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Econometrics 1st Edition Bruce Hansen - Solutions

Show (27.7).
Take themodel e S=1{X'y+u>0} Y = { X'+e if S=1 missing if S=0 02 021 (u) ~ N(0, (021 Show E[Y | X,S=1]=X'B+021(X'y). 1 })
A latent variable Y ¤ is generated bywhere X is scalar. Assume °0 È 0 and °1 È 0. The parameters are ¯,°0,°1. Find the log-likelihood function for the conditional distribution of Y given X. Y*Bo+XB1+e e|X~ N(0,0 (X)) (X) = Yo+X y Y = max(Y*,0).
For the truncated conditional mean (27.3) propose a NLLS estimator of (¯,¾).
For the censored conditional mean (27.2) propose a NLLS estimator of (¯,¾).
Take themodelLet b¯ denote the OLS estimator for ¯ based on an available sample.(a) Suppose that an observation is in the sample only if X1 È 0 where X1 is an element of X. Is b¯consistent for ¯? Obtain an expression for its probability limit.(b) Suppose that an observation is in the sample
Take themodelIn this model, we say that Y is capped from above. Suppose you regress Y on X. Is OLS consistent for ¯?Describe the nature of the effect of the mis-measured observation on the OLS estimator. Y* = X'+e e~N (0,0) Y* if Y* T Y = missing if Y*>T
Derive (27.2) and (27.3). Hint: Use Theorems 5.7 and 5.8 of Probability and Statistics for Economists.
Use the Koppelman dataset. Estimate a general multinomial probit model similar to that reported in Table 26.1 but with the following modifications. For each case report the estimated coefficients and standard errors for the cost and time variables, the log-likelihood, and describe how the results
Use the Koppelman dataset. Estimate a mixed logit model similar to that reported in Table 26.1 but with the following modifications. For each case report the estimated coefficients and standard errors for the cost and time variables, the log-likelihood, and describe how the results change.(a)
Use the Koppelman dataset. Estimate a nested logit model similar to those reported in Table 26.1 but with the following modifications. For each case report the estimated coefficients and standard errors for the cost and time variables, the log-likelihood, and describe how the results change.(a)
Use the Koppelman dataset. Estimate conditional logit models similar to those reported in Table 26.1 but with the following modifications. For each case report the estimated coefficients and standard errors for the cost and time variables, the log-likelihood, and describe how the results change.(a)
Use the cps09mar dataset and the subset of women. Estimate a nested logit model for marriage status as a function of age. Describe how you decide on the grouping of alternatives.
Use the cps09mar dataset and the subset of women with ages up to 35. Estimate a multinomial logit model for marriage status as linear functions of age and education. Interpret your results.
Use the cps09mar dataset and the subset of men. Estimate a multinomial logit model for marriage status similar to Figure 26.1 as a function of age. How do your findings compare with those for women?
Take the nested logit model. For groups j and `, showthat the ratio Pj /P` is independent of variables in the other groups. What does this mean?
Take the nested logit model. If k and ` are alternatives in the same group j , show that the ratio Pjk/Pj` is independent of variables in the other groups. What does this mean?
In the conditional logit model with no alternative-invariant regressorsW showthat (26.11)implies Pj (x)/P`(x) Æ exp³¡xj ¡x`¢0°´.
Show (26.11).
In the conditional logit model find an estimator for AMEj j .
Show that Pj (w,x) in the conditional logit model (26.8) only depends on the coefficient differences ¯j ¡¯J and variable differences xj ¡xJ .
For the conditional logit model (26.8) show that the marginal effects are (26.9) and (26.10).
Show that (26.8) holds for the conditional logit model.
For the multinomial logit model (26.2) show that the marginal effects equal (26.4).
Show that Pj (x) in the multinomial logit model (26.2) only depends on the coefficient differences ¯j ¡¯J .
For the multinomial logit model (26.2) show that 0 · Pj (x) · 1 and PJ jÆ1 Pj (x) Æ 1.
Replicate the previous exercise but with the subset of women. Interpret the results.
Use the cps09mar dataset and the subset of men. Set Y as in the previous question.Estimate a binary choice model for Y as a possibly nonlinear function of age, a linear function of education, and including indicators for Black individuals and for Hispanic individuals. Report the coefficient
Use the cps09mar dataset and the subset of women with a college degree. Set Y Æ 1 if marital equals 1, 2, or 3, and set Y Æ 0 otherwise. Estimate a binary choice model for Y as a possibly nonlinear function of age. Describe the motivation for the model you use. Plot the estimated response
Replicate the previous exercise but with the subset of women. Interpret the results.
