## Understanding the limits of statistical inference

6 July, 2017 at 18:35 | Posted in Statistics & Econometrics | 1 Comment

This is indeed an instructive video on what *statistical* inference is all about.

But we have to remember that economics and statistics are two quite different things, and as long as economists cannot identify their statistical theories with real-world phenomena there is no real warrant for taking their statistical inferences seriously.

Just as there is no such thing as a ‘free lunch,’ there is no such thing as a ‘free probability.’ To be able at all to talk about probabilities, you have to specify a model. If there is no chance set-up or model that generates the probabilistic outcomes or events -– in statistics one refers to any process where you observe or measure as an experiment (rolling a die) and the results obtained as the outcomes or events (number of points rolled with the die, being e. g. 3 or 5) of the experiment -– there, strictly seen, is no event at all.

Probability is a relational element. It always must come with a specification of the model from which it is calculated. And then to be of any empirical scientific value it has to be shown to coincide with (or at least converge to) real data generating processes or structures –- something seldom or never done in economics.

And this is the basic problem!

If you have a fair roulette-wheel, you can arguably specify probabilities and probability density distributions. But how do you conceive of the analogous ‘nomological machines’ for prices, gross domestic product, income distribution etc? Only by a leap of faith. And that does not suffice. You have to come up with some really good arguments if you want to persuade people into believing in the existence of socio-economic structures that generate data with characteristics conceivable as stochastic events portrayed by probabilistic density distributions! Not doing that, you simply conflate statistical and economic inferences.

And even worse — some economists using statistical methods think that algorithmic formalisms somehow give them access to causality. That is, however, simply not true. Assuming ‘convenient’ things like faithfulness or stability,is to assume what has to be proven. Deductive-axiomatic methods used in statistics do no produce evidence for causal inferences. The real casuality we are searching for is the one existing in the real-world around us. If there is no warranted connection between axiomatically derived statistical theorems and the real-world, well, then we haven’t really obtained the causation we are looking for.

## Econometric alchemy

2 July, 2017 at 21:09 | Posted in Statistics & Econometrics | Comments Off on Econometric alchemyThus we have “econometric modelling”, that activity of matching an incorrect version of [the parameter matrix] to an inadequate representation of [the data generating process], using insufficient and inaccurate data. The resulting compromise can be awkward, or it can be a useful approximation which encompasses previous results, throws light on economic theory and is sufficiently constant for prediction, forecasting and perhaps even policy. Simply writing down an “economic theory”, manipulating it to a “condensed form” and “calibrating” the resulting parameters using a pseudo-sophisticated estimator based on poor data which the model does not adequately describe constitutes a recipe for disaster, not for simulating gold! Its only link with alchemy is self-deception.

## Heterogeneity and the flaw of averages

29 June, 2017 at 00:11 | Posted in Statistics & Econometrics | Comments Off on Heterogeneity and the flaw of averagesWith interactive confounders explicitly included, the overall treatment effect β0 + β′zt is not a number but a variable that depends on the confounding effects. Absent observation of the interactive compounding effects, what is estimated is some kind of average treatment effect which is called by Imbens and Angrist (1994) a “Local Average Treatment Effect,” which is a little like the lawyer who explained that when he was a young man he lost many cases he should have won but as he grew older he won many that he should have lost, so that on the average justice was done. In other words, if you act as if the treatment effect is a random variable by substituting βt for β0 + β′zt , the notation inappropriately relieves you of the heavy burden of considering what are the interactive confounders and finding some way to measure them. Less elliptically, absent observation of z, the estimated treatment effect should be transferred only into those settings in which the confounding interactive variables have values close to the mean values in the experiment. If little thought has gone into identifying these possible confounders, it seems probable that little thought will be given to the limited applicability of the results in other settings.

Yes, indeed, regression-based averages is something we have reasons to be cautious about.

Suppose we want to estimate the average causal effect of a dummy variable (T) on an observed outcome variable (O). In a usual regression context one would apply an ordinary least squares estimator (OLS) in trying to get an unbiased and consistent estimate:

O = α + βT + ε,

where α is a constant intercept, β a constant ‘structural’ causal effect and ε an error term.

The problem here is that although we may get an estimate of the ‘true’ average causal effect, this may ‘mask’ important heterogeneous effects of a causal nature. Although we get the right answer of the average causal effect being 0, those who are ‘treated’ ( T=1) may have causal effects equal to -100 and those ‘not treated’ (T=0) may have causal effects equal to 100. Contemplating being treated or not, most people would probably be interested in knowing about this underlying heterogeneity and would not consider the OLS average effect particularly enlightening.

