By Edward Greenberg

This concise textbook is an advent to econometrics on the graduate or complex undergraduate point. It differs from different books in econometrics in its use of the Bayesian method of records. This method, unlike the frequentist method of records, makes specific use of previous info and relies at the subjective view of chance, which takes likelihood concept as employing to all occasions within which uncertainty exists, together with uncertainty over the values of parameters.

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**Sample text**

If we denote the true value of θ by θ0 , it can be shown that ¯ |y) → l(θ ¯ 0 |y). 1 Properties of Posterior Distributions 27 Accordingly, for large n, the posterior distribution collapses to a distribution with all its probability at θ0 . This property is similar to the criterion of consistency in the frequentist literature and extends to the multiparameter case. Finally, we can use these ideas to say something about the form of the posterior distribution for large n. To do this, take a second-order Taylor series approximation of l(θ |y) around θˆ , the MLE of θ : ˆ − n (θ − θˆ )2 [−l¯ (θˆ |y)] l(θ |y) ≈ l(θ|y) 2 ˆ − n (θ − θˆ )2 , = l(θ|y) 2v where l¯ (θˆ |y) = (1/n) k l (θˆ |yk ) and v = [−l¯ (θˆ |y)]−1 .

It will tend to become large if M1 is the true model and small if M2 is true. The second term shows that the log Bayes factor penalizes models with larger numbers of parameters, where the penalty is log(n) times the difference in the number of parameters divided by two. We return to the coin-tossing example to illustrate the use of Bayes factors for model comparison. To specify two competing models, consider the following variation on our basic experiment. A coin is tossed m times by Michaela and then tossed m times by Lila.

The frequentist approach to model comparison makes use of hypothesis tests. In this approach, the null hypothesis H0 is rejected in favor of the alternative hypothesis HA if the value of a statistic computed from the data falls in the critical region. The critical region is usually specified to set the probability that H0 is rejected when it is true at a small value, where the probability is computed over the distribution of the statistic. As mentioned before, this calculation depends on values of the statistic that were not observed.