Use the cps09mar dataset and the subset ofmen. Set Y Æ 1 if the individual is a member of a labor union (union=1) and Y Æ 0 otherwise. Estimate a probitmodel as a linear function of age, education, and indicators for Black individuals and forHispanic individuals. Report the coefficient estimates
Take the heteroskedastic nonparametric binary choice model Y ¤Æm(X)Åe e j X »N¡0,¾2 (X)¢Y Æ Y ¤1©Y ¤È 0ª.The observables are {Yi ,Xi : i Æ 1, ...,n}. The functionsm(x) and ¾2(x) are nonparametric.(a) Find a formula for the response probability .(b) Arem(x) and ¾2(x) both identified?
Take the endogenous probit model of Section 25.12.(a) Verify equation (25.16).(b) Explain why " is independent of e2 and Y2.(c) Verify that the conditional distribution of Y ¤1 is N¡¹(µ) ,¾2"¢.
Show how to use NLLS to estimate a probit model.
Show (25.14). In the logit model show that ¡ the right hand side of (25.14) simplies to Y ¡¤¡X0¯¢¢2.
Find the first-order condition for the probitMLE b¯probit.
Find the first-order condition for the logit MLE b¯logit.
Find the first-order condition for ¯0 from the population maximization problem (25.8).
For the normal distribution ©(x) verify that(a) hprobit(x) Æ d dx log©(x) Æ ¸(x) where ¸(x) Æ Á(x)/©(x).(b) Hprobit(x) Æ ¡ d2 dx2 log©(x) Æ ¸(x) (x Å¸(x)).Exercise 25.7(a) Verify equations (25.6) and (25.7).(b) Verify the assertion that H(x) È 0 implies thatHn(¯) È 0 globally in
For the logistic distribution ¤(x) Æ¡1Åexp(¡x)¢¡1 verify that(a) d dx¤(x) Æ¤(x)(1¡¤(x)).(b) hlogit(x) Æ d dx log¤(x) Æ 1¡¤(x).(c) Hlogit(x) Æ ¡ d2 dx2 log¤(x) Æ¤(x) (1¡¤(x)) .(d)¯¯Hlogit(x)¯¯· 1.
Verify (25.5), that ¼(Y j X) ÆG¡Z0¯¢.
Show (25.1) and (25.2).
Jackson estimates a logit regression where the primary regressor is measured in dollars.Julie esitmates a logit regression with the same sample and dependent variable, but measures the primary regressor in thousands of dollars. What is the difference in the estimated slope coefficients?
Emily estimates a probit regression setting her dependent variable to equal Y Æ 1 for a purchase and Y Æ 0 for no purchase. Using the same data and regressors, Jacob estimates a probit regression setting the dependent variable to equal Y Æ 1 if there is no purchase and Y Æ 0 for a purchase.What
Using the cps09mar dataset estimate similarly to Figure 24.6 the quantile regressions for log wages on a 5th- order polynomial in experience for college-educated Black women. Repeat for college-educated white women. Interpret your findings.
Take the Duflo, Dupas, and Kremer (2011) dataset DDK2011 and the subsample of students for which tracking=1. Estimate linear quantile regressions of totalscore on percentile (the latter is the student’s test score before the school year). Calculate standard errors by clustered bootstrap. Do the
Using the cps09mar dataset take the sample of Hispanic women with education 11 years or higher. Estimate linear quantile regression functions for log wages on education. Interpret.
Using the cps09mar dataset take the sample of Hispanicmen with education 11 years or higher. Estimate linear quantile regression functions for log wages on education. Interpret your findings.
Take the treatment response setting of Theorem 24.5. Suppose h(0,X2,U) Æ 0, meaning that the response variable Y is zero whenever there is no treatment. Show that Assumption 24.1.3 is not necessary for Theorem 24.5.
Show under correct specification that ¿ Æ E£X X0Ã2¿¤satisfies the simplification ¿ Æ¿(1¡¿)Q.
Show (24.19).
Suppose X1 and X2 are binary. Find Q¿[Y j X1,X2].
Suppose X is binary. Show that Q¿[Y j X] is linear in X.
Prove (24.14) in Theorem 24.2.
Prove (24.13) in Theorem 24.2.
You are interested in estimating the equation Y Æ X0¯Åe. You believe the regressors are exogenous, but you are uncertain about the properties of the error. You estimate the equation both by least absolute deviations (LAD) and OLS. A colleague suggests that you should prefer the OLS estimate,
Take themodel Y Æ X0¯Åe where the distribution of e given X is symmetric about zero.(a) Find E[Y j X] and med[Y j X].(b) Do OLS and LAD estimate the same coefficient ¯ or different coefficients?(c) Under which circumstances would you prefer LAD over OLS? Under which circumstances would you
Define Ã(x) Æ ¿¡1{x Ç 0}. Let µ satisfy E£Ã(Y ¡µ)¤Æ 0. Is µ a quantile of the distribution of Y ?
Prove (24.5) in Theorem 24.1.
Prove (24.4) in Theorem 24.1.
