Econometrics Field Examination
Department of Economics University of California, Berkeley
August, 2007
Answer any three of the four questions given below. Allocate approximately
equal time to each question. Clearly label your written answers with the corre-
sponding question number and part letter.
1. A simple version of the geometric distributed lag model is
yt=αxt+βyt−1+ut,
where ytand xtare observable scalar random variables and utis unob-
served. Suppose that utis MA(1),
ut=εt+θεt−1,
and that the exogenous variable xtis AR(1),
xt=γxt−1+ηt.
Here εtand ηtare mutually independent and i.i.d. normal random vari-
ables, with zero means and (nonnegative) variances σ2and τ2,respectively,
and it is assumed that |β|<1 and |θ|<1.
(a) Show that ythas a univariate ARMA(p,q ) representation for certain
values of pand q, and calculate the best linear prediction of yT+1
given the infinite past of ytand xt,i.e., derive an algebraic expression
for
ˆyT+1 =P[yT+1|(yT, xT),(yT−1, xT−1), ...]≡PT[yT+1].
Your expression should involve only the observable {(yt,xt)}and the
unknown parameters (α, β, γ, σ 2, τ2).
(b) Given a sample of observations for t= 1, ..., T from this process –
with ysfixed and known and εs= 0 for s≤0 – carefully explain how
the Gauss-Newton method can be used to obtain nonlinear least-
squares estimates of the parameter vector δ≡(α, β, θ)0.Be specific,
indicating how to proceed step-by-step. Also, derive a general ex-
pression for the asymptotic distribution of the resulting estimator
ˆ
δ.
1
