Midterm 06
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yerosanabraham83Midterm 06
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yerosanabraham83Econometrics II Mid-Term Exam 2006
October 23, 2009
INSTRUCTIONS: You should attempt all of the questions. You have 1hour 40 minutes for Part A of the
exam and there are 100 points so you should apportion your time accordingly. You may not use any notes,
your computer, refer to any books, or confer with anyone while the exam is in progress. Please insure that
you explain all the steps in your answer legibly.
Part A
1. Show that the variance of the random variable X1 E[X1 jX2 ] cannot be greater than the variance
of X1 , and that the two variances are equal if X1 and X2 are independent. You may assume that
E[X1 ] = 0: [10 points]
2. Consider two linear regressions, one restricted and the other unrestricted:
y = X + u;
y = X + Z + u:
Show that, in the case of mutually orthogonal regressors, with X 0 Z = 0; the estimates of from the
two regressions are identical. [10 points]
3. Consider the linear regression model
yi = 1 + 2 xi2 + 3 xi3 + ui :
Explain how you could estimate this model subject to the restriction that 2 + 3 = 1 by running a
regression that imposes the restriction. Also, explain how you could estimate the unrestricted model in
such a way that the value of one of the coe¢ cients would be zero if the restriction held exactly for your
data. [10 points]
4. Consider the linear regression model
2
y = X + u; u N (0; I); E[ujX] = 0;
where X is an n k matrix. If 0 denotes the true value of ; how is the quantity (y 0 Mx y) = 20
distributed? Explain your reasoning. Use this result to derive a test of the null hypothesis that = 0 :
Is this a one-tailed test or a two-tailed test? [15 points]
5. Resolve the following paradox: the asymptotic approximation to the true distribution of b and s2 implies
that we treat s2 as a constant when we draw inferences about but treat s2 as a normally distributed
random variable when we draw inferences about 2 : (NB I am using Hayashi’s notation.) [15 points]
1
6. Consider the time series regression model
yt = xt + ut
where xt is stationary and ergodic and
ut = ut 1 + vt ; j j<1
with vt white noise. Explain how and can be estimated by a least squares regression under the null
hypothesis that = 1: Describe how the parameters can be estimated from another regression without
imposing the null hypothesis. How can the estimates from this second approach be used to the test the
restriction = 1:[20 points]
7. Let the matrix of stationary and ergodic predetermined variables X be partitioned as [X1 ; X2 ] where
X1 and X2 are n d matrices with
11 12
E[Xi Xi0 ] xx = 0 :
21 22
We are interested in estimating
yi = x0i;1 + "i
where x0i;1 is the i0 th. row of X1 . We can establish that the variables in X2 are predetermined but
not those in X1 : We also know that gi xi;2 "i is a martingale di¤ erence sequence with E[gi gi0 ] = S:
Derive the asymptotic distribution of the estimator
~ = [X 0 X1 ] 1
X20 Y:
2
Suppose that 12 = 0:What will the estimates ~ tell us about the values of ? [20 points]