Applied ProblemsBootstrapQuantile RegressionBayesian and Quasi-Bayesian EstimatorsIntroduction to MCMC:Theory ProblemsProblem Set 3: Bootstrap, Quantile Regression and MCMC Methods MIT 14.385, Fall 2007 Due: Wednesday, 07 November 2007, 5:00 PM 1 Applied Problems Instructions: The page indications given below give you an upper bound on what you should write. Answers should be very brief. Hand in clear, annotated code. You are strongly encouraged to work in groups, but the write-ups should be individual. 1.1 Bootstrap 1. Explain briefly what the bootstrap is, why it works when it works and how it can fail. (1/2 page max, informal reasoning is fine) 2. Find an empirical paper that uses the bootstrap. Explain why they use the bootstrap. (1/4 page max) Write “pseudo-code” ( a brief outline of an algorithm) for how you would reproduce the bootstrap used in this paper. (1/2 page max) 3. What is the bootstrap bias correction method? How does it work? State the pseudo-code for bias correction. (1/2 page max, informal reasoning is fine; hint: read Horowitz) 4. Two-step estimators have messy standard errors, which makes bootstrap inference appealing. Consider the censored regression 2-step estimator of PSet 2, Q1.b.iii. Bootstrap the stan-dard errors. Compare these standard errors to those generated for you by the Stata canned command. Cite as: Victor Chernozhukov, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 1� 1.2 Quantile Regression 1. Theory. Briefly answer the following questions. Fundamentals: (a) (1/4 page max) Consider the location-scale model Y = X�α + (X�γ) u where u is independent of x and has distribution function Fu (u) Write down the con-ditional quantile function of y given x in the form QY (τ|x) = x �β (τ) Characterize β (τ ) as a function of τ (e.g. monotonicity if any ...). How does the answer change if (x�γ) = 1? (b) (1/4 page max) Show that E[Y |x] = QY (τ |x) dτ is of the form x�β, where β is what? (c) (1/4 page max) Consider a general linear quantile model, QY (τ|x) = x�β (τ ) corresponds to the random coefficient model Y = X�β(U ). Can we have slope coefficients β (τ) that behave non-monotonically in τ? Recall Doksum’s quantile treatment effect example. (d) (1/2 page max) Using (a)-(b), give a brief conceptual evaluation of the linear quantile regression model. Compare it with the classical linear model, and state the advantages and disadvantages of the two. Refer to at least one empirical example discussed in class where the difference in models is important. E.g. refer to Doksum’s guinea pig example. (e) (1 page max) Quantile equivariance to monotone transformations Suppose the conditional quantile function of y given x is in the form QY (τ|x) = x �β (τ) Now consider the following models of data transformation Cite as: Victor Chernozhukov, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 2i. y ∗ = f(y) = max {y, 0} ii. y ∗ = f(y) = 1 {y > 0} iii. y ∗ = f(y) = exp (y) For each of the above, write down the conditional quantile function for the trasnformed variable conditional on x: QY ∗ (τ|x) = f (x �β (τ )) Hint: quantile models have a remarkable property — equivariance to monotone trans-formations, which we (should) observe in these examples. (Useful reference: http://www.econ.uiuc.edu/~roger/research/rq/rq.pdf + lecture notes). Is the same property true of the conditional mean function, i.e. is E[Y ∗|x] = f[E[Y |x]] ? Why is this a problem? (f) (1/2 page) For each of the above three cases write down an estimators of β (τ ) that use data on (y ∗ , x) only. (Hint: Case (i) generates the censored quantile regression estimator, case (ii) the maximum score / perception / single-layer neural network estimator, and case (iii) a nonlinear quantile regression estimator.) (g) (1/4 page maximum) Summarize the findings above very briefly, and say why they are important for applications to duration models and censored regression models. 2. (1 page, excluding figures.). Empirical: Reproduce or closely replicate Figures 1, Table 1, then Figure 2 of Koenker’s “Vignette” (posted in Recitation Materials section on alternatively Roger Koenker’s website). You may use any software you like, but is probably just easier to use R (which is a great statistical freeware). As usual, you should briefly explain what the canned commands are doing. The Engel data are available from within R or on the 3. For this question, use the data in “Penn46.ascii”. See Section 5 of Koenker and Billias, “Quantile Regression for Duration Data,” Empirical Economics, 2001, for a description of the Cite as: Victor Chernozhukov, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 3 MIT ServerMIT Server.program and the data. For part (a), review W. Newey’s lecture note on duration models or Wooldridge 20.1-20.2. (a) (1 page) Conventional duration models usually boil down to an empirical relation of the following kind: h (Ti) = xi� β + ui (1) where Ti is a “survival time” (here, the duration of an unemployment spell), h () is monotonic and ui is iid with CDF Fu. (i) Show that if we assume a Cox proportional hazard model, we can write the duration of unemployment in the form in (a) above with h (T ) = log Λ0 (T ) where Λ0 (t) is the integrated baseline hazard. (ii) Show that in the special case of a Weibull baseline hazard, we have log Λ0 (T ) = γ log T − α (iii) Combine your results from (i) and (ii) to show that, under the assumptions of Cox proportional hazard and Weibull baseline hazard, the log of the unemployment duration satisfies the following equation: log Ti = xi� β + ui This is a special case of the model called the Accelerated Failure Time (AFT) model. (Note: AFT models do not make parametric assumptions on the error term). (b) (1/2 page) Show that an implication of the model (1) is that the covariates only shift the location of the distribution, not the