Random experiments
The starting point of Spanos’ econometric doctrine is the concept of a random experiment E, which he characterizes as an idealized representation of the dgp that satisfies the three conditions of the following definition.
12.3.1 Definition
A random experiment is an experiment E that satisfies the following conditions:
(1) all possible distinct outcomes are known a priori;
(2) in any particular trial the outcome is not known a priori;
(3) it can be repeated under identical conditions.
Examples of a random experiment are: the throwing of fair coin, or a lottery repeated under similar circumstances. The former definition of random experiment - nay, the very word ‘experiment’ - suggests that the mechanism producing the outcomes is under the control of the observer. For, otherwise, it would be impossible to repeat the process “under identical conditions”. That is why some authors prefer to characterize a random experiment as a process by which something uncertain is observed. Sometimes the observer is only a passive observer of a process that behaves in a random way, without being able to influence in the least the outcome of the process. Accordingly, I will introduce the more general concept of a random process modifying clause (3) as follows: it repeats itself under analogous conditions. When these conditions are under the control of the observer the process is an experiment.
On the other hand, the random process (or its description) does not have to be idealized. For instance, the set of outcomes of throwing a fair coin is exactly described as consisting of two elements, namely head and tail. Thus, the sample space usually is not an idealized construct, especially when it is finite. The set-theoretical operations are not idealized either, as they have a very clear and intuitive meaning.
Actually, Definition 12.3.1 is suited for the relative-frequency interpretation of probability, which requires the repetition of the same experiment. The problem is that it is not uncommon to find in economics that probabilities have to be obtained out of single-case occurrences of singular events, without any possibility of repetition. The reason why philosophers like Karl Popper and Patrick Suppes moved away from the relative-frequency to consider the propensity interpretation of the probability of singular events as fundamental or primary is the single-case problem.
Since we find in the economic realm many random phenomena occurring only once, the propensity interpretation seems more appropriate for econometrics. The conditional probability structures are more appropriate for propensity representations of probability, and more fundamental than the usual probability spaces defined by the Kolmogorov axioms. Nevertheless, it will be convenient to begin by recalling the definition of these spaces.
12.4