1 x, is not equal to zero: ij = y/x, where 7 is

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1 Statistics and Economtrics August 1997 Instructlonq: Answer all flve 15) questions. Polnt totals are given in parentheses. The parts vlthln each questlon recelve equal velght. You may use a calculator. but only for computatlons -- not for storage or retrieval of information. In estlmated equatlons. standard errors appear in parentheses below estlmated coefflclents. Unless othervlse stated. carry out statlstical signlflcance tests at the 5% level. The statlstical tables you need are attached (15 points) Suppose that XI and X2 are independent random variables dram from dlfferent populatlons. Each has expected value p (irhere j~ can be any 2 real number), but Var(X1) = u: and Var(X,) a u,. A general class of estlmators of p 1s given by. p = alx1' + where al and a, are constants. m (1) Show that a, + a, = 1 1s needed for p to be unbiased for j ~. A 2 2 (11) Flnd the varlance of p as a function of dl, a2. ul. u2. (ill) Imposing the restriction a, a 1 - a, from part. (1). show that the value of a1 that minimize^ Var(p) 1s a; 1 ud(ul + u21. EXplaln why this makes lntuitlve sense (20 points) Assume that the followlng simple regression model (without an intercept) satisfies the Gauss-Markov assumptions: Yt 3 BXt + Ut, t11.2,. -/:,.--\, (all quantities are scalars). Treat the xt a11 derivations. (1) Write down the expression for the OLS of 8. What are the finite sample statistical properties of 87 (You do not need to prove these. (11) Coaclider a second - estlutor of B under the assumption that the -1 " - saraple a v o w of x, n 1 x, is not equal to zero: ij = y/x, where 7 is t-1 a average of y. Saw that is unblassd for 8. Find ~ar(ij). (iv) Do you prefer j or a as an estimator for B? Ekplain.

2 3. (20 points) Answer each of the follouing questions. Provide a brief discussion or calculation, as appropriate. (if Agree or Disagree: A model causes the OLS estimator fi to be biased." (ii) The follouing equation relates the per c_""'; "Multicollinearity in the multiple regression fund invested in stocks (pcrcstck) to income in dollar$, education, and age: -. worker's pension years of percstck = incow + ' 1.79 educ age (21.54) (.00017) (1.06) (0.60) If we redefine income to be measured 1n thousands of dollars and reestimate the equation by OLS, what would happen to the coefficients. standard errors. and, R-squar Agm or Disagree: *When the WS estimator is unbippd; so is the /--' -'- ffe.asiblp CLS estimator." Ll The folloulng equations were estimated using annual data on the U.S. economy for the years I -. I The variable inven is the level of inventories, CDP is gross domestic product, and r3 is the interest rate on three month treasury bills. The symbol "A" denotes first differences. Is there evidence of serial correlation in the errors of the inventory equation? -lain..'. "v)/:~~ree ju or ~1sagree:- "Heteroskedasticity in the multiple regression model cause,s F statistics to no longer have an F distribution."

3 "-I"------' 4. (30 points) Using data on 935 working men in the U.S., the following equations vere estimated by OLS. The variable wage is monthly salary. 3duc is years of education. exper is years of experience. married is a binary variable indicating marital status, and IQ is the Lntelllgence quotient. [In the population, the average IQ score is 109 and its standard deviation is log(wage1 = educ exper (0.13) (.007) ( exper married.. ( ) (.041) R2 =.154, 2' =.lsl. log(wage) = educ krried (0.09) (.006). (.042) ' R = -126,' R =.I J ' i%? $ -@ a - (9 log(wage) = 5.03 educ exper ( (.003) married + (.041). F~ -., Q,."d R* =.18S, R' =.I81.?>\: (il Interpret the coefficient on educ in equation (4.1). Is the coefficient on education statistically different from zero? (11) Using equation (4.11, at what level of experience does the next a year of experience actually reduce log(wage)? Comment on your finding. (lii) Test the following hypothesis: Holding myrj&illt~us and education fixed, years of experience has no effect on wage.,"../, ( iv) ~Gidlng education and experience levels fixed, what is the wage preaium for married nn? (You should state this as a percent.1 (v) Wh8t do you make of the fact that the return to education is about 2 percentage points lower in equation (4.3) than in (4.113 (vi) Apw or disagree with the following statement. and explain why: "While IQ is statistically significant in (4.31, it has a small practical LK~ on wage. -

4 5. (15 points) Suppose a population is represented by the simple regression model (without an intercept) = Bx + u. While we can collect data on x, ue do not directly observe y. Instead, we have a measure of y. call it y. Define the measurement error as e = y - Both u and e have zero means. (i) Write y in terms of x, B. u. and e. What is the error term in the model relating y to x? (ii) Let be the OLS cstimator from regressing y on x (without an intercept), using a random sample of site n. - Assuming that x and u are uncorrelated, find the probability limit (plim) of B. What factors determine a the asymptotic bias of d? (iii) In the equation from part (i). explain why a constant (that is. z, 1 for all observations i ) is a valid instrumental variable for x provided E(x) * 0. Wlat is the I V estimator in this. case?

5 ~ -- ~ ~ \ ' ! i I : I U m (.i , , ,259, ' -257, , ,255-.2M,254.a ,706, , , , , , W.6&.683 : a m 1, W l.a W ! W ' , Nw: lt=ana~aar~zt-aruun&rborhuit. For25~dbwdom(bL).#r >2.080)*O.MSrndPct < ort >2.060~rO.O8. So-: B i o e 'IhC3a k,&duiaau. Vd I. Editd by E. S. Ramon and H. 0. Hurby, 3rd dtion. 1M. Reprinted with tha pamirrion of tbr Bionutn)u lhrrtm. ~

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