Concordia ECON 421 Final Exam -Consider the simple linear

ECONOMETRICS IECON 421/521Final Exam, Fall 2014December 5, 2014 – 1400 – 1700Room: H507Instructor: Prosper Dovonon NB: Read the questions carefully. Write clear and complete answers. Show your work and get credit. Partialanswers get partial credit. No programmable calculator, No cell phone, no notes.Question 1 [30 points/110] (Note that (i) and (ii) below are independent questions.)(i) Consider the?simple linear regression without intercept (y = ?x + u) and show that the OLS estimator ?ˆnxi yiof ? is ?ˆ = ?i=1n2 .i=1 xi sh isar stued dvi y reaC soou urcrs eeH wer aso…. (ii) Suppose that the population model determining y is y = ?0 + ?1 x1 + ?2 x2 + ?3 x3 + u, and this model satisfies the Gauss-Markov assumptions. However, we estimate the model that omits x3 .Let ?˜0 , ?˜1 , ?˜2 be the OLS estimators from the regression of y on x1 and x2 .Show that: n? rˆi1 xi3i=1˜E(?1 |X) = ?1 + ?3 ?,n2rˆi1 (1) i=1 where rˆi1 are the OLS residuals from the regression of x1 on x2 including an intercept and X includes x1 ,x2 and x3 . [Hint: Recall that ?˜1 is equal to the OLS estimator of the regression of yi on rˆi1 without anintercept. Then plug the relevant expression of yi in the obtained expression for ?˜1 .](iii) Derive the conditional bias (E(?˜1 |X) ? ?1 ) of ?˜1 and comment on its expression?Question 2 [30 points/110]Assume that we would like to test the rationality of assessments of housing prices.(i) In the simple regression model price = ?0 + ?1 assess + u, Th the assessment is rational if ?1 = 1 and ?0 = 0. The estimated equation is (the estimated standard errorsare in parenthesis):[ = ?14.47 +.976 assessprice(16.27) (.049)n = 88, SSR = 165, 644.51, R2 = .820.First, test the hypothesis that H0 : ?0 = 0 against a two-sided alternative. Then test H0 : ?1 = 1 againsta two-sided alternative. What do you conclude? (ii) Consider the joint hypothesisthat ?0 = 0 and ?1 = 1. Under?this hypothesis, explain why the SSR in the?nnrestricted model is i=1 (pricei ? assessi )2 . Consider that i=1 (pricei ? assessi )2 = 209, 448.99 in thesample and carry out an F -test of the joint hypothesis.(iii) Now test H0 : ?2 = 0, ?3 = 0, and ?4 = 0 in the modelprice = ?0 + ?1 assess + ?2 lotsize + ?3 sqrf t + ?4 bdrms + u.The R2 from estimating this model using the same 88 houses is .829.(iv) If the variance of price changes with assess, lotsize, or bdrms, what can you say about the F -test frompart (iii)?Question 3 [30 points/110]Consider a simple time series model where the explanatory variable has a classical measurement error:ytxt = ?0 + ?1 x?t + ut= x?t + et , (2) where ut has zero mean and is uncorrelated with x?t and et . We observe yt and xt only. Assume that et haszero mean and is uncorrelated with x?t and that x?t also has a zero mean.(i) Write x?t = xt ? et and plug into (2). Show that the error in the new equation, say, vt is negativelycorrelated with xt if ?1 > 0. What does this imply about the OLS estimator of ?1 from the regression ofyt on xt ? sh isar stued dvi y reaC soou urcrs eeH wer aso…. (ii) Assume that Cov(xt , ut ) = 0 and derive the probability limit of the OLS estimator ?˜1 of ?1 from theregression of yt on xt obtained in (1). Comment this result.(iii) In addition to the previous assumptions, assume that ut and et are uncorrelated with all past values ofx?t and et ; in particular, with x?t?1 and et?1 . Show that E(xt?1 vt ) = 0, where vt is the error term in themodel from part (i).(iv) Are xt and xt?1 likely to be correlated? Explain. (v) What do parts (ii) and (iii) suggest as a useful strategy for consistently estimating ?0 and ?1 (Propose aconsistent estimator and give the steps for its derivation derivation.)Question 4 [20 points/110]Consider a time series yt with t = 1, . . . , T and the regression: yt = ?0 + ?1 yt?1 + ut . Assume that ut = ?ut?1 + et with 0 < |?| < 1 and et ? i.i.d. and that E (et |ut?1 , ut?2 , . . .) = E (et |yt?1 , yt?2 , . . .) = 0. (i) Assume that ?1 ? ?= 1 and derive Cov(yt , ut ) in terms of ?u2 ? V ar(ut ), ?1 and ?. [Hint: In the process ofyour derivation, you shall consider that Cov(yt , ut ) does not depend on t]. Th (ii) Derive the probability limit of ?ˆ1 , the OLS estimator of ?1 , in terms of ?1 , ?, ?u2 and ?y2 ? V ar(yt ). Is ?ˆ1consistent? Why is this expected?