III Econometrics Tauchen Candidate Midterm Problems
Fall 2007
1. Let {Xn } be a sequence of random variables such that Xn = 0 with probablility 1 n an with probablility n
where 0 < n < 1 for all n, and an as n . 1-A. Show that n 0 Xn 0. 1-B. Does n 0 E(Xn ) 0?
2 1-C. Find a condition on {an } such that Xn 0.
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L
1-D. (Only counts for 2 points because it requires knowledge of the Borel-Cantelli as lemma.) Does n < Xn 0? n=1
2. Let yt be a 0, 1 binary random variable. Suppose {yt }n is iid, and P r(yt = 1) = 0 . t=1 Assume there exists a constant (0, 1 ) such that 0 (, 1 ). The data are 2 1 {yt }n . Let Ln () denote the log likelihood function, and Gn () = n Ln () is the t=1 mean log likelihood function. 2-A. Determine G() E [Gn ()] 2-B. Show directly that Gn () G() uniformly in probability over the interval (, 1 ). 2-C. Explain why you would expect n Gn (0 ) to obey a central limit theorem. 2-D. Check if information matrix equivalence holds.
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Econometrics III, Fall 2007 3. Suppose yt is a binary 0,1 variable such that P r(yt = 1|xt ) = (0 x) where for simplicity 0 is a scalar, xt is scalar, and (z) = ez 1 + ez
2
is the logit function. Suppose the observed sequence {yt , xt }n is iid and xt follows t=1 some distribution F (x). If it is simpler, you may assume a density f (x) = F (x) if you want to, but you do not need to. The estimation method is maximum likelihood with summands g(yt , xt , ), t = 1, 2, . . . , n being the log probability of observation t. 3-A. What is Gn () for this problem? Determine the function G() such that Gn () G(). 3-B. Verify information equality:
P
d E g(y, x, 0 ) d
2
= E
d2 g(y, x, 0 ) d2
where the expectation is with respect to the joint distribution of y, x.
Econometrics III, Fall 2007 4. Consider the linear regression model y t = xt 0 +
t
3
where yt is scalar, xt is K 1, 0 is the true value of , and assume that xt is iid, t is iid, E( t |xt ) = 0) although is not necessarily independent of xt , and all required moments exist. Consider the M-estimator based on maximizing Gn () = where gt () = (yt xt )2 . The M-estimator is obviously the OLS estimator = (X X)1 X y in the usual notation. 4-A. The direct estimate of the variance matrix I is 1 n gt () I= gt () n t=1 Determine the probability limit of I as n ; give the main ideas of the asymptotic arguments. 4-B. The direct estimate of the hessian matrix is
n 2 = 1 H gt () n t=1
1 n gt () n t=1
Determine the probability limit of H as n ; give the main ideas of the asymptotic arguments. 4-C. Given 4-A. and 4-B., what is the sandwich form of the asymptotic variance matrix? Find a condition on xt and t under which information matrix equivalence holds.
Econometrics III, Fall 2007
(2)
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5. Let rt denote the one-period interest rate for loans between t and t + 1. Let ft denote the forward rate of interest at time t for loans between t + 2 and t + 3. The value of (2) can ft be observed at time t because it can be calculated from the interest rates at time t for loans between t and t + 2 and loans between t and t + 3. Under one theory of interest rates the forward rate is an unbiased predictor of future interest rates ft
(2)
= E (rt+2 |It )
(2)
where the information set is generated by {rtj , ftj }, j 0 and possibly other variables. An economist aims to test this theory by considering the model rt+2 = + ft , where = 0, = 1 if the unbiased hypothesis holds. The error function is et () = rt+2 ft = ,
(2) (2) (2)
and the estimation method is GMM using as instruments xt the variables {rtj , ftj }L . j=0 5-A. Given a weighting matrix Wn write down the GMM estimation problem. 5-B. Are the GMM errors serially correlated or is there a need for a weighted covariance (also called HAC) matrix estimate? 5-C. Briey outline a two-step GMM procedure where in the rst step you estimate the optimal weighting matrix. 5-D. How could you use the GMM results to do a two-sided test of H0 : = 1 ?
Econometrics III, Fall 2007
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6. The exponential distribution has density ey , where y 0, and > 0 is the parameter. The mean is 1/ and the variance is 1/2 . Let {yt }n be iid exponential with t=1 true value 0 > 0. Consider the M-estimator dened by maximum likelihood. 6-A. What are I and H for this model? What is the maximum likelihood estimator? Explain how to use the asymptotic approximations to set a 95 percent condence interval for 0 . 6-B. Describe how you could use the Laplace Likelihood-MCMC approach to generate an estimator asymptotically equivalent to the maximum likelihood estimator. Would the calculations eectively compute the maximum likelihood estimator along the way? Explain. 6-C. Determine the various t...