An Introduction to Econometric Theory

12Random Regressors

12.1 Conditional Probability

At this point, get ready for another major change of gear. Chapters 7 to 11 have explored the properties and implications of the classical regression model, although not before explaining how this framework has major limitations from the point of view of modelling economic phenomena. What the present chapter aims to show is that a fairly minor modification of the statistical framework can provide a new context for the results assembled up to now, one much more appropriate to economics, and most of the results will hold virtually unchanged. However, some additional statistical theory is needed to make sense of the new approach. This needs to be developed with some care.

In elementary probability theory, Bayes' Rule is a well‐known formula for assigning a probability to an unobserved event when another event has been observed to occur. This is denoted , and the rule is

(12.1)

Simply enough, if we know has occurred, then the probability of , given ...