By Gary Koop
This quantity within the Econometric workouts sequence includes questions and solutions to supply scholars with worthy perform, as they try to grasp Bayesian econometrics. as well as many theoretical routines, this booklet includes routines designed to strengthen the computational instruments utilized in sleek Bayesian econometrics. The latter half the publication comprises workouts that exhibit how those theoretical and computational talents are mixed in perform, to hold out Bayesian inference in a large choice of types well-known by way of econometricians. Aimed essentially at complicated undergraduate and graduate scholars learning econometrics, this ebook can also be important for college students learning finance, advertising and marketing, agricultural economics, company economics or, extra typically, any box which makes use of facts. The ebook additionally comes outfitted with a helping site containing all of the proper info units and MATLAB desktop courses for fixing the computational workouts.
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Extra info for Bayesian Econometric Methods (Econometric Exercises)
The Bayesian solution to the randomness of the loss function is similar to the frequentist solution: Take its expectation before minimization. The expectation, however, is with respect to the posterior distribution θ|y, and not the sampling distribution y|θ used to obtain the risk function. The Bayesian prescription is equivalent to the principle usually advocated for economic agents acting in a world of uncertainty: Using all available information, choose actions so as to maximize expected utility, or equivalently, minimize expected loss.
12) can be written as θ1 = µ + η, where η ∼ N (0, h−1 ). Substituting the equation for our prior into the equation for y gives y = ιT µ + + ιT η, which implies y ∼ N ιT µ, θ2−1 IT + h−1 ιT ιT , or equivalently, p(y) = φ y|ιT µ, [θ2−1 IT + h−1 ιT ιT ] . 25) are in agreement, though if we have done our derivations correctly, they must agree. 25). 24) involving the normal density for y. 24) will also fall out of this derivation, though we suppress them here for clarity. 25) involves a term of the form exp −[1/2](y − ιT µ) [θ2−1 IT + h−1 ιT ιT ]−1 (y − ιT µ) .
The normalizing constant will serve to update the component weights ω m . 14 (Conjugate multivariate normal analysis) Consider a random sample Yt (t = 1, 2, . . , T ) from a N (µ, Σ) distribution, where µ is M × 1 and Σ is an M × M positive deﬁnite matrix. Deﬁne 1 y= T T yt t=1 and T (yt − y)(yt − y) . S= t=1 Suppose m and µ are both M × 1, T > 0 and ω > M are both scalars, and S is an M × M positive deﬁnite matrix. Consider the natural conjugate priors for the following three cases. (a) Suppose µ is unknown, Σ−1 is known, and the prior distribution for µ is multivariate normal with prior density p(µ) = φ(µ|µ, Σ).