A Simple Linear Trend Model with AR(1) Disturbances. This is based on Kr¨amer (1982).

(a) Consider the following simple linear trend model Yt = α + βt + ut where ut = ρut−1 + t with |ρ| < 1, t ∼ IID(0, σ2 ) and var(ut) = σ2 u = σ2 /(1 − ρ2). Our interest is focused on the estimates of the trend coefficient, β, and the estimators to be considered are OLS, CO (assuming that the true value of ρ is known), the first-difference estimator (FD), and the Generalized Least Squares (GLS), which is Best Linear Unbiased

(BLUE) in this case.

In the context of the simple linear trend model, the formulas for the variances of these estimators reduce to V (OLS) =12σ2{−6ρT+1[(T − 1)ρ − (T + 1)]2 − (T 3 − T )ρ4

+2(T 2 − 1)(T − 3)ρ3 + 12(T 2 + 1)ρ2 − 2(T 2 − 1)(T + 3)ρ

+(T 3 − T )}/(1 − ρ2)(1 − ρ)4(T 3 − T )2 V (CO) =12σ2(1 − ρ)2(T 3 − 3T 2 + 2T ), V (FD) =2σ2(1 − ρT−1)/(1 − ρ2)(T − 1)2, V (GLS) =12σ2/(T − 1)[(T − 3)(T − 2)ρ2 − 2(T − 3)(T − 1)ρ + T (T + 1)].

(b) Compute these variances and their relative efficiency with respect to the GLS estimator for T = 10, 20, 30, 40 and ρ between −0.9 and 0.9 in 0.1 increments.

(c) For a given T , show that the limit of var(OLS)/var(CO) is zero as ρ → 1. Prove that var(FD) and var(GLS) both tend in the limit to σ2 /(T − 1) < ∞ as ρ → 1. Conclude that var(GLS)/var(FD) tend to 1 as ρ → 1. Also, show that lim

ρ→1

[var(GLS)/var(OLS)] =

5(T 2 + T )/6(T 2 + 1) < 1 provided T >3.

(d) For a given ρ, show that var(FD) = O(T−2) whereas the variance of the remaining estimators is O(T−3). Conclude that lim T→∞

[var(FD)/var(CO)] = ∞ for any given ρ.