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1、Introduction to Econometrics 3e (Stock)Chapter 12 Instrumental Variables Regression12.1 Multiple Choice1) Estimation of the IV regression modelA) requires exact identification.B) allows only one endogenous regressor, which is typically correlated with the error term.C) requires exact identification
2、or overidentification.D) is only possible if the number of instruments is the same as the number of regressors.Answer: C2) Two Stage Least Squares is calculated as follows; in the first stage:A) Y is regressed on the exogenous variables only. The predicted value of Y is then regressed on the instrum
3、ental variables.B) the unknown coefficients in the reduced form equation are estimated by OLS, and the predicted values are calculated. In the second stage, Y is regressed on these predicted values and the other exogenous variables.C) the exogenous variables are regressed on the instruments. The pre
4、dicted value of the exogenous variables is then used in the second stage, together with the instruments, to predict the dependent variable.D) the unknown coefficients in the reduced form equation are estimated by weighted least squares, and the predicted values are calculated. In the second stage, Y
5、 is regressed on these predicted values and the other exogenous variables.Answer: B3) The conditions for a valid instruments do not include the following:A) each instrument must be uncorrelated with the error term.B) each one of the instrumental variables must be normally distributed.C) at least one
6、 of the instruments must enter the population regression of X on the Zs and the Ws.D) perfect multicollinearity between the predicted endogenous variables and the exogenous variables must be ruled out.Answer: B4) The IV regression assumptions include all of the following with the exception ofA) the
7、error terms must be normally distributed.B)E(wz | WiWrz-) = 0.C) Large outliers are unlikely: the Xs, Ws, Zs, and Ys all have nonzero, finite fourth moments.D) (Xj/,., Xki，WiZif, . Zmi，Yj) are i.i.d. draws from their joint distribution.Answer: A3) Describe the consequences of estimating an equation
8、by OLS in the presence of an endogenous regressor. How can you overcome these obstacles? Present an alternative estimator and state its properties.Answer: In the case of an endogenous regressor, there is correlation between the variable and the error term. In this case, the OLS estimator is inconsis
9、tent. To get a consistent estimator in this situation, instrumental variable techniques, such as TSLS, should be used. If one or more valid instruments can be found, meaning that the instrument must be relevant and exogenous, then a consistent estimator can be derived. The relevance of instruments c
10、an be tested using the rule of thumb (a first-stage F-statistic of more than 10 in the TSLS estimator). The exogeneity of the instruments can be tested using the /-statistic. The test requires that there is at least one more instrument than endogenous regressors, i.e., that the equation is overident
11、ified. In large samples the sampling distribution of the TSLS estimator is approximately normal, so that statistical inference can proceed as usual using the f-statistic, confidence intervals, or joint hypothesis tests involving the F-statistic. However, inference based on these statistics will be m
12、isleading in the case where instruments are not valid.4) Write an essay about where valid instruments come from. Part of your explorations must deal with checking the validity of instruments and what the consequences of weak instruments are.Answer: In order for instruments to be valid, they have to
13、be relevant and exogenous. To find valid instruments, two approaches are typically used. First economic theory can serve as a guide. In the case of simultaneous causality in a market, for example, theory predicts shifts in one curve but not the other as a result of changes in an instrumental variabl
14、e. The second approach focuses on shifts in the endogenous regressor that is caused by an exogenous source of variation* in the variable resulting from a random phenomenon. The textbook uses the example of an earthquake which changes student teacher ratios as students in affected areas have to be re
15、distributed.To check the validity of instruments, there is the rule of thumb to determine whether or not an instrument is weak. It states that the F-statistic in the first stage of the TSLS procedure should exceed 10. Instrument exogeneity can be tested only in the case of overidentification. If the
16、re are more instruments than endogenous regressors, then the /-statistic can be calculated. The null hypothesis of exogeneity will be rejected, in essence, if the TSLS residuals are correlated with the instruments.If instruments are weak, then the TSLS estimator is biased and statistical inference d
17、oes not yield reliable confidence intervals even in large samples.5) You have estimated a government reaction function, i.e., a multiple regression equation, where a government instrument, say the federal funds rate, depends on past government target variables, such as inflation and unemployment rat
18、es. In addition, you added the previous periods popularity deficit of the government, e.g. the (approval rating of the president - 50%), as one of the regressors. Your idea is that the Federal Reserve, although formally independent, will try to expand the economy if the president is unpopular. One o
19、f your peers, a political science student, points out that approval ratings depend on the state of the economy and thereby indirectly on government instruments. It is therefore endogenous and should be estimated along with the reaction function. Initially you want to reply by using a phrase that inc
