## A different way to solve quadratic equations

28 Sep, 2022 at 12:25 | Posted in Statistics & Econometrics | Leave a comment.

This one is for Linnea, my youngest daughter, who has now begun studying mathematics at my university đ

## Statistical models and the assumptions on which they build

21 Sep, 2022 at 19:47 | Posted in Statistics & Econometrics | Leave a commentEvery method of statistical inference depends on a complex web of assumptions about how data were collected and analyzed, and how the analysis results were selected for presentation. The full set of assumptions is embodied in a statistical model that underpins the method … Many problems arise however because this statistical model often incorporates unrealistic or at best unjustified assumptions …

The difficulty of understanding and assessing underlying assumptions is exacerbated by the fact that the statistical model is usually presented in a highly compressed and abstract formâif presented at all. As a result, many assumptions go unremarked and are often unrecognized by users as well as consumers of statistics. Nonetheless, all statistical methods and interpretations are premised on the model assumptions; that is, on an assumption that the model provides a valid representation of the variation we would expect to see across data sets, faithfully reflecting the circumstances surrounding the study and phenomena

occurring within it.

If anything, the common abuse of statistical tests underlines how important it is not to equate science with statistical calculation. All science entails human judgment, 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 once famously wrote:

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 âinformative,â 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.

## Support Vector Machines (student stuff)

20 Sep, 2022 at 10:42 | Posted in Statistics & Econometrics | Leave a comment.

## Avoiding statistical ‘dichotomania’

19 Sep, 2022 at 11:49 | Posted in Statistics & Econometrics | 2 CommentsWe are calling for a stop to the use of P values in the conventional, dichotomous way â to decide whether a result refutes or supports a scientific hypothesis …

The rigid focus on statistical significance encourages researchers to choose data and methods that yield statistical significance for some desired (or simply publishable) result, or that yield statistical non-significance for an undesired result, such as potential side effects of drugs â thereby invalidating

conclusions …Again, we are not advocating a ban on P values, confidence intervals or other statistical measures â only that we should not treat them categorically. This includes dichotomization as statistically significant or not, as well as categorization based on other statistical measures such as Bayes factors.

One reason to avoid such âdichotomaniaâ is that all statistics, including P values and confidence intervals, naturally vary from study to study, and often do so to a surprising degree. In fact, random variation alone can easily lead to large disparities in P values, far beyond falling just to either side of the 0.05 threshold …

We must learn to embrace uncertainty. One practical way to do so is to rename confidence intervals as âcompatibility intervalsâ and interpret them in a way that avoids overconfidence. Specifically, we recommend that authors describe the practical implications of all values inside the interval, especially the observed effect (or point estimate) and the limits. In doing so, they should remember that all the values between the intervalâs limits are reasonably compatible with the data, given the statistical assumptions used to compute the interval. Therefore, singling out one particular value (such as the null value) in the interval as âshownâ makes no sense.

In its standard form, a significance test is not the kind of âsevere testâ that we are looking for in our search for being able to confirm or disconfirm empirical scientific hypotheses. This is problematic for many reasons, one being that there is a strong tendency to accept the null hypothesis since it canât be rejected at the standard 5% significance level. In their standard form, significance tests bias against new hypotheses by making it hard to disconfirm the null hypothesis.

And as shown over and over again when it is applied, people have a tendency to read ânot disconfirmedâ as âprobably confirmed.â Standard scientific methodology tells us that when there is only say a 10 % probability that pure sampling error could account for the observed difference between the data and the null hypothesis, it would be more ‘reasonable’ to conclude that we have a case of disconfirmation. Especially if we perform many independent tests of our hypothesis and they all give about the same 10 % result as our reported one, I guess most researchers would count the hypothesis as even more disconfirmed.

Most importantly â we should never forget that the underlying parameters we use when performing significance tests areÂ *model constructions*. Our P values mean next to nothing if the model is wrong.

