## The difference between statistical and causal assumptions

24 Jun, 2019 at 19:57 | Posted in Statistics & Econometrics | Leave a commentThere are three fundamental differences between statistical and causal assumptions. First, statistical assumptions, even untested, are testable in principle, given sufficiently large sample and sufficiently fine measurements. Causal assumptions, in contrast, cannot be verified even in principle, unless one resorts to experimental control. This difference is especially accentuated in Bayesian analysis. Though the priors that Bayesians commonly assign to statistical parameters are untested quantities, the sensitivity to these priors tends to diminish with increasing sample size. In contrast, sensitivity to priors of causal parameters … remains non-zero regardless of (nonexperimental) sample size.

Second, statistical assumptions can be expressed in the familiar language of probability calculus, and thus assume an aura of scholarship and scientific re- spectability. Causal assumptions, as we have seen before, are deprived of that honor, and thus become immediate suspect of informal, anecdotal or metaphysical thinking. Again, this difference becomes illuminated among Bayesians, who are accustomed to accepting untested, judgmental assumptions, and should therefore invite causal assumptions with open arms—they don’t. A Bayesian is prepared to accept an expert’s judgment, however esoteric and untestable, so long as the judgment is wrapped in the safety blanket of a probability expression. Bayesians turn extremely suspicious when that same judgment is cast in plain English, as in “mud does not cause rain” …

The third resistance to causal (vis-a-vis statistical) assumptions stems from their intimidating clarity. Assumptions about abstract properties of density functions or about conditional independencies among variables are, cognitively speaking, rather opaque, hence they tend to be forgiven, rather than debated. In contrast, assumptions about how variables cause one another are shockingly transparent, and tend therefore to invite counter-arguments and counter-hypotheses.

Pearl’s seminal contributions to this research field is well-known and indisputable. But on the ‘taming’ and ‘resolve’ of the issues, yurs truly however has to admit that (under the influence of especially David Freedman and Nancy Cartwright) I still have some doubts on the reach, especially in terms of realism and relevance, of his ‘do-calculus solutions’ for social sciences in general and economics in specific (see here, here, here and here). The distinction between the causal — ‘interventionist’ — E[Y|do(X)] and the more traditional statistical — ‘conditional expectationist’ — E[Y|X] is crucial, but Pearl and his associates, although they have fully explained why the first is so important, have to convince us that it (in a relevant way) can be exported from ‘engineer’ contexts where it arguably easily and universally apply, to socio-economic contexts where ‘manipulativity’ and ‘modularity’ are not perhaps so universally at hand.

## Why statistics does not give us causality

24 Jun, 2019 at 12:28 | Posted in Statistics & Econometrics | 4 CommentsIf contributions made by statisticians to the understanding of causation are to be taken over with advantage in any specific field of inquiry, then what is crucial is that the right relationship should exist between statistical and subject-matter concerns …

Where the ultimate aim of research is not prediction

per sebut rather causal explanation, an idea of causation that is expressed in terms of predictive power — as, for example, ‘Granger’ causation — is likely to be found wanting. Causal explanations cannot be arrived at through statistical methodology alone: a subject-matter input is also required in the form of background knowledge and, crucially, theory …Likewise, the idea of causation as consequential manipulation is apt to research that can be undertaken primarily through experimental methods and, especially to ‘practical science’ where the central concern is indeed with ‘the consequences of performing particular acts’. The development of this idea in the context of medical and agricultural research is as understandable as the development of that of causation as robust dependence within applied econometrics. However, the extension of the manipulative approach into sociology would not appear promising, other than in rather special circumstances … The more fundamental difficulty is that, under the — highly anthropocentric — principle of ‘no causation without manipulation’, the recognition that can be given to the action of individuals as having causal force is in fact peculiarly limited.

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 causality we are searching for is the one existing in the real world around us. If there 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.

