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Introductory Econometrics A Modern Approach 6th Edition Jeffrey M. Wooldridge - Solutions

Let a be an n 3 1 nonrandom vector and let u be an n 3 1 random vector with E1uur 2 5 In. Show that E3tr1auu9a92 4 5 gn i51a2 i .
Let A be an n 3 n symmetric, positive definite matrix. Show that if P is any n 3 n nonsingular matrix, then PrAP is positive definite.
(i) Show that if A is an n 3 n symmetric, positive definite matrix, then A must have strictly positive diagonal elements.(ii) Write down a 2 3 2 symmetric matrix with strictly positive diagonal elements that is not positive definite.
(i) Use the definition of inverse to prove the following: if A and B are n 3 n nonsingular matrices, then 1AB2 21 5 B21 A21.(ii) If A, B, and C are all n 3 n nonsingular matrices, find 1ABC2 21 in terms of A21, B21, and C21.
(i) Use the properties of trace to argue that tr1ArA2 5 tr1AAr 2 for any n 3 m matrix A.(ii) For A 5 c 2 0 2 1 0 3 0d , verify that tr1ArA2 5 tr1AAr 2.
Let X be any n 3 k matrix. Show that XrX is a symmetric matrix.
If A and B are n 3 n diagonal matrices, show that AB 5 BA.
(i) Find the product AB using A 5 c 2 2 1 7 24 5 0d , B 5 £0 1 6 1 8 0 3 0 0S.(ii) Does BA exist?
The new management at a bakery claims that workers are now more productive than they were under old management, which is why wages have “generally increased.” Let Wb i be Worker i’s wage under the old management and let Wa i be Worker i’s wage after the change. The difference is Di ; Wa i 2
You are hired by the governor to study whether a tax on liquor has decreased average liquor consumption in your state. You are able to obtain, for a sample of individuals selected at random, the difference in liquor consumption (in ounces) for the years before and after the tax. For person i who is
Let Y denote a Bernoulli(u) random variable with 0 , u , 1. Suppose we are interested in estimating the odds ratio, g 5 u/11 2 u 2, which is the probability of success over the probability of failure.Given a random sample 5Y1,c, Yn6, we know that an unbiased and consistent estimator of u is Y, the
For positive random variables X and Y, suppose the expected value of Y given X is E1Y0X2 5 uX. The unknown parameter u shows how the expected value of Y changes with X.(i) Define the random variable Z 5 Y/X. Show that E1Z2 5 u. [Hint: Use Property CE.2 along with the law of iterated expectations,
Let Y denote the sample average from a random sample with mean m and variance s2. Consider two alternative estimators of m: W1 5 3 1n 2 12/n4Y and W2 5 Y/2.(i) Show that W1 and W2 are both biased estimators of m and find the biases. What happens to the biases as n S `? Comment on any important
This is a more general version of Problem C.1. Let Y1, Y2,c, Yn be n pairwise uncorrelated random variables with common mean m and common variance s2. Let Y denote the sample average.(i) Define the class of linear estimators of m by Wa 5 a1Y1 1 a2Y2 1 c 1 anYn, where the ai are constants. What
Let Y1, Y2, Y3, and Y4 be independent, identically distributed random variables from a population with mean m and variance s2. Let Y 5 1 41Y1 1 Y2 1 Y3 1 Y4 2 denote the average of these four random variables.(i) What are the expected value and variance of Y in terms of m and s2?(ii) Now, consider
(i) Let X be a random variable taking on the values 21 and 1, each with probability 1/2. Find E1X2 and E1X2 2.(ii) Now let X be a random variable taking on the values 1 and 2, each with probability 1/2. Find E1X2 and E11/X2.(iii) Conclude from parts (i) and (ii) that, in general, E3g1X2 4 2 g3E1X2
Suppose that at a large university, college grade point average, GPA, and SAT score, SAT, are related by the conditional expectation E1GPA0SAT2 5 .70 1 .002 SAT.(i) Find the expected GPA when SAT 5 800. Find E1GPA0SAT 5 1,4002. Comment on the difference.(ii) If the average SAT in the university is
(Requires calculus) Let X denote the prison sentence, in years, for people convicted of auto theft in a particular state in the United States. Suppose that the pdf of X is given by f 1x2 5 11/92x2, 0 , x , 3.Use integration to find the expected prison sentence.
