# The Monte Carlo Hypothesis

Fisher (1925) argued that business cycles could not be predicted because they resembled cycles observed by gamblers in an honest casino in that the periodicity, rhythm, or pattern of the past is of no help in predicting the future. Slutsky (1937) also believed that business cycles had the form of a chance function.

The Monte Carlo (MC) hypothesis, as formulated by McCulloch (1975), is that the probability of a reversal occurring in a given month is a constant which is independent of the length of time elapsed since the last turning point. The alternative (business cycle) hypothesis is that the probability of a reversal depends on the length of time since the last turning point.

The implication of the MC hypothesis is that random shocks are sufficiently powerful to provide the dominant source of energy to an econometric model which would probably display heavy dampening in their absence. The simulations with large scale econometric models in the early 1970s showed that random shocks are normally not sufficient to overcome the heavy dampening typical in these models and to produce a realistic cycle. Instead serially correlated shocks are required.12 If shocks were in fact serially correlated the gambler (forecaster) could exploit knowledge of the error process in forming predictions and we would move away from the honest MC casino. The need to use autocorrelated shocks could alternatively indicate that the propagation model is dynamically miss pacified.

McCulloch (1975) notes that if the MC hypothesis is true then the probability of a reversal in a given month is independent of the last turning point. Using as data NBER reference cycle turning points, McCulloch tests to see if the probability of termination is equal for 'young' and 'old' expansions (contractions). Burns and Mitchell (1946) did not record specific cycle11 expansions and contractions not lasting at least fifteen months, measured from peak to peak or trough to trough. The probability of reversal is therefore less for very young expansions (contractions) than for median or old expansions (contractions), and McCulloch (1975) disregards months in which the probability of reversal has been reduced.

A contingency table test, based on the asymptotic Chi-squared distribution of the likelihood ratio, with 'young' and 'old' expansions (contractions) as the two classes, is performed. Since the sample is not large, the total number of expansions being twenty-five, McCulloch feels that it is more appropriate to use a small sample distribution than the asymptotic Chi-squared distribution. The small sample distribution is calculated subject to the number of old expansions equaling the number of young expansions. Results are reported for the United States, the United Kingdom, France and Germany. In order to facilitate a test of whether post-war government intervention had been successful in prolonging expansions and curtailing contractions, two periods are analysed for the United States.14 In both periods the test statistic is insignificant, according to both the small sample and asymptotic Chi-squared distribution cases. Thus the implication is that the probability of termination of young and old expansions is the same for both expansions and contractions and that US government intervention had had no effect. For France the null hypothesis cannot be rejected for expansions or contractions, and a similar result is derived for Germany. In the United Kingdom, however, it is not rejected for contractions but it is rejected, at the 5 per cent significance level, for expansions in both the asymptotic and small sample distribution cases. The hypothesis would not have been rejected for the United Kingdom at 2.5 per cent significance level and McCulloch suggests that the significant statistic can be ignored anyway, since it is to be expected under the random hypothesis. He concludes that the MC hypothesis should be accepted.

McCulloch (1975) also notes that a lot of information is forfeited by working with NBER reference data rather than raw data, and that consequently tests performed using actual series are potentially more powerful. He assumes that economic time series follow a second order autoregressive process with a growth trend and fits "such processes to logs of annual US real income, consumption and investment data for the period 1929-73, in order to see if parameter values which will give stable cycles result. The required parameter ranges are well known for such processes (see Box and Jenkins 1970, for example).

McCulloch points out that one cannot discount the possibility of first order autocorrelation in his results but the regressions do, in many cases, indicate that stable cycles exist. He concludes that, due to the potential bias from autocorrelation, no conclusions can be drawn from this approach with regard to cyclically. The period is, however, calculated for each series that had point estimates indicating the presence of a stable cycle. These series were log real income, the change in log real income, log real investment and the change in log real investment, and log real consumption. The required parameter values were not achieved for the change in log real consumption and quarterly log real income and the change in log real income. Further, a measure of dampening used in physics, the Q statistic, is also calculated, and it indicates that the cycles that have been discovered are so damped that they are of little practical consequence.

Finally, McCulloch notes that spectral analytic results, especially those of Howrey (1968), are at variance with his results. His conclusion is that the spectral approach is probably inappropriate for the analysis of economic time series due to their non-stationarity, the absence of large samples, and their sensitivity to seasonal smoothing and data adjustment. (See section 4.3 for further discussion.)

Anderson (1977) also tested the MC hypothesis. The method employed is to subdivide the series into expansionary and contractionary phases; analyse the density functions for duration times between troughs and peaks, and peaks and troughs; and then compare the theoretical distribution, associated with the MC hypothesis, with the actual distributions generated by the time-spans observed. The MC hypothesis implies that the time durations of expansionary and contractionary phases will be distributed exponentially with constant parameters, * a *and P respectively. A Chi-squared goodness of fit test is performed to see if the actual (observed) distribution of phase durations is according to the discrete analogue of the exponential distribution, the geometric distribution.

