Statistical Analysis of the Business Cycle

Samuelson's thesis advocated the methodology of Operationalism. He used this methodology to argue that economic theory should be about generat­ing testable propositions. However, while at Harvard, with the exception of his paper on unemployment written with Russ Nixon, he neither engaged in empirical work nor wrote about how propositions might be tested.

Samuelson's first empirical project at MIT, on the business cycle, was supported by money donated by businessman Roger Babson. Babson, an MIT graduate from the 1890s, had become famous as the author of the “Babsonchart,” a device used to predict the ups and downs of the stock mar­ket.¹ In 1929, at a time when most analysts were predicting a continued rise in stock prices, Babson forecast a decline; and after the Great Crash, he became a celebrity, his methods seemingly having been vindicated.² The principle behind the Babsonchart was Isaac Newton's third law: that every force produces an equal and opposite reaction. Believing that the future could be predicted on the basis of past trends, he interpreted Newton's law as jus­tifying the line denoting normal business activity. The further business grew above this line, the greater the chance of a reaction that would take it below the line. Babson had learned about Newton while a student at MIT in the 1890s, and later made a donation to his alma mater for the study in econom­ics of Newton's third law. Ralph Freeman proposed that part of this money be used to support a project led by Samuelson.

Samuelson's research proposal, which Freeman sent to the MIT trea­surer on December 23, was clearly written with Babson's ideas in mind.³ It began by praising “the triumph of Newtonian mechanics,” arguing that its achievement was to formulate second-order differential equations that could fully determine the motion of a physical system. Though he criticized the approach as “primitive and rudimentary” in comparison with “the more advanced physical sciences,” he pandered to Babson by saying that the notion of the depth of a depression being directly related to the height of the pre­vious boom was an application of Newton's third law. His proposal was to make use of recent advances in statistics to go beyond such simple theories, implicitly claiming that he would build upon Babson's work.

Recent advances in mathematics, statistics, and economic theory open up for the first time the possibility of empirical determination of struc­tural relations in economic time series. These take the form of sto­chastic linear difference equations which yield as solutions damped or undamped harmonic series with coefficients varying according to some probability distribution.

Having constructed a theory in which cycles could be generated by a second- order difference equation, Samuelson was now proposing to estimate the coefficients in such a model, describing his project in terms developed by Ragnar Frisch during the 1930s. Though he had introduced his project in a way so as to impress Babson, Newtonian ideas were completely irrelevant to what he wanted to do. To undertake such an ambitious task—estimating such models was in their infancy—he needed a trained assistant, for which Freeman proposed to use $500 of the Babson funds.

Samuelson approached Oskar Lange, at Chicago, about whether he could recommend anyone, and in December 1940, Lange provided Samuelson with the names of four people who had a knowledge of mathematical statistics, including that of Leonid Hurwicz.⁴ Two years younger than Samuelson, Hurwicz had been born in Russia, he had studied law in Warsaw, and after studying economics in London (with Dennis Robertson, a former collabora­tor with Keynes) and Geneva, he had come to the United States in 1940, where he was attending lectures in Chicago.

He has an excellent mind, and is in my opinion, the best of the candi­dates on this list. He has quite a background in mathematical statis­tics, and has quite extensive knowledge of analysis. Before becoming an economist he was a theoretical physicist. He also did numerical work in experimental physics. He is really one of the very best I have had among students. In addition, he needs a job very badly, because he has no income at all.⁵

Samuelson later said he decided to take Hurwicz as he was the one in greatest financial need. Lacking any income other than support from the cousins with whom he was staying in Chicago, Hurwicz was willing to take the position even though it was for a single semester, from January to June 1941.

Later memories of the research the two of them undertook vary. Hurwicz remembered undertaking statistical work on how companies set prices.⁶ He cited the example of a coffin manufacturer who started with the cost of a coffin, multiplied it by 3, and added $50. Such investigations were common at this time, both in the United States, where there was great interest in companies’ pricing policies as a possible explanation for the Depression, and in Britain, where the Oxford Economics Research Group sought to explain the day-to-day setting of prices, a context in which it was believed that profit maximization made no sense. In contrast, Samuelson remembered Hurwicz as having worked explicitly on the business cycle: “We did early spectral analysis of Frickey’s aggregate U.S. output for the time slot 1865—1935.” Samuelson wrote,

When I say “we” I do not refer to Leo and Paul only. Instead I can still see in my mind’s eye Leo, whip in one hand, slide rule in the other, marshaling his crew of mostly young female National Youth Administration galley-slave computers. Parallel computer com­putation thus merits a marble marker at the northwest corner of Massachusetts Avenue and Memorial Drive.

In those days, a computer was a person, and a computer laboratory was a room in which rows of people sat doing calculations, which had to be broken down into components, each of which was worked out by a different per­son (the parallel computing) before the results were put together. This was Samuelson’s first experience with directing a team of researchers.

Frickey, one of Samuelson’s Harvard teachers, was concerned with devel­oping indicators of the business cycle, writing a series of papers in the 1930s.^a

a. See chapter 6 this volume. Spectral analysis of his data—trying to identify the periodicity or frequency of cycles—would have been a natural extension of his work. It seems plau­sible that Samuelson, who had studied Fourier analysis and other techniques used for spectral analysis in the natural sciences, might have wanted to apply more advanced mathematical techniques to the problem, especially given that he had an assistant trained in physics. In modern econometrics, spectral analysis and structural estimation are generally seen as alternatives, but in 1940, estimation methods were much more fluid. Samuelson may well have been trying to use methods that would have been a more rigorous exten­sion of the methods of his Harvard teachers, in order to estimate structural models.