## How to teach econometrics

16 Sep, 2022 at 15:09 | Posted in Statistics & Econometrics | Leave a commentOne way to change the sad state of econometrics — an industry with a huge output but no sales — would perhaps be to follow Ed Leamer’s advice on how to teach it …

To take my students on a journey toward real intelligence, I have given them the following assignment: Your boss walks into your office and says, âI read in the newspaper this morning that interest rates are going to increase this year. Is this right, and do I need to worry about it? I want you to give a presentation to the Board in four weeks.â OMG! What to do; what to do?? These students will need to know regression assumptions and regression properties, but if all they know is mathematical theorems regarding properties of estimators, they will have a hard time taking the first and most important step, which is to better define the question. Whatâs the industry? What is the time frame? What are the decision options? After that comes a search of the Web for theory and data. And then the data analysis, not carried out by preprogrammed rules, but designed to fit the special circumstances.

## Causal inference — what the machine cannot learn

12 Sep, 2022 at 16:55 | Posted in Statistics & Econometrics | 3 Comments.

The central problem with the present âmachine learningâ and âbig dataâ hype is that so many â falsely â think that they can get away with analyzing real-world phenomena without any (commitment to) theory. But â data never speaks for itself. Without a prior statistical set-up, there actually are no data at all to process.

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

If we wanted highly probable claims, scientists would stick toââ low-level observables and not seek generalizations, much less theories with high explanatory content. In this dayâ of fascination with Big data’s ability to predictâ what book I’ll buy next, a healthy Popperian reminder is due: humans also want to understand and to explain. We want bold ‘improbable’ theories. I’m a little puzzled when I hear leading machine learners praise Popper, a realist, while proclaiming themselves fervid instrumentalists. That is, they hold the view that theories, rather than aiming at truth, are just instruments for organizing and predicting observable facts. It follows from the success of machine learning, Vladimir Cherkassy avers, thatâ “realism is not possible.” This is very quick philosophy!

Quick indeed!

## Econometric fundamentalism

8 Sep, 2022 at 19:36 | Posted in Statistics & Econometrics | Leave a commentThe wide conviction of the superiority of the methods of the science has converted the econometric community largely to a group of fundamentalist guards of mathematical rigour. It is often the case that mathemical rigour is held as the dominant goal and the criterion for research topic choice as well as research evaluation, so much so that the relevance of the research to business cycles is reduced to empirical illustrations. To that extent, probabilistic formalization has trapped econometric business cycle research in the pursuit of means at the expense of ends.

Once the formalization attempts have gone significantly astray from what is needed for analysing and forecasting the multi-faceted characteristics of business cycles, the research community should hopefully make appropriate ‘error corrections’ of its overestimation of the power of a priori postulated models as well as its underestimation of the importance of the historical approach, or the ‘art’ dimension of business cycle research.

## Econometric FUQs

5 Sep, 2022 at 11:20 | Posted in Statistics & Econometrics | Comments Off on Econometric FUQsIf you can’t devise an experiment that answers your question in a world where anything goes, then the odds of generating useful results with a modest budget and nonexperimental survey data seem pretty slim. The description of an ideal experiment also helps you formulate causal questions precisely. The mechanics of an ideal experiment highlight the forces you’d like to manipulate and the factors you’d like to hold constant.

Research questions that cannot be answered by any experiment are FUQs: fundamentally unidentified questions.

One of the limitations of economics is the restricted possibility to perform experiments, forcing it to mainly rely on observational studies for knowledge of real-world economies.

But still â the idea of performing laboratory experiments holds a firm grip on our wish to discover (causal) relationships between economic âvariables.âIf we only could isolate and manipulate variables in controlled environments, we would probably find ourselves in a situation where we with greater ârigourâ and âprecisionâ could describe, predict, or explain economic happenings in terms of âstructuralâ causes, âparameterâ values of relevant variables, and economic âlaws.â

Galileo Galileiâs experiments are often held as exemplary for how to perform experiments to learn something about the real world. Galileoâs heavy balls dropping from the tower of Pisa, confirmed that the distance an object falls is proportional to the square of time and that this law (empirical regularity) of falling bodies could be applicable outside a vacuum tube when e. g. air existence is negligible.