The heterogeneity problem does not just turn up as an *external* validity problem when trying to ‘export’ regression results to different times or different target populations. It is also often an *internal* problem to the millions of OLS estimates that economists produce every year.

## What is a statistical model?

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

To 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.

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.

## Simpson’s paradox

21 June, 2017 at 08:29 | Posted in Statistics & Econometrics | Comments Off on Simpson’s paradox

From a more theoretical perspective, Simpson’s paradox importantly shows that causality can never be reduced to a question of statistics or probabilities, unless you are — miraculously — able to keep constant *all* other factors that influence the probability of the outcome studied.

To understand causality we always have to relate it to a specific causal *structure*. Statistical correlations are *never* enough. No structure, no causality.

Simpson’s paradox is an interesting paradox in itself, but it can also highlight a deficiency in the traditional econometric approach towards causality. Say you have 1000 observations on men and an equal amount of observations on women applying for admission to university studies, and that 70% of men are admitted, but only 30% of women. Running a logistic regression to find out the odds ratios (and probabilities) for men and women on admission, females seem to be in a less favourable position (‘discriminated’ against) compared to males (male odds are 2.33, female odds are 0.43, giving an odds ratio of 5.44). But once we find out that males and females apply to different departments we may well get a Simpson’s paradox result where males turn out to be ‘discriminated’ against (say 800 male apply for economics studies (680 admitted) and 200 for physics studies (20 admitted), and 100 female apply for economics studies (90 admitted) and 900 for physics studies (210 admitted) — giving odds ratios of 0.62 and 0.37).

Econometric patterns should never be seen as anything else than possible clues to follow. From a critical realist perspective it is obvious that behind observable data there are real structures and mechanisms operating, things that are — if we really want to understand, explain and (possibly) predict things in the real world — more important to get hold of than to simply correlate and regress observable variables.

Math cannot establish the truth value of a fact. Never has. Never will.

## Logistic regression (student stuff)

21 June, 2017 at 08:25 | Posted in Statistics & Econometrics | Comments Off on Logistic regression (student stuff)

And in the video below (in Swedish) yours truly shows how to perform a logit regression using Gretl:

## Causality matters!

20 June, 2017 at 10:29 | Posted in Statistics & Econometrics | 1 Comment

Causality in social sciences — and economics — can never solely be a question of statistical inference. Causality entails more than predictability, and to really in depth explain social phenomena require theory. Analysis of variation — the foundation of all econometrics — can never in itself reveal how these variations are brought about. First when we are able to tie actions, processes or structures to the statistical relations detected, can we say that we are getting at relevant explanations of causation.

Most facts have many different, possible, alternative explanations, but we want to find the best of all contrastive (since all real explanation takes place relative to a set of alternatives) explanations. So which is the best explanation? Many scientists, influenced by statistical reasoning, think that the likeliest explanation is the best explanation. But the likelihood of x is not in itself a strong argument for thinking it explains y. I would rather argue that what makes one explanation better than another are things like aiming for and finding powerful, deep, causal, features and mechanisms that we have warranted and justified reasons to believe in. Statistical — especially the variety based on a Bayesian epistemology — reasoning generally has no room for these kinds of explanatory considerations. The only thing that matters is the probabilistic relation between evidence and hypothesis. That is also one of the main reasons I find abduction — inference to the best explanation — a better description and account of what constitute actual scientific reasoning and inferences.

For more on these issues — see the chapter “Capturing causality in economics and the limits of statistical inference” in my On the use and misuse of theories and models in economics.

In the social sciences … regression is used to discover relationships or to disentangle cause and effect. However, investigators have only vague ideas as to the relevant variables and their causal order; functional forms are chosen on the basis of convenience or familiarity; serious problems of measurement are often encountered.

Regression may offer useful ways of summarizing the data and making predictions. Investigators may be able to use summaries and predictions to draw substantive conclusions. However, I see no cases in which regression equations, let alone the more complex methods, have succeeded as engines for discovering causal relationships.

Some statisticians and data scientists think that algorithmic formalisms somehow give them access to causality. That is, however, simply not true. Assuming ‘convenient’ things like faithfulness or stability is not to give proofs. It’s to assume what has to be proven. Deductive-axiomatic methods used in statistics do no produce evidence for causal inferences. The real casuality we are searching for is the one existing in the real-world around us. If their is no warranted connection between axiomatically derived theorems and the real-world, well, then we haven’t really obtained the causation we are looking for.

