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Introductory Econometrics A Modern Approach 3rd Edition Jeffrey M. Wooldridge - Solutions

C15.5 Use the data in CARD.RAW for this exercise. (i) In Table 15.1, the difference between the IV and OLS estimates of the return to education is economically important. Obtain the reduced form residuals, vˆ2, from (15.32). (See Table 15.1 for the other variables to include in the regression.)
C15.4 Use the data in INTDEF.RAW for this exercise. A simple equation relating the three-month T-bill rate to the inflation rate (constructed from the Consumer Price Index) is i3t 0 1inft ut . (i) Estimate this equation by OLS, omitting the first time period for later comparisons. Report
C15.3 Use the data in CARD.RAW for this exercise. (i) The equation we estimated in Example 15.4 can be written as log(wage) 0 1educ 2exper … u, where the other explanatory variables are listed in Table 15.1. In order for IV to be consistent, the IV for educ, nearc4, must be
C15.2 The data in FERTIL2.RAW includes, for women in Botswana during 1988, information on number of children, years of education, age, and religious and economic status variables. (i) Estimate the model children 0 1educ 2age 3age2 u by OLS, and interpret the estimates. In particular,
C15.1 Use the data in WAGE2.RAW for this exercise. (i) In Example 15.2, using sibs as an instrument for educ, the IV estimate of the return to education is .122. To convince yourself that using sibs as an IV for educ is not the same as just plugging sibs in for educ and running an OLS regression,
15.11 Consider a simple time series model where the explanatory variable has classical measurement error: yt 0 1x t * ut xt xt * et , (15.58) where ut has zero mean and is uncorrelated with xt * and et . We observe yt and xt only. Assume that et has zero mean and is uncorrelated with xt
15.10 In a recent article, Evans and Schwab (1995) studied the effects of attending a Catholic high school on the probability of attending college. For concreteness, let college be a binary variable equal to unity if a student attends college, and zero otherwise. Let CathHS be a binary variable
15.9 Suppose that, in equation (15.8), you do not have a good instrumental variable candidate for skipped. But you have two other pieces of information on students: combined SAT score and cumulative GPA prior to the semester. What would you do instead of IV estimation?
15.8 Suppose you want to test whether girls who attend a girls’ high school do better in math than girls who attend coed schools. You have a random sample of senior high school girls from a state in the United States, and score is the score on a standardized math test. Let girlhs be a dummy
15.7 The following is a simple model to measure the effect of a school choice program on standardized test performance (see Rouse [1998] for motivation): score 0 1choice 2 faminc u1,where score is the score on a statewide test, choice is a binary variable indicating whether a student
15.6 (i) In the model with one endogenous explanatory variable, one exogenous explanatory variable, and one extra exogenous variable, take the reduced form for y2, (15.26), and plug it into the structural equation (15.22). This gives the reduced form for y1: y1 0 1z1 2z2 v1. Find the j
15.5 Refer to equations (15.19) and (15.20). Assume that u x, so that the population variation in the error term is the same as it is in x. Suppose that the instrumental variable, z, is slightly correlated with u: Corr(z,u) .1. Suppose also that z and x have a somewhat stronger correlation:
15.4 Suppose that, for a given state in the United States, you wish to use annual time series data to estimate the effect of the state-level minimum wage on the employment of those 18 to 25 years old (EMP). A simple model is gEMPt 0 1gMINt 2gPOPt 3gGSPt 4gGDPt ut , where MINt is
15.3 Consider the simple regression model y = B0 + B1X + u
15.2 Suppose that you wish to estimate the effect of class attendance on student performance, as in Example 6.3. A basic model is stndfnl 0 1atndrte 2 priGPA 3ACT u, where the variables are defined as in Chapter 6. (i) Let dist be the distance from the students’ living quarters to
15.1 Consider a simple model to estimate the effect of personal computer (PC) ownership on college grade point average for graduating seniors at a large public university: GPA 0 1PC u, where PC is a binary variable indicating PC ownership. (i) Why might PC ownership be correlated with u?