20、ludes the words money neutrality but are worried about a lengthy debate. Instead you state that as an economist, you are not concerned about government approval ratings, and that government approval ratings are determined outside your (the economic) model. Does your whim make the regressor exogenous
21、? Why or why not?Answer: In general, the question of whether or not a variable is endogenous or exogenous depends on its correlation with the error term, not on the size of the underlying model. The point to make is that just because a variable is endogenous does not imply that its determinants have
22、 to be modeled. If the purpose of the exercise is to eventually simulate the model for policy purposes, then the feedback envisioned by the political science student is potentially important. However, if the aim is simply to forecast the behavior of the government reaction function, then the issue o
23、f endogeneity or exogeneity is only relevant for questions regarding the type of estimator to be used. Of course, if a regressor is endogenous, then instrumental variable techniques must be used to ensure desirable properties of the estimator.6) You have been hired as a consultant to estimate the de
24、mand for various brands of coffee in the market. You are provided with annual price data for two years by U.S. state and the quantities sold. You want to estimate a demand function for coffee using this data. What problems do you think you will encounter if you estimated the demand equation by OLS?A
25、nswer: Answers will differ by student. However, the following points should be mentioned: (i) there will be simultaneous equation bias because quantity and price are determined simultaneously in the market, (ii) If this is the case, then the OLS estimator will not be consistent, (iii) In that case,
26、IV estimation should be used to get a consistent estimator of the demand elasticity or response to a price increase. (iv) This brings up the question of a valid instrument. It is not clear that students will come up with an easy answer, but their deliberations should be insightful. One possible inst
27、rument is the price (change) from a previous year, which most likely will be highly correlated with this year*s price (change) but not with the error term in the equation. (v) There should be some discussion on the other factors determining coffee demand, although some of these can be ignored if the
28、re is data for two periods and the data is differenced (fixed effects).7) Studies of the effect of minimum wages on teenage employment typically regress the teenage employment to population ratio on the real minimum wage or the minimum wage relative to average hourly earnings using OLS. Assume that
29、you have a cross section of United States for two years. Do you think that there are problems with simultaneous equation bias?Answer: For OLS not to be consistent, there would have to be omitted variable bias or simultaneous equation bias. The former can be dealt with by differencing the data, if yo
30、u assume that most other factors are being held constant. If the minimum wage does not change between the two periods, i.e. it is constant, then this will bring further problems with the interpretation, since the variation in the RHS variable only comes from the denominator. In many ways, the questi
31、on should come down to the correlation between minimum wages and the error term in the equation. Students may argue that minimum wages are set by the legislature or, more recently, by ballot, and are therefore exogenous. A more nuanced discussion may point out that neither the legislature nor the el
32、ectorate will raise minimum wages in time periods of low employment (a recession although the 2008 and 2009 raises will contradict this statement to some extent; however, these were decided in 2006/2007 when the economy was booming). There may be further problems because of the denominator of the mi
33、nimum wage variable, either the CPI or AHE, both of which are potentially correlated with teenage employment. The point here is for the student to think about the problem at hand and to point out various obstacles to getting a good estimate of the elasticity/response of employment from a minimum wag
34、e increase.12.3 Mathematical and Graphical Problems1) To analyze the year-to-year variation in temperature data for a given city, you regress the daily high temperature (Temp) for 100 randomly selected days in two consecutive years (1997 and 1998) for Phoenix. The results are (heteroskedastic-robust
35、 standard errors in parenthesis):Tcnip 15.63 + 0.80 x Tcntp ;成 R2 = 0.65, SER = 9.63(0.10)(a) Calculate the predicted temperature for the current year if the temperature in the previous year was 40F, 78F, and 100F. How does this compare with you prior expectation? Sketch the regression line and comp
36、are it to the 45 degree line. What are the implications?(b) You recall having studied errors-in-variables before. Although the web site you received your data from seems quite reliable in measuring data accurately, what if the temperature contained measurement error in the following sense: for any g
37、iven day, say January 28, there is a true underlying seasonal temperature (X), but each year there are different temporary weather patterns w) which result in aAtemperature X different from X. For the two years in your data set, the situation can be described as follows:Subtracting X1997 from X1998,
38、 you get X1998 = 1997 + 叼998 一 4997 Hence the population parameter for the intercept and slope are zero and one, as expected. It is not difficult to show that the OLS estimator for the slope is inconsistent, where9 人(ycy Y + cy. .IVAs a result you consider estimating the slope and intercept by TSLS.
