Econ226_IAB

Econ226_IAB - Econ 226: Bayesian and Numerical Methods...

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1 Econ 226: Bayesian and Numerical Methods • Course requirements: two exams (based primarily on lectures) • Slides: sometimes, not always, check web page night before • Office hours: Tuesdays 9:30-10:30 a.m. • Theme: Bayesian econometrics and applications of numerical Bayesian methods Why Bayesian? (1) Allows us to incorporate information in addition to that in the sample (a) VARs (b) short time series, measurement error (c) lag lengths, nonstationarity (2) Integrate DSGEs with VARs (3) Clean solution to many otherwise tricky questions (4) Analyze models that are intractable using classical methods I. Bayesian Econometrics A. Introduction
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2 Consider simple model: y t ± 6 ²/ t t ± 1,2,. .., T / t ~ N ± 0, @ 2   i.i.d. @ 2 known, want to estimate 6 Estimate (MLE ± OLS ± GMM): § 6 ± T " 1 ! t ± 1 T y t § 6 ~ N ± 6 , @ 2 / T   Beginning student wants to say, “there is a 95% probability that 6 is in the interval § 6 o 1.96 @ / T Classical statistician says: “No, no, no! 6 is the true value. It either equals 5 or it doesn’t. There is no probability statement about 6 . “What is true is that if we use this procedure to construct an interval in thousands of different samples, in 95% of those samples, our interval will contain the true 6 .”
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3 Suppose we observe a sample mean of 5 and know that σ / T = 1 , and then ask the classical statistician: “Do you know the true 6 ?” “No.” “Choose between these options. Option A: I give you $5 now. Option B: I give you $10 if the true 6 is in the interval between 2.0 and 3.5.” “I’ll take the $5, thank you.” “How about these? Option A: I give you $5 now. Option B: I give you $10 if the true 6 is between -5.0 and +10.0.” “OK, I’ll take option B.” “Option A: I generate a uniform number between 0 and 1. If the number is less than = , I give you $5. Option B: I give you $5 if the true 6 is in the interval (2.0,4.5). The value of = is 0.2” “Option B.” “How about if = = 0.8?” “Option A.”
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4 If the statistician’s choices among such comparisons satisfy certain axioms of rationality, then there will exist a unique = ' such that he chooses Option A whenever = ± = ' and Option B whenever = ² = ' .Wem i g h t interpret this = ' as the statistician’s (subjective) probability that 6 is in the interval ± 2.0,4.0   . Bayesian idea: before seeing the data ± y 1 , y 2 ,..., y T   , the statistician hadsomesub ject iveprobab i l ity beliefs about the value of 6 , called the “prior distribution.” Suppose we represent these beliefs with a probability distribution, p ± 6   , called the “prior distribution.” For example, 6 ~ N ± m , A 2   . p ± 6   ³ 1 2 = A exp ¡ " ± 6 " m   2 / ± 2 A 2  ¢
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5 m represents our “best guess” as to the value of 6 before seeing data A 2 represents our confidence in this guess– small A , very confident We think of the usual likelihood function as the probability of the data given fixed values for 6 and @ : p ± y | 6 ; @   ±
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Econ226_IAB - Econ 226: Bayesian and Numerical Methods...

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