What is a statistical model?

24 June, 2017 at 14:05 | Posted in Statistics & Econometrics | 1 Comment

My critique is that the currently accepted notion of a statistical model is not scientific; rather, it is a guess at what might constitute (scientific) reality without the vital element of feedback, that is, without checking the hypothesized, postulated, wished-for, natural-looking (but in fact only guessed) model against that reality. To be blunt, as far as is known today, there is no such thing as a concrete i.i.d. (independent, identically distributed) process, not because this is not desirable, nice, or even beautiful, but because Nature does not seem to be like that … As Bertrand Russell put it at the end of his long life devoted to philosophy, “Roughly speaking, what we know is science and what we don’t know is philosophy.” In the scientific context, but perhaps not in the applied area, I fear statistical modeling today belongs to the realm of philosophy.

science_medal9_hTo make this point seem less erudite, let me rephrase it in cruder terms. What would a scientist expect from statisticians, once he became interested in statistical problems? He would ask them to explain to him, in some clear-cut cases, the origin of randomness frequently observed in the real world, and furthermore, when this explanation depended on the device of a model, he would ask them to continue to confront that model with the part of reality that the model was supposed to explain. Something like this was going on three hundred years ago … But in our times the idea somehow got lost when i.i.d. became the pampered new baby.

Rudolf Kalman

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 Haavelmo’s domain of probability theory and sample space of infinite populations — just as Fisher’s ‘hypothetical infinite population,’ von Mises’ ‘collective’ or Gibbs’ ‘ensemble’ — 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 so the way social scientists — including economists and econometricians — often uncritically and without arguments have come to simply assume that one can apply probability distributions from statistical theory on their own area of research, is not acceptable.

This importantly also means that if you cannot show that data satisfies all the conditions of the probabilistic nomological machine — including e. g. the distribution of the deviations corresponding to a normal curve — then the statistical inferences used, lack sound foundations.

Trying to apply statistical models outside overly simple nomological machines like coin tossing and roulette wheels, scientists run into serious problems, the greatest being the need for lots of more or less unsubstantiated — and sometimes wilfully hidden — assumptions to be able to make any sustainable inferences from the models. Much of the results that economists and other social scientists present with their statistical/econometric models depend to a substantial part on the use of mostly unfounded ‘technical’ assumptions.

Making outlandish statistical assumptions does not provide a solid ground for doing relevant social science. It is rather a recipe for producing fiction masquerading as science.

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1 Comment

  1. A fair coin or roulette wheel is a significant engineering achievement. It has to be constructed with directed and precise effort, guided not incidentally by a model idea of what is to be achieved and how it is to be achieved.
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    I have done quality control in a factory. We would consider getting errors, aka deviations from a control value within (hopefully) the bounds of a range of tolerance, into a Gaussian pattern our mission. We called that getting the production process under control. It did not just happen; we made it (a semblance of i.i.d.) happen.
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    And, sometimes, despite best efforts, it did not happen.
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    I am inclined to believe, most of the economically interesting behavior and institutional structure in our industry had to do with either creating and achieving models and associated control values or with dealing with the consequences when nature and ignorance (aka uncertainty) took over, and consequential mistakes (not to be confused with well-behaved and inconsequential errors of a process under control) were made, in a decidedly unGaussian fashion.
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    Economists really do need a theory of production. Then, maybe, they can get statistics into context.


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