Blatt (1980) notes that the Frisch-type econometric modelling of business cycles (see section 1.4) is dominant. Such models involve a linear econometric model which is basically stable but is driven into recurrent, but not precisely periodic, oscillations by shocks that appear as random disturbance terms in the econometric equations. Blatt (1978) had demonstrated that the econometric evidence which appeared to lead to the acceptance of linear, as opposed to nonlinear, propagation models was invalid (section 1.4). Blatt (1980) aims to show that all Frisch-type models are inconsistent with the observed facts as presented by Burns and Mitchell (1946). The qualitative feature or fact on which Blatt (1980) concentrates is the pronounced lack of symmetry between the ascending and descending phases of the business cycle. Typically, and almost universally, Blatt observes, the ascending portion of the cycle is longer and has a lower average slope than the descending portion. Blatt claims that this is only partly due to the general, but not necessarily linear, long-term trend towards increasing production and consumption. Citing Burns and Mitchell's evidence concerning data with the long-term trend removed, he notes that a great deal of asymmetry remains after detrending and argues that no one questions the existence of the asymmetry. De Long and Summers (1986a) subsequently do, however, as will be seen below.

Blatt (1980) points out that if the cyclical phases are indeed asymmetric, then the cycle cannot be explained by stochastic, Frisch-type, linear models. Linear deterministic models can only produce repeated sinuousidal cycles, which have completely symmetric ascending and descending phases, or damped or explosive, but essentially symmetric, cycles. Cycles produced by linear stochastic models will be less regular but nevertheless will be essentially symmetric in the sense that there will be no systematic asymmetry (see also section 3.3). Frisch-type models consequently do not fit the data which demonstrates systematic asymmetry. To complement Burns and Mitchell's findings, Blatt (1980) assesses the statistical significance of asymmetry in a detrended US pig-iron production series using a test implied by symmetry theorems in the paper and finds that the symmetry hypothesis can be rejected with a high degree of confidence. He concludes that the asymmetry between the ascending and descending phases of the cycle is one of the most obvious and pervasive facts about the entire phenomenon, and that one would have to be a statistician or someone very prejudiced in favour of Frisch-type modelling to demand explicit proof of the statistical significance of the obvious.

Neftci (1984) also examines the asymmetry of economic time series over the business cycle. Using unemployment series, which have no marked trend, he adopts the statistical theory of finite-state Markov processes to investigate whether the correlation properties of the series differ across phases of the cycle. He notes that the proposition that econometric time series are asymmetric over different phases of the business cycle appears in a number of major works on business cycles. Neftci presents a chart showing that the increases in US unemployment have been much sharper than the declines in the 1960s and 1970s and his statistical tests, which compare the sample evidence of consecutive declines and consecutive increases in the time series, offer evidence in favour of the asymmetric behaviour of the unemployment series analysed in the paper.

Neftci (1984) then discusses the implications of asymmetry in macroeconomic time series for econometric modelling. Firstly, in the presence of asymmetry the probabilistic structure of the series will be different during upswings and downswings and the models employed should reflect this by incorporating nonlinearities to allow 'switches' in optimising behaviour between phases. Secondly, although the implication is that nonlinear econometric or time series models should be employed, it may be possible to approximate these models, which are cumbersome to estimate, with linear models in which the innovations have asymmetric densities. Further work is required to verify this conclusion, he notes.

De Long and Summers (1986a) also investigate the proposition of business cycle asymmetry. They note that neither the econometric models built in the spirit of the Cowles Commission nor the modern time series vector autoregressive (VAR) models are entirely able to capture cyclical asymmetries. Consequently, they argue, if asymmetry is fundamentally important then standard linear stochastic techniques are deficient and the NBER-type traditional business cycle analysis may be a necessary component of empirical business cycle analysis. The question of asymmetry is therefore one of substantial methodological importance.

De Long and Summers undertake a more comprehensive study than Neftci (1984) using pre- and post-war US data and post-war data from five other OECD nations. They find no evidence of asymmetry in the GNP and industrial production series. For the United States only, like Neftci (1984) they find some asymmetry in the unemployment series. They conclude that asymmetry is probably not a phenomenon of first order importance in understanding business cycles.

