# The Frisch-Slutsky Hypothesis

Econometric analysis of business cycles has tended to concentrate on testing various versions of the hypothesis arising out of the work of Frisch (1933) and Slutsky (1937). Frisch (1933) postulated that the majority of oscillations were free oscillations - the structure of the system determining the length and dampening characteristics of the cycle and external (random) impulses determining the amplitude. As noted in section 1.2, such systems can produce regular fluctuations from an irregular (random) cause. If Frisch is correct then cycle analysis can proceed to tackle two separate problems: the propagation problem, which involves modelling the dynamics of the system; and the impulse problem, which involves the identification of the sources and effects of shocks and modelling the shock-generating process. Frisch believed that the solution of the propagation problem would be a system providing cyclical oscillations, in response to shocks, which converge on a new equilibrium.

As an approximation to the solution of the 'propagation problem, Frisch derives a macrodynamic system of mixed difference and differential equations based on the theory of Aftalion (1927). The model solutions have the properties sought by Frisch, namely a primary, a secondary and a tertiary cycle with a trend and, most importantly, the cycles are damped.

Frisch's approach is clearly a useful one but unfortunately many students of economic cycles have forgotten that he tried to solve the 'propagation problem' prior to tackling the 'impulse problem. The testing of the Frisch hypothesis often involves deriving a shock-generating mechanism with sufficient energy to produce cycles from an econometric model and thus gives undue attention to the solution of the impulse problem and inadequate attention to the solution of the propagation problem, i.e. dynamic specification. Frisch regarded his model as a first approximation, pointing to the work of Fisher (1925) and Keynes (1936) as sources of ideas for improvement. A systematic testing of various solutions to the propagation problem is noticeably lacking in the literature. Frisch's hypothesis that the propagation model should have damped, rather than self-sustaining, cycles has not been adequately tested. Questions that remain unanswered include the following. What degree of dampening, if any, should be expected? What are the relative roles of endogenous cycles and external shocks? Or, alternatively, to what extent is the cycle free or forced? It is to be noted that even if self-sustaining (endogenous) cycles are postulated, shocks will have a role to play in that they will add irregularity; so a solution to the impulse problem is still required. The role of the impulse model will of course differ in such cases from that attributed to it by Frisch, which was the excitement of free (damped) oscillations generated by the propagation model.

Frisch proposed two types of solution to the impulse problem. First, expose the system to a stream of erratic shocks to provide energy; second, following Schumpeter (1934), use innovations as a source of energy. The result of the former, Frisch finds, is a cycle that varies within acceptable limits in its period and amplitude. The dynamic system thus provides a weighting system that allows the effects of random shocks to persist. Frisch suggests that erratic shocks may not provide the complete solution to the impulse problem and assumes that inventions accumulate continuously but are put into practical use (as innovations) on a large scale only during certain phases of the cycle, thus providing the energy to maintain oscillations. The resulting cycle he calls an automaintained cycle. Frisch illustrates with a description of a pendulum and a water tank, with water representing inventions. A valve releases the water for practical use at certain points in the swing of the pendulum (economy), thus providing energy. Frisch notes that the model could lead to continuous swings or even increasing oscillations, in which case a dampening mechanism would be needed. He seemed to have in mind here something that reduces and slows movement, such as automatic stabilisers, rather than Hicksian ceilings and floors (Hicks 1950). Frisch regarded these two types of solution as possibly representing equally important aspects of the cycle.

Frisch (1933), therefore, provides two possible solutions to the impulse problem: the Frisch I hypothesis that exogenous, purely random, shocks provide energy to a system (propagation model), with a damped cyclical solution, to produce the cycles observed in the economy; and the Frisch II hypothesis that the shocks are provided by the movement of the economic system and these shocks supply the necessary energy to keep the otherwise damped oscillations from dying out. These shocks are released systematically, but whether they are regarded as exogenous or endogenous depends on whether or not a theory of innovations is included in the model. It should be noted that the Frisch I hypothesis is a bit loose in the sense that the random shocks could apply to equation error terms, exogenous variables or parameters; and shocks to each have different rationalisations and, therefore, imply subhypotheses. Further, these various types of shock are not mutually exclusive.

