Stylized Facts of Business Cycles in the OECD Countries

Transcription

1 European Regional Science Association 36th European Congress ETH Zurich, Switzerland August 1996 Jordi Pons, Ernest Pons and Jordi Suriñach Department of Econometrics, Statistics and Spanish Economy University of Barcelona Spain Stylized Facts of Business Cycles in the OECD Countries ABSTRACT: This paper investigates the basic stylized facts of business cyles in the OECD countries using quarterly data from 1970 to In the last years the study of business cycles, on both theoretical and empirical levels, has been again in the forefront of research in economics. The purpose of this paper is to determine if the OECD business cycles are indeed in the terminology of Lucas (1977). Lucas concluded that because the coherence and phase characteristics of many economic time series appeared to be the same across countries, business cycles are all alike. The methodology used is based on spectral analysis. This approach attempts to identify cycles in the frequency domain. Spectral analysis provides direct and relevant information about leads and lags between economic time series and determines if a substantial degree of coherence at a particular frequency or over a band of frequencies does indeed exist. Key words. business-cycle, spectral-analysis, frequency-domain.

2 I. Introduction The stylized facts of business cycles were in the forefront of research in macroeconomics in the first half of the twentieth century. Leading article of this literature is the work of Burns and Mitchell (1946) 1. The seminal contribution of Burns and Mitchell was influential because it provided a comprehensive catalogue of the empirical features of the business cycles of developed countries, notable, the United States. In recent years, most industrial countries experienced pronounced cyclical behaviour in an important number of economic indicators. Since the late sixties and early seventies the study of cycles has experienced a regeneration of research effort, on both theoretical and empirical levels. All economies experience recurrent fluctuations in economic activity that persist for periods of several quarters to several years. Further, there is a definite tendency for the business cycles of developed countries to move together. The challenge to theory is to develop consistent explanation for these phenomena. On the theoretical level, renewed interest in business cycle theory stems from an important article by Lucas (1977). Lucas concluded that business cycles are all alike, because the coherence and phase characteristics of many economic time series appeared to be the same across countries. In an attempt to explain this apparent stylized fact, there has been a proliferation of theoretical business cycle models. Lucas drew attention to a key business-cycle fact: outputs of broadly-defined sectors move together. Some important articles in this area are by Kydland and Prescott (1982, 1990 and 1991) and Long and Plosser (1983). The empirical studies of business cycles have been underpinned by two alternatives methodologies. The first approach develops the original methodology of Burns and Mitchell (1946). This method consists of identifying indicators and classifying these indicators as leading, coincident and lagging by examining the timing and consistency of the turning points of important economic time series. The second approach attempts to identify cycles in the frequency domain. This approach consist of characterising business cycles identifying the presence of peaks in the spectrum and determinying if a substantial degree of coherence at 1 Surveys of more recent work on this field are Zarnowitz (1992) and Niemira and Klein (1994). 1

3 either a particular frequency or over a band of frequencies does indeed exist (see Layton, 1986 and Martin,1987 and 1990). Perhaps the major limitation of these approaches is that they focus on either time or frequency domain characteristics, but not on both 2. In particular, the purpose of this paper is to investigate the basic stylized facts of business cycles in the OECD countries using GDP quarterly data from 1970 to 1994, applying the analysis in time and frequency domain. The present paper contribute to this literature by applying a methodology based on spectral analysis. This approach attempts to identify cycles in the frequency domain. Spectral analysis provides direct and relevant information about leads and lags between countries and determines if a substantial degree of coherence at a particular frequency or over a band of frequencies does indeed exist. The series used are the GDP of OECD, Europe, European Union, Australia, Canada, France, Germany, Italy, Japan, Norway, Spain, Switzerland, United Kingdom and United States 3. The empirical focus of the paper is on isolating cyclic fluctuations in economic time series, defined as cycles in the data between specified frequency bands. The scheme of the paper is as follows. Section II summarises the methodology of the spectral analysis. Section III presents and discusses the selected stylized facts. Section IV presents the leads and lags between OECD contries. The main conclusions of the paper are presented in Section V. II. Methodology The spectral analysis describes the cycle in terms of a frequency and amplitude. The frequency is defined as the inverse of the cycle lenght, whereas amplitude is the range between peak and trough values. Granger and Hatanaka (1964) and Priestley (1981) argue that the series must be stationary; the mean and variance of the series must remain constant over 2 See Bowden and Martin (1992 and 1994). 3 Each series is tested for a stochastic trend by using the augmented Dickey-Fuller and Phillips-Perron unit root tests. The results show that the null hypothesis of non-stationarity cannot be rejected. 2

