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
2020疫情作文素材(|,)E y x z x z βγ=+, 2(|,)Var y x z σ=.
where interest lies in the vector
1K ×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 β
based on N independent and identically distributed observations: (1) ˆβ
(obtained along with ˆγ) 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 large
numbers and central limit theorem), ˆ))β
βββ−−%−)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 is
symmetrically distributed: . Then 3
()K K E x μ−0=K β is the partial effect of K x on (|)E y x
evaluated 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.
冰箱冷冻室结冰的原因和解决方法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 rental units 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 is
01)+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 ceteris
paribus effect on monthly rents. State the alternative that there is an effect. (b) What signs do you expect for
1β and 2β?
(c) The equation estimated using 1990 data from RENTAL.RAW for 64 college towns is
ˆlog(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 =. So
1(|)(')(')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 , (|,)v
y 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 v
x L E y x z x L v x L E y x z x =+=+=%%
赞扬老师的句子Solution 2 :
五一快乐文案(a) let then (,)w x z =(|)E y w w δ=. Since 2
(|)Var y w σ=,
2
ˆ)[(')]E w w δ
δσ1
−−= where
ˆˆˆ(',)'δβ
财务会计与管理会计γ=K . Importantly, because , is block diagonal, with the upper (')0E x z =(')E w w K ×block gives
2
1
ˆ)[(')E x x β
βσ]−−= Next, we need to
find )β
β−%. 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=.
描写风景优美的成语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 ==2
2
(|,)(|,)E u x z Var y x z σ
==. Unless
is constant, the equation 2(|)E z x y x v β=+ generally violates the homoskedasticity assumption.
So, without further assumption,
1
2
)[(')](')[(')]E x x E v x x E x x β
β1
−−−=%
Now we can show ˆ))β
β−−−%ββ
is always positive semidefinite by writing ˆ))β
ββ−−−%β1−0 1212[(')](')[(')][(')]E x x E v x x E x x E x x σ−−=−
2
2
(')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 K
E y x x x βγμ∂
=+−∂ Hence
(|)|K K x K K
E 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 x
But, 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. So
3β is statistically different from zero at the 1% level.
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