The Method of Investigation

Beveridge ascribes crises to industrial competition, May to the disproportion between 19-20 the increase in wages and in productivity, Hobson to oversaving, Aftalion to the diminishing marginal utility of an increasing supply of commodities. Bouniatian to overcapitalization, Spiethoff to overproduction of industrial equipment and underproduction of complementary goods, Hull to high costs of construction, Lescure to declining prospects of profits, Veblen to a discrepancy between anticipated profits and current capitalization, Sombart to the unlike rhythm of production in the organic and inorganic realms, Carver to the dissimilar price fluctuation of producers' and consumers' goods, Fisher to the slowness with which interest rates are adjusted to changes in the price level.

One seeking to understand the recurrent ebb and flow of economic activity characteristic of the present day finds these numerous explanations both suggestive and perplexing. All are plausible, but which is valid? None necessarily excludes all the others, but which is the most important? Each may account for certain phenomena; does any one account for all the phenomena? Or can these rival explanations be combined in such a fashion as to make a consistent theory which is wholly adequate?

There is slight hope of getting answers to these questions by a logical process of proving and criticizing the theories. For whatever merits of ingenuity and consistency they may possess, these theories have slight value except as they give keener insight into the phenomena of business cycles.

Some Comments

Elmer Working called in 1927 the attention of economists to some facts well known in their substance to the price-theory but sometimes neglected in the statistical work. He stressed the fact that the mere existence of a statistical series of different prices and quantities of a commodity sold in the market does not exclude changes of demand and supply curves. If neither of these curves remains stable, then a curve ‘fitted’ statistically to the ‘scatter-diagram’ of the market necessarily reflects not only the shapes of the demand and supply curves but also the kind of fluctuations they underwent during the observation period. Accordingly, H. Schultz and H. L. Moore, in their well-known works, tried to eliminate at least a part of the time-change by assuming a ‘routine of change’ – of demand only. J. Tinbergen preferred to eliminate, not the time, which is of course in itself not an economic cause, but a few definite factors supposed to be pertinent to supply and demand shifts of the particular commodity considered. W. Leontief attempted to derive a method of eliminating time-shifts, starting from the assumption that they must have an erratic character.

Now, R. Frisch introduced notions which enabled him to develop analytically E. Working's statements. By applying these notions to Leontief's methods he succeeded in giving to this method an elementary mathematical exposition which is considerably simpler and at the same time more general than the exposition given originally by W. Leontief and his mathematical collaborator, R. Schmidt.

Having briefly discussed the needs of the economic policy-maker, let us now turn our attention to the efforts of the econometric model-builder. In particular we have in mind models of the type constructed for the United States economy by Tinbergen, Klein, and Colin Clark. In all of these models an attempt is made to represent the economy by a set of equations for the most part linear in nature. The variables appearing in these equations are divided into two types: endogenous and exogenous. Each of these models contains as many equations as there are current endogenous variables. Each equation, other than purely definitional equations, is imagined to hold only in a statistical sense. The equations having been set up, their parameters are estimated so as to fit the model to some existing body of data. Various ways of doing this have been used, but are not relevant to the major interests in this paper. There are two additional characteristics of what has been done or not done to which we specifically wish to draw attention.

The specification of which variables are to be considered exogenous is either done on the basis of theoretical convenience from the standpoint of limiting the field of interest or is done on the basis of some a priori knowledge of unspecified source. In any case, the specification is not subjected to any test whatsoever.

Statistical Significance of Results

The reliability of results may be judged by statistical as well as economic criteria. In general, the figures used are not exact. They are often derived from samples, or otherwise more or less inadequate for the problem under consideration. In addition, a number of minor explanatory causes are omitted; this seems to be the chief reason why observed and calculated values of x1 in general do not coincide, and this lack of coincidence is responsible for a certain ambiguity in the results obtained. The question arises whether limits may be indicated for this uncertainty. As nothing is known about the factors omitted, it can be answered only if certain additional hypotheses are made.

Various methods of statistical testing have been worked out, using different hypotheses and leading, therefore, to different results. Some account of these methods will now be given. The non-mathematical reader should be warned that their comprehension will make somewhat greater demands on his attention than has the foregoing exposition of the method of multiple correlation analysis itself; and he may perhaps prefer to take the remainder of this chapter, together with Appendix A, on trust.

The classical method goes back to Laplace and Gauss. It will be considered here in the final form that has been given to it by Professor R. A. Fisher.

Introduction

The majority of the economic oscillations which we encounter seem to be explained most plausibly as free oscillations. In many cases they seem to be produced by the fact that certain exterior impulses hit the economic mechanism and thereby initiate more or less regular oscillations.

The most important feature of the free oscillations is that the length of the cycles and the tendency towards dampening are determined by the intrinsic structure of the swinging system, while the intensity (the amplitude) of the fluctuations is determined primarily by the exterior impulse. An important consequence of this is that a more or less regular fluctuation may be produced by a cause which operates irregularly. There need not be any synchronism between the initiating force or forces and the movement of the swinging system. This fact has frequently been overlooked in economic cycle analysis.

