Econometrics 1st Edition Bruce Hansen - Solutions

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Verify that (2.57) holds withm(X) defined in (2.6) when (Y ,X) have a joint density f (y,x).
Show (2.47)-(2.48), namely that for d(¯) Æ E h¡m(X)¡X0¯¢2 ithen¯ Æ argmin b2Rk d(b) Æ¡E£X X0¤¢¡1 E[Xm(X)] Æ¡E£X X0¤¢¡1 E[XY ] .Hint: To show E[Xm(X)] Æ E[XY ] use the law of iterated expectations.
Suppose that X Æ (1,X2,X3) where X3 Æ ®1 Å®2X2 is a linear function of X2.(a) Show that QX X Æ E£X X0¤is not invertible.(b) Use a linear transformation of X to find an expression for the best linear predictor of Y given X.(Be explicit, do not just use the generalized inverse formula.)
Let X be a random variable with ¹ Æ E[X] and ¾2 Æ var[X]. Define g¡x,¹,¾2¢ÆÃx ¡¹ ¡x ¡¹¢2¡¾2!.Show that E£g (X,m, s)¤Æ 0 if and only ifm Æ ¹ and s Æ ¾2.
Let X and Y have the joint density f¡x, y¢Æ 3 2¡x2 Å y2¢on 0 · x · 1, 0 · y · 1. Compute the coefficients of the best linear predictor Y Æ ®Å¯X Åe. Compute the conditional expectationm(x) ÆE[Y j X Æ x] . Are the best linear predictor and conditional expectation different?
Consider the intercept-only model Y Æ ®Åe with ® the best linear predictor. Show that® Æ E[Y ] .
True or False. If Y Æ X0¯Åe, E[e j X] Æ 0, and E£e2 j X¤Æ ¾2, then e is independent of X.
True or False. If Y Æ X0¯Åe and E[Xe] Æ 0, then E[e j X] Æ 0.
True or False. If Y Æ X0¯Åe and E[e j X] Æ 0, then e is independent of X.
True or False. If Y Æ X¯Åe, X 2 R, and E[Xe] Æ 0, then E£X2e¤Æ 0.
True or False. If Y Æ X¯Åe, X 2 R, and E[e j X] Æ 0, then E£X2e¤Æ 0.
Suppose you have two regressors: X1 is binary (takes values 0 and 1) and X2 is categorical with 3 categories (A,B,C).Write E[Y j X1,X2] as a linear regression.
Suppose that Y is discrete-valued, taking values only on the non-negative integers, and the conditional distribution of Y given X Æ x is Poisson:P£Y Æ j j X Æ x¤Æexp¡¡x0¯¢ ¡x0¯¢j j !, j Æ 0, 1, 2, ...Compute E[Y j X] and var[Y j X] . Does this justify a linear regression model of the
Show that the conditional variance can be written as ¾2(X) Æ E£Y 2 j X¤¡(E[Y j X])2.
Use Y Æm(X)Åe to show that var[Y ] Æ var[m(X)]Å¾2.
Show that ¾2(X) is the best predictor of e2 given X:(a) Write down the mean-squared error of a predictor h(X) for e2.(b) What does it mean to be predicting e2?(c) Show that ¾2(X) minimizes the mean-squared error and is thus the best predictor.
Suppose that the random variables Y and X only take the values 0 and 1, and have the following joint probability distribution X Æ 0 X Æ 1 Y Æ 0 .1 .2 Y Æ 1 .4 .3 Find E[Y j X] , E£Y 2 j X¤, and var[Y j X] for X Æ 0 and X Æ 1.
Prove Theorem2.4.4 using the law of iterated expectations.
If E[Y j X] Æ a ÅbX, find E[Y X] as a function of moments of X.
Find E[E[E[Y j X1,X2,X3] j X1,X2] j X1] .