Slutsky's (1937) work (see also Yule 1927) largely overlaps with that of Frisch (1933) and tends to confirm some of its major propositions, but there are some useful additional points made. Slutsky considers the possibility that a definite structure of connection of random fluctuations could form them into a system of more or less regular waves. Frisch (1933) demonstrated that this was possible. Slutsky distinguishes two types of chance series: those where probabilities are conditional on previous or subsequent values, i.e. autocorrelation within the series but not cross-correlation between series, which he calls coherent series; and those with independence of values in the sequence (i.e. no autocorrelation), which he calls incoherent series.

Slutsky derives a number of random series which are transformed by moving summation. We shall call the resulting series type I series. Slutsky then forms type II series by taking moving sums of type I series. Analysis of type I series shows that cyclical processes can be derived from the (moving) summation of random causes. Type II series display waves of a different order to those in type I series and, Slutsky notes, a similar degree of regularity to economic series. The type II series are subjected to Fourier analysis which reveals a regular long cycle. Slutsky also finds evidence of dampening and suggests the system consists of two parts: vibrations determined by initial conditions; and vibrations generated by disturbances. The disturbances, he suggests, accumulate enough energy to counter the dampening, and the vibrations ultimately have the character of a chance function, the process being described solely by the summation of random causes. Tests of whether the business cycle is adequately described as a summation of random causes, rather than by a complicated weighting of such random shocks through a 'propagation model' derived from economic theoretic considerations, were discussed in section 1.2.

It is common in stochastic simulations of econometric models to feed in autocorrelated shocks. Since this is essentially what has been done by Slutsky to yield type II series, which provide his best results, we may regard these as tests of the Slutsky hypothesis. In terms of Frisch's analysis, Slutsky hardly considered the propagation problem, using instead purely mechanical moving sums. His work is best regarded as a contribution to the solution of the impulse problem. One further point arises from the work of Slutsky, and related work by Yule (1927). This has become known as the 'Slutsky-Yule' effect (Sargent 1979, pp.248-51), which is that using moving averages to smooth data automatically generates an irregular periodic function. It is likely that a number of series, smoothed by the same moving average process, will show similar cycles. The Slutsky-Yule effect does not mean that cycles do not exist in economic series, but it does imply the need to be careful in dealing with series that have been smoothed or filtered, perhaps to eliminate trend or seasonal effects, since spurious cycles may be introduced. This problem is particularly relevant when tests of the 'long swing hypothesis' are considered, since it should be borne in mind that smoothing the series to eliminate shorter cycles could well have created longer cycles in the smoothed data. (See section 4.3 for further discussion.) This likelihood is demonstrated by Slutsky's finding that type II series had clearly identifiable long cycles whereas type I series did not.

It should be noted that the Frisch I hypothesis implies that economic oscillations are free (although damped), whereas Slutsky's hypothesis, that the cycle is formed by the summation of autocorrelated shocks, implies that oscillations are more likely to be forced. It is also possible that the method of summation or weighting implicit in the propagation model could, in the Slutsky case, impart significant cyclical features in addition to those 'forced' by the autocorrelated shocks.

The greater the dampening factor, the larger the shocks needed to produce a regular cycle. The problem is that it is always possible to produce random shocks that produce cycles if they are of the right size and occur with the required frequency. What is needed is an indication of a reasonable magnitude of shocks and the frequency with which they occur. If this 'reasonable' random shock series cannot produce acceptably realistic cycles then something is wrong.

Kalecki (1952) illustrates the point that, with heavier dampening, a cycle that was regular becomes irregular and of the same order and magnitude as that of the shock series. The erratic shocks used by Kalecki in his demonstration were from an even frequency distribution, i.e. shocks with large or small deviations from the mean occurring with equal frequency. Frisch (1933) and Slutsky (1937) also worked with shocks of even frequency. Random errors are, however, usually assumed to be subject to the normal frequency distribution, in accordance with the hypothesis that they themselves are sums of numerous elementary errors and such sums conform to the normal frequency distribution.

Kalecki observes that, whether or not random shocks in economic phenomena can be considered as sums of numerous elemental errors (random shocks), it seems reasonable to assume that large shocks have a smaller frequency than small shocks. Hence a normal frequency distribution of shocks will be more realistic than an even frequency distribution. Kalecki finds that the cycle generated by normally distributed shocks shows considerable stability with respect to changes in the basic equation which involve a substantial increase in dampening and, even with fairly heavy dampening, normally distributed shocks can generate fairly regular cycles from a linear equation.

