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Questions and Answers of Econometrics

Repeat the above exercise using the subsample of Hispanic men (n Æ 4547).
Take the cpsmar09 dataset and the subsample of Asian women (n Æ 1149). Estimate a Lasso linear regression of log(wage) on the following variables: education; dummies for education equalling 12, 13,
You have the continuous variables (Y ,X) with X ¸ 0 and you want to estimate a regression tree for E[Y j X]. A friend suggests adding a quadratic X2 to the variables for added flexibility. Does this
Repeat the previous question for Lasso regression. Show that the Lasso coefficient estimates b¯1 and b¯2 are individually indeterminate but their sum satisfies b¯1 Å b¯2 Æ b¯Lasso, the
Does ridge regression require that the columns of X linearly independent? Take a sample(Y ,X ). Create the augmented regressor set eX Æ (X ,X ) (add a duplicate of each regressor) and let ( b¯1,
Which estimator produces a higher regression R2, least squares or ridge regression?
Show that the ridge regression estimator can be computed as least squares applied to an augmented data set. Take the original data (Y ,X ). Add p 0’s to Y and p rows of p¸I p to X , apply least
Derive the conditional bias (29.8) and variance (29.9) of the ridge regression estimator.
Show that (29.7) is the Mallows criterion for ridge regression. For a definition of the Mallows criterion see Section 28.6.
Prove Theorem29.1. Hint: The proof is similar to that of Theorem 3.7.
Using the cps09mar dataset perform an analysis similar to that presented in Section 28.18 but instead use the sub-sample of Hispanic women. This sample has 3003 observations. Which models are
You estimate M linear regressions Y Æ X0 m ¯m Åem by least squares. Let e Ymi Æ X0 mi b¯m(¡i )be the predicted values from the leave-one-out regressions. Show that the JMA criterion equals n
You estimate M linear regressions Y Æ X0 m ¯m Åem by least squares. Let b Ymi Æ X0 mi b¯m be the fitted values.(a) Show that theMallows averaging criterion is the same as(b) Assume the models
Suppose you have two unbiased estimators bµ1 and bµ2 of a parameter vector bµ with covariance matrices V 1 and V 2. Take the goal of minimizing the unweighted mean squared error, e.g. trV 1 for
Prove Theorem 28.14 for the simpler case of the unadjusted (not positive part) Stein estimator eµ, V Æ I K and r Æ 0.Extra challenge: Show under these assumptions that wmse [0]=K-(q-2)Jq (AR) AR =
Under the assumptions of Theorem 28.11, show that b¸Æ bµ0V ¡1bµ ¡K is an unbiased estimator of ¸ Æ µ0V ¡1µ.
Verify Theorem 28.11, including (28.21), (28.22), and (28.23).
An economist estimates several models and reports a single selected specification, stating that “the other specifications had insignificant coefficients”. How should we interpret the reported
Forward Stepwise Regression. Verify the claim that for the case of AIC selection, step (b)of the algorithm can be implemented by identifying the regressor in the inactive set with the greatest
Backward Stepwise Regression. Verify the claim that for the case of AIC selection, step (b)of the algorithmcan be implemented by calculating the classical (homoskedastic) t-ratio for each active
Find the Mallows criterion for the weighted least squares estimator of a linear regression Yi Æ X0 i¯Åei with weights !i (assume conditional homoskedasticity).
Verify equations (28.1)-(28.2).
Take the DDK2011 dataset. Create a variable testscore which is totalscore standardized to have mean zero and variance one. The variable tracking is a dummy indicating that the students were tracked
Take the cps09mar dataset and the subsample of individuals with at least 12 years of education. Create wage=earnings/(hours£weeks) and lwage=log(wage).(a) Estimate a linear regression of lwage on
Take the CHJ2004 dataset. The variables tinkind and income are household transfers received in-kind and household income, respectively. Divide both variables by 1000 to standardize. Create the
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.
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
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
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
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
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
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
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
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
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,
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
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
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))
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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

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