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Field Examination: Econometrics
Dept. of Economics, UC Berkeley
August, 2011
Instructions. You have 180 minutes to answer THREE out of the following four questions. Please make your answers elegant, that is, clear, concise, and, above all, correct. Good luck!
Question 1
Suppose {(yt, xt)′^ : 1 ≤ t ≤ T }^ is an observed time series generated by the cointegrated system
yt = βxt + ut,
where
ut ∆xt
(^) ∼ i.i.d. N
with initial condition x 0 = 0. A researcher wants to estimate the scalar parameter β. As estimators of β, consider
βˆ =
∑T
∑tT=2^ ∆xt∆yt t=2 (∆xt)^2
and β˜ =
∑T
∑tT=1^ ∆xtyt t=1 ∆xtxt
- Find a Vˆ = Vˆ (x 1 ,... , xT ) , a function (only) of (x 1 ,... , xT ) , such that
βˆ − β
Vˆ ∼
N (0, 1).
- Characterize the limiting distribution (after appropriate centering and rescaling) of β.ˆ
- Find a V˜ = V˜ (x 1 ,... , xT ) , a function (only) of (x 1 ,... , xT ) , such that
β − β
V ∼
N (0, 1).
- Is β˜ asymptotically efficient relative to βˆ in the sense that limT →∞ Pr
V < Vˆ
Question 2
Let (Wt)t be an iid sequence of real valued random variables drawn from P. Let θ 0 ∈ Θ ⊆ R be the (true) parameter of a model, and let τ 0 ∈ T be a “nuisance” parameter with (T, || · ||) being a normed space. Both are identified by
E[m(W, θ 0 , τ 0 )] = 0,
where m : R × R × T → R is a known function. E.g., m is the score of a ML model, or a moment. Suppose θ 0 is in the interior of Θ; and, for now, lets consider the case where τ 0 is known to the econometrician, and let m(·, ·) ≡ m(·, ·, τ 0 ). Consider the following estimator: θˆ where
m¯T (θˆ) ≡ T −^1
∑^ T
t=
m(Wt, θˆ) = 0.
Assume θˆ is consistent, i.e., For all ǫ > 0, limT →∞ P
|θˆ − θ 0 | > ǫ
- Suppose the following conditions hold: 1 (a) m(W, ·) is differentiable for almost all W. (b)
T m¯T (θ 0 ) ⇒ N (0, V ), with V > 0. (c) sup|θ−θ 0 |<δ
∣ d^ m¯ dθT^ ( θ)− d[E[m dθ(W,θ^0 )]]
∣ = oP (1) with Γ ≡ d[E[m dθ(W,θ 0 )]]> 0. Show that √ T (θˆ − θ 0 ) ⇒ N (0, Γ−^1 V Γ−^1 ).
- Suppose m(W, ·) is non-smooth, but E[m(W, ·)] is. That is, suppose the following new condi- tions hold: (a) E[m(W, ·)] is differentiable for almost all W. (b) νT (θ 0 ) ⇒ N (0, V ), with V > 0. (c) sup|θ−θ 0 |<δ
∣ dE[ ¯m dθT (θ)]−^ d[E[m dθ(W,θ^0 )]]
∣ =^ oP (1) with Γ^ ≡^ d[E[m dθ(W,θ 0 )]]>^ 0. (d) {νT (θ) : θ ∈ Θ} is stochastic equicontinuous; i.e., sup|θ−θ 0 |<δ |νT (θ) − νT (θ 0 )| = oP (1). Where νT (·) ≡ √^1 T^ ∑Tt=1{m(Wt, ·) − E[m(W, ·)]}. (^1) If you think you need more regularity conditions to show the results; feel free to add them. However, adding unnecessary regularity conditions will be penalized.
Question 3
Consider the following panel data relationship for a sample of N individuals over T = 2 time periods:
E[yit|xi 1 , xi 2 ] ≡ μt(xi 1 , xi 2 ) = α(xi 1 , xi 2 ) + δ · 1 {t = 2} + g(xit)
for i = 1, ..., N and T = 1, 2. Here yit is a scalar dependent variable, xit is a scalar explanatory variable, α(xi 1 , xi 2 ) ≡ αi is an individual-specific ”fixed effect,” δ is a constant ”time effect” for period t = 2, and g(xit), the object of interest,embodies the contemporaneous effect of xit on the conditional mean of yit. Assume the functions α(·) and g(·) are very smooth (i.e., are continuously differentiable of arbi- trarily high order), that αi and g(xit) have all moments finite, that xi 1 and xit are jointly continuously distributed with positive (and very smooth) density on R^2 , and that the data are i.i.d. over the index i.
- Show that the functions α(·) and g(·) are not identified by this moment condition without a normalization or similar restrictions.
- Imposing the normalization g(0) = 0, show that the function g(·) and the time effect δ are identified under the moment restriction.
- For what (nonrandom) values of x 1 and x 2 is the function α(x 1 , x 2 ) identified under the moment condition and the normalization?
- Given a consistent nonparametric estimator ˆμt(xi 1 , xi 2 ) of the conditional expectation function μt(·), construct consistent estimators of δ and g(x). If the rate of convergence of ˆμt is the usual N 1 /^3 rate for two-dimensional nonparametric estimation problems, i.e.,
N 1 /^3 (ˆμ(x 1 , x 2 ) − μ(x 1 , x 2 )) = Op(1),
what is the rate of convergence of the constructed δˆ and ˆg(x)?
- Now suppose that xi 1 and xi 2 have identical marginal distributions, and that the alternative normalization E[g(xit)] = 0 is imposed (allowing g(0) 6 = 0). Show that a
N -consistent estimator of ˆδ exists under this normalization and the moment restriction. Also, construct a kernel estimator of g(x) that converges at the one-dimensional rate N 2 /^5 , and derive its asymptotic distribution. (You do not need to verify any regularity conditions for your estimator, but should correctly cite existing results on kernel regression.)
Question 4
Suppose {yt : 1 ≤ t ≤ T } is an observed time series generated by the model
yt = ρyt− 1 + εt,
where y 0 = 0, εt ∼ i.i.d. N (0, 1) , and ρ is an unknown parameter of interest. Consider the (unit root) testing problem
H 0 : ρ = 1 vs. H 1 : ρ < 1.
- Find the log likelihood function LT (ρ) and show that a sufficient statistic is given by
(ST , HT ) =
T
∑^ T
t=
yt− 1 ∆yt, (^) T^1
∑^ T
t=
y^2 t− 1
- Express the likelihood ratio statistic LRT = 2 [max¯ρ≤ 1 LT (¯ρ) − LT (1)] as a function of (ST , HT ).
- Suppose ρ = 1. Find the asymptotic distribution of LRT.
- Show that LRT can be written as a function of the t-type statistic
τT = ρˆ^ −^1 1 /
√∑T
t=1 y^2 t−^1
, ρˆ =
∑T
∑t=1T^ yt−^1 yt t=1 y^ t^2 −^1
Is the converse true?
