International Business Cycle Asymmetry and Time Irreversible Nonlinearities

Abstract

Using tests of time reversibility, this paper provides further statistical evidence on the long-standing conjecture in economics concerning the potentially asymmetric behaviour of output over the expansionary and contractionary phases of the business cycle. A particular advantage of this approach is that it provides a discriminating test that is instructive as to whether any asymmetries detected are due to asymmetric shocks to a linear model, or an underlying non-linear model with symmetric shocks, and in the latter case is informative as to the potential form of that nonlinear model. Using a long span of international per capita output growth data, the asymmetry detected is overwhelmingly consistent with the long standing perception that the output business cycle is characterized by steeper recessions and longer more gentle expansions, but the evidence for this form of business cycle asymmetry is weaker in the data adjusted for the influence of outliers associated with wars and other extreme events. Statistically significant time irreversibility is reported for the output growth rates of almost all of the countries considered in the full sample data, and there is evidence that this time irreversibility is of a form implying an underlying nonlinear model with symmetrically distributed innovations for 15 of the 22 countries considered. However, the time irreversibility test results for the outlier-trimmed full sample data reveal significant time irreversibility in output growth for around one half of the countries considered, predominantly in Northern Europe and North America, and of a form implying a nonlinear underlying model in only a further half of those cases.

Keywords:

• Time reversibility
• time irreversibility
• nonlinearity
• per capita output growth.

Notes

1. See Psaradakis (2000) for discussion of a number of commonly employed tests of asymmetry and nonlinearity.

2. For reviews of the earlier elements of this literature see Mittnik & Niu (1994) and Potter (1994).

3. For criticisms of and alternatives to the HP filter see, for example, Nelson & Kang (1981), Harvey & Jaeger (1993), Cogley & Nason (1995), Kaiser & Maravall (2000), Psaradakis & Sola (2003) and Christiano & Fitzgerald (2003)

4. Ramsey & Rothman (1996) offer the time path of a round projectile in (windless) flight as an intuitive example of a time reversible process, and the dispersal of ink in water as an intuitive example of a time irreversible process. The diffusion of technology provides an obvious further example of a time irreversible economic process.

5. See Mittnik & Niu (1994) and Psaradakis (2000) for further discussion of a number of commonly employed tests of asymmetry.

6. Note that for i = j = 1 this expression reduces to the statement that the autocovariance of a stationary series at lag k is necessarily equal to itself, reflecting the fact that the simple autocovariance function provides no relevant information concerning time reversibility. However, where one of the powers i, j is greater than one, the cross moment terms in these expression are ‘generalised autocovariances’ (Welsh & Jernigan, 1983), and the existence of a lag k for which these generalised covariances moments are not equal then provides a sufficient condition for time irreversibility.

7. For further details, see Theorem 2 of Ramsey & Rothman (1996).

8. Under this latter approach an estimate of the variance of the variance of γˆ₂, 1 (k) is calculated by fitting a linear autoregressive (AR) model to the data, obtaining an estimate of the innovations variance, and then simulating a series using the estimated AR coefficients and generating a Gaussian error process with zero mean and variance equal to that estimated in the preceding stage. Values of γˆ₂, 1 (k) are calculated for each such replication for N replications, where N = 100, permitting straightforward computation of the estimated variance using the replicated values for γˆ₂, 1 (k). If the process is truly linear Gaussian, and time reversible, this is an exact simulation procedure. If the series is truly nonlinear (Type I time irreversible), the linear model constitutes a local approximation to the unknown nonlinear model, but the procedure should nonetheless provide asymptotically unbiased estimates of the variance of γˆ₂, 1 (k) in the presence of uncorrelated innovations.

9. Note that it is a requirement of the TR test statistic that the data possess a finite sixth moment. Chen et al. (2000) have proposed an alternative to the TR test statistic which does not have any moment restrictions. However, as Chen et al. note, their test is not directly applicable to model residuals because it is a test of unconditional symmetry, and cannot therefore be used in order to discriminate between Type I and Type II time irreversibility.