Just prior to jury selection for O. J. Simpson’s murder trial in 1995, a poll found that about 20% of the adult population believed Simpson was innocent (after much of the physical evidence in the case had been revealed to the public). Ignore the fact that this 20% is an estimate based on a
For a randomly selected county in the United States, let X represent the proportion of adults over age 65 who are employed, or the elderly employment rate. Then, X is restricted to a value between zero and one.Suppose that the cumulative distribution function for X is given by F1x2 5 3x2 2 2x3 for
Let X be a random variable distributed as Normal(5,4). Find the probabilities of the following events:(i) P1X # 62.(ii) P1X . 42.(iii) P1 0X 2 50 . 12.
Consider the line y 5 b0 1 b1x.(i) Let 1x1, y1 2 and 1x2, y2 2 be two points on the line. Show that 1x, y2 is also on the line, where x 5 1x1 1 x2 2/2 is the average of the two values and y 5 1y1 1 y2 2/2.(ii) Extend the result of part (i) to n points on the line, 5 1xi, yi 2: i 5 1, p , n6.
Suppose that in a particular state a standardized test is given to all graduating seniors. Let score denote a student’s score on the test. Someone discovers that performance on the test is related to the size of the student’s graduating high school class. The relationship is quadratic:score 5
Suppose the yield of a certain crop (in bushels per acre) is related to fertilizer amount (in pounds per acre) as yield 5 120 1 .19!fertilizer.(i) Graph this relationship by plugging in several values for fertilizer.(ii) Describe how the shape of this relationship compares with a linear
Let grthemp denote the proportionate growth in employment, at the county level, from 1990 to 1995, and let salestax denote the county sales tax rate, stated as a proportion. Interpret the intercept and slope in the equation grthemp 5 .043 2 .78 sales tax.
In Example A.2, quantity of compact discs was related to price and income by quantity 5 120 2 9.8 price 1 .03 income. What is the demand for CDs if price 5 15 and income 5 200? What does this suggest about using linear functions to describe demand curves?
This question asks you to study the so-called Beveridge Curve from the perspective of cointegration analysis. The U.S. monthly data from December 2000 through February 2012 are in BEVERIDGE.(i) Test for a unit root in urate using the usual Dickey-Fuller test (with a constant) and the augmented DF
Use the data in MINWAGE.DTA for sector 232 to answer the following questions.(i) Confirm that lwage232t and lemp232t are best characterized as I(1) processes. Use the augmented DF test with one lag of gwage232 and gemp232, respectively, and a linear time trend. Is there any doubt that these series
Use the data in TRAFFIC2 for this exercise. These monthly data, on traffic accidents in California over the years 1981 to 1989, were used in Computer Exercise C11 in Chapter 10.(i) Using the standard Dickey-Fuller regression, test whether ltotacct has a unit root. Can you reject a unit root at the
This exercise also uses the data from VOLAT. Computer Exercise C11 studies the long-run relationship between stock prices and industrial production. Here, you will study the question of Granger causality using the percentage changes.(i) Estimate an AR(3) model for pcipt, the percentage change in
Use the data in VOLAT for this exercise.(i) Confirm that lsp500 5 log1sp5002 and lip 5 log1ip2 appear to contain unit roots. Use DickeyFuller tests with four lagged changes and do the tests with and without a linear time trend.(ii) Run a simple regression of lsp500 on lip. Comment on the sizes of
Use CONSUMP for this exercise.(i) Let yt be real per capita disposable income. Use the data through 1989 to estimate the model yt 5 a 1 bt 1 ryt21 1 ut and report the results in the usual form.(ii) Use the estimated equation from part (i) to forecast y in 1990. What is the forecast error?(iii)
Use the data in FERTIL3 for this exercise.(i) Graph gfr against time. Does it contain a clear upward or downward trend over the entire sample period?(ii) Using the data through 1979, estimate a cubic time trend model for gfr (that is, regress gfr on t, t 2, and t 3, along with an intercept).