## Why attractive people you date tend to be jerks

19 Jun, 2019 at 00:12 | Posted in Statistics & Econometrics | 1 CommentHave you ever noticed that, among the people you date, the attractive ones tend to be jerks? Instead of constructing elaborate psychosocial theories, consider a simpler explanation. Your choice of people to date depends on two factors, attractiveness and personality. You’ll take a chance on dating a mean attractive person or a nice unattractive person, and certainly a nice attractive person, but not a mean unattractive person … This creates a spurious negative correlation between attractiveness and personality. The sad truth is that unattractive people are just as mean as attractive people — but you’ll never realize it, because you’ll never date somebody who is both mean and unattractive.

## Substantive relevance — not ‘clever’ design — is what matters most in science

11 Jun, 2019 at 16:40 | Posted in Statistics & Econometrics | 1 Comment

If anything, Snow’s path-breaking research underlines how important it is not to equate science with statistical calculation. And that the value of ‘as-if’ random interventions and experiments ultimately depend on the degree to which they if shed light on substantive and interesting scientific questions.

All science entail human judgement, and using statistical models doesn’t relieve us of that necessity. And we should never forget that the underlying parameters we use when performing statistical tests are *model constructions*. And if the model is wrong, the value of our calculations is nil. As ‘shoe-leather researcher’ David Freedman wrote in *Statistical Models and Causal Inference*:

I believe model validation to be a central issue. Of course, many of my colleagues will be found to disagree. For them, fitting models to data, computing standard errors, and performing significance tests is “informative,” even though the basic statistical assumptions (linearity, independence of errors, etc.) cannot be validated. This position seems indefensible, nor are the consequences trivial. Perhaps it is time to reconsider.

## Causal inference and the rhetoric of imaginary populations (wonkish)

11 Jun, 2019 at 11:10 | Posted in Statistics & Econometrics | 4 CommentsThe most

expedientpopulation and data generation model to adopt is one in which the population is regarded as a realization of an infinite super population. This setup is the standard perspective in mathematical statistics, in which random variables are assumed to exist with fixed moments for an uncountable and unspecified universe of events …This perspective is tantamount to assuming a population machine that spawns individuals forever (i.e., the analog to a coin that can be flipped forever). Each individual is born as a set of random draws from the distributions of Y¹, Y°, and additional variables collectively denoted by S …

Because of its

expediency, we will usually write with the superpopulation model in the background, even though the notions of infinite superpopulations and sequences of sample sizes approaching infinity aremanifestly unrealistic.

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.

In *Statistical Models and Causal Inference: A Dialogue with the Social Sciences *David Freedman also touched on this fundamental problem, arising when you try to apply statistical models outside overly simple nomological machines like coin tossing and roulette wheels:

Lurking behind the typical regression model will be found a host of such assumptions; without them, legitimate inferences cannot be drawn from the model. There are statistical procedures for testing some of these assumptions. However, the tests often lack the power to detect substantial failures. Furthermore, model testing may become circular; breakdowns in assumptions are detected, and the model is redefined to accommodate. In short,

hiding the problems can become a major goal of model building.Using models to make predictions of the future, or the results of interventions, would be a valuable corrective. Testing the model on a variety of data sets – rather than fitting refinements over and over again to the same data set – might be a good second-best … Built into the equation is a model for non-discriminatory behavior: the coefficient d vanishes. If the company discriminates, that part of the model cannot be validated at all.

Regression models are widely used by social scientists to make causal inferences; such models are now almost a routine way of demonstrating counterfactuals.

However, the “demonstrations” generally turn out to depend on a series of untested, even unarticulated, technical assumptions.Under the circumstances, reliance on model outputs may be quite unjustified. Making the ideas of validation somewhat more precise is a serious problem in the philosophy of science. That models should correspond to reality is, after all, a useful but not totally straightforward idea – with some history to it. Developing appropriate models is a serious problem in statistics; testing the connection to the phenomena is even more serious …In our days, serious arguments have been made from data. Beautiful, delicate theorems have been proved, although the connection with data analysis often remains to be established. And

an enormous amount of fiction has been produced, masquerading as rigorous science.