De Long and Summers observe that the asymmetry proposition amounts to the assertion that downturns are brief and severe relative to trend and upturns are larger and more gradual. This implies that there should be significant skewness in a frequency distribution of periodic growth rates of output. They therefore calculate the coefficient of skewness," which should be zero for symmetric series, for the various time series. Overall they find little evidence of skewness in the US data. In the pre-war period they find slight positive skewness, which implies a rapid upswing and a slow downswing, the opposite of what is normally proposed. In the post-war period there is some evidence of the proposed negative skewness and in the case of annual GNP the negative skewness approaches statistical significance. Turning to data from other OECD countries, they find that skewness is only notably negative in Canada and Japan. There is no significant evidence of asymmetry in the United Kingdom, France or Germany.

De Long and Summers argue that the picture of recessions as short violent interruptions of the process of economic growth is the result of the way in which economic data is frequently analysed. The fact that NBER reference cycles display contractions that are shorter than expansions is a statistical artifact, they assert, resulting from the superposition of the business cycle upon an economic growth trend. The result is that only the most severe portions of the declines relative to trend will appear as absolute declines and thus as reference cycle contractions.21 Consequently, they argue, even a symmetric cycle superimposed upon a rising trend would generate reference cycles with recessions that were short and severe relative to trend even though the growth cycles (the cycle in detrended series) would be symmetric. Comparing the differences in length of expansions and contractions for nine post-war US NBER growth cycles they find them not to be statistically significant, in contrast to a similar comparison of seven NBER reference cycles. They conclude that once one has taken proper account of trend, using either a skewness-based approach or the NBER growth cycle dating procedure, little evidence remains of cyclical asymmetry in the behaviour of output. This of course assumes that detrending does not distort the cycle so derived and that the trend and growth are separable phenomena.

De Long and Summers finally turn to Neftci's (1984) findings for US unemployment series, which contradict their results. They argue that Neftci's statistical procedure is inadequate and proceed to estimate the skewness in US post-war unemployment data. They discover significant negative skewness and are unable to accept the null hypothesis of symmetry. None of the unemployment series from other OECD countries displayed significant negative skewness, however. They are therefore able to argue that it reflects special features of the US labour market and is not a strong general feature of business cycles.

De Long and Summers are, as a result, able to conclude that it is a reasonable first approximation to model business cycles as symmetrical oscillations around a rising trend and that the linear stochastic econometric and time series models are an appropriate tool for empirical analysis. They consequently call into question at least one possible justification for using NBER reference cycles to study macroeconomic fluctuations. They note that an alternative justification for the reference cycle approach stresses the commonality of the patterns of comovements (section 1.1) in variables across different cycles and that Blanchard and Watson (1986) challenge this proposition (see section 1.5).

Within the context of an assessment of NBER methodology, Neftci (1986) considers whether there is a well-defined average or reference cycle and whether or not it is asymmetric. His approach is to confront the main assertions of the NBER methodology, discussed in the previous section, with the tools of time series analysis; these imply that NBER methodology will have nothing to offer beyond the tools of conventional time series if covariance-stationarity is approximately valid and if (log) linear models are considered. If covariance-stationarity and/or linearity does not hold, the NBER methodology may have something to contribute if it indirectly captures any nonlinear behaviour in the economic time series.

From each time series under consideration Neftci derives for the local maxima and minima of each cycle, which measure implied amplitudes, and the length of the expansionary and contractionary phases. This data, he argues, should contain all the information required for a quantitative measure of NBER methodology. Neftci first examines correlations between the phase length and maxima and minima and then between these variables and major macroeconomic variables. If the length of a stage is important in explaining the length of subsequent stages then the phase processes should be autocorrelated and the NBER methodology would, by implication, potentially capture aspects of cyclical phenomena that conventional econometrics does not account for. To investigate such propositions Neftci uses an updated version of Burns and Mitchell's (1946) pig-iron series.

Neftci finds that the length of the upturn does affect the length of the subsequent downturn significantly but that the length of past downturns does not affect the length of subsequent upturns. Using the series for local maxima and minima, Neftci examines the relationship between the drop and the increase during upswings and finds a significant relationship between the two. Again the result is unidirectional because he finds that the size of the upswing has no effect on the subsequent drop. Introducing the paper, Brunner and Meltzer (1986) note that the latter result confirms the important finding described by Milton Friedman in the 44th Annual Report of the NBER and that the unidirectional correlations run in opposite directions for the lengths series and the drop and increase series (which might imply stationarity; see Rotemberg 1986).

Neftci regards the results as tentative, given the small numbers of observations employed, but nevertheless concludes that sufficient information apparently exists in the series derived to represent NBER methodology to warrant investigating the information more systematically. To do this Neftci defines a new variable that can express the state of the current business cycle without prior processing of the data. This is done to avoid the possibility that selecting the turning points after observing the realisation of a time series will bias any estimation procedures in favour of the hypothesis that the reference cycle contains useful additional information not reflected in the time series, or, as Neftci puts it, a cyclical time unit exists separately and independently of calendar time.

