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Unformatted text preview: Summer 2010 Examination EC309
Econometric Theory Suitable for all candidates Instructions to candidates
Time allowed: 3 hours This paper contains SEVEN questions. Answer ANY four. All questions will be given equal weight (25%). Weight for each subquestion is given in each question. Calculators are NOT allowed in this exam. © LSE 2010/130309 Page 1 of 5 1. Consider an IV regression model 7%: = 3325 + Ut
E (ztut) = 0, where (mg, 2;, at) are iid. (a) (5 points) Discuss the identiﬁability of the parameter [3. What conditions
are necessary? (b) (10 points) Suppose that the model is over—identiﬁed. Deﬁne the optimal
GMM estimator for E and obtain its asymptotic distribution. (c) (10 points) As a check for the validity of the model, one may perform the
over—identiﬁcation test. State the test statistic and derive its asymptotic
distribution. 2. Answer the following: (a) (10 points) State and prove the Jensen’s inequality. (b) (15 points) Let {Xn} be a sequence of random variables deﬁned on a com—
mon probability Space such that X _ 0 with probability 1 *1/71
“ 1 with probability l/n ' . State the deﬁnitions of almost sure convergence, convergence in probability, and mean squared error (L2) convergence and discuss the convergence of Xn for each sense. © LSE 2010 /EC309 Page 2 of 5 3. Consider a linear regression model yi = 552,5 + 5i
E (Eilﬂﬁi) : 0, where (yhrm) are iid. (a) (5 points) Further assume that E = 02. State the best linear unbi~
ased estimator with justiﬁcation. And obtain its asymptotic distribution. (b) (5 points) Now assume that 2:, is a scalar and E = a0 +oz1mf. Then, what is the best linear unbiased estimator? (c) (5 points) The estimator in (b) is known to rely on the unknown parameter
a = (a0,a1)’. Thus, we need to replace it with a consistent estimator. Propose such an estimator for a and justify your answer. (d) (10 points) Now we want to test if the model is indeed heteroskedastistic.
How would you perform the test? Be explicit about your hypothesis, test statistic, and critical values. 4. Consider the AR(1) model 2/1: = .11: + CHM—1 + Uta where {ut} is a sequence of iid variates with mean zero and variance 02 < 00. (a) (10 points) Discuss the weak and strict stationarity of {11:} providing a proper condition on the parameter values.
(b) (10 points) Obtain the limiting distribution of the'OLS estimator of (,u, a) . (c) (5 points) Propose a consistent estimator of 02 and verify its consistency. Then, illustrate how to construct a 95% conﬁdence interval for (it. © LSE 2010/EC309 Page 3 of 5 5. Consider a recursive SVAR model
BYt = Biyi—i + Ur, where B is a lower—triangular matrix whose diagonal elements are ones, Yt is a
p—dimensional strictly stationary time—series and at is iid with mean zero and a diagonal variance—covariance matrix A. (a) (10 points) Show that the recursive SVAR model is identiﬁed. (b) (10 points) Explain how to recover the parameters (B7B17A) from the
estimates of the reduced form model. (c) (5 points) Obtain the l—step forecast error variance of the p—th component
of Y}. 6. Assume that we observe an iid data (yi, 927;) _1 in where they are generated by i— a: r
yr : alt/6 + 5i lyi:1{y:>0}' Also assume that mi and Ej are independent for all 2' and j and s, is distributed 'syrnrnetrically around zero and continuously. (a) (5 points) Let the distribution function of 8% be denoted by F. Show that
Eelm) = F (93%) (b) (10 points) Consider the non—linear least squares (NLLS) to estimate )6.
Assuming that F is continuously differentiable and the NLLS estimator is
consistent, derive the asymptotic distribution of the estimator. (c) (10 points) Propose a consistent estimator of the asymptotic variance and establish its consistency. © LSE 2010/EC309 Page 4 of 5 7. Consider an autoregressive model yt = [I + mgr—1 + Uta Where {at} is an iid sequence with mean zero and variance 02 < 00. (a) (15 points) The statistical inference for a process with a unit root is differ—
ent from that for a stationary process. Describe the testing procedure for
the presence of a unit root in the process {gt}w Make sure that you state your null hypothesis, test statistic, and its asymptotic distribution. (b) (10 points) Now consider a vector of I (1) process, in which the presence
of cointegration become important. State the deﬁnition of cointegration.
Also erstplain the Granger representation of cointegration, which is a basis of
estimation of a cointegrated VAR model. And describe J ohansen’s testing procedure for the presence of cointegration under the null hypothesis of no—cointegration. © LSE 2010/E0309 Page 5 of 5  ...
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This note was uploaded on 09/12/2010 for the course GERAS 099876f taught by Professor Gtewewa during the Spring '09 term at Aberystwyth University.
 Spring '09
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