Palgrave Handbook of Econometrics: Applied Econometrics

John DiNardo 113

low sciences, there is a great deal of interest in “natural experiments.” Put another
way, one wants to try to draw a contrast between “experience” and “experiment.”
In the case of the former, statistical tools may or may not be particularly helpful,
and other methods for gaining insight might easily dominate. In the latter case,
one generally feels more hopeful that statistical reasoning might help.

3.4 A few points of agreement, then...

Statistics and probability, as we understand them today, got a surprisingly late start
in the (European) history of ideas.^22 Before the seventeenth century a major use
of the word “probability” in English was to describe a characteristic of an opinion
and dealt with the authority of the person who issued the opinion. “Thus [it could
be said] Livy had more of probability but Polybius had more of truth.” Or, “Such a
fact is probable but undoubtedly false,” relying on the implicit reference of what
is “probable” to authority or consensus (Barnouw, 1979).
A theme that will recur frequently is the notion thateverythingin metastatistics
is a topic of debate. As I discuss in section 3.4.2, even the definition of probability
is the subject of considerable debate. However, it will be helpful to have at least
some terminology to work with before enjoining the metaphysics.

3.4.1 Kolmogorov’s axioms

One place to begin is a review of a few of Kolmogorov’s axioms which Bayesians and
non-Bayesians (generally) accept, although they interpret the meaning of “prob-
ability” very differently. Though they can be defined with much more care and
generality, we will define them crudely for the discrete case:

1. Given a sample spaceof possible eventsA 1 ,A 2 ,...Aksuch that:

≡

∑k

i= 1

⋃

Ai for i=1, 2,...k.

2. The probability of an eventAiis a number which lies between 0 and 1.

0 <P(Ai)<1.

An event which cannot happen has a probability of zero, and a certain

outcome has a probability of 1.^23 Two events, A 1 andA 2 , are mutually

exclusive ifP(A 1

⋂

A 2 )=0.

3. For any two mutually exclusive events probability is additive:

P(A 1

⋃

A 2 )=P(A 1 )+P(A 2 ).