Econometrics I

Econometrics IProblem Set 51. (Wooldridge 4.5) In Section 4.5, we used as an example testing the rationality of assessmentsof housing prices. There, we used a log-log model in price and assess [see equation (4.47)].Here, we use a level-level formulation.(a) In the simple regression modelprice = β 0 + β 1 assess + uthe assessment is rational if β 1 = 1 and β 0 = 0. The estimated equation iŝprice = −14.47 + .976assess(16.27) (.049)n = 88, SSR = 165, 644, R 2 = 0.82.First, test the hypothesis that H 0 : β 0 = 0 against the two-sided alternative. Then,test H 0 : β 1 = 1 against the two-sided alternative. What do you conclude?(b) To test the joint hypothesis that β 0 = 0 and β 1 = 1, we need the SSR in the restrictedmodel. This amounts to computingn ∑i=1(price i − assess i ) 2 , where n = 88, since theresiduals in the restricted model are just price i − assess i . (No estimation is neededfor the restricted model because both parameters are specified under H 0 .) This turnsout to yield SSR =209,448.99. Carry out the F test for the joint hypothesis.(c) Now, test H 0 : β 2 = 0, β 3 = 0 and β 4 = 0 in the modelprice = β 0 + β 1 assess + β 2 lotsize + β 3 dqrft + β 4 bdrms + uThe R-squared from estimating this model using the same 88 houses is .829.1

(d) If the variance of price changes with assess, lotsize, sqrft, or bdrms, what can yousay about the F test from part (c)?2. (Wooldridge 4.8) In Problem 3.3, we estimated the equationŝleep = 3, 638.25 − .148totwrk − 11.13 − educ + 2.20age(112.28) (.017) (5.88) (1.45)n = 706, R 2 = 0.113.where we now report standard errors along with the estimates.(a) Is either educ or age individually significant at the 5% level against a two-sided alternative?Show your work.(b) Dropping educ and age from the equation givesŝleep = 3, 638.25 − .151totwrk(38.91) (.017)n = 706, R 2 = 0.103.Are educ and age jointly significant in the original equation at the 5% level? Justifyyour answer(c) Does including educ and age in the model greatly affect the estimated tradeoff betweensleeping and working?(d) Suppose that the sleep equation contains heteroskedasticity.What does this meanabout the tests computed in parts (a) and (b)?3. (Wooldridge 4.9) Are rent rates influenced by the student population in a college town?Let rent be the average monthly rent paid on rental units in a college town in the UnitedStates. Let pop denote the total city population, avginc the average city income, and pctstuthe student population as a percentage of the total population. One model to test for arelationship islog(rent) = β 0 + β 1 log(pop) + β 2 log(avginc) + β 3 pctstu + u2

(a) State the null hypothesis that size of the student body relative to the population hasno ceteris paribus effect on monthly rents. State the alternative that there is an effect.(b) What signs do you expect for β 1 and β 2 ?(c) The equation estimated using 1990 data from RENTAL.RAW for 64 college towns iŝ log(rent) = .043 + .066 log(pop) + .507 log(avginc) + .0056pctstu(.844) (.039) (.081) (.0017)n = 64, R 2 = 0.458.What is wrong with the statement:A 10% increase in population is associated withabout a 6.6% increase in rent?(d) Test the hypothesis stated in part (a) at the 1% level.4. (Wooldridge 5.1) In the simple regression model under MLR.1 through MLR.4, we arguedthat the slope estimator, ˜β 1 , is consistent for β 1 . Using ˜β 1 = ȳ− ˜β 1¯x, show that plim˜β 0 = β 0 .[You need to use the consistency of ˜β 1 and the law of large numbers, along with the factthat β 0 = E(y) − β 1 E(x 1 ).]5. (Wooldridge C4.3)6. (Wooldridge C4.6)3