C14.10 Use the data in AIRFARE.RAW for this exercise. We are interested in estimating the model log(fareit) t 1concenit 2log(disti ) 3[log(disti )]2 ai uit, t 1,..., 4, where t means that we allow for different year intercepts. (i) Estimate the above equation by pooled OLS, being
C14.9 The file PENSION.RAW contains information on participant-directed pension plans for U.S. workers. Some of the observations are for couples within the same family, so this data set constitutes a small cluster sample (with cluster sizes of two). (i) Ignoring the clustering by family, use OLS to
C14.8 Use the data in MATHPNL.RAW for this exercise. You will do a fixed effects version of the first differencing done in Computer Exercise C13.11. The model of interest is math4it 1y94t ... 5y98t 1log(rexppit) 2log(rexppi,t1) 1log(enrolit) 2lunchit ai uit, where the first
C14.7 Use the state-level data on murder rates and executions in MURDER.RAW for the following exercise. (i) Consider the unobserved effects model mrdrteit t 1execit 2unemit ai uit, where t simply denotes different year intercepts and ai is the unobserved state effect. If past
C14.6 Add the interaction term unionitt to the equation estimated in Table 14.2 to see if wage growth depends on union status. Estimate the equation by random and fixed effects and compare the results.
C14.5 (i) In the wage equation in Example 14.4, explain why dummy variables for occupation might be important omitted variables for estimating the union wage premium. (ii) If every man in the sample stayed in the same occupation from 1981 through 1987, would you need to include the occupation
C14.4 In Example 13.8, we used the unemployment claims data from Papke (1994) to estimate the effect of enterprise zones on unemployment claims. Papke also uses a model that allows each city to have its own time trend: log(uclmsit) ai ci t 1ezit uit, where ai and ci are both unobserved
14.6 Using the “cluster” option in the econometrics package Stata®, the fully robust standard errors for the pooled OLS estimates in Table 14.2—that is, robust to serial correlation and heteroskedasticity in the composite errors, {vit: t 1,...,T}—are obtained as se(ˆ educ) .011,
14.5 Suppose that, for one semester, you can collect the following data on a random sample of college juniors and seniors for each class taken: a standardized final exam score, percentage of lectures attended, a dummy variable indicating whether the class is within the student’s major,
14.4 In order to determine the effects of collegiate athletic performance on applicants, you collect data on applications for a sample of Division I colleges for 1985, 1990, and 1995. (i) What measures of athletic success would you include in an equation? What are some of the timing issues? (ii)
14.3 In a random effects model, define the composite error vit ai uit, where ai is uncorrelated with uit and the uit have constant variance u 2 and are serially uncorrelated. Define eit vit v¯i , where is given in (14.10). (i) Show that E(eit) 0. (ii) Show that Var(eit) u 2, t 1,
4.2 With a single explanatory variable, the equation used to obtain the between estimator is y¯i 0 1x¯i ai u¯i , where the overbar represents the average over time. We can assume that E(ai ) 0 because we have included an intercept in the equation. Suppose that u¯i is uncorrelated
14.1 Suppose that the idiosyncratic errors in (14.4), {uit: t 1,2, …, T}, are serially uncorrelated with constant variance, u 2. Show that the correlation between adjacent differences, uit and ui,t1, is .5. Therefore, under the ideal FE assumptions, first differencing induces negative
C13.12 Use the data in MURDER.RAW for this exercise. (i) Using the years 1990 and 1993, estimate the equation mrdrteit 0 1d93t 1execit 2unemit ai uit, t 1,2 by pooled OLS and report the results in the usual form. Do not worry that the usual OLS standard errors are inappropriate
C13.11 The file MATHPNL.RAW contains panel data on school districts in Michigan for the years 1992 through 1998. It is the district-level analogue of the school-level data used by Papke (2005). The response variable of interest in this question is math4, the percentage of fourth graders in a
C13.10 For this exercise, we use JTRAIN.RAW to determine the effect of the job training grant on hours of job training per employee. The basic model for the three years is hrsempit 0 1d88t 2d89t 1grantit 2granti,t1 3log(employit) ai uit. (i) Estimate the equation using first
C13.9 Use CRIME4.RAW for this exercise. (i) Add the logs of each wage variable in the data set and estimate the model by first differencing. How does including these variables affect the coefficients on the criminal justice variables in Example 13.9? (ii) Do the wage variables in (i) all have the
C13.8 VOTE2.RAW includes panel data on House of Representative elections in 1988 and 1990. Only winners from 1988 who are also running in 1990 appear in the sample; these are the incumbents. An unobserved effects model explaining the share of the incumbent’s vote in terms of expenditures by both