4 time. If the series are not stationary, as most economic time series tend not to be, a first or higher difference of the series would be necessary until the differenced series meet the criterion of a stationary mean and variance. To determine the lead or lag between pairs of economic indicators, two spectral statistics are used: coherence and phase. Coherence measures the proportion of variance explained by one of the series at a given frequency of the second series. This measure can take a value between 0 and 1; the concept is similar to the square of the correlation coefficient from a regression. Phase measures the time difference between two series in the frequency domain 4. Using a Fourier transform, one can express a stationary time series as a cyclical components of different frequencies. The spectrum of the time series which decomposes the series s total variance into variance attributed to different frequencies. One can interpret the spectrum as a density function. The area under the spectrum for an interval between two frequencies equals the proportion of total variance attributed to components with frequencies within the interval. We will also consider the coherence and phase between the GDP series of a country and each of the other country. It is not meaningful to consider spectra, coherence and phase for series that are not stationary. We will refer to the statistical procedure used for rendering a time series stationary as detrending. Given that we do not have a particular theoretical model in mind, and given the weak power of most tests for stochastic versus deterministic trends, we prefer to take an agnostic view towards detrending. Therefore, it s possible to transform the raw economic series into stationary series in two different ways. One way to remove a trend from the data is simply to take the rates of growth. Another way is to use the filter previously used by Hodrick and Prescott (1980) and many others (application of two-sided moving averages, first-differencing and removal of linear or quadratic time trends), known as the Whittaker-Henderson type filter. Finally, we apply the 4 In the frequency domain, Sargent (1987, p. 282) offers the following update of Burns and Mitchell s definition: "(...) the business cycle is the phenomenon of a number of important economic aggregates (such as GNP, unemployment, and layoffs) being characterized by high pairwise coherences at the low business cycle frequencies". 3

5 rates of growth 5. Many recent studies using a battery of such methods to measure business cycles. III. Stylized facts An appropriate starting point in an investigation of business cycles, is to determine if a substantial degree of coherence at either particular frequency or over a band or frequencies does indeed exist. We adopt a traditional a priori definition of business cycles, namely cyclical commovements between important macroeconomic variables with periods of around five years (Lucas, 1977). Defining the business cycle by frequency of fluctuations, it becomes natural to exploit spectral analysis. The spectra for the each series (rates of growth) are shown in figure 1. We have previously talked about the business cycle as fluctuations with periods around five years. We make this more precise by focusing on cycles with periods between six and thirty two quarters 6. The interval between six and thirty two quarters is marked by vertical lines in the spectra (the line corresponding to thirty two quarters is the left one, since the period (frequency) is decreasing (increasing) to the right). We specified that business cycles were cyclical components of no less than six quarters (eighteen months) in duration and fewer that thirty two quarters (eight years). Table 1 and figure 1 show that most of the spectral mass for all series is between six and thirty two quarters. We conclude that there is indeed some empirical support for business cycles periods between six and thirty two quarters: most of the series have considerable spectral mass in the corresponding frequency band. 5 Because different filters have different transfer functions (that is, they pass through cyclical components at differents frequencies to different degrees) choosing a filter is equivalent to choosing which commovements to emphasize. The application of the rates of growth has two effects. A first effect is to reduce the fluctuations in all series and a second effect is to induce a more regular cyclical pattern with larger number of distinguishable cycles. 6 This definition of the business cycle was suggested by the procedures and findings of NBER researchers like Burns and Mitchell (1946). See Baxter and King (1995) for illuminating surveys. 4

6 Looking at the coherences in table 2 and figure 2, we note that there is relatively high coherence between OECD and most countries for periods longer that six quarters. Finally, table 3 shows the commovements of the different countries, as measured by their correlation with OECD 7. This table displays the correlation coefficient between OECD and each other country lagged from one to five quarters, contemporaneous, and led from one to five quarters. Most of the correlation coefficients are positive, indicating commovements between the countries. For most countries, the contemporaneous correlation are the highest. Of these, the correlation coefficients for Europe and European Union are highest, followed by those for United States and Canada. The correlation coefficients for Norway are lower. IV. Leads and lags between OECD countries The present epigraph benefits from the tools providing with the spectral analysis of time series in order to obtain an estimation of the lag the economical evolution of some countries go through in comparison with some others. The estimation of the time lag by using the phase function presents an important problem since this function is not easy to be interpreted in practice. It is already known that a null and constant phase function corresponds to a contemporary relation, whereas a linear phase function with slope being equal to the d lag corresponds to lag relations of the X t =ay t-d type. Although, in general, the interpretation of the estimated phase function seems to be very difficult mainly because of various aspects: a) When the relation between two variables is more complex, the phase function can obtain a great deal of different forms depending moreover on the relation parameters. b) There is a certain indetermination in the phase function. When concerning an angle, it is not possible to distinguish a value from another being in time different from the former one in a certain number of entire rounds of a circumference. The series used for this paper also present those difficulties. Figure 3 shows the phase function estimated between the OECD variable (the total OECD GNP rates of growth of a 7 The rates of growth of that series are stationary. 5