Use the data in BARIUM for this exercise.(i) Estimate the linear trend model chnimpt 5 a 1 bt 1 ut, using the first 119 observations(this excludes the last 12 months of observations for 1988). What is the standard error of the regression?(ii) Now, estimate an AR(1) model for chnimp, again using all
Use the data in PHILLIPS to answer these questions.(i) Estimate the models in (18.48) and (18.49) using the data through 1997. Do the parameter estimates change much compared with (18.48) and (18.49)?(ii) Use the new equations to forecast unem1998; round to two places after the decimal. Which
Use INTQRT for this exercise.(i) In Example 18.7, we estimated an error correction model for the holding yield on six-month T-bills, where one lag of the holding yield on three-month T-bills is the explanatory variable. We assumed that the cointegration parameter was one in the equation hy6t 5 a 1
In testing for cointegration between gfr and pe in Example 18.5, add t 2to equation (18.32) to obtain the OLS residuals. Include one lag in the augmented DF test. The 5% critical value for the test is 24.15.
Use the data in VOLAT for this exercise.(i) Estimate an AR(3) model for pcip. Now, add a fourth lag and verify that it is very insignificant.(ii) To the AR(3) model from part (i), add three lags of pcsp to test whether pcsp Granger causes pcip. Carefully, state your conclusion.(iii) To the model in
Use the data in HSEINV for this exercise.(i) Test for a unit root in log1invpc2, including a linear time trend and two lags of Dlog1invpct 2.Use a 5% significance level.(ii) Use the approach from part (i) to test for a unit root in log1price2.(iii) Given the outcomes in parts (i) and (ii), does it
Use the data in WAGEPRC for this exercise. Problem 5 in Chapter 11 gave estimates of a finite distributed lag model of gprice on gwage, where 12 lags of gwage are used.(i) Estimate a simple geometric DL model of gprice on gwage. In particular, estimate equation(18.11) by OLS. What are the estimated
Let 5yt 6 be an I(1) sequence. Suppose that g^ n is the one-step-ahead forecast of Dyn11 and let f^n 5 g^ n 1 yn be the one-step-ahead forecast of yn11. Explain why the forecast errors for forecasting Dyn11 and yn11 are identical.
Suppose that yt follows the model
Let gMt be the annual growth in the money supply and let unemt be the unemployment rate. Assuming that unemt follows a stable AR(1) process, explain in detail how you would test whether gM Granger causes unem.
Using the monthly data in VOLAT, the following model was estimated:pcip 5 1.54 1 .344 pcip21 1 .074 pcip22 1 .073 pcip23 1 .031 pcsp21 1.562 1.0422 1.0452 1.0422 1.0132 n 5 554, R2 5 .174, R2 5 .168, where pcip is the percentage change in monthly industrial production, at an annualized rate, and
Suppose the process 5 1xt, yt 2: t 5 0, 1, 2, p6 satisfies the equations yt 5 bxt 1 ut and Dxt 5 gDxt21 1 vt, where E1ut 0It21 2 5 E1vt 0It21 2 5 0, It21 contains information on x and y dated at time t 2 1 and earlier, b 2 0, and 0g0 , 1 [so that xt, and therefore yt, is I(1)]. Show that these two
Suppose that 5yt 6 and 5zt 6 are I(1) series, but yt 2 bzt is I(0) for some b 2 0. Show that for any d 2b, yt 2 dzt must be I(1).
