νt −

νt−1)2/

T t=1 ν2t and its plim d∗ = 2(1 − ρ). Using part (d), show that plim (d − d∗) =

2ρ(1 − β2)

1 + βρ

= 2plim(βOLS

− β) obtained in part (a). See Nerlove and Wallis (1966). For various values of |ρ| < 1 and |β| < 1, tabulate d∗ and d and the asymptotic bias in part (d).