By Gary Koop

This quantity within the Econometric routines sequence includes questions and solutions to supply scholars with worthwhile perform, as they try and grasp Bayesian econometrics. as well as many theoretical routines, this ebook comprises routines designed to advance the computational instruments utilized in sleek Bayesian econometrics. The latter half the ebook includes routines that convey how those theoretical and computational talents are mixed in perform, to hold out Bayesian inference in a large choice of versions regularly occurring by way of econometricians. Aimed basically at complicated undergraduate and graduate scholars learning econometrics, this e-book can also be worthy for college kids learning finance, advertising, agricultural economics, company economics or, extra usually, any box which makes use of statistics. The ebook additionally comes built with a assisting web site containing the entire correct facts units and MATLAB laptop courses for fixing the computational workouts.

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11) 40 4 Frequentist properties of Bayesian estimators (e) It is well known that µ ˜ = Y is an unbiased estimator of µ with MSE(˜ µ) = Var(˜ µ) = σ2 . 12), MSE(ˆ µ) < MSE(˜ µ) iff T σ 2 + T 2 (µ − µ)2 T 2 < or equivalently, iff µ−µ √ σ/ T 2 < T −1 (T + 2T ). 14) In other words, the Bayesian point estimator has a smaller MSE than the sample mean provided the prior mean µ is “sufﬁciently close” to the unknown population mean µ, where √ closeness is measured relative to the sampling standard deviation σ/ T in Y .

25)]. Using this result, we can write the quadratic form in our exponential kernel as exp −[1/2](hT −1 )(y − ιT µ) [IT − (β/T )ιT ιT ](y − ιT µ) . With a little work, one can show that this quadratic form can be expressed as 2 (y − ιT µ) [IT − (β/T )ιT ιT ](y − ιT µ) = 2 (yt − µ) − (β/T ) t (yt − µ) t . 24)], we can also write (yt − µ)2 = νs2 + T (y − µ)2 , t and similarly, (yt − µ) = T (y − µ). t Putting these results together gives (y − ιT µ) [IT − (β/T )ιT ιT ](y − ιT µ) = vs2 + T (y − µ)2 − βT (y − µ)2 .

Suppose a prior density of the form σ −2 ∼ γ(s−2 , ν) is employed. Find the posterior for σ −2 . 16) gives the likelihood function 1 L(σ −2 ) = (2π)−T /2 (σ −2 )T /2 exp − σ −2 [νs2 + T (y − θ1 )2 ] , 2 and the prior density is 1 p(σ −2 ) ∝ (σ −2 )(ν−2)/2 exp − σ −2 νs2 . 2 2 Bayesian inference 21 Using Bayes’ theorem gives the posterior density p(σ −2 |y) ∝ p(σ −2 )L(σ −2 ) ∝ (σ −2 )(ν+T −2)/2 exp − ∝ (σ −2 )(ν−2)/2 exp − σ −2 2 [νs + νs2 + T (y − θ1 )2 ] 2 σ −2 2 νs . 28) and s2 = ν −1 [νs2 + νs2 + T (y − θ1 )2 ].