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．Suppose that in the linear model y x u β=+, (')0E x u =(where x contain unity).2(|)Var u x σ=, but .(|)()E u x E u ≠ (a) Is it true that 22(|)E u x σ=?(b) What relevance does part (a) have for OLS estimation?2. Consider the simple regression model 01y x u ββ=++and let z be a binary instrumental variablefor x . Use 111()(ˆ()(niii niii z z y y z z x x β==)−−=−−∑∑ to show that the IV estimator 1ˆβ can be written as: 1ˆβ = (1y – 0y )/(1x – 0x ) Where 0y and 0x are the sample averages of yi and xi over the part of the sample with and where 0i z =1y and 1x are the sample averages of yi and xi over the part of the sample with .1i z =3. For the classical normal regression model Y X βε=+with no constant term and Kregressors, what is 22/)/()R Klim [,]lim (1p F K n K p R n K −−−=, assuming that the true value ofβ is zero?4. In the discussion of the instrumental variables estimator we showed that the least squares estimator b is biased and inconsistent. Nonetheless, b does estimate something: 1lim p b Q θβ−==+γ. Derive the asymptotic covariance matrix of b, and show that b isasymptotically normally distributed.5. Prove that under the hypothesis that R q β=,the estimator2***()'(Y Xb Y Xb s n K J−−=−+)Where J is the number of restrictions, is unbiased for 2σ6． The following equation describes the median housing price in a community in terms of amount ofpollution (nox for nitrous oxide) and the average number of rooms in houses in the community (rooms ):012log()+log(nox)+rooms+u price βββ=(a) What are the probable signs of1β and 2β? What is the interpretation of 1β? Explain.(b) Why might nox [or more precisely, log(nox )] and rooms be negatively correlated?If this is the case, does the simple regression of log(price ) on log(nox ) produce an upward or a downward biased estimator of1β?(c) Using the data in HPRICE2.RAW, the following equations were estimated:ˆlog()11.71 1.043log(nox)price =−2506,0.264n R ==ˆlog()9.230.718log(nox)+0.306rooms price =−2506,0.264n R ==Is the relationship between the simple and multiple regression estimates of the elasticity of price withrespect to nox what you would have predicted, given your answer in part (b)? Does this mean that -.718 is definitely closer to the true elasticity than -1.043?Solution 1 :（a ）Not in general. The conditional variance can always be written as()()22var ||[(|)]u x E u x E u x =−If , then (|)0E u x ≠()()2var ||u x E u x ≠（b ）It could be that , in which case OLS is consistent, and is constant. But ,generally, the usual standard errors would not be valid unless (')0E x u =(var |u x 0)(|)E u x =.Solution 2.It is easiest to use 111(ˆ(niii niii z z y y z z x x β==−−=−−∑∑ but where we drop z . Remember, this is allowedbecause1(n ii z z =−∑()ix x − = 1(niii z x x =)−∑ and similarly when we replace x with y . So thenumerator in the formula for 1ˆβ is 111111()n nn i i i i i i i i z y y z y z y n y n y ===⎛⎞−=−=−⎜⎟⎝⎠∑∑∑ where n 1 =is the number of observations with z i = 1, and we have used the fact that/n 1 = 1nii z=∑1n i i i z y =⎛⎞⎜⎟⎝⎠∑1y , the average of the y i over the i with z i = 1. So far, we have shown that the numerator in 1ˆβ is n 1(1y – y ). Next, write y as a weighted average of the averages over the two subgroups:y = (n 0/n )0y + (n 1/n )1y ,where n 0 = n – n 1. Therefore,1y – y = [(n – n 1)/n ] 1y – (n 0/n ) 0y = (n 0/n ) (1y - 0y ).Therefore, the numerator of 1ˆβ can be written as(n 0n 1/n )(1y – 0y ).By simply replacing y with x , the denominator in 1ˆβ can be expressed as (n 0n 1/n )(1x – 0x ). When we take the ratio of these, the terms involving n 0, n 1, and n , cancel, leaving1ˆβ = (1y – 0y )/(1x – 0x ).Solution3:22/''[,](1)/()'/()/R K b X Xb F K n K KR n K e e n K −==−−− 11'(')'(')'/'/()X X X X X X X X K M n K εεεε−−=−Pr2'/()M n K εεσ−⎯⎯→ 1((')')tr X X X X K −=So 21[,]()dF K n K K Kχ−⎯⎯→Solution 4:To obtain the asymptotic distribution, write the result already in hand as11()(')'b Q X X X Q 1βγε−−=++−ε−We have established that 1lim p b Q θβ−==+γ. For convenience, let θβ≠,denote. Write the preceding in the form 1lim Q p βγ−+=b 11('/)('/)b X X n X n Q θεγ−−−=−Since lim('/)p X X n Q =lim()p b , the large sample behavior of the right hand side is the same as that of11lim('/)Q p X n Q θεγ−−−−=.That is, we may replace ('/)X X n with Q in our derivation. Then, we seek the asymptoticdistribution ofb )θ−which is the same as that of1111lim('/)][ni i i Q p X n Q Qx n ]εγε−−−=−=−∑γ From this point, the derivation is exactly the same as that when 0γ=, so there is no need to redevelop the result. We may proceed directly to the same asymptotic distribution we obtained before.The only difference is that the least squares estimator estimatesθ, not β.Solution 5:we know , and 11**''()'[(')']e e e e Rb q R X X R q −−=+−−(Rb )2(')()E e e n K σ=−.So under the hypothesis,21**(')()()'[(')()E e e n K E Rb q R X X R Rb σ=−+−11]']−−)q −1']R 211()['(')'[('(')'n K E X X X R R X X R X X X σεε−−=−+−−1'X X ')X − 211()[{[(')']()'(')'}]n K E tr R X X R R X X X X R −−−−=−+1ε1'σε1 2112(){[(')']('(')'}n K tr R X X R R X X X X X R σσ−−−=−+ 22(){}()J n K tr I n K J σσ=−+=−+Solution 6:(a) 1β < 0 because more pollution can be expected to lower housing values; note that1β isthe elasticity of price with respect to nox . 2β is probably positive because rooms roughly measures the size of a house. (However, it does not allow us to distinguish homes where each room is large from homes where each room is small.) (b) If we assume that rooms increases with quality of the home, then log(nox ) and rooms are negatively correlated when poorer neighborhoods have more pollution, something that is often true. We can use Table 3.2 to determine the direction of the bias. If2β > 0 and Corr(x 1,x 2) < 0, the simpleregression estimator 1β% has a downward bias. But because 1β < 0, this means that the simple regression, on average, overstates the importance of pollution. [E(1β%) is more negative than 1β.](c) This is what we expect from the typical sample based on our analysis in part (ii). Thesimple regression estimate, −1.043, is more negative (larger in magnitude) than the multiple regression estimate, −.718. As those estimates are only for one sample, we can never know which is closer to 1β. But if this is a “typical” sample,1β is closer to −.718.。