Use the data in CRIME1 to answer this question.(i) For the OLS estimates reported in Table 17.5, find the heteroskedasticity-robust standard errors.In terms of statistical significance of the coefficients, are there any notable changes?(ii) Obtain the fully robust standard errors—that is, those
Use the data set in ALCOHOL, obtained from Terza (2002), to answer this question. The data, on 9,822 men, includes labor market information, whether the man abuses alcohol, and demographic and background variables. In this question you will study the effects of alcohol abuse on employ, which is a
Use the data in HTV to answer this question.(i) Using OLS on the full sample, estimate a model for log(wage) using explanatory variables educ, abil, exper, nc, west, south, and urban. Report the estimated return to education and its standard error.(ii) Now estimate the equation from part (i) using
Use the data in CHARITY to answer these questions.(i) The variable respond is a binary variable equal to one if an individual responded with a donation to the most recent request. The database consists only of people who have responded at least once in the past. What fraction of people responded
Use the data in CPS91 for this exercise. These data are for married women, where we also have information on each husband’s income and demographics.(i) What fraction of the women report being in the labor force?(ii) Using only the data for working women—you have no choice—estimate the wage
Use the data in SMOKE for this exercise.(i) The variable cigs is the number of cigarettes smoked per day. How many people in the sample do not smoke at all? What fraction of people claim to smoke 20 cigarettes a day? Why do you think there is a pileup of people at 20 cigarettes?(ii) Given your
Use the data in APPLE for this exercise. These are telephone survey data attempting to elicit the demand for a (fictional) “ecologically friendly” apple. Each family was (randomly) presented with a set of prices for regular apples and the ecolabeled apples. They were asked how many pounds of
The file JTRAIN2 contains data on a job training experiment for a group of men. Men could enter the program starting in January 1976 through about mid-1977. The program ended in December 1977. The idea is to test whether participation in the job training program had an effect on unemployment
Use the MROZ data for this exercise.(i) Using the 428 women who were in the workforce, estimate the return to education by OLS including exper, exper2, nwifeinc, age, kidslt6, and kidsge6 as explanatory variables. Report your estimate on educ and its standard error.(ii) Now, estimate the return to
Use the data in RECID to estimate the model from Example 17.4 by OLS, using only the 552 uncensored durations. Comment generally on how these estimates compare with those in Table 17.6.
Refer to Table 13.1 in Chapter 13. There, we used the data in FERTIL1 to estimate a linear model for kids, the number of children ever born to a woman.(i) Estimate a Poisson regression model for kids, using the same variables in Table 13.1. Interpret the coefficient on y82.(ii) What is the
Consider a family saving function for the population of all families in the United States:sav 5 b0 1 b1inc 1 b2hhsize 1 b3educ 1 b4age 1 u, where hhsize is household size, educ is years of education of the household head, and age is age of the household head. Assume that E1u/inc,hhsize,educ,age2 5
Let mvpi be the marginal value product for worker i, which is the price of a firm’s good multiplied by the marginal product of the worker. Assume that log1mvpi 2 5 b0 1 b1xi1 1 p 1 bkxik 1 ui wagei 5 max1mvpi,minwagei 2, where the explanatory variables include education, experience, and so on,
(Requires calculus)(i) Suppose in the Tobit model that x1 5 log1z1 2, and this is the only place z1 appears in x.Show that'E1y0y . 0,x2'z1 5 1b1/z1 2 51 2 l1xb/s2 3xb/s 1 l1xb/s2 4 6, [17.52]where b1 is the coefficient on log1z1 2.(ii) If x1 5 z1, and x2 5 z 21, show that'E1y0y . 0,x2'z1 5 1b1 1
Let grad be a dummy variable for whether a student-athlete at a large university graduates in five years. Let hsGPA and SAT be high school grade point average and SAT score, respectively. Let study be the number of hours spent per week in an organized study hall. Suppose that, using data on 420
21.4 Consider the capital asset pricing model (CAPM) discussed in Section 2.10 (Eq. (2.34)) and its empirical counterpart, the market model given in Eq. (2.35). Suppose we estimate the market model for, say, 100 securities, as follows: Rit – rft = Bi (Rmt – rft) + uit where Rit = rate of return
21.1 Refer to the airlines’ cost data. Consider the following log-linear cost function: 12 3 4 lnTC B B Q B PF B LF u ln ln ln where ln stands for natural log. (a) Estimate individual log-linear cost function for each airline. (b) Estimate the SURE model of the log-linear cost function.
20.3 Use the patent data given in Table 12.1, which can be downloaded from the book’s website. Treating the number of patents granted in year 1991 as the dependent variable and the data on R&D expenditure for 1991 and the industry and country dummies as regressors, estimate the 20th, 60th and
20.2 For the wage data considered in this chapter, use the (natural) log of wage and estimate (a) An OLS regression (b) the 25th, 50th and 75th quantile regressions and compare your results.