The Frisch-Slutsky hypothesis, that the business cycle is the result of a series of shocks to a linear economic model, which imparts dampening effects, has formed the basis of post-war business cycle modelling. It is implicit in the Keynesian approach, as demonstrated by the simulation analysis of the large scale econometric models in the 1970s, as well as the New Classical approach. Lucas and Sargent (1978) have explicitly observed that their equilibrium theory of the cycle (Mullineux 1984, Ch. 3) is also based on the Frisch-Slutsky hypothesis. In the New Classical models, the 'impulse problem' is solved by the real and monetary shocks that result in unanticipated price changes. The shocks are assumed to be random and non-autocorrelated with constant mean and variance. In order to explain the persistence of the effects of the shocks and to provide a model of the cycle, the impulse model must be supplemented with a propagation model. The 'Lucas supply hypothesis' (Lucas 1972, 1973) introduces a positive (negative) supply response to unanticipated price increases (decreases), so that a random, non-autocorrelated series for output would be expected to result from random shocks feeding through to prices (Mullineux 1984, Ch. 4). Lucas (1975, 1977) explains why these output effects might persist and thereby provides a solution to the propagation problem for these models, allowing them to explain the observed autocorrelation in output series. In Lucas's (1975) model, persistence is introduced by employing a modified accelerator hypothesis. The positive supply response leads to an increase in capital stock which cannot instantly be reversed if the supply response was incorrect, in the sense that it was a response to a monetary rather than a real shock. It must be reduced over time, at the rate of depreciation.

When simulated, most econometric models, which are essentially linear or log linear, display stable growth rather than damped oscillations.28 Thus these models cannot explain cycles, in the Frischian manner, when bombarded with random shocks and certainly cannot explain the cycle endogenously. Serially correlated shocks are usually required to simulate the economy to any degree of accuracy. Blatt (1978) calls this the modified Frisch-Slutsky theory. One interpretation of serial correlation in the shocks is that it indicates dynamic misspecification and, in particular, insufficient lags. The success of autoregressive integrated moving average (ARIMA) models, whose strength is lag specification, in forecasting economic time series also points towards the conclusion that the weakness of large scale econometric models was in their lag structure.

Attempts to improve models by refining their lag structures could, however, lead to further misspecification if it is to nonlinearities that we should be looking to solve the propagation problem. Further, if the nonlinear approach is correct, then it may be necessary to replace the traditional trend (growth) and deviation from trend (cycle) analysis with an integrated theory of the dynamic development of the economy.

Attempts to improve models by refining their lag structures could, however, lead to further misspecification if it is to nonlinearities that One of the first problems to be resolved is whether a linear system can provide a reasonable approximation to the economy. (See also section 1.3 and Chapter 4.) If it can, then efforts should be made to improve dynamic specification, and the Frischian approach of seeking the solution to the propagation and impulse problems should be pursued. In the case of explosive rather than the damped cycles usually associated with the Frischian approach, it would also be necessary to consider "billiard table' or type I nonlinearities, such as ceilings and floors.

In specifying a model for testing a theory of the business cycle, it is necessary to consider the shock-generation mechanism or to solve the impulse problem because the dynamic path of the stochastic form of the model will differ from that of the deterministic form. The importance of the shock-generating mechanism will depend on the type of model being considered. It will be less important for a nonlinear model with a stable limit cycle solution (see section 2.3) than for a monotonically stable system.

To construct a cycle model one must first decide whether the principal active forces are endogenous or exogenous to the model. Haavalmo (1940) called the exogenous case an open model and the endogenous case a closed model. The choice between an open or closed model should ideally be made after a priori theory has allowed full dynamic specification of the model, which involves specification of nonlinearities and lags. The model would then be analysed using simulation and/or analytical techniques in order to determine whether maintained, damped or explosive oscillations were present. In the case of damped cycles, or monotonic dampening, it is necessary to assume an open model in order to simulate observed cycles, whereas for the maintained or explosive cycle cases shocks would accentuate the explosiveness and add irregularity. Type I nonlinearities would be required to contain the cycle, and the model would be essentially closed. A nonlinear model with a stable limit cycle, in which shocks simply add irregularity, is clearly a closed model.