The big problem is to decide or find out exactly for which objects air resistance (and other potentially âconfoundingâ factors) is ânegligible.â In the case of heavy balls, air resistance is obviously negligible, but how about feathers or plastic bags?

One possibility is to take the all-encompassing-theory road and find out all about possible disturbing/confounding factors â not only air resistance â influencing the fall and build that into one great model delivering accurate predictions on what happens when the object that falls is not only a heavy ball but feathers and plastic bags. This usually amounts to ultimately stating some kind of *ceteris paribus*Â interpretation of the âlaw.â

Another road to take would be to concentrate on the negligibility assumption and to specify the domain of applicability to be only heavy compact bodies. The price you have to pay for this is that (1) ânegligibilityâ may be hard to establish in open real-world systems, (2) the generalization you can make from âsampleâ to âpopulationâ is heavily restricted, and (3) you actually have to use some âshoe leatherâ and empirically try to find out how large is the âreachâ of the âlaw.â

In mainstream economics, one has usually settled for the âtheoreticalâ road (and in case you think the present ânatural experimentsâ hype has changed anything, remember that to mimic real experiments,Â exceedingly stringent special conditions standardly have to obtain).

In the end, it all boils down to one question â are there any Galilean âheavy ballsâ to be found in economics, so that we can indisputably establish the existence of economic laws operating in real-world economies?

As far as I can see there are some heavy balls out there, butÂ not even one singleÂ realÂ economic law.

Economic factors/variables are more like feathers than heavy balls â non-negligible factors (like air resistance and chaotic turbulence) are hard to rule out as having no influence on the object studied.

Galilean experiments are hard to carry out in economics, and the theoretical âanalogueâ models economists construct and in which they perform their âthought experimentsâ build on assumptions that are far away from the kind of idealized conditions under which Galileo performed his experiments. The ânomological machinesâ that Galileo and other scientists have been able to construct have no real analogues in economics. The stability, autonomy, modularity, and interventional invariance, that we may find between entities in nature, simply are not there in real-world economies. Thatâs are real-world fact, and contrary to the beliefs of most mainstream economists, they wonât go away simply by applying deductive-axiomatic economic theory with tons of more or less unsubstantiated assumptions.

By this, I do not mean to say that we have to discard all (causal) theories/laws building on modularity, stability, invariance, etc. But we have to acknowledge the fact that outside the systems that possibly fulfil these requirements/assumptions, they are of little substantial value. Running paper and pen experiments on artificial âanalogueâ model economies is a sure way of âestablishingâ (causal) economic laws or solving intricate econometric problems of autonomy, identification, invariance and structural stability â in the model world. But they are pure substitutes for the real thing and they donât have much bearing on what goes on in real-world open social systems. Setting up convenient circumstances for conducting Galilean experiments may tell us a lot about what happens under those kinds of circumstances. But â few, if any, real-world social systems are âconvenient.â So most of those systems, theories and models, are irrelevant for letting us know what we really want to know.

To solve, understand, or explain real-world problems you actually have to know something about them â logic, pure mathematics, data simulations or deductive axiomatics donât take you very far. Most econometrics and economic theories/models are splendid logic machines. But â applying them to the real world is a totally hopeless undertaking! The assumptions one has to make in order to successfully apply these deductive-axiomatic theories/models/machines are devastatingly restrictive and mostly empirically untestableâ and hence make their real-world scope ridiculously narrow. To fruitfully analyze real-world phenomena with models and theories you cannot build on patently and known to be ridiculously absurd assumptions. No matter how much you would like the world to entirely consist of heavy balls, the world is not like that. The world also has its fair share of feathers and plastic bags.