In Exercise 9.26, you estimated a cost function on a cross-section of electric companies.Consider the nonlinear specification logTC Æ ¯1 Å¯2 logQ Å¯3¡logPLÅlogPK ÅlogPF¢Å¯4 logQ 1Åexp¡¡¡logQ ¡°¢¢ Åe. (23.11)This model is called a smooth threshold model. For values of logQ much
The file RR2010 contains the U.S. observations from the Reinhart and Rogoff (2010). The data set has observations on real GDP growth, debt/GDP, and inflation rates. Estimate the model (23.4)setting Y as the inflation rate and X as the debt ratio.
The file PSS2017 contains a subset of the data from Papageorgiou, Saam, and Schulte(2017). For a robustness check they re-estimated their CES production function using approximated capital stocks rather than capacities as their input measures. Estimate the model (23.3) using this alternative
Suppose that Y Æ m(X,µ)Åe with E[e j X] Æ 0, bµ is the NLLS estimator, and bV the estimator of var£bµ¤. You are interested in the CEF E[Y j X Æ x] Æ m(x) at some x. Find an asymptotic 95%confidence interval form(x).
Take themodel Y Æ exp¡X0µ¢Åe with E[Ze] Æ 0, where X is k £1 and Z is `£1.(a) What relationship between ` and k is necessary for identification of µ?(b) Describe how to estimate µ by GMM.(c) Describe an estimator of the asymptotic covariance matrix.
Take themodel Y Æm(X,µ)Åe with e j X » N(0,¾2). Find theMLE for µ and ¾2.
Take themodel Y Æ ¯1 exp¡¯2X¢Åe with E[e j X] Æ 0.(a) Are the parameters (¯1,¯2) identified?(b) Find an expression to calculate the covariancematrix of the NLLS estimatiors ( b¯1, b¯2).
Take themodel Y Æ¯1¯2 Å¯3XÅe with E[e j X] Æ 0.(a) Are the parameters (¯1,¯2,¯3) identified?(b) If not, what parameters are identified? How would you estimate the model?
Take the model Y (¸) Æ ¯0Å¯1X Åe with E[e j X] Æ 0 where Y (¸) is the Box-Cox transformation of Y .(a) Is this a nonlinear regressionmodel in the parameters (¸,¯0,¯1)? (Careful, this is tricky.)
Take themodel Y Æ exp(µ)Åe with E[e] Æ 0.(a) Is the CEF linear or nonlinear in µ? Is this a nonlinear regression model?(b) Is there a way to estimate the model using linear methods? If so, explain howto obtain an estimator bµ for µ.(c) Is your answer in part (b) the same as the NLLS
set g (u) Æ 1¡cos(u).(a) Sketch g (u). Is g (u) continuous? Differentiable? Second differentiable?(b) Find the functions ½(Y ,X,µ) and Ã(Y ,X,µ).(c) Calculate the asymptotic covariance matrix.
For the estimator described in
set g (u) Æ 14 u4.(a) Sketch g (u). Is g (u) continuous? Differentiable? Second differentiable?(b) Find the functions ½(Y ,X,µ) and Ã(Y ,X,µ).(c) Calculate the asymptotic covariance matrix.
For the estimator described in
Take the model Y Æ X0µ Åe. Consider the m-estimator of µ with ½(Y ,X,µ) Æ g¡Y ¡X0µ¢where g (u) is a known function.(a) Find the functions ½(Y ,X,µ) and Ã(Y ,X,µ).(b) Calculate the asymptotic covariance matrix.
Take the model Y Æ X0µÅe where e is independent of X and has known density function f (e) which is continuously differentiable.(a) Show that the conditional density of Y given X Æ x is f¡y ¡x0µ¢.(b) Find the functions ½(Y ,X,µ) and Ã(Y ,X,µ).(c) Calculate the asymptotic covariance
Do a similar estimation as in the previous exercise, but using the dependent variable mort_age59_related_preHS (mortality due to HS-related causes in the 5-9 age group during 1959-1964, before the Head Start program was started).
Do a similar estimation as in the previous exercise, but using the dependent variable mort_age25plus_related_postHS (mortality due to HS-related causes in the 25+ age group).
Use the datafile LM2007 on the textbook webpage. Ludwig and Miller (2007) shows that similar RDD estimates for other forms of mortality do not display similar discontinuities. Perform a similar check. Estimate the conditional ATE using the dependent variable mort_age59_injury_postHS(mortality due
Use the datafile LM2007 on the textbook webpage. Replicate the baseline RDD estimate as reported in Table 21.1. This uses a normalized Triangular kernel with a bandwidth of h Æ 8. (If you use an unnormalized Triangular kernel (as used, for example, in Stata) this corresponds to a bandwidth of h Æ
Use the datafile LM2007 on the textbook webpage. Replicate the regresssion (21.5) using the subsample with poverty rates in the interval 59.1984§13.8 (as described in the text). Repeat with intervals of 59.1984§7 and 59.1984§20. Report your estimates of the conditional ATE and standard error.