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?

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:

With this approach, the investigator does not explicitly define a population that could in principle be studied, with unlimited resources of time and money. The investigator merely

assumesthat such a population exists in some ill-defined sense. And there is a further assumption, that the data set being analyzed can be treatedas ifit were based on a random sample from the assumed population.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.

In social sciences — including economics — it’s always wise to ponder C. S. Peirce’s remark that universes are not as common as peanuts …

## How to think about statistics

29 May, 2019 at 09:32 | Posted in Statistics & Econometrics | 1 Comment

If anything, Gelman’s talk underlines how important it is not to equate science with statistical calculation. All science entail human judgement, and using statistical models doesn’t relieve us of that necessity. Working with misspecified models, the scientific value of statistics is actually zero — even though you’re making valid statistical inferences! Statistical models are no substitutes for doing real science. Or as a famous German philosopher famously wrote 150 years ago:

There is no royal road to science, and only those who do not dread the fatiguing climb of its steep paths have a chance of gaining its luminous summits.

We should never forget that the underlying parameters we use when performing statistical tests are *model constructions*. And if the model is wrong, the value of our calculations is nil. As ‘shoe-leather researcher’ David Freedman wrote in *Statistical Models and Causal Inference*:

I believe model validation to be a central issue. Of course, many of my colleagues will be found to disagree. For them, fitting models to data, computing standard errors, and performing significance tests is “informative,” even though the basic statistical assumptions (linearity, independence of errors, etc.) cannot be validated. This position seems indefensible, nor are the consequences trivial. Perhaps it is time to reconsider.

All of this, of course, does also apply when we use statistics in economics. 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 too good to be true, they usually aren’t. And that goes for econometrics too. The snag is 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 is 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.

## Data without theory is always treacherous

18 May, 2019 at 15:50 | Posted in Statistics & Econometrics | Comments Off on Data without theory is always treacherousData without theory can lead to bogus inferences …

Before being comforted or alarmed, consider whether it makes sense to extrapolate. Is there a persuasive reason why the future can be predicted simply by looking at the past? Or is that wishful thinking? Or nothing at all? …

Remember that even random flips can yield striking, even stunning, patterns that mean nothing at all …

A statistical comparison of two things is similarly unpersuasive unless there is a logical reason why they should be related … Ask yourself whether the people who did the study thought before calculating.

The central problem with the present ‘machine learning’ and ‘big data’ hype is that so many — falsely — think that they can get away with analysing real-world phenomena without any (commitment to) theory. But — data never speaks for itself. Without a prior statistical set-up, there actually are no data at all to process. And — using a machine learning algorithm will only produce what you are looking for.

Machine learning algorithms always express a view of what constitutes a pattern or regularity. They are never theory-neutral.

Clever data-mining tricks are not enough to answer important scientific questions. Theory matters.

## Monte Carlo simulations and NHST

6 May, 2019 at 21:42 | Posted in Statistics & Econometrics | 1 CommentIn many social sciences p-values and null hypothesis significance testing (NHST) are often used to draw far-reaching scientific conclusions — despite the fact that they are as a rule poorly understood and that there exist altenatives that are easier to understand and more informative.

Not the least using confidence intervals (CIs) and effect sizes are to be preferred to the Neyman-Pearson-Fisher mishmash approach that is so often practised by applied researchers.

Running a Monte Carlo simulation with 100 replications of a fictitious sample having N = 20, confidence intervals of 95%, a normally distributed population with a mean = 10 and a standard deviation of 20, taking two-tailed p-values on a zero null hypothesis, we get varying CIs (since they are based on varying sample standard deviations), but with a minimum of 3.2 and a maximum of 26.1 we still get a clear picture of what would happen in an infinite limit sequence. On the other hand p-values (even though from a purely mathematical statistical sense more or less equivalent to CIs) vary strongly from sample to sample, and jumping around between a minimum of 0.007 and a maximum of 0.999 doesn’t give you a clue of what will happen in an infinite limit sequence! So, I can’t but agree with Geoff Cummings:

The problems are so severe we need to shift as much as possible from NHST … The first shift should be to estimation: report and interpret effect sizes and CIs … I suggest p should be given only a marginal role, its problem explained, and it should be interpreted primarily as an indicator of where the 95% CI falls in relation to a null hypothesised value.

