Palgrave Handbook of Econometrics: Applied Econometrics

John DiNardo 123

0

5

10

Posterior Distribution

0 .2 .4 .6 .8 1

x

0

5

10

0 .2 .4 .6 .8 1

x

Very informative prior

Less informative prior

Figure 3.3 Different prior distributions, same posterior

“no information” into a prior distribution.^35 Consider the problem of estimat-

ing the length of a square garden which has sides of length between 1 and 5

feet. Based on this information, it seems “natural” to say that there is a 0.5 prob-

ability that the garden has sides oflengthbetween 1 and 3 feet. Equivalently,

the information could be cast as saying that the area of the garden is between 1

and 25 square feet. In that case, it would appear just as natural to say that the

probability is 0.5 that theareaof the garden is between 1 and 13 square feet.

This natural assignment of probability, however, implies that the probability

is 0.5 that the length of the sides is between 1 and≈3.61 feet (

√

13). How-

ever, it would be personally inconsistent to believe both claims and there is no

principled method to reconcile the two different priors.

2. Even if a prior distribution is useful to the person holding it, it is not clear
   that it is useful to anyone else. LeCam (1977) observes that, for the bino-
   mial experiment, for arbitrary positive constantC, “if we follow the theory
   and communicate to another person a densityCθ^100 ( 1 −θ)^100 this person
   has no way of knowing whether (1) an experiment with 200 trials has taken
   place or (2) no experiment took place and this is simply ana prioriexpres-
   sion of opinion. Since some of us would argue that the case with 200 trials
   is more ‘reliable’ than the other, something is missing in the transmission of
   information.”

3.5 The importance of the data-generation process

3.5.1 An idealized hypothesis test

Ultimately we would like to return to the “introductory puzzle,” but before we do,
let us introduce some context. The value of hypothesis testing has been frequently
debated among non-Bayesians, but it may help to consider an idealized notion of
how it issupposedto be done – this version is from Kmenta (2000) – when wishing