If a cyclical variation is analysed from the point of view of a free oscillation, we have to distinguish between two fundamental problems: first, the propagation problem; second, the impulse problem. The propagation problem is the problem of explaining by the structural properties of the swinging system what the character of the swings would be in case the system was started in some initial situation. This must be done by an essentially dynamic theory, that is to say, by a theory that explains how one situation grows out of the foregoing.

Introduction

The present memorandum has been written rather hurriedly, and the text is therefore not as carefully polished as it ought to be in a manuscript ready for publication. It should, however, be clear enough to bring out my point of view.

The present memorandum does not discuss details of the various equations which Tinbergen has obtained and whose coefficients he has determined statistically. My main concern has been to discuss what equations of this type really mean, and to what extent they can be looked upon as ‘A Statistical Test of Business Cycle Theories’. (The title of one of the volumes which Tinbergen has presented for discussion.)

My conclusion is that the work which Tinbergen is now presenting is of paramount importance, perhaps the most important single step forward in Business Cycle Analysis of recent years. But I do not think that it can be looked upon as ‘A Test of Business Cycle Theories’. The question of what connection there is between the relations we work with in theory and those we get by fitting curves to actual statistical data is a very delicate one. I think it has never been exhaustively and satisfactorily discussed. Tinbergen in his work hardly mentions it. He more or less takes it for granted that the relations he has found are in their nature the same as those of theory.

In the September issue of the Economic Journal, Mr Keynes discusses critically the work done by Professor Tinbergen with the League of Nations on the ‘Statistical Testing of Business Cycle Theories’. We share Mr Keynes' opinion on some of the shortcomings of statistical procedures and even see a few further such shortcomings. Mr Keynes' criticism, however, seems to us to be going too far and to question the practicability of any attempts to test business cycle theory statistically. We do not think that the shortcomings in Professor Tinbergen's work are due to any logically inherent impossibility of verifying business cycle theory statistically. Since we are both in profound agreement with the economic theories of Mr Keynes, we are anxious to prevent the readers of the Economic Journal getting from Mr Keynes' review the impression that his theories are not capable of empirical and statistical verification. We believe, on the contrary, that many of the ideas expressed in the ‘General Theory of Employment’ are capable of such verification.

Some of the misunderstandings contained in Mr Keynes' review are due to the fact that it was written without the knowledge of Mr Tinbergen's second volume which has become available only a short time before the review was published (‘Business Cycles in the United States, 1919–1932’, Geneva, 1939). The second volume is the one concerned with trade cycle theories proper, while the first merely gives a few examples of the well-known method of multiple regression.

In this volume Professor Moore again makes use of his characteristic method, developed in his earlier volume on Laws of Wages. The method, in brief, is to derive economic laws inductively from statistics by means of the modern refined methods of the calculus of probabilities. The specific problem in the present instance is to derive the law of business cycles of expansion and depression from data as to rainfall, crops, and prices.

First, by an application of Fourier's formula to data as to rainfall in the Ohio valley and in Illinois, he finds that the annual rainfall obeys a compound cyclical law based on cycles of eight and thirty-three years. He then correlates the rainfall at the critical period of growth for each crop with the total yield and with the yield per acre of the principal staple crops. These in turn are correlated with prices of pig iron and with general prices. The laws which he derives from this analysis may be briefly stated as follows. The annual rainfall, as just stated, obeys a law of compound cycles of eight and thirty-three years’ duration. The yield of the great staple crops, both the gross yield and the yield per acre, obeys a similar law, presumably in the relation of cause and effect.

‘On Political Economy and Statistics’

It is, indeed, necessary to recognize a theory of statistics, dealing with what may be called the technique of the statistical method, that is to say, the conditions that statistical data must fulfill, the modes in which they are to be ascertained and collected, the manner of their arrangement and employment for purposes of reasoning, the criteria determining the validity of arguments based upon them, and the logical character of the conclusions established by their aid. But all this is really antecedent to the actual use of statistics for any particular purpose. The whole discussion constitutes, not a separate science, but a special branch or department of inductive logic or methodology – that is, of the science or art which treats of scientific method in general.

2 Statistics regarded as a method In seeking to define statistics regarded as a method, it 320 is convenient to adopt the somewhat clumsy phrase already quoted from Dr Mayr, and say that it is a scientific method based on the quantitative observation of aggregates. It is, in the first place, a method based on observation. It goes direct to facts, which it collects and systematically arranges. It is, in the second place, based on an observation of quantities. It deals with phenomena that are measurable, and hence capable of numerical expression. It is, in the third place, concerned with aggregates, as distinguished from individuals or units.

General Remarks on the Analysis of Time Series

The current assumption made in business cycle research is that a time series is composed from four component motions, which are defined in the following manner:

1. a) the trend: the general tendency of development over long time periods which reproduces the main course of the time series;

2. b) the cyclical fluctuations: these are wave-shaped movements that are superimposed on the trend:

3. c) the seasonal fluctuations: these are regular movements recurring yearly at the same time points:

4. d) the irregular fluctuations: these are composed chiefly of smaller fluctuations often appearing to have a chance character.