**[**In case you want to do your own Monte Carlo simulation, here’s one yours truly made using my favourite econometrics program Gretl:

nulldata 20

loop 100 –progressive

series y = normal(10,15)

scalar zs = (10-mean(y))/sd(y)

scalar df = $nobs-1

scalar ybar=mean(y)

scalar ysd= sd(y)

scalar ybarsd=ysd/sqrt($nobs)

scalar tstat = (ybar-10)/ybarsd

pvalue t df tstat

scalar lowb = mean(y) – critical(t,df,0.025)*ybarsd

scalar uppb = mean(y) + critical(t,df,0.025)*ybarsd

scalar pval = pvalue(t,df,tstat)

store E:\pvalcoeff.gdt lowb uppb pval

endloop**]**

## Publishing ‘hacked’ p-values — the chocolate hoax

30 Apr, 2019 at 17:19 | Posted in Statistics & Econometrics | Comments Off on Publishing ‘hacked’ p-values — the chocolate hoax

## Revisiting the foundations of randomness and probability

30 Apr, 2019 at 14:17 | Posted in Statistics & Econometrics, Theory of Science & Methodology | 5 CommentsRegarding models as metaphors leads to a radically different view regarding the interpretation of probability. This view has substantial advantages over conventional interpretations …

Probability does not exist in the real world. We must search for her in the Platonic world of ideals. We have shown that the interpretation of probability as a metaphor leads to several substantial changes in interpretations and justifications for conventional frequentist procedures. These changes remove several standard objections which have been made to these procedures. Thus our model seems to offer a good foundation for re-building our understanding of how probability should be interpreted in real world applications. More generally, we have also shown that regarding scientific models as metaphors resolves several puzzles in the philosophy of science.

Although yours truly has to confess of not being totally convinced that redefining probability as a metaphor is the right way to go forward on these foundational issues, Zaman’s article sure raises some very interesting questions on the way the concepts of randomness and probability are used in economics.

Modern mainstream economics relies to a large degree on the notion of probability. To at all be amenable to applied economic analysis, economic observations have to be conceived as random events that are analyzable within a probabilistic framework. But is it really necessary to model the economic system as a system where randomness can only be analyzed and understood when based on an *a priori* notion of probability?

When attempting to convince us of the necessity of founding empirical economic analysis on probability models, mainstream economics actually forces us to (implicitly) interpret events as random variables generated by an underlying probability density function.

This is at odds with reality. Randomness obviously is a fact of the real world (although I’m not sure Zaman agrees but rather puts also randomness in ‘the Platonic world of ideals’). Probability, on the other hand, attaches (if at all) to the world via intellectually constructed models, and *a fortiori* is only a fact of a probability generating (nomological) machine or a well constructed experimental arrangement or ‘chance set-up.’

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.

And this is the basic problem with economic data. 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.

We simply have to admit that the socio-economic states of nature that we talk of in most social sciences — and certainly in economics — are not amenable to analyze as probabilities, simply because in the real world open systems there are no probabilities to be had!

The processes that generate socio-economic data in the real world cannot just be assumed to always be adequately captured by a probability measure. And, so, it cannot be maintained that it even should be mandatory to treat observations and data — whether cross-section, time series or panel data — as events generated by some probability model. The important activities of most economic agents do not usually include throwing dice or spinning roulette-wheels. Data generating processes — at least outside of nomological machines like dice and roulette-wheels — are not self-evidently best modelled with probability measures.

If we agree on this, we also have to admit that much of modern neoclassical economics lacks sound foundations.

