The (re)search of the correct statistical model

If you have chosen to study in a Master program at the Toulouse School of Economics, you already have a good background in maths for analysing economic behaviour, choices and mechanisms.

When you study macroeconomic and microeconomic theory, the economic intuition is useful in order to overcome the mathematical burden. Although this approach works for theoretical economics, it barely holds for econometrics. In fact, as a graduate student, when you take early econometrics’ classes, you might wonder: “Why should I put myself through this?” and mainly: “What is the point of studying such a subject which is only a field of statistics, and apparently has no direct link to economics?”

All these questions are indeed legitimate and, like me, many PhD students have wondered exactly the same. However, as soon as you begin your own research, or when you start working in a consultancy company, you often realize the importance of this subject. In research a theory will never be powerful enough to explain all the elements of real life, which are often too complicated. You do not have enough information to provide a full description of the observed phenomena. Due to the lack of information the use of randomness and stochasticity seem to be inevitable.

In particular, imagine you are in a room alone. You have to toss a coin, and you want to know whether you will obtain heads or tails. If you knew the level of air pressure in that room, the exact force your finger provides to the coin, the intensity of the wind entering the window (and the impact it has on the trajectory of your coin), the precise slope of the desk where the coin is going to land, then you would be able to exactly predict, i.e. to say ex-ante, whether your coin will fall either heads or tails. Since you cannot know all these factors, you will have to rely on randomness, on stochasticity, and conclude that there is 50 per cent chance of observing either heads or tails. This example shows the real nature of stochasticity: “fate” does not actually exist and is just an artificial device, whose origins stem from our ignorance, and inability to observe aspects of reality.

In economics, the problem of observability can be thought of similarly: any economic theory that aims to analyse agents’ behaviour using a specific set of assumptions – like rationality, for instance – that only captures a limited number of aspects of the agents’ decision process, while there are a bunch of other elements that we cannot identify, will not be able to explain them.

 Therefore, imagine you want to analyse a consumer’s decision. What you first come up with is probably that such a decision will depend on income. However, since a single agent is also a human being with an irrational component, their consumption decisions might also depend on their mood, on whether their favourite team won the day before, or whether they had nice sex the previous night, and so on. As researchers, we have data on the first element, but not on others that cannot be screened. Such a problem is well captured in a statement by an eccentric English economist who lived through the 1929 economic crisis and World War II, who once said: “When an apple falls in physics you are almost always able to say where it is going to fall, while in economics you never know whether it is going to fall either on the right or the left.” In other words, the importance of unobserved components in economics, and social sciences in general, is great, maybe more relevant than in the exact sciences.

To sum up, when we have a theory saying, for example, that consumption is a function of income, we can never be sure about the sources other than income that might impact consumption. Accordingly, we can conclude that there are unobserved components influencing the variable that we want to study. Since these influential components are unobserved, we cannot predict their exact values. As a consequence, like the example of tossing a coin, we will rely on “randomness” and will assume that consumption is influenced by both income and an unobserved component, commonly denoted by the error term.

Now, this simple and straightforward reasoning can be considered as the logical basis for the application of Probability Theory and statistics for economics. Many researchers, after Ragnar Frisch and Trygve Haavelmo, have started to make intensive use of statistics in economics, which has been prompted by the development of computer software that are more user friendly and efficient in terms of computational capacities. Nonetheless, as often happens when a solution to a problem is put forward, it is useful to check whether the remedy is worse than the disease. Thus, it seems correct to wonder whether the use of statistics in economics is indeed justified?

Historically in mainstream Applied Econometrics, researchers often used to model economic relationships by assuming a parametric form. For instance, let’s look again at the consumption theory sketched above, considering consumption as a function of income. Researchers’ approach over the past years has been based on two points: first, they used to recognize that the theory might not be complete, so that consumption was influenced both by income and an unobserved component, the error term; second, they used to assume that consumption is a linear function of income. So, within this approach we are using a statistical model. However, is such a use of statistics correct and logically coherent?

Now, the controversial aspect of this approach is that we do not have any economic theory telling us that consumption should be linked to income by a linear function: our economic theory only tells us about the sign of these relationships but not about their form. So, why should a linear link be more correct that an exponential one? We cannot answer this question from an economic point of view, so the use of such a linear function is not justified at all, and the same holds for a logarithmic form, for a quadratic form, and for any other functional form we might impose within our consumption model.

