AAEC/ECON 5126 FINAL EXAM: SOLUTIONS

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1 AAEC/ECON 5126 FINAL EXAM: SOLUTIONS SPRING 2013 / INSTRUCTOR: KLAUS MOELTNER This exam is open-book, open-notes, but please work strictly on your own. Please make sure your name is on every sheet you re handing in. You have 120 minutes to complete this exam. You can collect a maximum of 50 points. Each question is scored as indicated below. Vectors are given in lower-case boldface. Matrices are written in upper-case boldface. Question I 20 points) Consider the Bayesian estimation of a CLRM without explanatory variables i.e. just a constant term). At the observation level this model can be written as y i = µ + ɛ i ɛ i n 0, σ 2) 1) Thus, the only parameters in this model are the population mean µ and variance σ 2. throughout that σ 2 is known. Assume You opt for a normal prior for µ, i.e. µ n µ 0, V 0 ), 2) where µ 0 and V 0 are the prior mean and variance, respectively. Note that V 0 is, of course, a scalar). a) 3 pts.) Write down the regression model for the full sample of n observations. b) 3 pts.) Write down the likelihood function for the full sample call it p y µ, σ 2) ). Since σ 2 is known, the conditional posterior of µ, p µ σ 2, y ) is the end-product for this analysis. Recall that for a CLRM with covariates, the moments for the conditional posterior can be expressed as V 1 = V X X ) 1 σ 2 µ 1 = V 1 V 1 0 µ X y ) 3) σ 2 where µ 1 and V 1 are the conditional posterior mean and variance, respectively. Date: May 11,

2 c) 7 pts.) Working from these expressions, derive the conditional posterior variance of µ for your model call it V 1 ). Show that it is always smaller than the prior variance V 0 for any n, V 0 > 0. d) 7 pts.) Derive the conditional posterior mean call it µ 1 ) and show that it can be written as a weighted average of the prior mean and the sample mean ȳ, with the weights summing to one. State the condition under which the sample mean will receive a larger weight than the prior mean. Elaborate on the effect of the prior variance V 0 and the sample size n on the relative weight of the sample mean. 2

3 Solutions: a) y = iµ + ɛ ɛ n 0, σ 2 I ) b) p y µ, σ 2) = 2π) n/2 σ 2) n/2 exp 1 2σ 2 y iµ) y iµ) ) c) V 1 = V i i ) 1 σ = n ) 1 V 0 σ 2 = σ 2 ) V 0 σ 2 + nv 0 d) µ 1 =V 1 V 1 0 µ i y ) = σ 2 σ 2 ) n µ0 i=1 V 0 σ 2 + y ) i + nv 0 V 0 σ 2 = σ 2 ) ) nv0 σ 2 µ nv 0 σ 2 ȳ + nv 0 The sample mean ȳ receives more weight as long as n V 0 > σ 2. This is very likely in reality, especially with a vague large) value for V 0 and large sample size n. This is precisely in the spirit of Bayesian analysis: If you have a large sample and vague priors, data-based statistics will dominate your posterior moments. 3

4 Question II 30 points) Consider a population of workers, some of whom enroll voluntarily) in a job training program. Let y be the hourly wage of a worker measured for all workers after the job training period). As always, let w be the binary indicator for training w = 1 if trained, w = 0 otherwise). Let y 0 = y w = 0) be the outcome wage) in absence of training, and y 1 = y w = 1) the outcome if trained. Naturally, we can only observe one or the other for a given individual. Since participation in the program was voluntary, we can not rule out some sort of self-selection problem. Therefore, we will ex ante allow y and w to be correlated i.e. we can t invoke ignorability). Consider a sample of workers from this population, some of whom have been trained, some haven t. Let the two sample means be given as ȳ 1 and ȳ 0, respectively. Consider the difference between these sample means, call it m, as an estimator for the Average Treatment Effect on the Treated ATT). Note that E ȳ 0 ) = E y w = 0) and E ȳ 1 ) = E y w = 1). τ att = E y 1 y 0 w = 1). As always the ATT is given as a) 8 pts.) Derive the bias in m as an estimator of τ att, i.e. derive E m τ att ). Hint: The relationship y = y 0 + w y 1 y 0 ) might come in handy.) b) 3 pts.) Argue that the bias would vanish if we had ignorability of w with respect to E y 0 ) i.e. mean independence of y 0 and w). Explain what this would mean in the current context. c) 3 pts.) Now assume the bias is negative. What kind of selection story would that tell? More specifically, who would be more likely to enroll in the job market training? d) 3 pts.) Now assume the bias is positive. What kind of selection story would that tell? More specifically, who would be more likely to enroll in the job market training? Now consider the ATE, τ ate = E y 1 y 0 ). Also, let the Average Treatment Effect on the Controls i.e. the un-treated) be given as τ atc = E y 1 y 0 w = 0), and the propensity score as p = probw = 1). e) 3 pts.) Using the law of iterated expectations, show that the ATE can be expressed in terms of ATT, ATC, and the propensity score. f) 10 pts.) Show the bias of using your estimator m for the ATE, i.e. derive E m τ ate ). Your solution should be in terms of p, τ atc, τ att, and E y 0 w = 1) E y 1 w = 0). Argue that this bias only vanishes under ignorability independence) of both y 0 and y 1 from w. 4

5 Solutions: a) Note that we can write E y w = 1) = E y 0 w = 1) + E y 1 y 0 w = 1) and E y w = 0) = E y 0 w = 0). Thus: E m τ att ) =E y w = 1) E y w = 0) τ att = E y 0 w = 1) + E y 1 y 0 w = 1) E y 0 w = 0) τ att = E y 0 w = 1) E y 0 w = 0) + τ att τ att = E y 0 w = 1) E y 0 w = 0) 4) Therefore, mean independence of y 0 and w is sufficient for the bias to go to zero, since then E y 0 w = 1) E y 0 w = 0) = E y 0 ) E y 0 ) = 0. Here, this would mean that the expected earnings of job trainees, had they NOT enrolled in the program, would have been equal to the mean earnings of those that actually did not enroll. b) If the bias is negative, we have E y 0 w = 1) < E y 0 w = 0). This means that if those who did participate had NOT participated, their earnings would have been lower than for those that actually did not participate. In other words, only the weakest members of the labor force chose to enroll in the training program. Naturally this would bias downward the expected impact of the training program for the average member of the total population. c) The converse holds - only the strongest members of the labor force decided to participate in the training program. Their earnings would have been higher regardless of the training. This induces an upward bias on the expected impact of the training program for the average member of the total population. d) E y 1 y 0 ) =E w E y 1 y 0 w)) = E y 1 y 0 w = 1) p + E y 1 y 0 w = 0) 1 p) = p τ att + 1 p) τ atc 5) e) E m τ att ) = E y 0 w = 1) E y 0 w = 0) + τ att p τ att + 1 p) τ atc ) = E y 0 w = 1) E y 0 w = 0) + E y 1 w = 0) E y 1 w = 0) + τ att p τ att + 1 p) τ atc ) = E y 0 w = 1) E y 1 w = 0) + τ atc + τ att p τ att + 1 p) τ atc ) = 6) E y 0 w = 1) E y 1 w = 0) + 1 p) τ att + p τ atc This bias will only go to zero under full mean independence of y 0, y 1 and w. In that case we have τ ate = τ atc = τ att, and E m τ att ) = E y 0 y 1 )+1 p) τ ate +p τ ate = τ ate +τ ate = 0. 5