C13.7 Use GPA3.RAW for this exercise. The data set is for 366 student-athletes from a large university for fall and spring semesters. (A similar analysis is in Maloney and McCormick [1993], but here we use a true panel data set.) Because you have two terms of data for each student, an unobserved
C13.6 Use CRIME3.RAW for this exercise. (i) In the model of Example 13.6, test the hypothesis H0: 1 2. (Hint: Define 1 1 2 and write 1 in terms of 1 and 2. Substitute this into the equation and then rearrange. Do a t test on 1.) (ii) If 1 2, show that the differenced equation can be
C13.5 Use the data in RENTAL.RAW for this exercise. The data for the years 1980 and 1990 include rental prices and other variables for college towns. The idea is to see whether a stronger presence of students affects rental rates. The unobserved effects model is log(rentit) 0 0y90t
C13.4 Use the data in INJURY.RAW for this exercise. (i) Using the data for Kentucky, reestimate equation (13.12), adding as explanatory variables male, married, and a full set of industry and injury type dummy variables. How does the estimate on afchngehighearn change when these other factors are
C13.3 Use the data in KIELMC.RAW for this exercise. (i) The variable dist is the distance from each home to the incinerator site, in feet. Consider the model log( price) 0 0y81 1log(dist) 1y81log(dist) u. If building the incinerator reduces the value of homes closer to the site, what
C13.2 Use the data in CPS78_85.RAW for this exercise. (i) How do you interpret the coefficient on y85 in equation (13.2)? Does it have an interesting interpretation? (Be careful here; you must account for the interaction terms y85educ and y85female.) (ii) Holding other factors fixed, what is the
C13.1 Use the data in FERTILI.RAW for this exercise. (i) In the equation estimated in Example 13.1, test whether living envi- ronment at age 16 has an effect on fertility. (The base group is large city.) Report the value of the F statistic and the p-value. (ii) Test whether region of the country at
13.7 (i) Using the data in INJURY.RAW for Kentucky, the estimated equation when afchnge is dropped from (13.12) is log(durat) 1.129+.253 highearn + .198 afchnge-highearn (0.022) (.042) (.052) n = 5,626, R2 = .021. == Is it surprising that the estimate on the interaction is fairly close to that in
13.6 In 1985, neither Florida nor Georgia had laws banning open alcohol containers in vehi- cle passenger compartments. By 1990, Florida had passed such a law, but Georgia had not. (i) Suppose you can collect random samples of the driving-age population REVIEW in both states, for 1985 and 1990. Let
13.5 Suppose that we want to estimate the effect of several variables on annual sav- ing and that we have a panel data set on individuals collected on January 31, 1990, and January 31, 1992. If we include a year dummy for 1992 and use first differencing, can we also include age in the original
13.4 If we think that , is positive in (13.14) and that Au, and Aunem, are negatively correlated, what is the bias in the OLS estimator of B, in the first-differenced equation? (Hint: Review Table 3.2.)
13.3 Why can we not use first differences when two years (as opposed to panel data)? have independent cross sections in
13.2 Using the data in KIELMC.RAW, the following equations were estimated using the years 1978 and 1981: and == log(price) 11.49.547 nearinc+ .394 y81-nearing = (.26) (.058) n = 321, R2 = .220 (.080) log(price) 11.18+.563 y81.403 y81-neart (.27) (.044) n = 321, R2 337. TRALOR CLASSRO So have to in
13.1 In Example 13.1, assume that the average of all factors other than educ have remained constant over time and that the average level of education is 12.2 for the 1972 sample and 13.3 in the 1984 sample. Using the estimates in Table 13.1, find the estimated change in average fertility between
C12.12 Use the data in INVEN.RAW for this exercise; see also Computer Exercise C11.6. (i) Obtain the OLS residuals from the accelerator model Ainven, B BAGDP, + u, and use the regression , on ,-, to test for serial corre- lation. What is the estimate of p? How big a problem does serial cor-
C12.9 The file FISH.RAW contains 97 daily price and quantity observations on fish prices at the Fulton Fish Market in Manhattan. Use the variable log(avgprc) as the dependent variable. (i) Regress log(avgprc) on four daily dummy variables, with Friday as the base. Include a linear time trend. Is
C12.8 Use the data in TRAFFIC2.RAW for this exercise. (i) Run an OLS regression of prcfat on a linear time trend, monthly dummy variables, and the variables wkends, unem, spdlaw, and beltlaw. Test the errors for AR(1) serial correlation using the regression in equation (12.14). Does it make sense
C12.7 (i) For Example 12.4, using the data in BARIUM.RAW, obtain the itera- tive Cochrane-Orcutt estimates. (ii) Are the Prais-Winsten and Cochrane-Orcutt estimates similar? Did you expect them to be?