Most of the âidealizationsâ we find in mainstream economic models are not âcoreâ assumptions, but rather structural âauxiliaryâ assumptions. Without those supplementary assumptions, the core assumptions deliver next to nothing of interest. So to come up with interesting conclusions you have to rely heavily on those other â âstructuralâ â assumptions.

In physics, we have theories and centuries of experience and experiments that show how gravity makes bodies move. In economics, we know there is nothing equivalent. So instead mainstream economists necessarily have to load their theories and models with sets of auxiliary structural assumptions to get any results at all in their models.

So why then do mainstream economists keep on pursuing this modelling project?

Mainstream âas ifâ models are based on the logic of idealization and a set of tight axiomatic and âstructuralâ assumptions from which consistent and precise inferences are made. The beauty of this procedure is, of course, thatÂ *if*Â the assumptions are true, the conclusions necessarily follow. But it is a poor guide for real-world systems.

The way axioms and theorems are formulated in mainstream economics often leaves their specification without almost any restrictions whatsoever, safely making every imaginable evidence compatible with the all-embracing âtheoryâ â and theory without informational content never risks being empirically tested and found falsified. Used in mainstream âthought experimentalâ activities, it may, of course, âbe very âhandy,â but totally void of any empirical value.

Some economic methodologists have lately been arguing that economic models may well be considered âminimal modelsâ that portray âcredible worldsâ without having to care about things like similarity, isomorphism, simplified ârepresentationalityâ or resemblance to the real world. These models are said to resemble ârealistic novelsâ that portray âpossible worldsâ. And sure: economists constructing and working with those kinds of models learn things about what might happen in those âpossible worldsâ. But is that really the stuff real science is made of? I think not. As long as one doesnât come up with credible export warrants to real-world target systems and show *how*Â those models â often building on idealizations withÂ *known* to be false assumptions â enhance our understanding or explanations about the real world, well, they are just nothing more than just novels. Â Showing that something is possible in a âpossible worldâ doesnât give us a justified license to infer that it therefore also is possible in the real world. âThe Great Gatsbyâ is a wonderful novel, but if you truly want to learn about what is going on in the world of finance, I would recommend rather reading Minsky or Keynes andÂ *directly* confronting real-world finance.

Different models have different cognitive goals. Constructing models that aim for explanatory insights may not optimize the models for making (quantitative) predictions or deliver some kind of âunderstandingâ of whatâs going on in the intended target system. All modelling in science has tradeoffs. There simply is no âbestâ model. For one purpose in one context model A is âbestâ, for other purposes and contexts model B may be deemed âbestâ. Depending on the level of generality, abstraction, and depth, we come up with different models. But even so, I would argue that if we are looking for what I have called âadequate explanationsâ (Syll, *Ekonomisk teori och metod*, Studentlitteratur, 2005) it is not enough to just come up with âminimalâ or âcredible worldâ models.

The assumptions and descriptions we use in our modelling have to be true â or at least âharmlesslyâ false â and give a sufficiently detailed characterization of the mechanisms and forces at work. Models in mainstream economics do nothing of the kind.

Coming up with models that show how things mayÂ *possibly*Â be explained is not what we are looking for. It is not enough. We want to have models that build on assumptions that are not in conflict with known facts and that show how thingsÂ *actually*Â are to be explained. Our aspirations have to be more far-reaching than just constructing coherent and âcredibleâ models about âpossible worldsâ. We want to understand and explain âdifference-makingâ in the real world and not just in some made-up fantasy world. No matter how many mechanisms or coherent relations you represent in your model, you still have to show that these mechanisms and relations are at work and exist in society if we are to do real science. Science has to be something more than just more or less realistic âstory-tellingâ or âexplanatory fictionalism.â You have to provideÂ *decisive*Â empirical evidence that what you can infer in your model also helps us to uncover what actually goes on in the real world. It is not enough to present your students with epistemically informative insights about logically possible but non-existent general equilibrium models. You also, and more importantly, have to have a world-linking argumentation and showÂ *how* those models explain or teach us something about real-world economies. If you fail to support your models in that way, why should we care about them? And if you do not inform us about what are the real-world intended target systems of your modelling, how are we going to be able to value or test them? Without giving that kind of information it is impossible for us to check if the âpossible worldâ models you come up with actually hold also for the one world in which we live â the real world.