The decomposition of the time series into the above four components corresponds to the notion that the time series is to be considered as the effect of four groups of causes. The trend depends on secular causes, such as population increase, industrial and cultural development, improvement in the means of transport, and so on. The cyclical fluctuations originate in changes in economic activity and conditions of profitability. The seasonal fluctuations are brought about through the influence of the weather, or by social institutions, that, calendar like, fixed and yearly, regularly bring about similar fluctuations. Finally the irregular fluctuations arise according to causes which fall under none of the other three named groups.

The task of the analysis is the quantitative determination of the separate components corresponding to the above groups of causes.

Economic Generalizations and Reality

Statistical ‘Laws’ of Supply and Demand

2 As we have seen already, the generalizations which are deduced from this 98 observation are purely formal in character. If a certain good is scarce, then we know that its disposal must conform to certain laws. If its demand schedule is of a certain order, then we know that with alterations of supply its price must move in a certain way. But, as we have discovered already, there is nothing in this conception of scarcity which warrants us in attaching it to any particular commodity. Our a priori deductions do not provide any justification for saying that caviare is an economic good and carrion a disutility. Still less do they inform us concerning the intensity of the demand for caviare or the demand to be rid of carrion. From the point of view of pure Economics these things are conditioned on the one side by individual valuations, and on the other by the technical facts of the given situation. And both individual valuations and technical facts are outside the sphere of economic uniformity. To use Strigl's expressive phrase, from the point of view of economic analysis, these things constitute the irrational element in our universe of discourse.

But is it not desirable to transcend such limitations? Ought we not to wish to be in a position to give numerical values to the scales of valuation, to establish quantitative 99 laws of demand and supply? This raises, in a slightly different form, the questions we left unanswered at the conclusion of the last chapter.

Preface

This study is intended as a contribution to econometrics. It represents an attempt to supply a theoretical foundation for the analysis of interrelations between economic variables. It is based upon modern theory of probability and statistical inference. A few words may be said to justify such a study.

The method of econometric research aims, essentially, at a conjunction of economic theory and actual measurements, using the theory and technique of statistical inference as a bridge pier. But the bridge itself was never completely built. So far, the common procedure has been, first to construct an economic theory involving exact functional relationships, then to compare this theory with some actual measurements, and, finally, ‘to judge’ whether the correspondence is ‘good’ or ‘bad’. Tools of statistical inference have been introduced, in some degree, to support such judgements, e.g., the calculation of a few standard errors and multiple-correlation coefficients. The application of such simple ‘statistics’ has been considered legitimate, while, at the same time, the adoption of definite probability models has been deemed a crime in economic research, a violation of the very nature of economic data. That is to say, it has been considered legitimate to use some of the tools developed in statistical theory without accepting the very foundation upon which statistical theory is built. For no tool developed in the theory of statistics has any meaning – except, perhaps, for descriptive purposes – without being referred to some stochastic scheme.

This book deserves careful study by both economists and statisticians, for it makes an interesting statistical approach to an important economic problem. The problem is that of finding for a given commodity the laws of demand and supply, and hence the elasticities of demand and supply, and the solution involves the ingenious application of correlation, curve fitting, and other statistical devices to price, consumption, and output data for sugar, the commodity chosen to illustrate the method.

The first and fourth chapters have to do with the theoretical considerations relating to the problem, those relating to demand in chapter I and those relating to supply in chapter IV. The nature of neo-classical, mathematical, and statistical demand and supply curves is discussed, and the meaning to be attached to the terms ‘elasticity of demand’ and ‘elasticity of supply’ is explained. Considerable space is devoted to the distinction between arc elasticity and point elasticity, and it is shown how the difficulties arising from the use of percentages in determining elasticity may in some cases be removed by the use of logarithms. Careful consideration is given to the opinions of other students of the subject – Cournot, Marshall, Moore, Davenport, Wicksteed, Auspitz, Lieben, and others – and to the different types of demand and supply curves which these authorities have set forth and discussed.

It has several times been noted that time series commonly possess in many respects the characteristics of series of cumulated random numbers. The separate items in such time series are by no means random in character, but the changes between successive items tend to be largely random. This characteristic has been noted conspicuously in sensitive commodity prices. On the basis of the differences between chain and fixed-base index numbers King has concluded that stock prices resemble cumulations of purely random changes even more strongly than do commodity prices.

The fact that series commonly used as indexes of business activity closely resemble series obtainable by cumulating random numbers has given support to the theory that so-called business cycles result in large degree from cumulative effects of independent random influences bearing on the business situation – some favourably, some unfavourably.

In the discussion which follows, a series such as may be obtained by cumulating random numbers will for brevity and clarity be called usually a random-difference series, since it is the first differences of the series and not the items of the series itself which are random. The natural alternative term of cumulated random series is subject to misinterpretation as describing a series that is itself random.

Economic theory has fallen far short of recognizing the full implications of the resemblance of many economic time series to random-difference series; and methods of statistical analysis in general use have given these implications virtually no recognition.