## Some methodological perspectives on statistical inference in economics

from **Lars Syll**

Causal modeling attempts to maintain this deductive focus within imperfect research by deriving models for observed associations from more elaborate causal (‘structural’) models with randomized inputs … But in the world of risk assessment … the causal-inference process cannot rely solely on deductions from models or other purely algorithmic approaches. Instead, when randomization is doubtful or simply false (as in typical applications), an honest analysis must consider sources of variation from uncontrolled causes with unknown, nonrandom interdependencies. Causal identification then requires nonstatistical information in addition to information encoded as data or their probability distributions …

This need raises questions of to what extent can inference be codified or automated (which is to say, formalized) in ways that do more good than harm. In this setting, formal models – whether labeled ‘‘causal’’ or ‘‘statistical’’ – serve a crucial but limited role in providing hypothetical scenarios that establish what would be the case if the assumptions made were true and the input data were both trustworthy and the only data available. Those input assumptions include all the model features and prior distributions used in the scenario, and supposedly encode all information being used beyond the raw data file (including information about the embedding context as well as the study design and execution).

Overconfident inferences follow when the hypothetical nature of these inputs is forgotten and the resulting outputs are touted as unconditionally sound scientific inferences instead of the tentative suggestions that they are (however well informed) …

The practical limits of formal models become especially apparent when attempting to integrate diverse information sources. Neither statistics nor medical science begins to capture the uncertainty attendant in this process, and in fact both encourage pernicious overconfidence by failing to make adequate allowance for unmodeled uncertainty sources. Instead of emphasizing the uncertainties attending field research, statistics and other quantitative methodologies tend to focus on mathematics and often fall prey to the satisfying – and false – sense of logical certainty that brings to population inferences. Meanwhile, medicine focuses on biochemistry and physiology, and the satisfying – and false – sense of mechanistic certainty about results those bring to individual events.

Wise words from a renowned epidemiologist.

As long as economists and statisticians 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!

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!

The tool of statistical inference becomes available as the result of a self-imposed limitation of the universe of discourse. It is assumed that the available observations have been generated by a probability law or stochastic process about which some incomplete knowledge is available a priori …

It should be kept in mind that the sharpness and power of these remarkable tools of inductive reasoning are bought by willingness to adopt a specification of the universe in a form suitable for mathematical analysis.

Yes indeed — using statistics and econometrics to make inferences you have to make lots of (mathematical) tractability assumptions. And especially since econometrics aspires to explain things in terms of causes and effects, it needs loads of assumptions, such as e.g. invariance, additivity and linearity.

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. If not, they are of limited value to our explanations and predictions of real economic systems.

Unfortunately, real world social systems are usually 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 invariant, atomistic and additive. But — when causal mechanisms operate in the real world they mostly do it in ever-changing and unstable ways. If economic regularities obtain they do so as a rule only because we engineered them for that purpose. Outside man-made ‘nomological machines’ they are rare, or even non-existant.

So — if we want to explain and understand real-world economies we should perhaps be a little bit more cautious with using universe specifications “suitable for mathematical analysis.”

It should be kept in mind, when we evaluate the application of statistics and econometrics, that the sharpness and power of these remarkable tools of inductive reasoning are bought by willingness to adopt a specification of the universe in a form suitable for mathematical analysis.

As emphasised by Greenland, can causality in social sciences — and economics — 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.

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 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.

Lars, As a mathematician I would rather say that the problem lies with the interpretation of the inferences, not in the mathematics as such. Whatever assumptions one makes in whatever analysis need to be carried forward to the conclusion. This is not unique to the social sciences or to mathematics.

Many professionals are expected to make firm conclusions, and so they have to persuade us that what should be tentative hypotheses are actually ‘facts’. But surely the point of a sound education is to make us sceptical? But please don’t throw out the mathematical baby.

Dave, I think we are in agreement here. There is, of course, nothing wrong what so ever with mathematics per se. Applied in the right context it can be a wonderful instrument . But in science it ought NEVER to be regarded as self-evidently the only or best instrument of analysis to use.

Certainly mathematics cannot establish the truth of anything. But neither can

anything else. Truth cannot be established. The best humans can do (and this is

science) is to narrow down the options for describing and explaining events

around us. The more often we confirm again the relationship between certain

events and actors, the more “faith” we have that the relationship actually exists.

This is not truth. But it is useful and applicable to our living arrangements.

Mathematics may help in this work by providing a framework that helps us verify

relationships. But mathematics is clearly only one possible tool in carrying out

this work. Economics as primarily constituted for the last 200 years is a failure in

all these areas. That’s why it’s being widely rejected. Situated as a historical

study mathematics is useful and relevant but not ultimately determinant.