ECONOMET RICS P RELIM EXAM August 24, 2010 Department of Economics, Michigan State University

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1 ECONOMET RICS P RELIM EXAM August 24, 2010 Department of Economics, Michigan State University Instructions: Answer all four (4) questions. Be sure to show your work or provide su cient justi cation for your answers. Unless explicitly asked, do not worry about regularity conditions such as existence of moments or di erentiability in parameters. This exam is closed book. You may use a calculator and tables of relevant distributions are provided on additional pages of the exam. 1.(25 points) Let fx n g denote a sequence of random variables. X n is said to converge to X in mean square if lim n!1 E[(X n X) 2 ] = 0. X n is said to converge to X in probability if lim n!1 P (jx n Xj > ) = 0 for any > 0. It is well known that if X n converges to X in mean square then it follows that X n converges to X in probability. Let y 1 ; y 2 ; :::y n be a random sample from a population with mean and variance 2. Assume that 2 < 1. Let y denote the sample average of the y data. a) Derive formulas for E(y) and var(y): b) Is y an unbiased estimator of? Is y a consistent estimator of? Provide sketches of proofs for your answers. De ne an alternative estimator of as e = y + X n where X n is a discrete random variable de ned as X n = n with probability n 1 X n = 0 with probability 1 2n 1 X n = n with probability n 1 : Assume that X n is independent of the y data and assume that n > 1. c) Derive formulas for E(e) and var(e): d) Is e an unbiased estimator of? Is e a consistent estimator of? Provide sketches of proofs for your answers. e) Which is the better estimator of? Clearly explain what you mean by "better". 1

2 2. (25 points) Consider the following two regressions: (1) log(wage) = jc + 2 univ + 3 exper + u; (2) log(wage) = jc + 2 (jc + univ) + 3 exper + u; where wage is hourly wage, jc is the number of years spent in junior college (2 year college), univ is the number of years spent at university (4 year college) and exper is years of experience. 2 3 b 0 Let b = b b 7 denote the OLS estimator from regression (1). 2 5 b Let e = 6 4 e 0 e 1 e denote the OLS estimator from regression (2). e 3 a) Provide an economic interpretation of the parameters 1 and 2 in regression (2). b) Prove that e 1 = b b 1 2. c) Prove that the SSR (sum of squared residuals) and R 2 are the same for regressions (1) and (2). d) Prove that se( e 1 ) = se( b b 1 2 ). e) Prove that the t-statistic for testing H 0 : 1 2 in regression (1) is exactly equal to the t-statistic for testing H 0 : 1 0 in regression (2). Hint: Determine the 4 4 matrix of numbers, G; such that Z = XG where Z is the matrix of regressors from regression (2) (1; jc; jc + univ; exp er) and X is the matrix of regressors from regression (1) (1; jc; univ; exp er). 2

3 3. (25 points) You have used ordinary least squares (OLS) to estimate the housing demand function log H i = log P i + 2 log Y i + 3 D i + u i where H i is the i th family s units of housing consumption, P i is their unit price of housing, Y i is their income, and D i is a dummy variable equal to 1 if they live in an urban area and 0 if not. Based on 404 observations, your estimated equation is log H b i = 10:0 :7 log P i +:9 log Y i :1D i (1:0) (:3) (:3) (:1) with R 2 = :25. The numbers in parentheses are estimated standard errors, and Cbov( b 1 ; b 2 ) = :085. Assume that the ideal conditions pertain. a) Brie y interpret your estimates of 1, 2 and 3 : b) Test the null hypothesis 1 = 2 = 3 = 0 at the.01 signi cance level. State the relevant critical value for your test statistic. c) Using a.05 signi cance level, test the null hypothesis that 1 = 0 against the alternative 1 < 0. State the relevant critical value for your test statistic. d) Using a.10 signi cance level, test the null hypothesis 2 = 1 against the alternative 2 6= 1. Again state the relevant critical value. e) Increasing housing price and income in the same proportion has no e ect on housing demand if = 0. Using a.05 signi cance level, test the null hypothesis = 0 against the alternative = 0. State the relevant critical value. f) Suppose that the urban families in the sample typically face higher housing prices. Does the resulting correlation between D i and log P i bias least squares estimation of the s? Explain brie y. 3

4 4. (25 points) Consider the model Y t = X t + 1 X t X t X t 3 + ::: + u t where the X s are exogenous, 0 < < 1, and u t obeys the ideal conditions. This model represents a situation in which X has lagged e ects on Y, but the lagged e ects die out geometrically. a) At rst glance this model may seem intractable because it contains an in nite number of explanatory variables. This problem can be solved, though, by writing the equation for Y t 1, multiplying it through by, and subtracting the resulting equation from the equation for Y t. Show how this procedure re-expresses Y t as a regression function of only Y t 1 and X t. b) Is the regression function from part (a) estimated consistently by OLS? Explain brie y. c) Describe an alternative estimation procedure that consistently estimates the coe cients in the regression of Y t on Y t 1 and X t. d) Describe how you would use the resulting coe cient estimates from part (c) to produce consistent estimates of, 0, and 1. 4

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