C12.6 (i) In Computer Exercise C10.7, you estimated a simple relationship between consumption growth and growth in disposable income. Test the equation for AR(1) serial correlation (using CONSUMP.RAW). (ii) In Computer Exercise C11.7, you tested the permanent income hypoth- esis by regressing the
C12.5 Consider the version of Fair's model in Example 10.6. Now, rather than predict- ing the proportion of the two-party vote received by the Democrat, estimate a linear prob- ability model for whether or not the Democrat wins. (i) Use the binary variable demwins in place of demvote in (10.23) and
C12.3 (i) In part (i) of Computer Exercise C11.6, you were asked to estimate the accelerator model for inventory investment. Test this equation for AR(1) (ii) If you find evidence of serial correlation, reestimate the equation by Cochrane-Orcutt and compare the results.
C12.2 (i) Using the data in WAGEPRC.RAW, estimate the distributed lag model from Problem 11.5. Use regression (12.14) to test for AR(1) serial correlation. Reestimate the model using iterated Cochrane-Orcutt estimation. What is your new estimate of the long-run propensity? (iii) Using iterated CO,
C12.1 In Example 11.6, we estimated a finite DL model in first differences: Agfr, Yo + SoApe, + 8 Ape-1 + Ape,-2 Use the data in FERTIL3.RAW to test whether there is AR errors. serial correlation in the
12.6 In Example 12.8, we found evidence of heteroskedasticity in u, in equation (12.47). Thus, we compute the heteroskedasticity-robust standard errors (in []) along with the usual standard errors: return, .180+.059 return,-1 (.081) (.038) [.085] [.069] n = 689, R = .0035, R = .0020. What does
12.5 (i) In the enterprise zone event study in Computer Exercise C10.5, a regression of the OLS residuals on the lagged residuals produces ˆ .841 and se(ˆ) .053. What implications does this have for OLS? (ii) If you want to use OLS but also want to obtain a valid standard error for the EZ
12.4 True or False: “If the errors in a regression model contain ARCH, they must be serially correlated.”
12.3 In Example 10.6, we estimated a variant on Fair’s model for predicting presidential election outcomes in the United States. (i) What argument can be made for the error term in this equation being serially uncorrelated? (Hint: How often do presidential elections take place?) (ii) When the
12.2 Explain what is wrong with the following statement: “The Cochrane-Orcutt and Prais-Winsten methods are both used to obtain valid standard errors for the OLS estimates when there is a serial correlation.”
12.1 When the errors in a regression model have AR(1) serial correlation, why do the OLS standard errors tend to underestimate the sampling variation in the ˆ j ? Is it always true that the OLS standard errors are too small?
C11.10 Use all the data in PHILLIPS.RAW to answer this question. You should now use 56 years of data. (i) Reestimate equation (11.19) and report the results in the usual form. Do the intercept and slope estimates change notably when you add the recent years of data? (ii) Obtain a new estimate of
C10.11, except you should first difference the unemployment rate, too. Then, include a linear time trend, monthly dummy variables, the weekend variable, and the two policy variables; do not difference these. Do you find any interesting results? (iii) Comment on the following statement: “We
C11.9 Use the data in TRAFFIC2.RAW for this exercise. Computer Exercise C10.11 previously asked for an analysis of these data. (i) Compute the first order autocorrelation coefficient for the variable prcfat. Are you concerned that prcfat contains a unit root? Do the same for the unemployment rate.