Unlike McCulloch, Anderson does not follow Burns and Mitchell in ignoring expansions and contractions of less than fifteen months since, by definition, this precludes the most prevalent fluctuations under the MC hypothesis, namely the short ones. The seasonally adjusted series used are total employment, total industrial production and the composite index of five leading indicators (NBER) for the period 1945-75 in the United States. The phase durations for each series are calculated by Anderson and are consistent with the MC hypothesis. They are short. The differences in length between expansions and contractions is attributed to trend.

The null hypothesis that expansionary and contractionary phases are geometrically distributed with parameters * a' *and /3' was tested against the alternative that the phases are not geometrically distributed. The null hypothesis, and hence the MC hypothesis, could not be rejected. The hypothesis that the expansion and contraction phases were the same was also tested. The composite and unemployment indices showed no significant difference in the phase, but the hypothesis was rejected for the production series.

Savin (1977) argues that the McCulloch test based on NBER reference cycle data suffers from two defects. Firstly, because the variables constructed by McCulloch are not geometrically distributed, the test performed does not in fact test whether the parameters of two geometric distributions are equal and the likelihood ratio used is not a true likelihood ratio. Secondly, the criterion for categorising old and young cycles is random. The median may vary between samples and it is the median that forms the basis of the categorisation. An estimate of the population median is required in order to derive distinct populations of young and old expansions. Savin proposes to test the MC hypothesis by a method free from these criticisms. Like Anderson, he uses a Chi-squared goodness of fit test but he works with the NBER data used by McCulloch and concentrates on expansions. He too finds that the MC hypothesis cannot be rejected. McCulloch (1977) replied to Savin (1977), arguing that his constructed variables were indeed geometrically distributed and that the contingency table tests he had employed were more efficient than the goodness of fit test used by Savin.

Two methods have, therefore, been used to test the MC hypothesis: Chi-squared contingency table tests, as used by McCulloch, and Chi-squared goodness of fit tests, as used by Savin and Anderson. In both testing procedures there is some arbitrariness in choice of categories, and although Savin uses rules such as 'equal classes' or "equal probabilities' to select his classes, he ends up with an unreliable test.15

In view of these findings on the MC hypothesis, one might wonder whether further cycle analysis would be futile. The tests are, however, confined to hypotheses relating to the duration of the cycle alone. Most economists would also take account of the comovements that are stressed by both Burns and Mitchell (1946) and students of the cycle such as Lucas (1977) and Sargent (1979). There are, however, two sources of evidence that can stand against that of McCulloch, Savin and Anderson. Firstly, there are the findings from spectral analysis, the usefulness of which should be weighed in the light of the problems of applying spectral techniques to economic time series (see section 4.3). Secondly, there are the findings of the NBER, which will be considered in the next section.

As noted in the previous section, the NBER defines the business cycle as recurrent but not periodic. The variation of cycle duration is a feature accepted by Burns and Mitchell (1946), who classify a business cycle as lasting from one to ten or twelve years. It seems to be this range of acceptable period lengths that has allowed the test of the MC hypothesis to succeed. The approach pioneered by Burns and Mitchell was described by Koopmans (1947) as measurement without theory. It leaves us with a choice of accepting the MC hypothesis or accounting for the variability in duration. However, the sheer volume of statistical evidence on specific and reference cycles produced by the NBER and, perhaps most strikingly, the interrelationships between phases and amplitudes of the cycle in different series (comovements) should make us happier about accepting the existence of cycles and encourage us to concentrate on explaining their variation.

Koopmans (1947) categorises NBER business cycle measures into three groups. The first group of measures is concerned with the location in time and the duration of cycles. For each series turning points are determined along with the time intervals between them (expansion, contraction, and trough to trough duration of 'specific cycles'). In addition turning points, and durations, are determined for 'reference cycles'. These turning points are points around which the corresponding specific cycle turning points of a number of variables cluster. Leads and lags are found as differences between corresponding specific and reference cycle turning points. All turning points are found after elimination of seasonal variation but without prior trend elimination -using, as much as possible, monthly data and otherwise quarterly data. The second group of measures relates to movements of a variable within a cycle specific to that variable or within a reference cycle.17 The third group of measures expresses the conformity of the specific cycles of a variable to the business or reference cycle. These consist of ratios of the average reference cycle amplitudes to the average specific cycle amplitudes of the variable for expansions and contractions combined and indices of conformity.

