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1、Introduction to Econometrics, 3e (Stock)Chapter 17 The Theory of Linear Regression with One Regressor17.1 Multiple Choice1) All of the following are good reasons for an applied econometrician to learn some econometric theory, with the exception ofA) turning your statistical software from a nblack bo
2、x into a flexible toolkit from which you are able to select the right tool for a given job.B) understanding econometric theory lets you appreciate why these tools work and what assumptions are required for each tool to work properly.C) learning how to invert a 4x4 matrix by hand.D) helping you recog
3、nize when a tool will not work well in an application and when it is time for you to look for a different econometric approach.Answer: C2) Finite-sample distributions of the OLS estimator and t-statistics are complicated, unlessA) the regressors are all normally distributed.B) the regression errors
4、are homoskedastic and normally distributed, conditional on Xj,. X,C) the Gauss-Markov Theorem applies.D) the regressor is also endogenous.Answer: B3) If, in addition to the least squares assumptions made in the previous chapter on the simple regression model, the errors are homoskedastic, then the O
5、LS estimator isA) identical to the TSLS estimator.B) BLUE.C) inconsistent.D) different from the OLS estimator in the presence of heteroskedasticity.Answer: B4) When the errors are heteroskedastic, thenA) WLS is efficient in large samples, if the functional form of the heteroskedasticity is known.B)
6、OLS is biased.C) OLS is still efficient as long as there is no serial correlation in the error terms.D) weighted least squares is e仔icient.Answer: Aestimator.3) One of the earlier textbooks in econometrics, first published in 1971, compared estimation of a parameter to shooting at a target with a ri
7、fle. The bulls-eye can be taken to represent the true value of the parameter, the rifle the estimator, and each shot a particular estimate.* Use this analogy to discuss small and large sample properties of estimators. How do you think the author approached the n s condition? (Dependent on your view
8、of the world, feel free to substitute guns with bow and arrow, or missile.) Answer: Unbiasedness: the shots produce a scatter, but the center of the scatter is the bulls-eye. If the riffle produces a scatter of shots that is centered on another point, then the gun is biased.Efficiency: Requires comp
9、arison with other unbiased guns. Looking at the scatters produced by the shots, the smallest scatter is the one from the efficient gun.BLUE: Remove all guns which are not linear and/or biased. The gun among these remaining ones which produces the smallest scatter is the BLUE gun.Consistency: rz oo i
10、s the condition as you march towards the bulls-eye, i.e.z the distance becomes shorter as - 8. A shot fired from a consistent gun hits the bulls-eye with increasing probability as you get closer to the bulls-eye. Or, perhaps even better, you might want to substitute being very close to the bulls-eye
11、 for hitting the bulls-eye.”4) nI am an applied econometrician and therefore should not have to deal with econometric theory. There will be others who I leave that to. I am more interested in interpreting the estimation results.1 Evaluate. Answer: Being presented with regression output and interpret
12、ing these uncritically does not allow the applied econometrician to understand the limitations of the tool. As a result, the interpretation may be false as might be the case in rejecting hypotheses when standard statistical inference does not apply in the situation at hand. In particular, having kno
13、wledge of econometric theory allows the econometrician to check whether or not the assumptions, which are necessary for statistical properties to hold, apply in a given situation. Knowing when to apply and when not to apply certain techniques is essential in conducting statistical inference, such as
14、 hypothesis testing and using confidence intervals. If the applied econometrician understands the limitations of certain estimation techniques, such as OLS, then she will be able to look for alternative approaches rather than blindly applying techniques by pushing buttons1 in econometric software. T
15、he above statement therefore seems short-sighted.5) One should never bother with WLS. Using OLS with robust standard errors gives correct inference, at least asymptotically.* True, false, or a bit of both? Explain carefully what the quote means and evaluate it critically.Answer: WLS is a special cas
16、e of the GLS estimator. Furthermore, OLS is a special case of the WLS estimator. Both will produce different estimates of the intercept and the coefficients of the other regressors, and different estimates of their standard errors. WLS has the advantage over OLS, that it is (asymptotically) more eff
17、icient than OLS. However, the e仔iciency result depends on knowing the conditional variance function. When this is the case, the parameters can be estimated and the weights can be specified. Unfortunately in practice, as Stock and Watson put it, nthe functional form of the conditional variance functi
18、on is rarely known. Using an incorrect functional form for the estimation of the parameters results in incorrect statistical inference. The bottom line is that WLS should be used in those rare instances where the functional form is known, but not otherwise. Estimation of the parameters using OLS wit
19、h heteroskedasticity-robust standard errors, on the other hand, leads to asymptotically validinferences even for the case where the functional form of the heteroskedasticity is not known. It therefore seems that for real world applications the above statement is true.17.3 Mathematical and Graphical
20、Problems21) Consider the model Yj = g1Xj + uj, where 国=cX ： ej and all of the Xs and es are i.i.d. and distributed(a) Which of the Extended Least Squares Assumptions are satisfied here? Prove your assertions.(b) Would an OLS estimator of 01 be efficient here?(c) How would you estimate 肉 by WLS?Answe
21、r:(a) The extended least squares assumptions are:1. E(cXjei| Xj) = 0 (conditional mean zero) - this holds here since the Xs and es are i.i.d;2. (X工 Yf), i = 1,., n are independent and identically distributed (i.i.d.) draws from their joint distribution-this applies here;3. (Xz;劭)have nonzero finite
22、fourth moments - this follows from the normal distribution, which has moments of all orders.244. var(w/| Xf)= o (homoskedasticity) 一 this fails since var(uzj Xf) = X y ； and5. The conditional distribution of uj given Xf is normal (normal errors) - this holds since X工is perfectlynormal, so to speak.(
23、b) Since the model is heteroskedastic, WLS offers efficiency gains.22(c) You would weight each observation by 1/ X p i.e., regress Yf/X j on 1/X.2) (Requires Appendix material) This question requires you to work with Chebychevs Inequality.(a) State Chebychev s Inequality.(b) Chebychevs Inequality is
24、 sometimes stated in the form The probability that a random variable is further than k standard deviations from its mean is less than 1陵:Deduce this form. (Hint: choose 6 artfully.)(c) If X is distributed N(0,l), what is the probability that X is two standard deviations from its mean? Three? What is
25、 the Chebychev bound for these values?(d) It is sometimes said that the Chebychev inequality is not sharp. What does that mean?(a) Pr(|VM 6) 5) 62J f (iu)dw + J f (io)dw -00b,6), where the first equality is the definition of E(W2), the second equality holds because the range of integration divides u
26、p the real line, the first inequality holds because the term that was dropped is nonnegative, the second inequality holds because w2 over the range of integration, and the final equality holds by the definition of Pr(| W| 6 ). Substituting W=V-/.ty into the final expression, noting that E(W2)= (V - y)2 = var(V), and rearranging yields the inequality.8) Consider the simple regression model Yz =+ uj where Xf 0 for all i, and the conditional2variance is var(w/| X。= 6X . where 0 is a known constant