C11.8 Use the data in PHILLIPS.RAW for this exercise. (i) Estimate an AR(1) model for the unemployment rate. Use this equation to predict the unemployment rate for 2004. Compare this with the actual unemployment rate for 2004. (You can find this information in a recent Economic Report of the
C11.7 Use CONSUMP.RAW for this exercise. One version of the permanent income hypothesis (PIH) of consumption is that the growth in consumption is unpredictable. (Another version is that the change in consumption itself is unpredictable; see Mankiw [1994, Chapter 15] for discussion of the PIH.) Let
C11.6 Let invent be the real value inventories in the United States during year t, let GDPt denote real gross domestic product, and let r3t denote the (ex post) real interest rate on three-month T-bills. The ex post real interest rate is (approximately) r3t i3t inft , where i3t is the rate on
C11.5 (i) Add a linear time trend to equation (11.27). Is a time trend necessary in the first-difference equation? (ii) Drop the time trend and add the variables ww2 and pill to (11.27) (do not difference these dummy variables). Are these variables jointly significant at the 5% level? (iii) Using
C11.4 Use the data in PHILLIPS.RAW for this exercise, but only through 1996. (i) In Example 11.5, we assumed that the natural rate of unemployment is constant. An alternative form of the expectations augmented Phillips curve allows the natural rate of unemployment to depend on past levels of
C11.3 (i) In Example 11.4, it may be that the expected value of the return at time t, given past returns, is a quadratic function of returnt1. To check this possibility, use the data in NYSE.RAW to estimate returnt 0 1returnt1 2returnt 2 1 ut ; report the results in standard form. (ii)
C11.2 In Example 11.7, define the growth in hourly wage and output per hour as the change in the natural log: ghrwage log(hrwage) and goutphr log(outphr). Consider a simple extension of the model estimated in (11.29): ghrwaget 0 1goutphrt 2goutphrt1 ut. This allows an increase in
C11.1 Use the data in HSEINV.RAW for this exercise. (i) Find the first order autocorrelation in log(invpc). Now, find the autocorrelation after linearly detrending log(invpc). Do the same for log( price). Which of the two series may have a unit root? (ii) Based on your findings in part (i),
11.7 A partial adjustment model is yt * 0 1xt et yt yt1 (yt * yt1) at , where yt * is the desired or optimal level of y, and yt is the actual (observed) level. For example, yt * is the desired growth in firm inventories, and xt is growth in firm sales. The parameter 1 measures
11.6 Let hy6t denote the three-month holding yield (in percent) from buying a six-month T-bill at time (t 1) and selling it at time t (three months hence) as a three-month T-bill. Let hy3t1 be the three-month holding yield from buying a three-month T-bill at time (t 1). At time (t 1), hy3t1
11.5 For the U.S. economy, let gprice denote the monthly growth in the overall price level and let gwage be the monthly growth in hourly wages. [These are both obtained as differences of logarithms: gprice log( price) and gwage log(wage).] Using the monthly data in WAGEPRC.RAW, we estimate the
11.4 Let {yt : t 1,2,…} follow a random walk, as in (11.20), with y0 0. Show that Corr(yt ,yth) t/(t h) for t 1, h 0.
11.3 Suppose that a time series process {yt } is generated by yt z et , for all t 1,2,…, where {et } is an i.i.d. sequence with mean zero and variance se 2. The random variable z does not change over time; it has mean zero and variance sz 2. Assume that each et is uncorrelated with z. (i)
11.2 Let {et : t 1,0,1, …} be a sequence of independent, identically distributed random variables with mean zero and variance one. Define a stochastic process by xt et (1/2)et1 (1/2)et2, t 1,2,…. (i) Find E(xt ) and Var(xt ). Do either of these depend on t? (ii) Show that Corr(xt
11.1 Let {xt : t 1,2,…} be a covariance stationary process and define h Cov(xt ,xth) for h 0. [Therefore, 0 Var(xt ).] Show that Corr(xt ,xth) h/0.