2. Another popular approach which addresses issues similar to Bayesian model averaging is described in George and McCulloch (1993). This approach involves using the Normal linear regression model with independent Normal-Gamma prior (see Chapter 4), with one minor alteration. This alteration is in
1. The empirical illustration in this chapter used only one choice for gr and did not discuss the issue of prior sensitivity analysis. Do the empirical illustration over again for other choices for this prior hyperparameter. You may choose whatever values you want. Choices used by other researchers
4. The empirical illustration of the mixtures of Normals model (Section 10.3.6)used information criteria to select the number of elements in the mixture.(a) Write a program (or modify the one on the website associated with this book) which uses marginal likelihoods to choose the number of elements
3. Chapter 9 included empirical illustrations of the tobit and probit models (see Sections 9.3.1 and 9.4.1, respectively).(a) Re-do these empirical illustrations using semiparametric tobit and probit as described in the present chapter. Note: the empirical illustrations describe the artificial data
2. The additive version of the partial linear model may be too restrictive in some applications. This motivates interest in specifications which are more general, but less likely to suffer from the curse of dimensionality than the partial linear model. One such specification includes interaction
1. The empirical illustration in Chapter 3 (Section 3.9) used the Normal linear regression model with natural conjugate prior with the house price data set.Please refer to this illustration for data definitions.(a) Use the house price data set and the partial linear model to investigate whether the
4. (a) Derive the conditional posterior distributions necessary to set up a Gibbs sampler with data augmentation for the random coefficients panel probit model introduced (9.26).(b) Program up the Gibbs sampler described in part (a) and test your program using artificial data.(c) How would your
3. (a) Write a program which carries out posterior simulation for the simple probit model of Section 9.3 (or take the program from the website associated with this book and make sure you understand it) using an informative Normal prior for þ. Use artificial data to confirm your program is
2. (a) Derive the likelihood function for the probit model. Hint: Remember that the likelihood function is QN iD1 p.yi jþ; h/ and use (9.7).(b) Verify that the identification problem discussed in Section 9.4 occurs with the probit model.
1. Consider the tobit model with an unknown censoring point. All model details and assumptions are as in the body of the chapter (see (9.1) and (9.2)), except that (9.3) is replaced by yi D yŁi if yŁi > c yi D0 ifyŁi c and c is an unknown parameter known to lie in the interval .0; 1/.(a)
3. Use the methods of Section 8.3 for the general state space model to answer this question. Use either real data (e.g. the industrial production data from the empirical illustration in Section 8.3.2) or artificial data generated according to various data generating processes of your choice.(a)
2. Unit root testing with the local level model. To do this exercise, use either real data (e.g. the industrial production data from the empirical illustration in Section 8.3.2) or artificial data generated according to various data generating processes of your choice. Use the local level model
1. (a) Use the derivations in Section 8.2 and Chapter 6, Section 6.4 to obtain a posterior simulator for the local level model with independent Student-t errors. You may use whatever prior you wish for the model parameters(although the natural conjugate one of Section 6.4 will be the easiest).(b)
8. The empirical illustration for the stochastic frontier model (see Section 7.7.5)used an exponential inefficiency distribution (i.e. zi ¾ G.¼z ; 2/).(a) Repeat the empirical illustration assuming zi ¾ G.¼z ; 4/ and zi ¾ G.¼z ; 6/ using your answer to Exercise 6.(b) Write up computer code
7. (a) Write a computer program which uses your answer to Exercise 5 to carry out posterior simulation in the individual effects model with hierarchical prior and independent Student-t errors.(b) Using artificial data sets simulated as in the empirical illustration of Section 7.6, test your program
6. The posterior simulator for the stochastic frontier model (see Section 7.7.4)used an exponential inefficiency distribution (i.e. zi ¾ G.¼z ; 2/). Derive a posterior simulator for the case where zi ¾ G.¼z ; 4/ and zi ¾ G.¼z ; 6/.Note: The Gamma distribution with (known) degrees of freedom
5. Consider the individual effects model with hierarchical prior with likelihood given in (7.3), prior given in (7.7)–(7.11) and posterior conditionals given in(7.14) through (7.18). Extend these results to allow for independent Student-t errors (see Chapter 6, Section 6.4).
4. Panel data with T D 1 reduces to the cross-sectional data sets used in previous chapters. The pooled model is identical to the Normal linear regression model of Chapters 3 and 4. But the other models in this chapter differ from those in earlier chapters even if T D 1. Discuss Bayesian inference
3. Suppose you have an unbalanced panel. That is, for each individual data for Ti periods is available.(a) Describe how the posterior simulators for the individual effects models(see Section 7.3.3) would be altered by the unbalanced panel.(b) Describe how the posterior simulators for the random
2. Consider a model which combines some elements of the individual effects model and some elements of the pooled model. The intercept and some of the regression coefficients vary across individuals, but other regression coefficients do not. That is, yi D Xiþi C Wi C "i where the explanatory
1. Some general results for Normal linear hierarchical regression models (see Lindley and Smith (1972) for more details).Let y be an N-vector and 1, 2 and 3 be parameter vectors of length k1, k2 and k3, respectively. Let X, W and Z be known N ðk1, k1ðk2 and k2ðk3 matrices and C1, C2 and C3 be
5. For this exercise obtain any relevant time series data set (e.g. the Yankees data set of Section 6.5.3 or an artificially generated data set).(a) Write a program which carries out posterior simulation for the Normal linear regression model with AR(p) errors (or obtain from the website associated
4. Use the house price data set for this question. Remember that in Chapter 4 we carried out Bayesian inference using this data and a Normal linear regression model, in the present chapter we added heteroskedasticity of a known form(see Section 6.3.2) and independent Student-t errors (see Section
3. Derive a posterior simulator for the linear regression model with independent Student-t errors and AR(p) errors. (Hint: If you are having difficulty, the answer to this question is given in Chib, 1993.)
