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Econometrics 1st Edition Bruce Hansen - Solutions
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 ů2 logQ ů3¡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 Ư1¯2 ů3XÅe 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 ¯0ů1X 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
For the estimator described in Exercise 22.2 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 Exercise 22.2 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.
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 ů1x ů2x3 ů3 (x ¡¿1)21{x ¸ ¿1}ů4 (x ¡¿2)21{x ¸ ¿2}ů5 (x ¡¿3)21{x ¸ ¿3} .Find the inequality restrictions on the coefficients ¯j so thatmK (x) is concave.
Take the linear spline from the previous question. Find the inequality restrictions on the coefficients ¯j so thatmK (x) is concave.
Take the linear spline with three knots mK (x) Æ ¯0 ů1x ů2 (x ¡¿1)1{x ¸ ¿1}ů3 (x ¡¿2)1{x ¸ ¿2}ů4 (x ¡¿3)1{x ¸ ¿3} .Find the inequality restrictions on the coefficients ¯j so thatmK (x) is non-decreasing.
Take the estimated model Y Æ ¡1Å2X Å5(X ¡1)1{X ¸ 1}¡3(X ¡2)1{X ¸ 2}Åe.What is the estimated marginal effect of X on Y for X Æ 3?
We will consider a nonlinear AR(1) model for gdp growth rates Yt Æm(Yt¡1)Ået Yt Æ 100µµGDPt GDPt¡1¶4¡1(a) Create GDP growth rates Yt . Extract the level of real U.S. GDP (gdpc1) from FRED-QD and make the above transformation to growth rates.(b) Use Nadaraya-Watson to estimatem(x). Plot
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 Invest1993 dataset and the subsample of observations with Q · 5. (In the dataset Q is the variable vala.)(a) Use Nadaraya-Watson to estimate the regression of I on Q. (In the dataset I is the variable inva.)Plot with 95% confidence intervals.(b) Repeat using the Local Linear estimator.(c)
Take the cps09mar dataset and the subsample of individuals with education=20 (professional degree or doctorate), with experience between 0 and 40 years.(a) Use Nadaraya-Watson to estimate the regression of log(wage) on experience, separately for men and women. Plot with 95% confidence intervals.
Take the DDK2011 dataset and the subsample of boys who experienced tracking. As in Section 19.21 use the Local Linear estimator to estimate the regression of testscores on percentile but now with the subsample of boys. Plot with 95% confidence intervals. Comment on the similarities and differences
Prove Theorem 19.11: Show that when d ¸ 1 the AIMSE optimal bandwidth takes the form h0 Æ cn¡1/(4Åd) and AIMSE is O¡n¡4/(4Åd)¢.
Supposem(x) Æ ® is a constant function. Find the AIMSE-optimal bandwith (19.6) forNW estimation? Explain.
Suppose the true regression function is linearm(x) Æ ®Å¯x and we estimate the function using the Nadaraya-Watson estimator. Calculate the bias function B(x). Suppose ¯ È 0. For which regions is B(x) È 0 and for which regions is B(x) Ç 0? Now suppose that ¯ Ç 0 and re-answer the
Describe in words how the bias of the local linear estimator changes over regions of convexity and concavity inm(x). Does this make intuitive sense?
Show that (19.6) minimizes the AIMSE (19.5).
For kernel regression suppose you rescale Y , for example replace Y with 100Y . How should the bandwidth h change? To answer this, first address how the functions m(x) and ¾2(x) change under rescaling, and then calculate how B and ¾2 change. Deduce how the optimal h0 changes due to rescaling Y .
Use the datafile BMN2016 on the textbook webpage. The authors report results for liquor sales. The data file contains the same information for beer and wine sales. For either beer or wine sales, estimate diff-in-diff models similar to (18.7) and (18.8) and interpret your results. Some relevant
Use the datafile DS2004 on the textbook webpage. The authors argued that an exogenous police presence would deter automobile theft. The evidence presented in the chapter showed that car theft was reduced for city blocks which received police protection. Does this deterrence effect extend beyond the
Use the datafile CK1994 on the textbook webpage. Classical economics teaches that increasing the minimum wage will increase product prices. You can therefore use the Card-Krueger diffin-diff methodology to estimate the effect of the 1992 New Jersey minimum wage increase on product prices. The data
An economist is interested in the impact ofWisconsin’s 2011“Act 10” legislation on wages.(For background, Act 10 reduced the power of labor unions.) She computes the following statistics5 for average wage rates inWisconsin and the neighboring state ofMinnesota for the decades before and after
For the specification tests of Section 18.4 explain why the regression test for homogeneous treatment effects includes only N2 ¡1 interaction dummy variables rather than all N2 interaction dummies. Also explain why the regression test for equal control effects includes only N1 ¡1 interaction
Take the basic difference-in-difference model Yi t Æ µDi t Åui űt Å"i t .Instead of assuming that Di t and "i t are independent, assume we have an instrumental variable Zi t which is independent of "i t but is correlated with Di t . Describe how to estimate µ.Hint: Review Section 17.28.
