Since Andre Nataf stated what became known as Nataf's Theorem, in the 1940s, economists have to acknowledge that there are very narrow conditions for doing regressions with aggregate data. Most of them prefer not to think of it (they don't even teach it in grad courses), but this light oblivion can lead to grate mistakes.

In a 1946 paper, Laurence Klein complained that, “many of the mathematical models for newly constructed economic systems are very poorly related to the individual behavior of families and firms – which should form the basis of all theories of economic behavior.” In his article, published in Econometrica, Klein noted that L, the aggregate work in the Cobb-Douglas function, is a sum with work data from several companies (Σ li) and production does not grow in the same way when people are hired in each one of them. The relationship between work and production, calculated with aggregates, was different from the relation calculated with individual data. Klein wanted to find ways of aggregating data that would not produce this distortion. His alternative proposals for aggregation were very criticized until, in 1948, the French economist André Nataf demonstrated that, for data aggregation to be consistent, the terms of the production function analyzed in Klein’s example had to be mathematically independent. That phrase (“for aggregation to be consistent terms have to be independent”) is Nataf’s Theorem. It sounds silly but, if the theorem is valid (and it is), Central Bank’s inflation targeting models, Solow model’s productivity estimates, Input-Output matrices’ projections and all other models where relations are estimated from aggregate data are wrong. “Independent terms”, in Nataf’s Theorem, means that they have to be representable as sums, not as products: it means that an increase in the use of an input – in a production function – must not affect the relationship between the others.

In the early years of the twentieth century, in the United States, if you hired an extra worker in a factory, the country’s industrial output would grow proportionally to , where L and C are the total of workers and capital in the country’s factories.Don’t panic. The sentence above does not make any sense. But it is a good summary of aclassic paper published in 1928 by Charles Cobb and Paul Douglas.Cobb and Douglas estimated the relation between capital, labor, and production with series of aggregated data from the American economy.With their equation (the Cobb-Douglas Function), they intended to measure the contribution of capital and labor to production.It took 18 years for someone to complain. But finally, in 1946, a paper by Lawrence Klein pointed to the problem:“Many of the mathematical models for newly constructed economic systems are very poorly related to the individual behavior of families and firms – which should form the basis of all theories of economic behavior.”In his article, published in Econometrica, Klein complained that L, the aggregate work in the Cobb-Douglas function, is a sum with work data from several companies (Σ li) and production does not grow in the same way when people are hired in each one of them.The relationship between work and production, calculated with aggregates, was different from the relation calculated with individual data.Klein wanted to find ways of aggregating data that would not produce this distortion.His alternative proposals for aggregation were very criticized until, in 1948, the French economist André Nataf demonstrated that, for data aggregation to be consistent, the terms of the production function analyzed in Klein’s example had to be mathematically independent.That phrase (“for aggregation to be consistent terms have to be independent”) is Nataf’s Theorem.It sounds silly but, if the theorem is valid (and it is), Central Bank’s inflation targeting models, Solow model’s productivity estimates, Input-Output matrices’ projections and all other models where relations are estimated from aggregate data are wrong.“Independent terms”, in Nataf’ s Theorem, means that they have to be representable as sums, not as products: it means that an increase in the use of an input – in a production function – must not affect the relationship between the others.Imagine a country with several companies (I companies). The output of each firm (yi) has to do with the inputs it uses to produce: x1i, x2i, x3i … (where x1, x2 and x3 are quantities of electricity, paper, oil, etc.). dding the output and inputs used by all firms, we would have an aggregate equation Y = β1 * X1 + β2 * X2 + β3 * X3, where capital letters are the sums of these inputs for all firms and βs are the coefficients that the model estimates: they are the aggregate function (F).What Nataf’s Theorem says is that, for the aggregate model to make sense, it is necessary that, starting from the inputs per company (x1i and x2i in the table below), one arrives at the aggregate production (Y) by two ways:– applying the relation between inputs and production of each company (fi) to reach the production of each company (yi) and then adding the productions;– adding the inputs used by all companies (X1 and X2, in the table below) and then applying the aggregate function (F) to arrive at aggregate production (Y).If, by both ways, the estimated aggregate production is the same, aggregation is consistent.But the theorem says that this will only happen if the functions (for each firm and for the aggregate) are additive, that is, if changes in the quantity of one input (xi) do not affect the relation between the others – which rarely happens.There are many mathematical proofs of the theorem. The first is from Nataf himself. The one I find most intelligible is this one, by two of Dutch economists.But the only really strange thing for anyone who studies the theorem is that, 70 years after being refuted, aggregate models continue to be taught to economics students around the world (without any comment on aggregation problems).Armies of academics continue to replicate models like the ones Klein called new in the 1940s (he himself worked with many…).Meanwhile, Nataf’s Theorem has almost no references in today’s economics papers.

November -0001

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