Furthermore, there is another important drawback with the adoption of a particular functional form. This approach generates a relevant logical contradiction with respect to the lack of information problem outlined above, and that justifies the use of probability in economics. However, if in our mathematical model we include the elements that are not explained by our theory, i.e. the error component, for our modelling approach to be logically coherent, we will also have to recognize the presence of a fully unknown relationship between consumption and income, and therefore a general functional link between these two variables and not a linear one! In other words, we usually accept that our theory is lacking two things: first, it cannot explain what the impact of all the variables is on, say, consumption; second, it does not explain the functional form of our model, for instance that consumption must be a linear function of income. We justify the use of statistics by recognizing the first problem. So, why should we ignore the second issue and impose a linear form without any justification? On which logical basis do we recognize the first problem and exclude the second one? This reasoning highlights an important lack of logical coherence in the modelling approach imposing unjustified parametric restriction: these functional forms are not only unsupported by the theory, but imposing them also contradicts the logical foundations for the use of statistics in economics.

Now, such a problem finds its solution in a simple phrase – Nonparametric Econometrics. Even though, I admit, it might sound more scary than simple. Within this approach no restriction is imposed on the functional form, and the goal is to estimate functions rather than a finite number of parameters.

In order to sketch what this field is about, consider again our benchmark consumption theory, where consumption is a function of income. In Nonparametric Econometrics we assume that consumption is a general function of income, without imposing, say, linearity. This therefore represents a better general framework than parametric models for two fundamental reasons: first, it does not impose any restriction that is not supported by the theory; second, it overcomes the logical contradiction in the parametric approach outlined above, i.e. the one about imposing parametric restrictions and the foundations of the use of statistics in economics.

Nevertheless, the rule of “no free lunch” applies here as well, as the logical coherence and generality of the framework comes at the price of having more complex mathematical proof.

In all honesty, the first time I saw Nonparametric Econometrics in my life I was scared of it, and said to myself: ”I will never learn these proofs, they are just too complicated!” Nevertheless, the attraction of estimating functions, of building up a mathematical setting which was general and powerful enough that it could therefore provide the researcher with the correct insights from the reality, was a good enough incentive to learn about nonparametric economics. And it works! In fact, for my PhD I am doing research on the Nonparametric Theory of Test, and the aim of my work is to test functions. It is almost the same principle studied in our Master’s econometrics’ classes: we build up tests for a finite number of parameters, while in my research we build up tests for functions. And in particular, the test I am trying to build up can be seen as a statistical test for Bayesian Nash equilibrium in a free entry game. So, to explain it briefly, imagine you have a typical Game Theory model of Industrial Organization analysing the decision by firms on whether to enter or not to enter a market. You can verify whether this model is correct or not by applying a specific statistical test, which is the objective of my research.

Of course, as highlighted above, the tools used in Nonparametric Econometrics are sometimes different from those you might have already seen in standard statistics classes. In Nonparametric Econometrics you estimate the unknown function of your theoretical model – like the unknown function linking consumption to income – by another function, which depends on the data; but since the data is random, you will have to deal with a random function. This sometimes requires nonstandard tools from Probability Theory, like the Theory of Empirical Processes. This is a great subject that can be considered as an extension of the “standard” Probability Theory that many students might have already studied. In the “standard” Probability Theory you have random variables, so that after the realization of a random event you obtain a real value (1, 2 or 3, or whatever); in Empirical Process Theory, instead, you have random functions, so that after the realization of a random event you obtain a function (like an exponential, a parabola and so on). It is a fairly general theory and, accordingly, it may seem far from many applied problems. But it is the generality aspect that is responsible both the power and the broad range of application.

As regards my personal experience with it, I did not study Empirical Process Theory during my Master in Mathematical Economics and Econometrics at TSE, and usually this subject is not even studied in Statistics Masters, as it really is a field of Applied Mathematics. However, I have exploited the time I have spent the first year of the PhD (the DEEQA year) getting these tools and attending salsa classes, with great help of my advisor, Pascal Lavergne, who, however only helped me with Empirical Processes and not with salsa classes. Although it is sometimes complex – but rarely complicated – this is a great theory and this is the reason why it is so attractive to me now.

by Elia Lapenta

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