2. Discuss how posterior simulation could be done for all themodels in this chapter if they were modified to have nonlinear regression functions as described in Chapter 5.
1. (a) Section 6.2 contains results for the Normal linear regression model with error covariance matrix using the independent Normal-Gamma prior of(6.9) and (6.10). Show how posterior results would change if the natural conjugate Normal-Gamma prior of Chapter 3, Section 3.4 were used forþ and
6. Investigate the performance of Independence Chain and Random Walk Chain Metropolis–Hastings algorithms using artificial data. The artificial data should be generated from (5.1) using various values for and h. You should experiment with different candidate generating densities and different
5. Re-do this chapter’s empirical illustration using importance sampling using the ft .jbML;var.bML/; ¹/ as an importance function.
4. Use the house price data set and the Normal linear regression model which is partly nonlinear and partly linear described in Exercise 2. You may use whatever prior you wish for þ andprovided it is informative. Normal priors will be the most convenient.(a) Using your results from Exercise 2,
3. Use the house price data set and the Normal linear regression model with natural conjugate prior described in the Empirical Illustration to Chapter 3.4 Use the posterior predictive p-value approach (see Section 5.6) to see whether this model can reasonably capture the skewness and kurtosis in
2. Consider the Normal regression model which is partly linear and partly non-linear:yi = B1 + 2x12 + · ·· + xxik + f (xi,k+1. .. . . xi,k+p . Y ) + 81(a) Using a noninformative prior, derive the marginal posterior p(y ly).(b) Discuss how you might derive a posterior simulator if you wished to
1. Which of the following regression functions is intrinsically linear? For those which are intrinsically linear, write out a linear regression model in terms of(possibly transformed) dependent and explanatory variables and discuss what the properties of the regression errors are. Assume yi and xi
4. The purpose of this question is to learn about the properties of importance sampling in a very simple case. Assume you have a model which a single parameter, , and its posterior is N.0; 1/.(a) Write a program which calculates the posterior mean and standard deviation of using Monte Carlo
3. The purpose of this question is to learn about the properties of the Gibbs sampler in a very simple case. Assume that you have a model which yields a bivariate Normal posterior,12¾ N\]0 0½;]1 ²² 1½where j²j < 1 is the (known) posterior correlation between 1 and 2.(a) Write a program
2. For the Normal linear regression model with natural conjugate prior, the Bayes factor for comparing M1 : þi D 0 to M2 : þi 6D 0 (where the þi is a single regression coefficient and the same prior is used for h in both models) can be obtained from Chapter 3 (3.34). Alternatively, this Bayes
1. The Savage–Dickey density ratio.(a) Prove Theorem 4.1. (Hint: If you are having trouble with this problem, the proof is provided in Verdinelli and Wasserman, 1995.)(b) How would your answer change if the condition p. j! D !0; M2/ D p. jM1/ did not hold?
11. Using the house price data set described in the Empirical Illustration, carry out a prior sensitivity analysis using various priors of your choice. Are the results presented in the text robust to reasonable changes in the prior?
10. Repeat Exercise 8 (b) using Monte Carlo integration for various values of S.How large does S have to be before you reproduce the results of Exercise 8(b) to two decimal places?
9. Repeat Exercise 8 for different choices of N and h to investigate the role of sample size and error size in Bayesian estimation, model comparison and prior sensitivity for the Normal linear regression model.
8. (a) Generate an artificial data set of size N D 100 for the Normal linear regression model with an intercept and one other explanatory variable.Set the intercept to 0, the slope coefficient to 1:0 and h D 1:0. Generate the explanatory variable by taking random draws from the U.0; 1/distribution
7. Multicollinearity. Consider the Normal linear regression model with likelihood, prior and posterior as described Sections 3.3, 3.4 and 3.5, respectively.Assume in addition that Xc D 0 for some non-zero vector of constants c.Note that this is referred to as a case of perfect multicollinearity. It
6. Ellipsoid Bound Theorems. Consider the Normal linear regression model with natural conjugate prior, þ; h ¾ NG.þ; V ; s32; ¹/:(a) Show that for any choice of V, the posterior mean of þ must lie in the ellipsoid:.þ4þave/0X0X.þ4þave/ .bþ4þ/0X0X.bþ4þ/4 whereþave D 12.bþ C þ/(b) For
5. Problems with Bayes Factors using Noninformative Priors. Consider comparing two models using Bayes factors based on noninformative priors in the setup of Section 3.6.2. The Bayes factor can be obtained using (3.34) for the two models:(a) Consider a noninformative prior created by setting ¹j D
4. Suppose you have a Normal linear regression model with partially informative natural conjugate prior, where prior information is available only on J k linear combinations of the regression coefficients, and the prior for h is the noninformative one of (3.24). Thus, Rþjh ¾ N.r; h1Vr /, where
3. Show that (3.13) can be written in the following alternative ways:¹s2 D ¹s2 C ¹s2 C .bþ þ/0[.X0X/VV1].bþ þ/D ¹s2 C .y Xþ/0.y Xþ/ C .þ þ/0V1.þ þ/
2. Assuming the prior for þ and h is NG.þ; V; s2; ¹/, derive the posterior forþ and h and, thus, show that the Normal-Gamma prior is a conjugate prior for this model.