## Donald Rubin on the history of causal inference

3 Sep, 2022 at 13:18 | Posted in Statistics & Econometrics | 1 Comment.

## Propensity score analysis — some critical remarks

1 Sep, 2022 at 17:39 | Posted in Statistics & Econometrics | Comments Off on Propensity score analysis — some critical remarksOur findings suggest that researchers need comprehensive knowledge of model assumptions and knowledge of plausible causal structure. From prior research, sources of selection bias must be understood. Substantive knowledge of plausible causal structure typically includes the theory of change of an intervention program being evaluated, which determines the covariates that should be used in the model predicting propensity scores and in the outcome analysis. Sample reduction after running a propensity score model is a key issue and should always be considered … Our findings suggest that it is of paramount importance to understand the assumptions of propensity score models and attend to potential violations of these assumptions. This requires both methodological and substantive knowledge …

Finally, this study supports a methodological caution made repeatedly by experienced observational researchers: OLS regression is a poor and ill-advised analytic approach in the presence of endogeneity or selection bias.

## History of econometric causality

26 Aug, 2022 at 12:53 | Posted in Statistics & Econometrics | 2 Comments.

## Randomization strategies (student stuff)

26 Aug, 2022 at 12:07 | Posted in Statistics & Econometrics | Comments Off on Randomization strategies (student stuff).

## Synthetic control methods (student stuff)

24 Aug, 2022 at 16:27 | Posted in Statistics & Econometrics | Comments Off on Synthetic control methods (student stuff).

## Causal inference methods

21 Aug, 2022 at 11:24 | Posted in Statistics & Econometrics | Comments Off on Causal inference methods.

## Econometrics — the danger of calling your pet cat a dog

19 Aug, 2022 at 13:24 | Posted in Statistics & Econometrics | Comments Off on Econometrics — the danger of calling your pet cat a dogThe assumption of additivity and linearity means that the outcome variable is, in reality, linearly related to any predictors … and that if you have several predictors then their combined effect is best described by adding their effects together …

This assumption is the most important because if it is not true then even if all other assumptions are met, your model is invalid because you have described it incorrectly. It’s a bit like calling your pet cat a dog: you can try to get it to go in a kennel, or to fetch sticks, or to sit when you tell it to, but don’t be surprised when its behaviour isn’t what you expect because even though you’ve called it a dog, it is in fact a cat. Similarly, if you have described your statistical model inaccurately it won’t behave itself and there’s no point in interpreting its parameter estimates or worrying about significance tests of confidence intervals: the model is wrong.

Econometrics fails miserably over and over again — and not only because of the additivity and linearity assumption

Another reason why it does, is that the error term in the regression models used is thought of as representing the effect of the variables that were omitted from the models. The error term is somehow thought to be a ‘cover-all’ term representing omitted content in the model and necessary to include to ‘save’ the assumed deterministic relation between the other random variables included in the model. Error terms are usually assumed to be orthogonal (uncorrelated) to the explanatory variables. But since they are unobservable, they are also impossible to empirically test. And without justification of the orthogonality assumption, there is, as a rule, nothing to ensure identifiability.

Nowadays it has almost become a self-evident truism among economists that you cannot expect people to take your arguments seriously unless they are based on or backed up by advanced econometric modelling. So legions of mathematical-statistical theorems are proved — and heaps of fiction are being produced, masquerading as science. The rigourâ of the econometric modelling and the far-reaching assumptions they are built on is frequently not supported by data.

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