Field Examination: Econometrics
Department of Economics
University of California, Berkeley
August, 2009
Answer THREE of the following four questions (one hour each):
1. Suppose that economic theory suggests that a latent dependent variable y
isatis…es a classical linear
model
y
i=x0
i0+"i;
but that you do not observe y
iover its entire range. Instead, you observe a random sample of size
nof yiand xi;where yiis a transformed version of y
i,
yii(y
i)
= 0 if y
i0;
=y
iif 0< y
iLi;
=Liif Li< y
iUi;and
=y
iif Ui< y
i:
That is, the latent variable y
iis observed unless it is less than zero or in the interval (Li; Ui);where
the threshold variables Liand Ui> Li>0are assumed known for all i.
(a) Assuming that "iis normally distributed with zero mean and unknown variance 2
0;and is inde-
pendent of xi;derive the form of the average log-likelihood function for the unknown parameters
of this problem and the form of the asymptotic distribution of the corresponding maximum like-
lihood estimator.
(b) Suppose that the parametric form of the error distribution is unknown. Propose a pn-consistent
estimator of 0, imposing a suitable stochastic restriction on the conditional distribution of "i
given xi;and without imposing a scale normalization on 0:If possible, give an expression for
the asymptotic distribution of your estimator.
(c) Now suppose that y
iis never observed, but only the range that it falls into is observed. More
speci…cally, the dependent variable yiis now de…ned as
yiti(y
i)
= 0 if y
i0;
= 1 if 0< y
iLi;
= 2 if Li< y
iUi;and
= 3 if Ui< y
i:
Describe an alternative consistent estimator of 0under a semiparametric restriction on the
conditional distribution of the errors given the regressors. Is a scale normalization on 0needed,
or are all the components of 0(including the scale) identi…able under your restriction?
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