Explain why equation (21.4) estimated on the subsample for which jX ¡cj · h is identical to a local linear regression with a Rectangular bandwidth.
Show that (21.1) is obtained by taking the conditional expectation as described.
Suppose treatment occurs for D Æ 1{c1 · X · c2} where both c1 and c2 are in the interior of the support of X. What treatment effects are identified?
We have described the RDD when treatment occurs for D Æ 1{X ¸ c}. Suppose instead that treatment occurs for D Æ 1{X · c}. Describe the differences (if any) involved in estimating the conditional ATE µ.
The AL1999 dataset is from Angrist and Lavy (1999). It contains 4067 observations on classroom test scores and explanatory variables including those described in Section 20.30. In Section 20.30 we report a nonparametric instrumental variables regression of reading test scores (avgverb) on classize,
The CHJ2004 dataset is from Cox, Hansen and Jimenez (2004). As described in Section 20.6 it contains a sample of 8684 urban Phillipino households. This paper studied the crowding-out impact of a family’s income on non-governmental transfers. Estimate an analog of Figure 20.2(b) using polynomial
Take the DDK2011 dataset (full sample). Use a quadratic spline to estimate the regression of testscore on percentile.(a) Estimate five models: (1) no knots (a quadratic); (2) one knot at 50; (3) two knots at 33 and 66;(4) three knots at 25, 50 & 75; (5) knots at 20, 40, 60, & 80. Plot the five
The RR2010 dataset is from Reinhart and Rogoff (2010). It contains observations on annual U.S. GDP growth rates, inflation rates, and the debt/gdp ratio for the long time span 1791-2009. The paper made the strong claim that GDP growth slows as debt/gdp increases, and in particular that this
Take the cps09mar dataset (full sample).(a) Estimate quadratic spline regressions of log(wage) on education. Estimate four models: (1) no knots (a quadratic); (2) one knot at 10 years; (3) three knots at 5, 10, and 15; (4) four knots at 4, 8, 12, & 16. Plot the four estimates. Intrepret your
Take the cps09mar dataset (full sample).(a) Estimate quadratic spline regressions of log(wage) on experience. Estimate four models: (1) no knots (a quadratic); (2) one knot at 20 years; (3) two knots at 20 and 40; (4) four knots at 10, 20, 30,& 40. Plot the four estimates. Intrepret your
Continuing the previous exercise, compute the cross-validation function (or alternatively the AIC) for polynomial orders 1 through 8.(a) Which order minimizes the function?(b) Plot the estimated regression function along with 95% pointwise confidence intervals.
Take the cps09mar dataset (full sample).(a) Estimate a 6th order polynomial regression of log(wage) on education. To reduce the ill-conditioned problem first rescale education to lie in the interval [0,1].(b) Plot the estimated regression function along with 95% pointwise confidence intervals.
Continuing the previous exercise, compute the cross-validation function (or alternatively the AIC) for polynomial orders 1 through 8.(a) Which order minimizes the function?(b) Plot the estimated regression function along with 95% pointwise confidence intervals.
Take the cps09mar dataset (full sample).(a) Estimate a 6th order polynomial regression of log(wage) on experience. To reduce the ill-conditioned problem first rescale experience to lie in the interval [0,1] before estimating the regression.(b) Plot the estimated regression function along with 95%
Take the NPIV approximating equation (20.35) and error eK .(a) Does it satisfy E[eK j Z] Æ 0?(b) If L Æ K can you define ¯K so that E[ZK eK ] Æ 0?(c) If L È K does E[ZK eK ] Æ 0?
Does rescaling Y or X (multiplying by a constant) affect the CV(K) function? The K which minimizes it?
You estimate the polynomial regression model:b mK (x) Æ b¯0 Å b¯1x Å b¯2x2 Å¢ ¢ ¢Å b¯p xp.You are interested in the regression derivativem0(x) at x.(a) Write out the estimator b m0 K (x) of m0(x).(b) Is b m0 K (x) is a linear function of the coefficient estimates?(c) Use Theorem 20.8 to
Consider spline estimation with one knot ¿. Explain why the knot ¿ must be within the sample support of X. [Explain what happens if you estimate the regression with the knot placed outside the support of X].
Take the quadratic spline with three knots mK (x) Æ ¯0 Å¯1x Å¯2x3 Å¯3 (x ¡¿1)21{x ¸ ¿1}Å¯4 (x ¡¿2)21{x ¸ ¿2}Å¯5 (x ¡¿3)21{x ¸ ¿3} .Find the inequality restrictions on the coefficients ¯j so thatmK (x) is concave.

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