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Field Examination: Econometrics
Department of Economics
University of California, Berkeley
January, 2008
Instructions: Answer THREE of the following four questions (one hour each).
- Consider estimation of a parametric model that you can easily simulate but you cannot easily compute
the likelihood or moment functions. Let y ( 0
; X; V ) be the data generating function of a dependent
variable Y given the population parameter 0
2 R and conditional on a covariate X. V is a random
variable with a known and easily computable p.d.f. f V
(a) Describe a method for simulating V; stating any additional requirements.
(b) Given an i.i.d. sample f(X i
; Y
i )g
n
i=
and the OLS linear Ötted coe¢ cients
^
n
P
n
i=
X
i
X
n
Y
i
P
n
i=
X
i
X
n
2
; ^
n
Y
n
^
n
X
n
where
X
n
P
n
i=
X
i
=n and
Y
n
P
n
i=
Y
i
=n; consider an estimator
^
n
that solves
^
n
P
n
i=
X
i
X
n
y
^
n
; X
i
; v i
P
n
i=
X
i
X
n
2
where fv i g
n
i=
is a set of simulations of V.
i. State su¢ cient conditions for
^
n
to be a consistent estimator and explain the role of each
condition.
ii. State su¢ cient conditions for
p
n
^
n
0
to be asymptotically normal.
iii. How could you use ^ n
to improve the e¢ ciency of the estimator?
(c) Suggest an alternative to using the OLS linear Ötted coe¢ cients and motivate your suggestion.
- Consider an econometric model
y = g (x
; ) + ";
where x
is an (unobserved) ideal explanatory variable and " is a mean-zero, normally distributed
disturbance that is independent of x
: Suppose one observes
x = x
where is a normally distributed, mean-zero measurement error that is independent of x
: Suppose
one also observes an instrument w that is normally distributed, conditioned on and x
; and is
conditionally independent of with a conditional mean that is a linear increasing function of x
:
Discuss methods for estimating when
(a) g () is linear in x
;
(b) g () is linear in ;
(c) g () is a non-linear function of x
and that is fully determined once is speciÖed; and
(d) g () is a non-linear function that is not fully speciÖed by (the semi-parametric case).
- Consider the ARMA(1; 1) process
y 1
1
y t = y t 1
t
t 1 ; (t = 2; : : : ; n)
where the " t are i:i:d: N (0; 1) random variables independent of the N
2
random variable :
(a) For what value of!
2
is the process fy t
g stationary?
(b) Explain briezy how the Kalman Ölter can be exploited to simplify the calculation of the exact
likelihood function. (Do not derive the Ölter; just explain how its output can be used.)
(c) Explain brieáy how approximate maximum likelihood estimators of and can be obtained by
nonlinear least squares. What approximations are employed?