19.8 Consider the model given in Eqs. (19.35) and (19.36). Obtain data on the crime rate, law enforcement spending and Gini coeffi cient for any country of your choice, or for a group of countries, or for a group of states within a country and estimate the two equations by OLS. How would you use IV
19.7 In his article, “Instrumental-Variable Estimation of Count Data Models: Applications to Models of Cigarette Smoking Behavior”, Review of Economics and Statistics (1997, pp. 586–93), John Mullahy wanted to fi nd out if a mother’s smoking during pregnancy adversely aff ected her
19.6 Continue with the wage function discussed in the text. Th e raw data contains information on several variables besides those included in Eq. (19.39). For example, there is information on marital status (single, married, and divorced), ASVAB scores on arithmetic reasoning and word knowledge,
19.5 Return to the wage regression discussed in the text. Empirical evidence shows that the wage–work experience (wexp) profi le is concave – wages increase with work experience, but at a diminishing rate. To see if this is the case, one can add the wexp2 variable to the wage function Eq.
19.4 Verify Eq. (19.29).
19.2 Verify Eq. (19.11).
18.6 Th e book by Klein and Moeschberger gives several data sets from the fi elds of biology and health.13 Th ese data can be accessed from the website of the book. Pick one or more data sets from this book and estimate the hazard function using one or more probability distributions discussed in
18.5 Th e Kleinbaum text cited in this chapter gives several data sets on survival analysis in Appendix B. Obtain one or more of these data sets and estimate appropriate SA model(s) so that you are comfortable in dealing with duration models.
18.4 See Table 18.10 on the companion website.12 In a cancer drug trial, 28 patients were given a drug (drug =1) and 20 patients received a placebo (drug = 0). Th e age distribution of the patients ranged from 47 to 67 years. Th e objective of this exercise is to analyze time until death, measured
18.3 Table 18.9 gives data on 14 people aged 15 and older on the following variables: Minutes: time spent running on a treadmill, in minutes Age: age in years Weight: weight in pounds Gender: 1 for female, 0 for male Censored: 0 if censored, 1 if not censored.(a) What is the expected relationship
18.2 Which of the regressors given in Section 18.1 are time-variant and which are time-invariant? Suppose you treat all the regressors as time-invariant. Estimate the exponential, Weibull and PH survival models and comment on your results.
18.1 Using Durat as the dependent variable, estimate an OLS regression in relation to the regressors given in Table 18.1 and interpret your results. How do these results compare with those obtained from the exponential, Weibull and PH models?
A common method for estimating Engel curves is to model expenditure shares as a function of total expenditure, and possibly demographic variables. A common specification has the form sgood 5 b0 1 b1ltotexpend 1 demographics 1 u, where sgood is the fraction of spending on a particular good out of
Use the entire panel data set in AIRFARE for this exercise. The demand equation in a simultaneous equations unobserved effects model is log1passenit 2 5 ut1 1 a1log1fareit 2 1 ai1 1 uit1, where we absorb the distance variables into ai1.(i) Estimate the demand function using fixed effects, being
For this exercise, use the data in AIRFARE, but only for the year 1997.(i) A simple demand function for airline seats on routes in the United States is log1passen2 5 b10 1 a1log1fare2 1 b11log1dist2 1 b12 3log1dist2 42 1 u1, where passen 5 average passengers per day, fare 5 average airfare, and
Use the data set in FISH, which comes from Graddy (1995), to do this exercise. The data set is also used in Computer Exercise C9 in Chapter 12. Now, we will use it to estimate a demand function for fish.(i) Assume that the demand equation can be written, in equilibrium for each time period, as
Refer to Example 13.9 and the data in CRIME4,(i) Suppose that, after differencing to remove the unobserved effect, you think Dlog1polpc2 is simultaneously determined with Dlog1crmrte2; in particular, increases in crime are associated with increases in police officers. How does this help to explain
Use the data in CEMENT for this exercise.(i) A static (inverse) supply function for the monthly growth in cement price (gprc) as a function of growth in quantity (gcem) is gprct 5 a1gcemt 1 b0 1 b1gprcpet 1 b2febt 1 p 1 b12dect 1 us t,
Use the Economic Report of the President (2005 or later) to update the data in CONSUMP, at least through 2003. Reestimate equation (16.35). Do any important conclusions change?