In the open model case it is also necessary to consider whether the driving force is itself cyclical, resulting in 'forced oscillations', or whether the cycle is the result of the way the system responds to non-oscillatory stimulating forces, i.e. 'free oscillations'. In the cases of damped cycles or monotonic convergence, it is clear from the previous discussion of the Slutsky-Yule effect that an open model can generate business cycles. For many of the large scale macroeconometric models, random shock simulations proved inferior to autocorrelated shock simulations. The resulting cycles were consequently forced oscillations, the driving force coming from the imposed error structure. In view of the fact that the presence of autocorrelation can be viewed as indicative of dynamic misspecification, it is not clear whether these oscillations should really be viewed as 'forced', on an otherwise monotonically stable system, or whether the system, with correct dynamic specification, would produce its own, perhaps damped, cycles that would be stimulated by random shocks to produce 'free' oscillations.

It is also to be noted that forced oscillations could arise from the exogenous variable generating process, a possibility largely ignored in the model simulation exercises of the early 1970s. With multiplicative errors, the exogenous variable generating process might even produce free oscillations, another virtually untested hypothesis.

Thus, especially where it is believed that the oscillations are forced, but also in the Frischian case, where random shocks solve the impulse problem and the model the propagation problem, it is essential to have a model of shock generation and also a model of exogenous variable generation. Further, to achieve a degree of realism, closed cycle models need to be analysed in stochastic form so that in this case too shock generation should be considered.

In order to formulate a theory in the forced oscillation case, it is essential to decide where the source of energy originates. Once the probable source of energy is located, and if we believe the dynamic specification of the model to be correct, it is not safe to assume that we can simply choose a (possibly ARIMA) process to generate the energy (impulses) to our propagation model that best simulates observed cycles. We ought to have a rationalisation for the forcing elements. In other words an impulse model is required. This is difficult to derive because the errors could represent omitted variables - which are omitted because they are unobservable, not believed to be relevant, or due to considerations of model size. If a propagation model cannot generate an acceptable cycle when hit by random shocks we should look at it critically, unless we have good a priori reasons to expect ARIMA generated shocks of a particular degree, given the risk that the autocorrelated error shocks could represent misspecification.

It is necessary to decide whether the shocks are to be applied via the error term or the exogenous variables. If they are then more attention must be paid to the prediction of exogenous variables. It seems desirable that, instead of trend predictions for exogenous variables in simulation experiments, ARIMA processes should be used to derive optimal linear forecasts based on past observations of the exogenous variables. The only relevant information in forecasting a truly exogenous variable should be its own past history. Sims (1980) opened up the whole question of the erogeneity and endogeneity of variables. He challenged the a priori approach to this choice and suggested that the division of variables, between exogenous and endogenous, should be based on causality tests and, at minimum, the a priori choice should be checked in this way. There seems to be some ambiguity in the choice of endogenous and exogenous variables which results from the size of model to be considered. For example, government policy variables have often been treated as exogenous because no government objective function is included in standard econometric models. If a government objective function is included, however, policy variables become endogenous. Further, in concentrating on economic factors, it is common to treat non-economic factors such as weather and demographic trends as exogenous. There are, however, scientists who regard these variables as endogenous to their models. Thus in some cases it may be possible to utilise satellite models, for weather or population prediction for example, into which information can be fed to generate exogenous (to the economic model) variable processes.

Once these 'forecasts' for exogenous variables have been made they can be fed into the model, in place of linear trend predictions, in order to see how much energy is provided. The expected amount of energy can be gauged by comparing * ex post *simulations using true exogenous variables with those using trend generated exogenous variables instead. Further, the use of ARIMA forecasted exogenous variables for

*forecasting will introduce systematic or random shocks to the exogenous variables which are not provided by the trend extrapolation of exogenous variables. The size and nature of the random errors can be gauged by comparing*

**ex ante***simulations, using true exogenous variables, with ARIMA generated exogenous variables. Once the propagation model, with its ARIMA forecasts of exogenous variables, has been simulated it may be possible to decide how much additional energy is required to solve the impulse problem. It is then necessary to identify realistic sources of energy rather than impose on the equation errors the form that produces representative time paths.*

**ex post**Haitovsky and Wallace (1972) suggested adding error terms to exogenous variables and to parameters in simulation experiments. The latter introduces multiplicative errors if we assume the errors on the stochastic coefficients follow the same process, or have a common factor. The aim is to prevent overstating the error in the equation residuals. One rationale for parameter shocks is the introduction of errors to account for irrationality or erratic behaviour in decision-making by economic agents. Multiplicative, as well as additive, errors and therefore stochastic parameters should also be considered as part of the impulse model. The additive residual errors on the equations of a model are usually assumed to represent one of the following: measurement errors, aggregation errors, omitted variables or specification errors.