1. For the Normal linear regression model, show that the likelihood function in(3.3) can be written in terms of OLS quantities as in (3.7).
4. Bayesian inference in the Normal linear regression model: prior sensitivity.(a) Generate an artificial data set with þ D 2, h D 1 and N D 100 using the U.0; 1/ distribution to generate the explanatory variable.(b) Assume a prior of the form þ; h ¾ NG.þ; V ; s2; ¹/ with þ D 2; V D 1; s2 D
3. Generating artificial data sets. This is an important skill since they can be used to understand the properties of models and investigate the performance of a particular computer algorithm. Since you have chosen the values for parameters, you know roughly what answer you would hope your
2. For this question, assume the likelihood function is as described in Section 2.2 with known error precision, h = 1, and x¡ = 1 for i = 1 ,..., N.(a) Assume a Uniform prior for B such that B ~ U (@, y). Derive the posterior p(fly).(b) What happens to p(8|y) as a -> -co and y -> 00?(c) Use the
1. Prove the result in (2.8). Hint: This is a standard derivation proved in many other textbooks such as Poirier (1995, p. 527) or Zellner (1971, pp. 60-61), and you may wish to examine some of these references if you are having trouble.
4. Suppose that the posterior for a parameter, , is N.0; 1/:(a) Create a computer program which carries out Monte Carlo integration (see(1.12)) to estimate the posterior mean and variance of . (Note: Virtually any relevant computer package such as MATLAB or Gauss will have a function which takes
3. Let y D .y1; : : : ; yN /0 be a random sample, where p.yi j/ D( yi .1/yi if 0 yi 1 0 otherwise(a) Derive the posterior for assuming a prior ¾ U.0; 1/. Derive E.jy/.(b) Repeat part (a) assuming a prior of the form:p./ D 8>:0. Þ C þ/0. Þ/0. þ/Þ1.1/þ1 if 0 < < 1 0
2. Let y D .y1; : : : ; yN /0 be a random sample where p.yi j/ D fG.yi j; 2/.Assume a Gamma prior for : p./ D fG.j; ¹/:(a) Derive p.jy/ and E.jy/.(b) What happens to E.jy/ as ¹ ! 0? In what sense is such a prior ‘noninformative’?
1. Decision Theory. In this book, we usually use the posterior mean as a point estimate. However, in a formal decision theoretic context, the choice of a point estimate of is made by defining a loss function and choosing the point estimate which minimizes expected loss. Thus, if C.e; / is the
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, 14, 15, 16, 18, and 20; experience/40 in powers from 1 to 9; dummies for marriage categories
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 make sense?
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 coefficients from the Lasso regression of Y on X .
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, b¯2)be the ridge regression coefficients for the regression of Y on eX . Showthat b¯1 Æ b¯2 Æ 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 squares, and show that this equals b¯ridge.
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 selected by BIC, AIC, CV and FIC? (The precise information criteria you examine may be limited depending on
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 (Yi-wi-W221-..- WMYMi). i=1
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 are nested with M the largest model. If the previous criterion wereminimized over w in the probability
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 bµ1. Assume that bµ1 and bµ2 are uncorrelated.(a) Show that the optimal weighted average estimator
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 = O'R (R'R) R'O.
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 parameter estimates and t-ratios?
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 absolute correlation with the residual from step (a).Hint: This is challenging. First show that the goal
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 regressor and find the regressor with the smallest absolute t-ratio.Hint: Use the relationship between
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 (separated by initial test score). The varible percentile is the student’s percentile in the
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 education and education^2. Interpret the results.(b) Suppose the wage data had been capped about
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 regressor Dincome=(income-1)£1{income È 1}.(a) Estimate a linear regression of tinkind on income and

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Econometrics 1st Edition Bruce Hansen - Solutions

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