ECONOMETRICS FIELD EXAM May 10,2013 Department ofeconomics, Michigan State University

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1 ECONOMETRICS FIELD EXAM May 10,2013 Department ofeconomics, Michigan State University Instructions: Answer all four (4) questions. Be sure to show your work or provide sufficient justification for your answers. Unless explicitly asked, do not worry about regularity conditions such as existence of moments. You may use your notes and books. 1. (25 points) Consider, for a random draw i, the simple regression model Yi == a + {3x; + Uj E(uI) 0, COV(Xi' Ui) 0 Var(ui) O'~ > 0, Var(xi) O'i > 0 Rather than observe Xj you observe Zi == XI + VI, where Vi has zero mean, variance 0';, and is uncorre1ated with Xi and Ui. (a) Assuming random sampling, let aand ft denote the OLS estimators from the regression Yi on 1, Zi using a sample of size n. How come ft is inconsistent for {3? (b) Derive an expression for the probability limit of ft. (Do not just write down an expression. You need to derive it.) (c) Suppose you are interested only in testing Ho : {3 O. How would you proceed? (d) Suppose that you somehow know 0';. Find a consistent estimator of{3. (e) Suppose you can observe a second measure ofxi, Wj = Xi + rl, where ri is uncorrelated with Xi, Ui, and Vi. Given a random sample on (Yi,Zi, Wi), propose a consistent estimator of{3. 1

2 2. (25 points) Consider a panel data model with an endogenous explanatory variable'yit2, and additive unobserved heterogeneity: Yill = alyil2 + ZitlOI + en + UUI E(UiIllzil,..., ZiT) 0 where Zit == (Ziti,Zit2) and Zil2 is a time-varying scalar. Assume that this is a balanced panel, and that you have large N and small T. (a) (5 points) Write down a linear reduced form for Yit2 that depends on Zit, with an additive heterogeneity term. What is the rank condition, stated in terms ofthe parameters of the reduced form, for estimating a I and 01 by fixed effects 2SLS? How would you test whether this condition fails? (b) Suppose that Yil2 is binary and where Wit (Zit,Zi). Is the FE2SLS estimator that uses Zit as IVs for (Yit2,Zitl) inconsistent because it uses a linear reduced form for Yil2? Explain. Ifthe FE2SLS estimator is consistent, explain how to test the null hypothesis that Yit2 is exogenous. (c) Assuming the setup from part (b), suppose you think that 112 0, and so you use pooled probit ofyit2 on 1, Wit to estimate V'2 and ;2' Let <1>it2 denote the probit fitted values. In a second step, you estimate by fixed effects. Discuss the asymptotic properties ofthis procedure. Can you think of a better way to use the <1>it2 in a second step? Explain. 2

3 (d) Now write ail at + ril with at = E(ail), and assume the model is Also assume that Cil = '1'1 + Zi'il + gil Yil2 ::;: 1['1'2 + Wit'i 2 + eu2 > 0] and define Viti = gil + Uitl. Assume that (ril, Vi/J,eU2) is independent ofzi, E(ril\eit2) = Pleit2, E(VitIieit2) = rlejt2, and eu2 - Normal(O, O. Find E(YildYit2,Zi) and use it to obtain a control function method for estimating a t and 0 I. (e) In the estimation from part (d), describe in detail how you would obtain a valid confidence interval for a 1. How would you test the null hypothesis that Y2 is exogenous? 3

4 3. (25 points) Consider the following model YI = {3XI + 81 XI = Ilt + (I III = IlH + ~/, where Yt. xt,{3 and el are all scalars, et is iid(o,crd, (I is iid(o,cr~), and ~I is iid(o,crv; are all mutually independent. (a) Show that where al is iid(o,cr~) and 8 is obtained as a solution ofthe equation (q + 2) , where q = (cr~/cr~). What is the interpretation of q? (b) Show thatyi follows an A RIMA process. (c) What are the properties of '/J, the OLS estimator of {3 from estimating the equation YI {3XI + 81? (d) What are the properties of jj, the OLS estimator of {3 from estimating the equation (e) If you assume Gaussianity ofthe three white noise processes above, what estimator would be the most desirable? 4

5 4. (25 points) Provide an answer to each ofthe following questions. (a) Consider a simple regression, obtained from n obse,rvations: YI = a+jjxl+e" where {e/ : t = 1,...,n} are the OLS residuals. Suppose now you estimate, by OLS, the equation YI = 0 + rxi + q>et + errort. What are the values of8, y, and ijj? Justify your answer. (b) Suppose a trend~stationary time series {V, : t = 0, 1,2,... } is generated as Yt = a + [3t + Ut, where {Ut : t = 0,1,2... } is a white noise process with mean zero and variance (1~ > O. Let jj be the sample average of n first differences: n jj = n- 1 L: AYh t=l where AYt YI - Yt-l. Is jj consistent for [3? Is it In ~asymptotically normal? Justify your answers. (c) Consider a linear model for 1(0) time series data, Yt = Xt(3 + U" E(Ut!x,) 0, t = 1,...,n Suppose you think: {Ut} follows a stable AR(1) model with a known value ofo < p < 1. What is the appropriate rank: condition for the GLS estimator to be uniquely defined? Assuming this condition, is the GLS estimator generally unbiased for (3 (when you condition on X, the matrix of all observations on {Xt})? Explain. 5

6 (d) For a balanced panel with Ttime periods and a binary response YiI, suppose P(Yu Ilx;) = 1 - exp[ - exp('" + Xit~ + ii~)], where Xi =: r- 2:: I Xir. 1 Given a cross section of size N, write down the pooled log-likelihood function. When would this be the same as the joint log-likelihood function (conditional on the Xi)? (e) For a response variable Y :::: 0, consider the model E(Ylx) = exp[x~ + Y(X~)2], where X is a 1 x K vector with unity as its first element and ~ is K x I. IfXK is a continuous variable, show how to consistently estimate the average partial effect ofxk on E(Ylx), assuming that i3 and r have been obtained by nonlinear least squares using a sample of size N. 6

ECONOMETRICS FIELD EXAM Michigan State University August 21, 2009

ECONOMETRICS FIELD EXAM Michigan State University August 21, 2009 Instructions: Answer all four (4) questions. Point totals for each question are given in parentheses; there are 100 points possible. Within