Statistics and econometrics — science building on fantasy worlds

28 Sep, 2021 at 11:04 | Posted in Statistics & Econometrics | 2 Comments

In econometrics one often gets the feeling that many of its practitioners think of it as a kind of automatic inferential machine: input data and out comes casual knowledge. This is like pulling a rabbit from a hat. Great — but first you have to put the rabbit in the hat. And this is where assumptions come into the picture.

The assumption of imaginary ‘super populations’ is one of the many dubious assumptions used in modern econometrics.

As social scientists — and economists — we have to confront the all-important question of how to handle uncertainty and randomness. Should we define randomness with probability? If we do, we have to accept that to speak of randomness we also have to presuppose the existence of nomological probability machines, since probabilities cannot be spoken of – and actually, to be strict, do not at all exist – without specifying such system-contexts. Accepting a domain of probability theory and sample space of infinite populations also implies that judgments are made on the basis of observations that are actually never made!

Infinitely repeated trials or samplings never take place in the real world. So that cannot be a sound inductive basis for a science with aspirations of explaining real-world socio-economic processes, structures or events. It’s not tenable.

And as if this wasn’t enough, one could — as we’ve seen — also seriously wonder what kind of ‘populations’ these statistical and econometric models ultimately are based on. Why should we as social scientists — and not as pure mathematicians working with formal-axiomatic systems without the urge to confront our models with real target systems — unquestioningly accept models based on concepts like the ‘infinite super populations’ used in e.g. the ‘potential outcome’ framework that has become so popular lately in social sciences?

The theory requires that the data be embedded in a stochastic framework, complete with random variables, probability distributions, and unknown parameters. However, the data often arrive without benefit of randomness. In such cases, the investigators may still wish to separate effects of “the causes they wish to study or are trying to detect” from “accidental occurrences due to the many other circumstances which they cannot control.” What can they do? Usually, they follow Fisher (1922) into a fantasy world “by constructing a hypothetical infinite population, of which the actual data are regarded as constituting a random sample.” Unfortunately, this fantasy world is often harder to understand than the original problem which lead to its invocation.

David Freedman & David Lane

Of course one could treat observational or experimental data as random samples from real populations. I have no problem with that (although it has to be noted that most ‘natural experiments’ are not based on random sampling from some underlying population — which, of course, means that the effect-estimators, strictly seen, only are unbiased for the specific groups studied). But probabilistic econometrics does not content itself with that kind of populations. Instead, it creates imaginary populations of ‘parallel universes’ and assume that our data are random samples from that kind of  ‘infinite super populations.’

But this is actually nothing else but hand-waving! And it is inadequate for real science. As David Freedman writes:

These are convenient fictions … Nevertheless, reliance on imaginary populations is widespread. Indeed regression models are commonly used to analyze convenience samples … The rhetoric of imaginary populations is seductive because it seems to free the investigator from the necessity of understanding how data were generated.

Modelling assumptions made in statistics and econometrics are more often than not made for mathematical tractability reasons, rather than verisimilitude. That is unfortunately also a reason why the methodological ‘rigour’ encountered when taking part of statistical and econometric research to a large degree is nothing but deceptive appearance. The models constructed may seem technically advanced and very ‘sophisticated,’ but that’s usually only because the problems here discussed have been swept under the carpet. Assuming that our data are generated by ‘coin flips’ in an imaginary ‘superpopulation’ only means that we get answers to questions that we are not asking. The inferences made based on imaginary ‘superpopulations,’ well, they too are nothing but imaginary. We should not — as already Aristotle noted — expect more rigour and precision than the object examined allows. And in social sciences — including economics and econometrics — it’s always wise to ponder C. S. Peirce’s remark that universes are not as common as peanuts …