ECONOMETRICS FIELD EXAM Michigan State University August 21, 2009

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1 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 a question, each part receives equal weight. You may use a calculator, but only for computations - not for storage or retrieval of information. You must show all your work to get credit for your solutions. Be sure to show your work or provide sufficient justification for your answers. You may use your notes and books. 1. (25 points) Consider the simple regression without intercept where the scalar regressor is non-stochastic and is strictly positive for all i. If this model satisfies all the ideal conditions (including normality of ui), the ordinary least squares estimator, is the best unbiased estimator of P. A. Suppose we treat the regressors as nonrandom and assume the ideal conditions hold except that 2 ~i N N(0, u xi). i. (5 points) Given the heteroskedasticity of ui, is POLS still unbiased? Is it still best unbiased? ii. (5 points) Show that the best unbiased estimator of P is and derive its variance.

2 B. Now suppose the ideal conditions hold except that, instead of being normally distributed, ui follows the discrete Poisson distribution with probability function with,b > 0. It can be shown then that e-p~i (,Bxi)Yi f (yi) = for yi = O,1,2,... pi! It therefore is still appropriate to say that with E(u~) = 0 except that now ui is non-normal with var(ui) =,Bxi. i. (5 points) Derive the GLS estimator of,b. ii. (5 points) In general, in a regression model in which ui is non-normal and het- eroskedastic (but the ideal conditions otherwise are satisfied), is the GLS estimator of,b necessarily best unbiased? Is it necessarily asymptotically efficient? Is it necessarily best linear unbiased? iii. (5 points) Returning to the specifics of this problem, derive the maximum likelihood estimator of,b in the Poisson case. (For full credit, you should check the second order A A A condition.) Compare,BMLE to,bgls. IS PGLS asymptotically efficient in this case?

3 2. (25 points) Let yt be a covariance stationary univariate time series with population spectrum given by 1 sy(w) = - ( COS(W)) ( COS(W)). 27T (a) (5 points) Determine the frequencies of the cycles of yt that contribute the most to the variation of yt. Assuming that yt is observed yearly, give a real world interpretation to these frequencies/cycles. (b) (5 points) Derive the autocovariance function of yt. Does the autocovariance function depend on t? (c) (5 points) Compute the impulse response function of yt for 10 periods assuming a shock of magnitude 1 occurs at time 0. (d) (5 points) Suppose we observe yt for t = 1,2,...,183 and we compute the sample average of yt to be jj = Let p = E(yt). Test the null hypothesis, Ho : p = 0, against the alternative hypothesis, HI : p > 0, at the 3% significance level. Carefully describe your test statistic and fully justify your choice of critical value. Be sure to explain why any asymptotic approximations you use are valid. (e) (5 points) Define the univariate time series process, xt, as where L is the lag operator and h(l) is a filter. Determine the form of h(l) such that xt is a white noise process. Determine the frequencies of the cycles of xt that contribute the most to the variation of xt.

4 3. (25 points) Let yit be a fractional response variable taking on values strictly between zero and one, that is, 0 < yit < 1 for all i and t. Suppose that, for a random draw i (and for a balanced panel of length T), where ci is unobserved heterogeneity and the {uit : t = 1,..., T) are unobserved shocks. Assume that, for each t, uit has zero mean and is independent of (xil,-2,..., X~T, ci) Assume that all elements of xit have some variation over time. Assume that T is small relative to N, and so all asymptotic statements are with fked T and N growing. (a) (5 points) Show that the log-odds, wit = log[yit/(l - yit)], can be written as a linear function of xit, ci, and uit. How would you estimate P using this equation? (b) (5 points) Let ti = jji - zip, where is the estimate from part (a) and the overbar denotes averages across time. Consider the following predictor of yit: 6. zt exp(xd + ti) exp(xitp + ti) Do you think this is a "good" predictor of yit? Provide a careful explanation. (c) (5 points) If you knew ci rather than having to estimate it, would the predictor in (3.2) with ci in place of ti be a "good" predictor? Explain. (d) (5 points) Now assume that ci = $J + tt + ai where ai has zero mean and is independent of (xil, xi2,..., xit). HOW would you estimate $J and t? (e) (5 points) Under the same assumptions as in part (d) (and all previous assumptions), suggest a consistent estimator of E(yit lxit = %O, = fo) for given values (%O, f O). (Hint: This will entail averaging out some residuals.)

5 4. (25 points) Provide an answer for each of the following five questions. You must support any "agree/disagree7' answer with a careful explanation. (a) (5 points) Suppose that the causal effect of xi on yi can be expressed by the regression model Yi li PI + P2xi where E~ N N (0,~:). Suppose you run a controlled experiment in which xf, your intended value of the explanatory variable, is independent of ~i. Unfortunately, calibration error in your laboratory instruments causes the actual value, xi, to deviate from the intended value, xf, according to xi = xf + vi, where vi N N(0,o;) is independent of the intended value. Because it is impossible for you to observe the actual value xi, you apply OLS to the regression of yi on the intended value xf instead. Does this yield consistent estimation of PI and P2? Justify your answer. (b) (5 points) Do you agree or disagree with the following claim? "The impulse response function of a VAR is only well defined when the VAR process is covariance stationary because of the direct link between the autocovariance function and the parameters of the moving average representation. This is why it is important to conduct unit root tests on the individual series before estimating an impulse response function for a VAR." (c) (5 points) Consider the simple linear trend model given by Ei, where ct is a mean zero white noise process with variance 0: < co. Assume that 141 < 1 and 101 < 1. Let p be the OLS estimate of 8. Derive the asymptotic distribution of the appropriately centered and scaled p. [Note: you are NOT being asked to prove that p is consistent]. Is this asymptotic distribution of p Gaussian (normal)? Why or why not? (d) (5 points) For a response variable y with E(y2) < co and a 1 x K vector x with E(xj2) < co, the population R-squared is defined as pelx = Var(xp)/Var(y), where /3 is the vector in the linear projection L(yJ 1, x) = a + xp. Partition x into xl and x2 where xl is a 1 x Kl vector and xg is a 1 x K2 vector and assume these are uncorrelated. Show that pix = pelx, + PE~X,

6 (e) (5 points) For hourly workers, let yi be the number of hours during a week spent working for a wage for a random draw i from the working-age population. Does a two-limit Tobit model with corners at zero and 168 seem reasonable as a model for yi? Explain.

ECONOMETFUCS FIELD EXAM Michigan State University May 11, 2007

ECONOMETFUCS FIELD EXAM Michigan State University May 11, 2007 Instructions: Answer all four (4) questions. Point totals for each question are given in parenthesis; there are 100 points possible. Within