Burns and Mitchell (1946) are well aware of the limitations of their approach which result from its heavy reliance on averages. In Chapter 12 of their book they tackle the problem of disentangling the relative importance of stable and irregular features of cyclical behaviour, analysing the effects that long cycles may have had on their averages. In Chapter 11 they analyse the effects of secular changes. The point that comes out of these two investigations is that irregular changes in cyclical behaviour are far larger than secular or cyclical changes (see also section 4.3). They observe that this finding lends support to students who believe that it is futile to strive after a general theory of cycles. Such students, they argue, believe that each cycle is to be explained by a peculiar combination of conditions prevailing at the time, and that these combinations of conditions differ endlessly from each other at different times. If these episodic factors are of prime importance, averaging will merely cancel the special features. Burns and Mitchell try to analyse the extent to which the averages they derive are subject to such criticisms, which are akin to a statement of the MC hypothesis.

They accept that business activity is influenced by countless random factors and that these shocks may be very diverse in character and scope. Hence each specific and reference cycle is an individual, differing in countless ways from any other. But to measure and identify the peculiarities, they argue, a norm is required because even those who subscribe to the episodic theory cannot escape having notions of what is usual or unusual about a cycle. Averages, therefore, supply the norm to which individual cycles can be compared. In addition to providing a benchmark for judging individual cycles, the averages indicate the cyclical behaviour characteristic of different activities. Burns and Mitchell argue that the tendency for individual series to behave similarly in regard to one another in successive business cycles would not be found if the forces that produce business cycles had only slight regularity. As a test of whether the series move together, the seven series chosen for their analysis are ranked according to durations and amplitudes, and a test for ranked distributions is used.18 Durations of expansions and contractions are also tested individually and correlation and variance analysis is applied. They find support for the concept of business cycles as roughly concurrent fluctuations in many activities. The tests demonstrate that although cyclical measures of individual series usually vary greatly from one cycle to another, there is a pronounced tendency towards repetition of relationships among movements of different activities in successive business cycles. Given these findings, Burns and Mitchell argue that the tendency for averages to conceal episodic factors is a virtue. The predictive power of NBER leading indicators provides a measure of whether information gained from cycles can help to predict future cyclical evolution and consequently allows an indirect test of the MC hypothesis.

Evans (1967) concluded that some valuable information could be gained from leading indicators since the economy had never turned down without ample warning from them and they had never predicted false upturns in the United States (between 1946 and 1966). For further discussion of the experience of forecasting with NBER indicators see Daly (1972).

Largely as a result of the work of the NBER a number of 'stylised' or qualitative facts about relationships between economic variables, particularly their pro-cyclicality or anti-(counter) cyclicality, have increasingly become accepted as the minimum that must be explained by any viable cycle theory prior to detailed econometric analysis. Lucas (1977), for example, reviews the main qualitative features of economic time series which are identified with the business cycle. He accepts that movements about trend in GNP, in any country, can be well described by a low order stochastic difference equation and that these movements do not exhibit uniformity of period or amplitude. The regularities that are observed are in the comovements among different aggregate time series. The principal comovements, according to Lucas, are as follows:

1. Output changes across broadly defined sectors move together in the sense that they exhibit high conformity or coherence.

2. Production of producer and consumer durables exhibits much more amplitude than does production of non-durables.

3. Production and prices of agricultural goods and natural resources have lower than average conformity.

4. Business profits show high conformity and much greater amplitude than other series.

5. Prices generally are pro-cyclical.

6. Short-term interest rates are pro-cyclical while long-term rates are only slightly so.

7. Monetary aggregates and velocities are pro-cyclical.

Lucas (1977) notes that these regularities appear to be common to all decentralised market economies, and concludes that business cycles are all alike and that a unified explanation of business cycles appears to be possible. Lucas also points out that the list of phenomena to be explained may need to be augmented in an open economy to take account of international trade effects on the cycle. Finally, he draws attention to the general reduction in amplitude of all series in the post-war period (see section 1.5 for further discussion). To this list of phenomena to be explained by a business cycle theory, Lucas and Sargent (1978) add the positive correlation between time series of prices (and/or wages) and measures of aggregate output or employment and between measures of aggregate demand, like the money stock, and aggregate output or employment, although these correlations are sensitive to the method of detrending. Sargent (1979) also observes that 'cycle' in economic variables seems to be neither damped nor explosive, and there is no constant period from one cycle to the next. His definition of the 'business cycle' (see section 1.1) also stresses the comovements of important aggregate economic variables. Sargent (1979, Ch. XI) undertakes a spectrum analysis of seven US time series and discovers another 'stylised fact' to be explained by cycle theory, that output per man-hour is markedly pro-cyclical. This cannot be explained by the application of the law of diminishing returns since the employment/capital ratio is itself pro-cyclical.