It is clear from the previous discussion that, if errors are applied to the exogenous variable generating process and parameter values (multiplicative errors), a clearer picture emerges of what energy is required from the equation errors to solve the impulse problem. It has been noted above that an estimate of the error process on exogenous variables can be generated using simulations. It is not clear how the error process on parameters is to be determined. The simplest solution is to assume errors are random. If the propagation model, with its exogenous variable generating process and stochastic parameters, still requires autocorrelated shocks to the equations in order to generate realistic simulations, then misspecification is a strong possibility and, in the absence of strong a priori reasons to expect autocorrelated shocks, an attempt should be made to identify it. It is not too difficult, using simulations experiments, to get a good idea of the ARIMA error process required to solve the impulse problem. This could be used if misspecification is not identifiable; if the error process is believed to represent common factors of lag polynomials so that the model is specified in its most efficient estimation form; or if it is believed to be due to the omission of immeasurable variables.

Zarnowitz (1972) suggests that we might expect some autoregression (AR) in the errors as a result of structural change and that the cyclical aspects of the simulations would probably be strengthened by application of autocorrelated shocks not only to the equations with endogenous variables, but also to exogenous variables. He notes that wars, policy actions and technical change (innovations), * inter alia, *would frequently result in autocorrelated 'autonomous' shocks to the economy. The simulations in Hickman (ed.) (1972), for example, reveal a neglect of exogenous variable generation and shocks. Exceptions are the work of the Adelmans (1959) and of the OBE group. The OBE group found that cycles were increased in amplitude and showed absolute declines in GNP lasting three to five quarters when shocks were applied to the exogenous variables. This result suggests that movements commonly considered exogenous in large scale models may play a crucial role in the determination of business cycles. The review in section 2.2 will also show that AR shocks to equations have been used with some success and analysis of the forecasting performance of a number of models suggests that the errors may be AR due to dynamic misspecification of the lag structure.

Hickman (1972) points out that in broadening the class of shocks to include perturbations to exogenous variables and autocorrelated errors in the equations, the role of the model as a cycle maker is diminished. If the real roots dominate the cyclical ones and the lag structure does not propagate cycles from serially independent impulses, the model becomes simply a multiplier mechanism for amplifying the various shocks. There is still an impulse response mechanism but the cycles are inherent in the impulses rather than the responses. This could be the correct position, in which case we should model, as accurately as possible, the shock-generating process by analysing carefully the effects of innovations and other sources of shocks in order to solve the impulse problem. Alternatively, if the autocorrelation in the errors does in fact represent misspecification of the structural model, more attention should be paid to the solution of the propagation problem, but a model of shock generation will still be required.

If the propagation model is believed to be 'correct' in that it forecasts well when subjected to AR errors, then omitted variables are likely to be the source of the AR errors. By definition large scale econometric models will be misspecified representations of the real world economy because the aim of a model is to explain the main features of the real world without being unmanageably large. To prevent the model becoming too large certain variables must be omitted by choice; yet other potential influences on the chosen endogenous variables are probably omitted as a result of our view of the world through the narrow blinkers of economic analysis, which prevent consideration of political, sociological, demographic and other factors. As a first approximation an ARIMA process can be used to generate the errors but identification of the likely sources of the errors is essential. For example, the period after a major war could be treated as a special period, and the time series could be divided into policy periods, technological periods and so on. De Leeuw (1972) suggested a systematic historical investigation of the role of identifiable exogenous influences. Further, to the extent that some external events (e.g. wars and oil crises) have a general impact on the economy, allowance should be made, in simulation experiments, for covariation between disturbance terms on exogenous variables and stochastic equations.

In order to test competing cycle theories each one has to be put into a testable form. This involves careful specification of the deterministic part of the model to solve the propagation problem. If the model is linear then careful attention must be paid to lag specification at this stage, and a priori theoretical considerations based on microeconomic foundations should be utilised, as far as possible, in order to avoid * ad hoc *dynamic specification. A fully specified cycle model should also have a well specified exogenous variable and shock-generating functions, representing a solution to the impulse problem. For nonlinear models with stable limit cycles, the role of the shock-generating mechanism will be different because the shocks merely introduce the required irregularity to simulated cycles. In this case the role of the deterministic model is not really one of propagating exogenous shocks. It is one of producing endogenous cycles.

In order to test competing cycle hypotheses it will be necessary to consider various alternative cycle-generating or propagation models, exogenous variable generating models and shock-generating models; all of which would be linked by an overall model.