Suppose you test a highly confirmed hypothesis, for example, that the price elasticity of demand is negative. What would you do if the computer were to spew out a positive coefficient? Surely you would not claim to have overthrown the law of demand … Instead, you would rerun many variants of your regression until the recalcitrant computer finally acknowledged the sovereignty of your theory …
Only the naive are shocked by such soft and gentle testing … Easy it is. But also wrong, when the purpose of the exercise is not to use a hypothesis, but to determine its validity …
Econometric tests are far from useless. They are worth doing, and their results do tell something … But many economists insist that economics can deliver more, much more, than merely, more or less, plausible knowledge, that it can reach its results with compelling demonstrations. By such a standard how should one describe our usual way of testing hypotheses? One possibility is to interpret it as Blaug [The Methodology of Economics, 1980, p. 256] does, as ‘playing tennis with the net down’ …
Perhaps my charge that econometric testing lacks seriousness of purpose is wrong … But regardless of the cause, it should be clear that most econometric testing is not rigorous. Combining such tests with formalized theoretical analysis or elaborate techniques is another instance of the principle of the strongest link. The car is sleek and elegant; too bad the wheels keep falling off.
Because I was there when the economics department of my university got an IBM 360, I was very much caught up in the excitement of combining powerful computers with economic research. Unfortunately, I lost interest in econometrics almost as soon as I understood how it was done. My thinking went through four stages:
1.Holy shit! Do you see what you can do with a computer’s help.
2.Learning computer modeling puts you in a small class where only other members of the caste can truly understand you. This opens up huge avenues for fraud:
3.The main reason to learn stats is to prevent someone else from committing fraud against you.
4.More and more people will gain access to the power of statistical analysis. When that happens, the stratification of importance within the profession should be a matter of who asks the best questions.
Disillusionment began to set in. I began to suspect that all the really interesting economic questions were FAR beyond the ability to reduce them to mathematical formulas. Watching computers being applied to other pursuits than academic economic investigations over time only confirmed those suspicions.
1.Precision manufacture is an obvious application for computing. And for many applications, this worked magnificently. Any design that combined straight line and circles could be easily described for computerized manufacture. Unfortunately, the really interesting design problems can NOT be reduced to formulas. A car’s fender, for example, can not be describe using formulas—it can only be described by specifying an assemblage of multiple points. If math formulas cannot describe something as common and uncomplicated as a car fender, how can it hope to describe human behavior?
2.When people started using computers for animation, it soon became apparent that human motion was almost impossible to model correctly. After a great deal of effort, the animators eventually put tracing balls on real humans and recorded that motion before transferring it to the the animated character. Formulas failed to describe simple human behavior—like a toddler trying to walk.
Lately, I have discovered a Swedish economist who did NOT give up econometrics merely because it sounded so impossible. In fact, he still teaches the stuff. But for the rest of us, he systematically destroys the pretensions of those who think they can describe human behavior with some basic Formulas.
Wonder who that Swedish guy is …
- Always, but always, plot your data.
- Remember that data quality is at least as important as data
- Always ask yourself, “Do these results make economic/common sense”?
- Check whether your “statistically significant” results are also
- Be sure that you know exactly what assumptions are used/needed to obtain
the results relating to the properties of any estimator or test that you
- Just because someone else has used a particular approach to analyse a
problem that looks like yours, that doesn’t mean they were right!
- “Test, test, test”! (David Hendry). But don’t forget that “pre-testing”
raises some important issues of its own.
- Don’t assume that the computer code that someone gives to you is
relevant for your application, or that it even produces correct results.
- Keep in mind that published results will represent only a fraction of the
results that the author obtained, but is not publishing.
- Don’t forget that “peer-reviewed” does NOT mean “correct results”, or
even “best practices were followed”.
Some variants of ‘data mining’ can be classified as the greatest of the basement sins, but other variants of ‘data mining’ can be viewed as important ingredients in data analysis. Unfortunately, these two variants usually are not mutually exclusive and so frequently conflict in the sense that to gain the benefits of the latter, one runs the risk of incurring the costs of the former.
Hoover and Perez (2000, p. 196) offer a general definition of data mining as referring to “a broad class of activities that have in common a search over different ways to process or package data statistically or econometrically with the purpose of making the final presentation meet certain design criteria.” Two markedly different views of data mining lie within the scope of this general definition. One view of ‘data mining’ is that it refers to experimenting with (or ‘fishing through’) the data to produce a specification … The problem with this, and why it is viewed as a sin, is that such a procedure is almost guaranteed to produce a specification tailored to the peculiarities of that particular data set, and consequently will be misleading in terms of what it says about the underlying process generating the data. Furthermore, traditional testing procedures used to ‘sanctify’ the specification are no longer legitimate, because these data, since they have been used to generate the specification, cannot be judged impartial if used to test that specification …
An alternative view of ‘data mining’ is that it refers to experimenting with (or ‘fishing through’) the data to discover empirical regularities that can inform economic theory … Hand et al (2000) describe data mining as the process of seeking interesting or valuable information in large data sets. Its greatest virtue is that it can uncover empirical regularities that point to errors/omissions in theoretical specifications …
In summary, this second type of ‘data mining’ identifies regularities in or characteristics of the data that should be accounted for and understood in the context of the underlying theory. This may suggest the need to rethink the theory behind one’s model, resulting in a new specification founded on a more broad-based understanding. This is to be distinguished from a new specification created by mechanically remolding the old specification to fit the data; this would risk incurring the costs described earlier when discussing the first variant of ‘data mining.’