In regression (18.1) with T Æ 2 and N Æ 2 suppose the time variable is omitted. Thus the estimating equation is Yi t Æ ¯0 ů1Statei ŵDi t Å"i t .where Di t ÆStateiTimet is the treatment indicator.(a) Find an algebraic expression for the least squares estimator bµ.(b) Show that bµ is a
In the text it was claimed that in a balanced sample individual-level fixed effects are orthogonal to any variable demeaned at the state level.(a) Show this claim.(b) Does this claim hold in unbalanced samples?(c) Explain why this claim implies that the regressions Yi t Æ ¯0 ů1Statei
Use the datafile Invest1993 on the textbook webpage. You will be estimating the model Di t Æ ®Di ,t¡1 ů1Ii ,t¡1 ů2Qi ,t¡1 ů3CFi ,t¡1 Åui Å"i t .The variables are debta, inva, vala, and cfa in the datafile. See the description file for definitions.(a) Estimate the above regression
Use the datafile Invest1993 on the textbook webpage. You will be estimating the panel AR(1) Di t Æ ®Di ,t¡1 Åui Å"i t for D Ædebt/assets (this is debta in the datafile). See the description file for definitions.(a) Estimate the model using Arellano-Bond twostep GMMwith clustered standard
This exercise uses the same dataset as the previous question. Blundell and Bond (1998)estimated a dynamic panel regression of log employment N on log real wages W and log capital K. The following specification1 used the Arellano-Bond one-step estimator, treatingWi ,t¡1 and Ki ,t¡1 as
In this exercise you will replicate and extend the empirical work reported in Arellano and Bond (1991) and Blundell and Bond (1998). Arellano-Bond gathered a dataset of 1031 observations from an unbalanced panel of 140 U.K. companies for 1976-1984 and is in the datafile AB1991 on the textbook
In Section 17.33 verify that in the just-identified case the 2SLS estimator b¯2sls simplifies as claimed: b¯1 and b¯2 are the fixed effects estimator. b°1 and b°2 equal the 2SLS estimator froma regression of bu on Z1 and Z2 using X 1 as an instrument for Z2.
Take the fixed effects model Yi t Æ Xi t¯1 Å X2 i t¯2 Åui Å"i t . A researcher estimates the model by first obtaining the within transformed ˙ Yi t and ˙X i t and then regressing ˙ Yi t on ˙X i t and ˙X 2 i t . Is the correct estimation method? If not, describe the correct fixed effects
(a) Show (17.61).(b) Show (17.62).(c) For eV fe defined in (17.60) show E£eV fe j X¤ÆV fe.
(a) For b¾2i defined in (17.59) show E£b¾2i j X i¤Æ ¾2i.(b) For eV fe defined in (17.58) show E£eV fe j X¤ÆV fe.
Show (17.57).
Develop a version of Theorem 17.2 for the differenced estimator b¯¢. Can you weaken Assumption 17.2.3? State an appropriate version which is sufficient for asymptotic normality.
Verify that b¾2"defined in (17.37) is unbiased for ¾2" under (17.18), (17.25) and (17.26).
In Section 17.14 it is described how to estimate the individual-effect variance ¾2 u using the between residuals. Develop an alternative estimator of ¾2 u only using the fixed effects error variance b¾2"and the levels error varianceb¾2eÆ n¡1PN iÆ1 Pt2Si be2 i t where bei t Æ Yi t ¡X0 i t
Show that when T Æ 2 the differenced estimator equals the fixed effects estimator.
Show (17.28).
Show (17.24).
Show that var£˙X i t¤· var[Xi t ].
Is E["i t j Xi t ] Æ 0 sufficient for b¯fe to be unbiased for ¯? Explain why or why not.
(a) Show (17.11) and (17.12).(b) Show (17.13).
For each of the following monthly pairs from FRED-MD test the hypothesis of no cointegration using the Johansen trace test. For each, you need to consider the VAR order p and the trend specification.(a) 3-month treasury interest rate (tb3ms) and 10-year treasury interest rate (gs10). Note: In the
For each of the series in the previous exercise implement the KPSS test of stationarity.For each, you need consider the lag truncation M and the trend specification.
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