When economists and econometricians — often uncritically and without arguments — simply assume that one can apply probability distributions from statistical theory on their own area of research, they are really skating on thin ice.

This importantly also means that if you cannot show that data satisfies *all* the conditions of the probabilistic nomological machine, then the statistical inferences made in mainstream economics lack sound foundations!

## Significance testing and the real tasks of social science

30 Mar, 2019 at 09:56 | Posted in Statistics & Econometrics | Comments Off on Significance testing and the real tasks of social scienceAfter having mastered all the technicalities of regression analysis and econometrics, students often feel as though they are masters of the universe. I usually cool them down with required reading of Christopher Achen’s modern classic *Interpreting and Using Regression*. It usually gets them back on track again, and they understand that

no increase in methodological sophistication … alter the fundamental nature of the subject. It remains a wondrous mixture of rigorous theory, experienced judgment, and inspired guesswork. And that, finally, is its charm.

And in case they get too excited about having learned to master the intricacies of proper significance tests and p-values, I ask them to also ponder on Achen’s warning:

Significance testing as a search for specification errors substitutes calculations for substantive thinking. Worse, it channels energy toward the hopeless search for functionally correct specifications and diverts attention from the real tasks, which are to formulate a manageable description of the data and to exclude competing ones.

## Econometric beasts of bias

8 Mar, 2019 at 15:16 | Posted in Statistics & Econometrics | 2 CommentsIn an article posted earlier on this blog — What are the key assumptions of linear regression models? — yours truly tried to argue that since econometrics doesn’t content itself with only making ‘optimal’ predictions,” but also aspires to explain things in terms of causes and effects, econometricians need loads of assumptions — and that most important of these are **additivity** and **linearity**.

Let me take the opportunity to elaborate a little more on why I find these assumptions of such paramount importance and ought to be much more argued for — on both epistemological and ontological grounds — if at all being used.

Limiting model assumptions in economic science always have to be closely examined since if we are going to be able to show that the mechanisms or causes that we isolate and handle in our models are stable in the sense that they do not change when we ‘export’ them to our ‘target systems,’ we have to be able to show that they do not only hold under *ceteris paribus* conditions and *a fortiori* only are of limited value to our understanding, explanations or predictions of real economic systems.

Econometrics may be an informative tool for research. But if its practitioners do not investigate and make an effort of providing a justification for the credibility of the assumptions on which they erect their building, it will not fulfil its tasks. There is a gap between its aspirations and its accomplishments, and without more supportive evidence to substantiate its claims, critics will continue to consider its ultimate argument as a mixture of rather unhelpful metaphors and metaphysics. Maintaining that economics is a science in the ‘true knowledge’ business, yours truly remains a sceptic of the pretences and aspirations of econometrics. So far, I cannot really see that it has yielded very much in terms of relevant, interesting economic knowledge.

The marginal return on its ever higher technical sophistication in no way makes up for the lack of serious under-labouring of its deeper philosophical and methodological foundations that already Keynes complained about. The rather one-sided emphasis of usefulness and its concomitant instrumentalist justification cannot hide that neither Haavelmo, nor the legions of probabilistic econometricians following in his footsteps, give supportive evidence for their considering it “fruitful to believe” in the possibility of treating unique economic data as the observable results of random drawings from an imaginary sampling of an imaginary population. After having analyzed some of its ontological and epistemological foundations, I cannot but conclude that econometrics, on the whole, has not delivered ‘truth.’ And I doubt if it has ever been the intention of its main protagonists.

