上海财⼤研究⽣⾼级计量经济学期末试卷上海财经⼤学研究⽣试题命题纸（20 --20 学年第学期）课程名称：计量经济学（I ）（经济学院各专业）（B ）命题教师：周亚虹1. Given a random variables y, and random vectors x, z. The linear projection of y on x is defined as ：.1(|)(')(')L y x xE x x E x y ?=(a) Let , show that (|)u y L y x =?(')0E x u = (b) show that ((|,)|)(|)L L y x z x L y x =(c) show that ((|,)|) (|)L E y x z x L y x =2．Let y and z be random scalars, and x be a 1K ×random vector, where one element of x can be unity to allow for a nonzero intercept. Consider the population model(|,)E y x z x z βγ=+, 2(|,)Var y x z σ=.where interest lies in the vector1K ×β. To rule out trivialities, assume that 0γ≠. In addition,assume that x and z are orthogonal in the population: (')0E x z =.Consider two estimators of βbased on N independent and identically distributed observations: (1) ?β(obtained along with ?γ) is from the regression of y on x and z; (2) β% is from the regression of y on x. Both estimators are consistent for β. (along with the standard rank conditions)(a) Show that , without and additional assumptions(except those needed to apply the law of largenumbers and central limit theorem), ?))ββββ??%?)is always positive semidefinite (and usually positive definite). Therefore-from the standpoint of asymptotic analysis-it is always better to include variables in a regression model that are uncorrelated with the variables of interest. (b) Consider the special case where 2(K K z x µ=?,where()K K E x µ≡, and K x issymmetrically distributed: . Then 3()K K E x µ?0=K β is the partial effect of K x on (|)E y xevaluated at K K x µ=2)K . Is it better to estimate the average partial effect with or without(K z x µ=?1K × included as a regressor ?3. Suppose that b is the least squares coefficient vector in the regression of Y on X and that c is any other vector. Prove that the difference on the two sums of squared residuals is()Y Xc '()()'()()''(Y Xc Y Xb Y Xb c b X X c b =??)Prove that this difference is positive.4. Suppose the regression model is i i y x i αβε=++, ()(1/)exp(/)0i f ελελ=?>.This is rather a peculiar model in that all of the disturbances are assumed to be positive. Note that the disturbances have ()i E ελ=. Show that the least squares constant term is unbiased but the intercept is biased.5． Are rent rates influenced by the student population in a college town? Let rent be the average monthly rent paid on rentalunits in a college town in the United States. Let pop denote the total city population, avginc the average city income, and pctstu the student population as a percentage of the total population. One model to test for a relationship is01)+log(rent β23log(pop)+log(avginc)+pctstu+u =βββ(a) State the null hypothesis that size of the student body relative to the population has no ceterisparibus effect on monthly rents. State the alternative that there is an effect. (b) What signs do you expect for1β and 2β?(c) The equation estimated using 1990 data from RENTAL.RAW for 64 college towns islog(rent)0.043=+0.066log(pop)+0.507log(avginc)+0.0056pctstu (.844) (.039) (.081) (.0017) 264,0.458n R ==What is wrong with the statement: “A 10% increase in population is associated with about a 6.6%increase in rent”?(d) Test the hypothesis stated in part (a) at the 1% level.Solution 1: (a) By definition,1(')['((|))](')(')(')(')0E x u E x y L y x E x y E x x E x x E x y ?=?=?=(b) let (|,), by part (a), v y L y x z =?(')0E x v =. So1(|)(')(')0L v x xE x x E x v ?==(|)((|,)|)((|,)|)(|)((|,)|)L y x L L y x z v x L L y x z x L v x L L y x z x =+=+= (c) let , (|,)vy E y x z =?%(|,)0E v x z =%. Hence 1(|)(')(')0L v x xE x x E x v ?==%% (|)((|,)|)((|,)|)(|)((|,)|)L y x L E y x z vx L E y x z x L v x L E y x z x =+=+=%%Solution 2 :(a) let then (,)w x z =(|)E y w w δ=. Since 2(|)Var y w σ=,2)[(')]E w w δδσ1= where(',)'δβγ=K . Importantly, because , is block diagonal, with the upper (')0E x z =(')E w w K ×block gives21)[(')E x x ββσ]??= Next, we need tofind )ββ?%. It is helpful to write y x v β=+,where v z u γ=+ and . Because (|,)u y E y x z ≡?(')0E x z = and ,(')E x u 0=('E x v )0=. Further,222222(|)(|)(|)2(|)(|)E v x E z x E u x E zu x E z x 2,γγγ=++=σ+where we use and (|,)(|,)0E zu x z zE u x z ==22(|,)(|,)E u x z Var y x z σ==. Unlessis constant, the equation 2(|)E z x y x v β=+ generally violates the homoskedasticity assumption.So, without further assumption,12)[(')](')[(')]E x x E v x x E x x ββ1=%Now we can show ?))ββ%ββis always positive semidefinite by writing ?))βββ%β1?0 1212[(')](')[(')][(')]E x x E v x x E x x E x x σ??=?22(')E z x x γ=≥(b) If 2()K K z x µ=?, . Further, 2(|)(|,)()K K E y x E y x z x x βγµ==+?(|)2()K K K KE y x x x βγµ?=+?? Hence(|)|K K x K KE y x x µβ=?=?. If , using the conclusion of part (a), it is better to estimate the average partial effect with(')0E x z =2(K K z x )µ=? included as a regressor .Solution 3:Write c as . Then, the sum of squared residuals based on c is()b c b +?()'()()'()()''()2()''(Y Xc Y Xc Y Xb Y Xb c b X X c b c b X Y Xb ??=??+??+??)=i xBut, the third term is zero, as . Therefore,2()''()2()''0c b X Y Xb c b X e ??=?()'()'()''()Y Xc Y Xc e e c b X X c b =??The right hand side is necessarily positive. This confirms what we knew at the outset, least squares is least squares. Solution 4:We could write the regression as**()()i i i i y x αλβελαβε=+++?=++Then, we know the *()i E ελ=, and that it is independent of i x . Therefore, the second form of the model satisfies all of our assumptions for the classical regression. Ordinary least squares will give unbiased estimators of*α and β. As long as λ is not zero, the constant term will differ from α.Solution 5:(a) H 0:3β = 0. H 1:3β ≠ 0.(b) Other things equal, a larger population increases the demand for rental housing, which should increase rents. The demand for overall housing is higher when average income is higher, pushing up the cost of housing, including rental rates.(c) The coefficient on log(pop ) is an elasticity. A correct statement is that “a 10% increase in population increases rent by .066(10) = .66%.”(d) With df = 64 – 4 = 60, the 1% critical value for a two-tailed test is 2.660. The t statistic is about 3.29, which is well above the critical value. So3β is statistically different from zero at the 1% level.。