C10.12 (i) Estimate equation (10.2) using all the data in PHILLIPS.RAW and report the results in the usual form. How many observations do you have now? (ii) Compare the estimates from part (i) with those in equation (10.14). In particular, does adding the extra years help in obtaining an estimated
C10.11 The file TRAFFIC2.RAW contains 108 monthly observations on automobile accidents, traffic laws, and some other variables for California from January 1981 through December 1989. Use this data set to answer the following questions. (i) During what month and year did California's seat belt law
C10.10 Consider the model estimated in (10.15); use the data in INTDEF.RAW. (i) Find the correlation between inf and def over this sample period and comment. (ii) Add a single lag of inf and def to the equation and report the results in the usual form. (iii) Compare the estimated LRP for the effect
C10.9 Use the data in VOLAT.RAW for this exercise. The variable rsp500 is the monthly return on the Standard & Poor’s 500 stock market index, at an annual rate. (This includes price changes as well as dividends.) The variable i3 is the return on three-month T-bills, and pcip is the percentage
C10.8 Use the data in FERTIL3.RAW for this exe (i) Add pe,-3 and pe,-4 to equation (10.19). Test for joint significance of these lags. (ii) Find the estimated CCSLER propensity and its standard error in the model from part (i). Compare these with those obtained from equation (10.19). (iii) Estimate
C10.7 Use the data set CONSUMP.RAW for this exercise. (i) Estimate a simple regression model relating the growth in real per capita consumption (of nondurables and services) to the growth in real per capita disposable income. Use the change in the logarithms in both cases. Report the results in the
C10.6 Use the data in FERTIL3.RAW for this exercise. (i) Regress gfr, on t and t and save the residuals. This gives a detrended gfr, say, gf (ii) Regress gf, on all of the variables in equation (10.35), including t and t. Compare the R-squared with that from (10.35). What do you con- clude? (iii)
C10.5 Use the data in EZANDERS.RAW for this exercise. The data are on monthly unemployment claims in Anderson Township in Indiana, from January 1980 through November 1988. In 1984, an enterprise zone (EZ) was located in Anderson (as well as other cities in Indiana). (See Papke [1994] for details.)
C10.4 Use the data in FERTIL3.RAW to verify that the standard error for the LRP in equation (10.19) is about .030.
C10.3 Add the variable log(prgnp) to the minimum wage equation in (10.38). Is this variable significant? Interpret the coefficient. How does adding log(prgnp) affect the esti- mated minimum wage effect?
C10.2 Use the data in BARIUM.RAW for this exercise. (i) Add a linear time trend to equation (10.22). Are any variables, other than the trend, statistically significant? (ii) In the equation estimated in part (i), test for joint significance of all variables except the time trend. What do you
C10.1 In October 1979, the Federal Reserve changed its policy of targeting the money supply and instead began to focus directly on short-term interest rates. Using the data in INTDEF.RAW, define a dummy variable equal to 1 for years after 1979. Include this dummy in equation (10.15) to see if there
10.7 In Example 10.4, we wrote the model that explicitly contains the long-run propen- sity, 60, as gfr, = a + ope, + 8(pe-1 - pe,) + 2(pe-2-pe,) + u,, where we omit the other explanatory variables for simplicity. As always with multiple regression analysis, 0, should have a ceteris paribus
10.6 In Example 10.4, we saw that our estimates of the individual lag coefficients in a distributed lag model were very imprecise. One way to alleviate the multicollinearity prob- lem is to assume that the 8, follow a relatively simple pattern. For concreteness, consider a model with four lags:
10.5 Suppose you have quarterly data on new housing starts, interest rates, and real per capita income. Specify a model for housing starts that accounts for possible trends and seasonality in the variables.
10.4 When the three event indicators befile6, affile6, 1afdec6 are dropped from equa- tion (10.22), we obtain R2 = .281 and R2 = .264. Are the event indicators jointly signif- icant at the 10% level?
10.3 Suppose y, follows a second order FDL model: y = +82 +8-1 + 822-2+u.that Let z* denote the equilibrium value of z, and let y* be the equilibrium value of y,, such y* = a + 8oz* + 8z* + 8z*. Show that the change in y*, due to a change in 2*, equals the long-run propensity times the change in
10.2 Let gGDP, denote the annual percentage change in gross domestic product and let int, denote a short-term interest rate. Suppose that gGDP, is related to interest rates by gGDP, = a + 8,int, + 8,int,-1 + u,, == where u, is uncorrelated with int, int,-1, and all other past values of interest
10.1 Decide if you agree or disagree with each of the following statements and give a brief explanation of your decision: (i) PRO Like cross-sectional observations, we can assume that most time series observations are independently distributed. The OLS estimator in a time series regression is

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