Our admiration for technical virtuosity should not blind us to the fact that we have to have a cautious attitude towards probabilistic inferences in economic contexts. Science, as Keynes said, should help us penetrate to “the true process of causation lying behind current events” and disclose “the causal forces behind the apparent facts.” We should look out for causal relations, but econometrics can never be more than a starting point in that endeavour since econometric (statistical) explanations are not explanations in terms of mechanisms, powers, capacities or causes. Firmly stuck in an empiricist tradition, econometrics is only concerned with the measurable aspects of reality, But there is always the possibility that there are other variables – of vital importance and although perhaps **unobservable** and **non-additive**, not necessarily epistemologically inaccessible – that were not considered for the model. Those who were can hence never be guaranteed to be more than potential causes, and not real causes. 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. A perusal of the leading econom(etr)ic journals shows that most econometricians still concentrate on fixed parameter models and that 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.

Real world social systems are not governed by stable causal mechanisms or capacities. As Keynes wrote in his critique of econometrics and inferential statistics already in the 1920s (emphasis added):

The atomic hypothesis which has worked so splendidly in Physics breaks down in Psychics.

We are faced at every turn with the problems of Organic Unity, of Discreteness, of Discontinuity – the whole is not equal to the sum of the parts, comparisons of quantity fails us, small changes produce large effects, the assumptions of a uniform and homogeneous continuum are not satisfied.Thus the results of Mathematical Psychics turn out to be derivative, not fundamental, indexes, not measurements, first approximations at the best; and fallible indexes, dubious approximations at that, with much doubt added as to what, if anything, they are indexes or approximations of.

The kinds of ‘laws’ and relations that econometrics has established, are laws and relations about entities in models that presuppose causal mechanisms being **atomistic **and** additive**. When causal mechanisms operate in real world social target systems they only do it in ever-changing and unstable combinations where the whole is more than a mechanical sum of parts. If economic regularities obtain they do it (as a rule) only because we engineered them for that purpose. Outside man-made ‘nomological machines’ they are rare, or even non-existant. Unfortunately that also makes most of the achievements of econometrics – as most of contemporary endeavours of mainstream economic theoretical modeling – rather useless.

## Econometrics — the path from cause to effect

7 Mar, 2019 at 18:58 | Posted in Statistics & Econometrics | 2 Comments

In their book — *Mastering ‘Metrics: The Path from Cause to Effect —* **Joshua D. Angrist** and **Jörn-Steffen Pischke** write:

Our first line of attack on the causality problem is a randomized experiment, often called a randomized trial. In a randomized trial, researchers change the causal variables of interest … for a group selected using something like a coin toss. By changing circumstances randomly, we make it highly likely that the variable of interest is unrelated to the many other factors determining the outcomes we want to study. Random assignment isn’t the same as holding everything else fixed, but it has the same effect. Random manipulation makes

other things equalhold on average across the groups that did and did not experience manipulation. As we explain … ‘on average’ is usually good enough.

Angrist and Pischke may “dream of the trials we’d like to do” and consider “the notion of an ideal experiment” something that “disciplines our approach to econometric research,” but to maintain that ‘on average’ is “usually good enough” is an allegation that in my view is rather unwarranted, and for many reasons.

First of all it amounts to nothing but hand waving to *simpliciter* assume, without argumentation, that it is tenable to treat social agents and relations as homogeneous and interchangeable entities.

Randomization is used to basically allow the econometrician to treat the population as consisting of interchangeable and homogeneous groups (‘treatment’ and ‘control’). The regression models one arrives at by using randomized trials tell us the average effect that variations in variable X has on the outcome variable Y, without having to explicitly control for effects of other explanatory variables R, S, T, etc., etc. Everything is assumed to be essentially equal except the values taken by variable X.

In a usual regression context one would apply an ordinary least squares estimator (OLS) in trying to get an unbiased and consistent estimate:

Y = α + βX + 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”( X=1) may have causal effects equal to – 100 and those “not treated” (X=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.

Limiting model assumptions in economic science always have to be closely examined since if we are going to be able to show that the mechanisms or causes that we isolate and handle in our models are stable in the sense that they do not change when we “export” them to our “target systems”, we have to be able to show that they do not only hold under *ceteris paribus* conditions and *a fortiori* only are of limited value to our understanding, explanations or predictions of real economic systems.