The variable introduced is a counting process whose value at any time indicates the number of periods lapsed since the last turning point if the time series exhibits strong cyclically but no trend. When a positive trend is present, however, the variable will be a forty-five degree line and when the series is strongly asymmetric, with large jumps being followed by gradual declines, then the variable will have a negative trend with occasional upward movements. It can, therefore, capture some of the nonlinear characteristics of the series. Counting variables were derived from various macroeconomic time series and included in vector and univariate autoregressions. The major findings from the vector autoregressions were the following. The counting variable significantly affects the rate of unemployment in all cases. It shows little feedback into nominal variables such as prices and money supply. The fact that the counting variable helps explain the variation in unemployment, which has no trend, implies that information about the stage of the cycle reflected in the variable in the absence of trend - carries useful additional information. Since the counting variable is a nonlinear transformation of the unemployment series, the implication is that the NBER methodology may capture some nonlinear stochastic properties of the economic time series which are unexploited in the standard linear stochastic framework. The univariate autoregressions for major macroeconomic time series included lagged values of the counting variable and a time trend. For most of the macroeconomic variables the counting variable was significant and in many cases strongly so.

Neftci then considers the reasons for the significance of his findings that cyclical time units carry useful additional information. The first possibility he identifies is that turning points may occur suddenly and it may be important for economic agents to discover these sudden occurrences (Neftci 1982). The second is that the derivative of the observed processes has different (absolute) magnitudes before and after turning points. In other words, there is asymmetry as discovered by Neftci (1984) but disputed by De Long and Summers (1986a) (see discussion above). Thirdly, the notion of trend may be more complex than usually assumed in econometric analysis. It may for example be non-deterministic (see section 4.3); consequently it may be useful to work with cyclical time units rather than standard calendar time. From a different perspective one could argue that the stage of the business cycle may explicitly enter into a firm's or even a consumer's decision-making process.23 If cyclical time unit, or average or reference cycle, can be consistently defined and successfully detected, then macroeconomic time series can be transformed to eliminate business cycles and highlight any remaining periodicity, or long cycles, in the trend component (see section 4.3).

The phase-averaging of data employed by Friedman and Schwartz (1982) and criticized by Hendry and Ericsson (1983) is a procedure that uses a cyclical time unit. Phase-averaging entails splitting a time series into a number of consecutive business cycles after a visual inspection of a chart of the series. The time series are then averaged over the selected phases of the cycle and the behaviour of the process during a phase is replaced by the average. Usually only the expansionary and contractionary phases are selected; consequently the whole cycle will be replaced by two points of observation. (See Neftci 1986, p.40, for a formal discussion.) The procedure effectively converts calendar time data into cyclical time unit observations. Following Hendry and Ericsson (1983), Neftci concurs that if a traditional linear stochastic econometric model with a possibly nonlinear trend is the correct model, then the application of phase-averaging, which is like applying two complicated nonlinear filters that eliminate data points and entail a loss of information, would be inappropriate, even if there was a cyclical time unit. Consequently, phase-averaging can be justified only if a linear econometric model is missing aspects of the cyclical phenomena which, if included, would provide some justification of phase-averaging. Neftci (1986) notes that users of phase-averaging24 would reject the insertion of deterministic, rather than stochastic, trends in linear econometric models. In fact, Neftci argues, phase-averaging can be seen as a method of using the cyclical time unit to isolate a stochastic trend in economic time series.

Neftci concludes that the introduction of the counting variable, which effectively involves a nonlinear transformation of the data, improves explanatory power and indicates that this was the result of the presence of (stochastic) nonlinearities. It therefore appears that nonlinear time series analysis will contribute to future analysis of the business cycle. Commenting on Neftci (1986), Rotemberg (1986) expresses concern about the general applicability of Neftci's procedure for identifying the stochastic trend. In series with trends where a growth cycle is present it is difficult to date local maxima and minima without first detrending, as the NBER has discovered in the post-war period. One possible way round the problem, he suggests, is to use series without trends, such as unemployment, to date the peaks and troughs and then use these dates to obtain phase-averages in other series. Since the timing of peaks and troughs in different series will vary stochastically, it would be important to analyse 'clusters' of peaks and troughs in detrended series to arrive at appropriate dates.

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