Use the data in CONSUMP for this exercise.(i) In Example 16.7, use the method from Section 15-5 to test the single overidentifying restriction in estimating (16.35). What do you conclude?(ii) Campbell and Mankiw (1990) use second lags of all variables as IVs because of potential data measurement
Use the data in OPENNESS for this exercise.(i) Because log(pcinc) is insignificant in both (16.22) and the reduced form for open, drop it from the analysis. Estimate (16.22) by OLS and IV without log(pcinc). Do any important conclusions change?(ii) Still leaving log(pcinc) out of the analysis, is
Use MROZ for this exercise.(i) Reestimate the labor supply function in Example 16.5, using log(hours) as the dependent variable. Compare the estimated elasticity (which is now constant) to the estimate obtained from equation (16.24) at the average hours worked.(ii) In the labor supply equation from
Use SMOKE for this exercise.(i) A model to estimate the effects of smoking on annual income (perhaps through lost work days due to illness, or productivity effects) is log1income2 5 b0 1 b1cigs 1 b2educ 1 b3age 1 b4age2 1 u1, where cigs is number of cigarettes smoked per day, on average. How do you
How big is the effect of per-student school expenditures on local housing values? Let HPRICE be the median housing price in a school district and let EXPEND be per-student expenditures. Using panel data for the years 1992, 1994, and 1996, we postulate the model lHPRICEit 5 ut 1 b1lEXPENDit 1
For a large university, you are asked to estimate the demand for tickets to women’s basketball games.You can collect time series data over 10 seasons, for a total of about 150 observations. One possible model is lATTENDt 5 b0 1 b1lPRICEt 1 b2WINPERCt 1 b3RIVALt 1 b4WEEKENDt 1 b5t 1 ut,
Consider a linear probability model for whether employers offer a pension plan based on the percentage of workers belonging to a union, as well as other factors:pension 5 b0 1 b1percunion 1 b2avgage 1 b3avgeduc 1 b4percmale 1 b5percmarr 1 u1.(i) Why might percunion be jointly determined with
A simple model to determine the effectiveness of condom usage on reducing sexually transmitted diseases among sexually active high school students is infrate 5 b0 1 b1conuse 1 b2percmale 1 b3avginc 1 b4city 1 u1, where infrate 5 the percentage of sexually active students who have contractedvenereal
Suppose that annual earnings and alcohol consumption are determined by the SEM log1earnings2 5 b0 1 b1alcohol 1 b2educ 1 u1 alcohol 5 g0 1 g1log1earnings2 1 g2educ 1 g3log1price2 1 u2, where price is a local price index for alcohol, which includes state and local taxes. Assume that educ and price
In Problem 3 of Chapter 3, we estimated an equation to test for a tradeoff between minutes per week spent sleeping (sleep) and minutes per week spent working (totwrk) for a random sample of individuals. We also included education and age in the equation. Because sleep and totwrk are jointly chosen
Let corn denote per capita consumption of corn in bushels at the county level, let price be the price per bushel of corn, let income denote per capita county income, and let rainfall be inches of rainfall during the last corn-growing season. The following simultaneous equations model imposes the
Write a two-equation system in “supply and demand form,” that is, with the same variable yt(typically,“quantity”) appearing on the left-hand side:y1 5 a1y2 1 b1z1 1 u1 y1 5 a2y2 1 b2z2 1 u2.(i) If a1 5 0 or a2 5 0, explain why a reduced form exists for y1. (Remember, a reduced form
Use the data in CATHOLIC to answer this question. The model of interest is math12 5 b0 1 b1cathhs 1 b2lfaminc 1 b3motheduc 1 b4fatheduc 1 u, where cathhs is a binary indicator for whether a student attends a Catholic high school.(i) How many students are in the sample? What percentage of these
The data set in VOUCHER, which is a subset of the data used in Rouse (1998), can be used to estimate the effect of school choice on academic achievement. Attendance at a choice school was paid for by a voucher, which was determined by a lottery among those who applied. The data subset was chosen so
Use the data in HTV for this exercise.(i) Run a simple OLS regression of log(wage) on educ. Without controlling for other factors, what is the 95% confidence interval for the return to another year of education?(ii) The variable ctuit, in thousands of dollars, is the change in college tuition
The purpose of this exercise is to compare the estimates and standard errors obtained by correctly using 2SLS with those obtained using inappropriate procedures. Use the data file WAGE2.(i) Use a 2SLS routine to estimate the equation log1wage2 5 b0 1 b1educ 1 b2exper 1 b3tenure 1 b4black 1 u, where

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