39、 You think about an instrument and consider the temperature one month ahead of the observation in the previous year. Discuss instrument validity for this case.(c) The TSLS estimation result is as follows:Tenip 微=-6.24 + 1.07xT :聆:(0.06)Perform a t-test on whether or not the slope is now significantl
40、y different from one.Answer:(a) The three predicted temperatures will be 47.6, 78.0, and 95.6 respectively. The initial expectation should be that the temperature in 1998 is the same in 1997 for a given date. The regression line and the 45 degree line are sketched in the accompanying figure. The imp
41、lication is mean reversion: if the temperature was low (40 degrees), then it will also be low the following year, but not as low.Alternatively, if the temperature was high (100 degrees), then it will be high again, but not as high. If this prediction extrapolated into the future, then eventually all
42、 temperatures should be the same for all days. This obviously does not make sense.(b) For an instrument to be valid, two conditions have to hold. First, the instrument has to be relevant, and second, the instrument has to be exogenous. If temperatures in one month ahead can predict the current tempe
43、rature, as it certainly does in Phoenix, then the instrument is relevant or correlated with the current month*s temperature. If in addition, whatever caused the temperature in the current month to deviate from its long-term value is only a temporary phenomenon, such as a weather system created by a
44、storm in the Pacific, then next month*s temperature should not be correlated with this event. Hence the instrument would be exogenous.(c) The f-statistic is 1.17, and hence you cannot reject the null hypothesis that the slope equals one.2) Consider the following population regression model relating
45、the dependent variable Yj and regressor Xi，Yj = ,0 + PlXj + ui，z = 1, n.Xi = Y/ + Ziwhere Z is a valid instrument for X.(a) Explain why you should not use OLS to estimate(b) To generate a consistent estimator for 01，what should you do?(c) The two equations above make up a system of equations in two
46、unknowns. Specify the two reduced form equations in terms of the original coefficients. (Hint: substitute the identity into the first equation and solve for Y. Similarly, substitute Y into the identity and solve for X.)(d) Do the two reduced form equations satisfy the OLS assumptions? If so, can you
47、 find consistent estimators of the two slopes? What is the ratio of the two estimated slopes? This estimator is called Indirect Least Squares/ How does it compare to the TSLS in this example?Answer:(e) Substitution of the first equation into the identity shows that X is correlated with the error ter
48、m. Hence estimation with OLS results in an inconsistent estimator.A 2SLS $ZY(f) The instrumental variable estimator is consistent and in this case is 0 】= 互*, Adventurous students will derive this estimator along the lines shown in Appendix 10.2.(。Yi 0 + 肉(X + Zi) + ujXi = (6o + 肉 X，+ u/) + Zior(1-肉
49、)丫产优+ 口必+质(1 肉)X产 f0 + Zz +所HenceYj= 71Q + 7l2Zf + viXi = 713 + 714Z/ + V2i,。肉11where 710 = 3= 7，n2 = 77，冗4 = and vy = V2i =(d) Since Z is a valid instrument by assumption, it must be uncorrelated with the error term and hence SyzAusing OLS results in a consistent estimator. 一 714Szz Syz 二 which is identical to the TSLSSxz SZZSzzestimator.3) Here are some example