Real world social systems are not governed by stable causal mechanisms or capacities. The kinds of “laws” and relations that econometrics has established, are laws and relations about entities in models that presuppose causal mechanisms being atomistic and additive. When causal mechanisms operate in real world social target systems they only do it in ever-changing and unstable combinations where the whole is more than a mechanical sum of parts. If economic regularities obtain they do it (as a rule) only because we engineered them for that purpose. Outside man-made “nomological machines” they are rare, or even non-existant. Unfortunately that also makes most of the achievements of econometrics – as most of contemporary endeavours of mainstream economic theoretical modeling – rather useless.

Remember that a model is not the truth. It is a lie to help you get your point across. And in the case of modeling economic risk, your model is a lie about others, who are probably lying themselves. And what’s worse than a simple lie? A complicated lie.

Sam L. Savage The Flaw of Averages

When Joshua Angrist and Jörn-Steffen Pischke in an earlier article of theirs [“The Credibility Revolution in Empirical Economics: How Better Research Design Is Taking the Con out of Econometrics,” *Journal of Economic Perspectives, *2010] say that

anyone who makes a living out of data analysis probably believes that heterogeneity is limited enough that the well-understood past can be informative about the future

I really think they underestimate the heterogeneity problem. It 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 regression estimates that economists produce every year.

But when the randomization is purposeful, a whole new set of issues arises — experimental contamination — which is much more serious with human subjects in a social system than with chemicals mixed in beakers … Anyone who designs an experiment in economics would do well to anticipate the inevitable barrage of questions regarding the valid transference of things learned in the lab (one value of z) into the real world (a different value of z) …

Absent observation of the interactive compounding effects z, 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 …

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.

Evidence-based theories and policies are highly valued nowadays. Randomization is supposed to control for bias from unknown confounders. The received opinion is that evidence based on randomized experiments therefore is the best.

More and more economists have also lately come to advocate randomization as the principal method for ensuring being able to make valid causal inferences.

I would however rather argue that randomization, just as econometrics, promises more than it can deliver, basically because it requires assumptions that in practice are not possible to maintain.

Especially when it comes to questions of causality, randomization is nowadays considered some kind of “gold standard”. Everything has to be evidence-based, and the evidence has to come from randomized experiments.

But just as econometrics, randomization is basically a deductive method. Given the assumptions (such as manipulability, transitivity, separability, additivity, linearity, etc.) these methods deliver deductive inferences. The problem, of course, is that we will never completely know when the assumptions are right. And although randomization may contribute to controlling for confounding, it does not guarantee it, since genuine ramdomness presupposes infinite experimentation and we know all real experimentation is finite. And even if randomization may help to establish average causal effects, it says nothing of individual effects unless homogeneity is added to the list of assumptions. Real target systems are seldom epistemically isomorphic to our axiomatic-deductive models/systems, and even if they were, we still have to argue for the external validity of the conclusions reached from within these epistemically convenient models/systems. Causal evidence generated by randomization procedures may be valid in “closed” models, but what we usually are interested in, is causal evidence in the real target system we happen to live in.

When does a conclusion established in population X hold for target population Y? Only under very restrictive conditions!

Angrist’s and Pischke’s “ideally controlled experiments” tell us with certainty what causes what effects — but only given the right “closures”. Making appropriate extrapolations from (ideal, accidental, natural or quasi) experiments to different settings, populations or target systems, is not easy. “It works there” is no evidence for “it will work here”. Causes deduced in an experimental setting still have to show that they come with an export-warrant to the target population/system. The causal background assumptions made have to be justified, and without licenses to export, the value of “rigorous” and “precise” methods — and ‘on-average-knowledge’ — is despairingly small.

## Random walks (student stuff)

7 Mar, 2019 at 00:07 | Posted in Statistics & Econometrics | 2 Comments

## Pólya urn models mathematics

27 Feb, 2019 at 09:11 | Posted in Statistics & Econometrics | Comments Off on Pólya urn models mathematics

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