## Proving gender discrimination using randomization (student stuff)

17 June, 2017 at 10:06 | Posted in Statistics & Econometrics | Comments Off on Proving gender discrimination using randomization (student stuff)

## Ed Leamer and the pitfalls of econometrics

16 June, 2017 at 18:09 | Posted in Statistics & Econometrics | 1 CommentEd Leamer’s Tantalus on the Road to Asymptopia is one of my favourite critiques of econometrics, and for the benefit of those who are not versed in the econometric jargon, this handy summary gives the gist of it in plain English:

Most work in econometrics and regression analysis is — still — made on the assumption that the researcher has a theoretical model that is ‘true.’ Based on this belief of having a correct specification for an econometric model or running a regression, one proceeds as if the only problem remaining to solve have to do with measurement and observation.

When things sound to good to be true, they usually aren’t. And that goes for econometric wet dreams too. The snag is, as Leamer convincingly argues, that there is pretty little to support the perfect specification assumption. Looking around in social science and economics we don’t find a single regression or econometric model that lives up to the standards set by the ‘true’ theoretical model — and there is pretty little that gives us reason to believe things will be different in the future.

To think that we are being able to construct a model where all relevant variables are included and correctly specify the functional relationships that exist between them, is not only a belief without support, but a belief *impossible* to support.

The theories we work with when building our econometric regression models are insufficient. No matter what we study, there are always some variables missing, and we don’t know the correct way to functionally specify the relationships between the variables.

*Every* regression model constructed is misspecified. There are always an endless list of possible variables to include, and endless possible ways to specify the relationships between them. So every applied econometrician comes up with his own specification and ‘parameter’ estimates. The econometric Holy Grail of consistent and stable parameter-values is nothing but a dream.

In order to draw inferences from data as described by econometric texts, it is necessary to make whimsical assumptions. The professional audience consequently and properly withholds belief until an inference is shown to be adequately insensitive to the choice of assumptions. The haphazard way we individually and collectively study the fragility of inferences leaves most of us unconvinced that any inference is believable. If we are to make effective use of our scarce data resource, it is therefore important that we study fragility in a much more systematic way. If it turns out that almost all inferences from economic data are fragile, I suppose we shall have to revert to our old methods …

A rigorous application of econometric methods in economics really presupposes that the phenomena of our real world economies are ruled by stable causal relations between variables. Parameter-values estimated in specific spatio-temporal contexts are *presupposed* to be exportable to totally different contexts. To warrant this assumption one, however, has to convincingly establish that the targeted acting causes are stable and invariant so that they maintain their parametric status after the bridging. The endemic lack of predictive success of the econometric project indicates that this hope of finding fixed parameters is a hope for which there really is no other ground than hope itself.

The theoretical conditions that have to be fulfilled for regression analysis and econometrics to really work are nowhere even closely met in reality. Making outlandish statistical assumptions does not provide a solid ground for doing relevant social science and economics. Although regression analysis and econometrics have become the most used quantitative methods in social sciences and economics today, it’s still a fact that the inferences made from them are invalid.

Econometrics — and regression analysis — is basically a deductive method. Given the assumptions (such as manipulability, transitivity, separability, additivity, linearity, etc) it delivers deductive inferences. The problem, of course, is that we will never completely know when the assumptions are right. Conclusions can only be as certain as their premises — and that also applies to econometrics and regression analysis.

## The fundamental econometric dilemma

27 May, 2017 at 10:20 | Posted in Statistics & Econometrics | Comments Off on The fundamental econometric dilemmaMany thanks for sending me your article. I enjoyed it very much. I am sure these matters need discussing in that sort of way. There is one point, to which in practice I attach a great importance, you do not allude to. In many of these statistical researches, in order to get enough observations they have to be scattered over a lengthy period of time; and for a lengthy period of time it very seldom remains true that the environment is sufficiently stable. That is the dilemma of many of these enquiries, which they do not seem to me to face. Either they are dependent on too few observations, or they cannot rely on the stability of the environment. It is only rarely that this dilemma can be avoided.

Letter from J. M. Keynes to T. Koopmans, May 29, 1941

Econometric patterns should never be seen as anything else than possible clues to follow. Behind observable data there are real structures and mechanisms operating, things that are — if we really want to understand, explain and (possibly) predict things in the real world — more important to get hold of than to simply correlate and regress observable variables.

Math cannot establish the truth value of a fact. Never has. Never will.

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