The issue here is how should the model specification be chosen? As usual, Leamer (1996, p. 189) has an amusing view: “As you wander through the thicket of models, you may come to question the meaning of the Econometric Scripture that presumes the model is given to you at birth by a wise and beneficent Holy Spirit.”
In practice, model specifications come from both theory and data, and given the absence of Leamer’s Holy Spirit, properly so.
Brad DeLong wonders why Cliff Asness is clinging to a theoretical model that has clearly been rejected by the data …
There’s a version of this in econometrics, i.e. you know the model is correct, you are just having trouble finding evidence for it. It goes as follows. You are testing a theory you came up with, but the data are uncooperative and say you are wrong. But instead of accepting that, you tell yourself “My theory is right, I just haven’t found the right econometric specification yet. I need to add variables, remove variables, take a log, add an interaction, square a term, do a different correction for misspecification, try a different sample period, etc., etc., etc.” Then, after finally digging out that one specification of the econometric model that confirms your hypothesis, you declare victory, write it up, and send it off (somehow never mentioning the intense specification mining that produced the result).
Too much econometric work proceeds along these lines. Not quite this blatantly, but that is, in effect, what happens in too many cases. I think it is often best to think of econometric results as the best case the researcher could make for a particular theory rather than a true test of the model.
Mark touches the spot — and for the sake of balancing the overly rosy picture of econometric achievements given in the usual econometrics textbooks today, it may also be interesting to see how Trygve Haavelmo, with the completion (in 1958) of the twenty-fifth volume of Econometrica, assessed the the role of econometrics in the advancement of economics. Although mainly positive of the “repair work” and “clearing-up work” done, Haavelmo also found some grounds for despair:
We have found certain general principles which would seem to make good sense. Essentially, these principles are based on the reasonable idea that, if an economic model is in fact “correct” or “true,” we can say something a priori about the way in which the data emerging from it must behave. We can say something, a priori, about whether it is theoretically possible to estimate the parameters involved. And we can decide, a priori, what the proper estimation procedure should be … But the concrete results of these efforts have often been a seemingly lower degree of accuracy of the would-be economic laws (i.e., larger residuals), or coefficients that seem a priori less reasonable than those obtained by using cruder or clearly inconsistent methods.
There is the possibility that the more stringent methods we have been striving to develop have actually opened our eyes to recognize a plain fact: viz., that the “laws” of economics are not very accurate in the sense of a close fit, and that we have been living in a dream-world of large but somewhat superficial or spurious correlations.
And as the quote below shows, Frisch also shared some of Haavelmo’s — and Keynes’s — doubts on the applicability of econometrics:
I have personally always been skeptical of the possibility of making macroeconomic predictions about the development that will follow on the basis of given initial conditions … I have believed that the analytical work will give higher yields – now and in the near future – if they become applied in macroeconomic decision models where the line of thought is the following: “If this or that policy is made, and these conditions are met in the period under consideration, probably a tendency to go in this or that direction is created”.
Peter Dorman is one of those rare economists that it is always a pleasure to read. Here his critical eye is focused on economists’ infatuation with homogeneity and averages:
You may feel a gnawing discomfort with the way economists use statistical techniques. Ostensibly they focus on the difference between people, countries or whatever the units of observation happen to be, but they nevertheless seem to treat the population of cases as interchangeable—as homogenous on some fundamental level. As if people were replicants.
You are right, and this brief talk is about why and how you’re right, and what this implies for the questions people bring to statistical analysis and the methods they use.
Our point of departure will be a simple multiple regression model of the form
y = β0 + β1 x1 + β2 x2 + …. + ε
where y is an outcome variable, x1 is an explanatory variable of interest, the other x’s are control variables, the β’s are coefficients on these variables (or a constant term, in the case of β0), and ε is a vector of residuals. We could apply the same analysis to more complex functional forms, and we would see the same things, so let’s stay simple.
What question does this model answer? It tells us the average effect that variations in x1 have on the outcome y, controlling for the effects of other explanatory variables. Repeat: it’s the average effect of x1 on y.
This model is applied to a sample of observations. What is assumed to be the same for these observations? (1) The outcome variable y is meaningful for all of them. (2) The list of potential explanatory factors, the x’s, is the same for all. (3) The effects these factors have on the outcome, the β’s, are the same for all. (4) The proper functional form that best explains the outcome is the same for all. In these four respects all units of observation are regarded as essentially the same.