This research project appears to have been unsuccessful, for it did not lead to any publications. Correspondence with the MIT treasurer makes it clear that it was understood the results of their work “would be presented in print at a later date” and that funds from the Babson grant had been reserved for this purpose, but no trace of such a printed report exists.⁸ Furthermore, when Samuelson had to write a report on his use of the Babson money, he wrote to Hurwicz, who had by then returned to Chicago, asking whether he might list a paper Hurwicz had forthcoming in Econometrica as stemming from work done at MIT.⁹ He asked him specifically whether he would be willing to insert in the paper a footnote thanking Babson.¹⁰ Hurwicz’s foot­note described his work as having arisen from “interpreting the results of two business-cycle studies,” one at MIT and one at Chicago.¹¹ While this was related to the subject of Samuelson’s research project, it was based pri­marily on a paper that Hurwicz had written before he went to MIT, “The Phenomenon of Hysteresis in the Correlation of Time Series.”¹² It went beyond Samuelson’s model of the cycle, in that it analyzed stochastic models of fluctuations (Samuelson’s earlier model had no stochastic terms), and it tackled problems relating to estimation, but it was entirely theoretical, con­taining no data analysis.

A few months after the Babson-funded project ended, Samuelson wrote a paper that did involve the analysis of statistical data: “A Statistical Analysis of the Consumption Function.”¹³ While there is no direct evidence that it was a spin-off from the business cycle research project, it would have been natural for them to have estimated a consumption function, for this was one of the two equations they would have to have estimated if the structural model was the multiplier-accelerator model that Samuelson had published the previous year.

Samuelson’s paper began by surveying the literature on consumption and income, classified according to the methods used: from household budget data; from time series of national income; and “more or less plausible rough” estimates, such as those of Kahn, Keynes, and J. M. Clark.¹⁴ Whereas Hansen had adopted the first method, Samuelson proposed to test the relationship using time series data on consumption and income. He started with data for the period 1921—35 produced by Simon Kuznets (omitting data for 1919—20 on grounds that they reflected anomalous wartime events) and added data that the National Resources Planning Board had produced for 1936—39. In order to find what he called “a reversible analytical relationship” rather than just a description of the historical data, he needed to adjust for price changes so as to relate real consumption expenditure to real income. Kuznets had used a complicated procedure to obtain his deflated series, but Samuelson discovered that using the wage earner’s cost of living index produced by the Bureau of Labor Statistics provided similar results, so he used this simpler method to obtain the series he used for his analysis.

Samuelson fitted a least squares regression line to the data, relating consumption just to income. However, though this apparently fitted the data for the whole period, he noted that “the deviations from the line of best fit are not randomly distributed.”¹⁵ He deduced this not from a statistical test but from close observation of the data. If errors were not random, it implied that the least squares regression line was unsatisfac­tory from a statistical point of view, and so he then tested the hypoth­esis that a secular trend was operating by including time as an additional factor. He was thus testing the hypothesis that there was a consumption function relating consumption to income that shifted up or down at a constant rate. This worked for Kuznets’s original data, but when the addi­tional four years were added, the coefficient on the time variable was no longer significantly different from zero “in a sampling sense.”¹⁶ In other words, though including the extra variable inevitably made the equation fit the data better, the improvement was not sufficient to justify including it. Unlike his earlier observation that the errors were not random, based only on visual inspection of the data, this involved calculating test statistics.⁶

b. In the 1950s, it was Milton Friedman who developed the permanent-income theory of consumption, and Albert Ando and Franco Modigliani, authors of the life-cycle theory, who came to be associated with such ideas. These ideas were, however, already circulating in the early 1940s. See also figures 16.1 and 20.1, this volume, and the associated discussion.

Samuelson then explored whether the problem with the simple least squares consumption function could be explained by changes in business saving. The rationale for this was that this part of national income was not received by households, so it should not affect their consumption. To do this, he estimated consumption as a function, not of national income (“income produced”) but of income received by households (national income minus business saving). This produced a marginal propensity to consume of 1.06, suggesting an unstable system. However, even with such a marginal propen­sity to consume, the system would not be unstable once account was taken of business saving. Moreover, he argued, retained corporate earnings should be reflected in equity prices, reducing the need for individuals to save out of income received. This meant that there was no reason to expect there would be a close connection between income received and consumption.

In this paper, Samuelson showed great familiarity with the data and the method by which they were calculated. He was also familiar with the idea that aggregate relationships could be tested statistically, and that such tests should be employed in working out the best specification for an empirical model. The tests he used were both informal (noting that the residuals from his regression equation did not appear to be random) and formal (noting that the coefficient on the time trend was not significantly different from zero). It has been claimed that his work “represented the first published diagnostic use of residual analysis in the econometrics of the consumption function.”¹⁷ The paper is important because it shows that, having written a thesis in which he argued that the task of economic theory was to derive testable predictions and to show the methods by which this could be achieved, he was now turning to the problem of testing theory. That is, he was ceasing to be just a mathematical economic theorist. Clearly, Hansen, in whose book the paper was published, was the major influence on this work, but the transition fits with his move into the more technical world of MIT. Not only did MIT provide the research grant that enabled him to recruit Hurwicz, but also he had moved to an envi­ronment where the emphasis was on solving difficult practical problems.