10. Note that whilst Type I time irreversibility implies nonlinearity, the converse is not necessarily true, since there exist stationary nonlinear processes that are time reversible (e.g. Lewis et al., 1989). The TR test cannot therefore be considered as equivalent to a test for nonlinearity of unknown form. For more detailed discussion of this test procedure and its rationale, see Ramsey & Rothman (1996). On the uses and possible limitations of the TR test statistic as a guide to the model specification tool in application to nonlinear conditional mean and conditional variance model residuals, see Rothman (1999) and Belaire-Franch & Contreras (2002, 2003), and for a frequency domain variant of the TR test statistic based upon the bispectrum, see Hinich & Rothman (1998).

11. For further details of the dataset construction, which is based on the data available in Maddison (2001a, 2001b), as well as access to the data, see http://www.ggdc.net/EMaddison.

12. Previous vintages of this data set, covering the periods up to 1987 and 1994, have been considered in the context of the empirical assessment of theories of economic growth and convergence by Bernard & Durlauf (1995), Ben-David & Papell (1995), Jones (1995), Crafts & Mills (1996) and Mills & Crafts (2000). Mills & Crafts (2000), in particular, provide a detailed examination of the time series properties of the data up to 1994, and the interested reader is referred to them for further information.

13. Formal statistical tests concerning the (non-)stationarity of the data using conventional augmented Dickey–Fuller statistics, which confirm the appropriateness of this transformation for all of the series under investigation, are omitted here in the interests of conserving space but are available on request from the authors.

14. For further discussion of the effects of temporal aggregation on asymmetric time series dynamics see, for example, Bränäs & Ohlsson (1999).

15. Outlier adjustment for the observations identified by the means described in the text are conducted by the standard method of regressing the growth rate on a constant and dummy variables for the observations in question, and retrieving the regression residuals. Further details of the dates of outlier observations for each country are omitted here in the interests of conserving space, but are available on request from the authors.

16. Recall from the discussion in the third section that an appropriate linear specification is necessary in order for the significance of the standardized test statistic results to be calculated and the alternative hypothesis of time irreversibility to be tested against the null of time reversibility, as well as in discriminating between Type I and Type II time-irreversibility. In particular, the results reported employ an estimate of the variance of γˆ₂, 1 (k) calculated by Monte Carlo simulation using a linear autoregressive AR(p) model fitted to the data of order p determined by reference to the Akaike Information Criterion (AIC). Alternative results based on application of the Schwarz (Bayesian) Information Criterion (BIC), which provide qualitatively equivalent results are omitted here in the interest of conserving space, but are available from the authors on request.

17. Whilst the choice of K is essentially arbitrary, we select K = 5 following Ramsey & Rothman (1996), who argue that this is an appropriate value given the degrees of freedom typically available for economic time series. In the results of robustness exercises (not reported here), we found that adopting alternative values of K made no qualitative and little quantitative difference to the results reported in the fourth section.

18. These findings are broadly consistent with the results reported in Mills (2001), and particularly in the sense that once the presence of outliers in long run international output data has been accounted for, the evidence in favour of statistically significant cyclical asymmetry is reduced.

19. Commensurately, there would appear to be little justification for the application to long-run annual output data of the range of nonlinear models that have become popular amongst applied econometricians in recent years for the modelling of higher frequency business cycle indicators, such as quarterly industrial production and monthly unemployment data.

20. For the post-1950 period alone, there are yet fewer instances of test statistic significance, though the much smaller sample of annual observations must be borne in mind in appraising these results, with the output growth of less than one quarter of the countries being characterized by time irreversibility, and implying an underlying nonlinear model in two cases. For the vast majority of countries, the post-war evidence, whilst limited, is therefore suggestive of full time reversibility and the absence of any significant cyclical asymmetries or asymmetric dynamics.