Now what is permitted to differ across these observations? Simply the values of the x’s and therefore the values of y and ε. That’s it.
Thus measures of the difference between individual people or other objects of study are purchased at the cost of immense assumptions of sameness. It is these assumptions that both reflect and justify the search for average effects …
In the end, statistical analysis is about imposing a common structure on observations in order to understand differentiation. Any structure requires assuming some kinds of sameness, but some approaches make much more sweeping assumptions than others. An unfortunate symbiosis has arisen in economics between statistical methods that excessively rule out diversity and statistical questions that center on average (non-diverse) effects. This is damaging in many contexts, including hypothesis testing, program evaluation, forecasting—you name it …
The first step toward recovery is admitting you have a problem. Every statistical analyst should come clean about what assumptions of homogeneity are being made, in light of their plausibility and the opportunities that exist for relaxing them.
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.
Real world social systems are not governed by stable causal mechanisms or capacities. If economic regularities obtain they — as a rule — do it only because we engineered them for that purpose. Outside man-made “nomological machines” they are rare, or even non-existant. Unfortunately that also makes them 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
In Gretl it’s extremely simple to do this kind of bootstrapping. Run the regression and you get an output-window with the regression results. Click on Analysis at the top of the window and then on Bootstrap and select the options Confidence interval and Resample residuals. After having selected the coefficient for which you want to you get bootstrapped estimates, you just click OK and a window will appear showing the 95% confidence interval for the coefficient. It’s as simple as that!
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 in to the picture.
The assumption of imaginary “superpopulations” is one of the many dubious assumptions used in modern econometrics, and as Clint Ballinger has highlighted, this is a particularly questionable rabbit pulling assumption:
Inferential statistics are based on taking a random sample from a larger population … and attempting to draw conclusions about a) the larger population from that data and b) the probability that the relations between measured variables are consistent or are artifacts of the sampling procedure.
However, in political science, economics, development studies and related fields the data often represents as complete an amount of data as can be measured from the real world (an ‘apparent population’). It is not the result of a random sampling from a larger population. Nevertheless, social scientists treat such data as the result of random sampling.
Because there is no source of further cases a fiction is propagated—the data is treated as if it were from a larger population, a ‘superpopulation’ where repeated realizations of the data are imagined. Imagine there could be more worlds with more cases and the problem is fixed …
What ‘draw’ from this imaginary superpopulation does the real-world set of cases we have in hand represent? This is simply an unanswerable question. The current set of cases could be representative of the superpopulation, and it could be an extremely unrepresentative sample, a one in a million chance selection from it …
The problem is not one of statistics that need to be fixed. Rather, it is a problem of the misapplication of inferential statistics to non-inferential situations.
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 Haavelmo’s domain of probability theory and sample space of infinite populations – just as Fisher’s “hypothetical infinite population, of which the actual data are regarded as constituting a random sample”, von Mises’s “collective” or Gibbs’s ”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.
As David Salsburg once noted – in his lovely The Lady Tasting Tea - on probability theory:
[W]e assume there is an abstract space of elementary things called ‘events’ … If a measure on the abstract space of events fulfills certain axioms, then it is a probability. To use probability in real life, we have to identify this space of events and do so with sufficient specificity to allow us to actually calculate probability measurements on that space … Unless we can identify [this] abstract space, the probability statements that emerge from statistical analyses will have many different and sometimes contrary meanings.
Just as e. g. John Maynard Keynes and Nicholas Georgescu-Roegen, Salsburg is very critical of the way social scientists – including economists and econometricians – uncritically and without arguments have come to simply assume that one can apply probability distributions from statistical theory on their own area of research:
Probability is a measure of sets in an abstract space of events. All the mathematical properties of probability can be derived from this definition. When we wish to apply probability to real life, we need to identify that abstract space of events for the particular problem at hand … It is not well established when statistical methods are used for observational studies … If we cannot identify the space of events that generate the probabilities being calculated, then one model is no more valid than another … As statistical models are used more and more for observational studies to assist in social decisions by government and advocacy groups, this fundamental failure to be able to derive probabilities without ambiguity will cast doubt on the usefulness of these methods.
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.
In his great book Statistical Models and Causal Inference: A Dialogue with the Social Sciences David Freedman also touched on these fundamental problems, arising when you try to apply statistical models outside overly simple nomological machines like coin tossing and roulette wheels (emphasis added):
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 Haavelmo’s “infinite population”, Fisher’s “hypothetical infinite population”, von Mises’s “collective” or Gibbs’s ”ensemble”?
Of course one could treat our observational or experimental data as random samples from real populations. I have no problem with that. 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 populations.
But this is actually nothing else but hand-waving! And it is inadequate for real science. As David Freedman writes in Statistical Models and Causal Inference (emphasis added):
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 assumes that such a population exists in some ill-defined sense. And there is a further assumption, that the data set being analyzed can be treated as if it 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 …