Introduction to Bayesian Econometrics by Edward Greenberg

By Edward Greenberg

This concise textbook is an creation to econometrics on the graduate or complicated undergraduate point. It differs from different books in econometrics in its use of the Bayesian method of statistics. This process, not like the frequentist method of information, makes particular use of past details and relies at the subjective view of chance, which takes likelihood conception as using to all occasions during which uncertainty exists, together with uncertainty over the values of parameters.

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Example text

Yn ) have been observed, and we wish to predict the outcome of the next toss, yn+1 . From the Bayesian viewpoint, we can compute P (yn+1 = 1|y), the probability that the next toss results in a head, given the data previously observed; the probability of a tail is one minus this probability. We compute this probability by making use of the identity P (yn+1 = 1|y) = f (yn+1 = 1, θ |y) dθ = P (yn+1 = 1|θ, y)π (θ |y) dθ = P (yn+1 = 1|θ )π (θ |y) dθ. Notice carefully what we have done. Following the rules of probability, we write P (yn+1 = 1|y) as an integral in which θ is first introduced into and then integrated out of the joint density f (yn+1 = 1, θ |y).

Why choose the beta distribution? First, it is defined in the relevant range. 1. Depending on the choice of α and β, this prior can capture beliefs that indicate θ is centered at 1/2, or it can shade θ toward zero or one; it can be highly concentrated, or it can be spread out; and, when both parameters are less than one, it can have two modes. The shape of a beta distribution can be understood by examining its mean and variance: α αβ E(θ ) = , Var(θ ) = . 2 α+β (α + β) (α + β + 1) From these expressions you can see that the mean is 1/2 if α = β, a larger α (β) shades the mean toward 1 (0), and the variance decreases as α or β increases.

Yn ) , X is the n × K matrix of covariates,   x11 x12 ... x1K   x21 x22 ... x2K    X= . , . .  .. ..    xn1 xn2 ... xnK and u = (u1 , . . , un ) . Inference in this model finds point estimates for the unknown parameters β and σ 2 , constructs interval estimates for the parameters, compares models that contain different sets of covariates, and predicts a value of yi for a given set of covariate values. The first covariate xi1 is often set equal to 1 for all observations, in which case β1 is called the intercept.

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