7 quarter with respect to the same quarter of the former year) and the EUR variable (referred to the whole OECD European countries) 8. Figure 3. Phase between OECD and Europe. Figure 4. Phase between Germany and Spain. The fact that this phase function is so close to the value 0 refers to the almost equality of both series being then their relation contemporary. A statistical test based on the phase function as it follows below can confirm that the relation between two variables is contemporary. By using a spectral window of convenient properties, an estimation of the individual and cross spectral densities of both variables can be obtained. Noting each variable for X and Y and the estimated spectral densities for f X,f Y and f XY, consistent estimators of the coherence and the phase function being: ˆκ XY (ω) f XY (ω) f X (ω)f Y (ω) 1/2 (1) and ˆφ XY (ω) tan 1 Im f XY (ω) Re f XY (ω) (2) considering that the phase angle must be taken in the [-π,π] interval. Whenever the coherence is strictly positive, the phase function will asymptotically spread itself as a normal one: 8 The phase function is expressed in sexagesimal degrees to highlight that we are dealing with angles. 6

8 ˆφ XY (ω) AN φ XY (ω), n ˆα 2 XY(ω) 2 a ˆκ XY (ω) 2 (3) where a m 2 depends on the W k ponderations and the m width of the spectral window used: a 2 m k <m W 2 k and α XY (w) being the width of the cross spectral density between both variables. If the value of the coherence is replaced by expression (1), and α XY (w) by a consistent estimator: ˆα XY (ω) f XY (ω) (5) the null hypothesis can be contrasted considering that the phase function is null when comparing the following statistical value: Z XY (ω) ˆφ XY (ω) 2 ˆκ XY (ω) 2 a 2 m ˆα 2 XY(ω) ˆκ XY (ω) 2 1 (6) with the table value of a normal distribution. Figure 4 shows when the null hypothesis of a contemporary relation can be refused by using this test. When using this contrast in the case of the OECD and Europe, the null hypothesis saying that the relation is contemporary cannot be rejected. But when comparing other variables, the interpretation might be less clear 9. Figure 4 presents the phase estimated between the quarter evolution of Germany and Spain. Apparently, in this figure the relation between both variables can no longer be accepted as contemporary. Precisely, when implementing the former test both evolutions are refused to be simultaneous. Through the sign of the phase function, it can also be deduced that the German evolution leads the Spanish one. And then another problem must be faced. How to know wether this lag consists of one, two or even more quarters 9 Table 4 shows countries in which the null hypothesis of a contemporary relation can be rejected. 7

9 Figure 5. Phase between OECD(+1) and Europe. Figure 6. Phase between OECD(-1) and Europe. When a phase function is linear, the interpretation is then clear. But focusing on our case, neither the estimated phase functions are linear (both for the case of those countries and for other variables), nor has it sense to think that the relations between the variables seem so simple to present such a clear lag. Anyhow, it is worthwhile asking oneself which the dominant lag is. In most cases, the phase function can give an answer to this question. As far as the OECD and Europe comparison is concerned, only ask ourselves what effect would leading or lagging one of both variables a quarter have in the estimated phase function before considering it. If the relation is mostly contemporary, then the new phase functions should respectively indicate a lag of 1 and -1. Figures 5 and 6 enclose these new phase functions. Both estimations cooroborate the already mentioned hypothesis in the sense that the relation between both variables has neither a lag nor a lead of a quarter. In order to complete the analysis, it would be also interesting to estimate the phase when a lag or a lead of two quarters of the OECD variables is previously imposed. Figures 7 and 8 show the new phase functions. Figure 7. Phase between OECD(+2) and Europe. Figure 8. Phase between OECD(-2) and Europe. 8

10 In the case of the OECD and Europe those movements allow the rejection of a likelihood two-quarter lag. In addition, it is clear that when moving one of both series more than two quarters, the phase functions will adopt behaviours every time further from a constant value; consequently the hipothesis is corroborated in the sense that the relation between both variables is contemporary. On the other hand, in the case of Germany and Spain those movements provide with some information. As it can be seen in Figure 9 where the variable enclosing the German evolution leads one quarter, then the phase function reaches the value 0 so that now a contrast of the phase function nullity does not allow the rejection of such hypothesis. If the DEU variable leads instead of lags a quarter (Figure 10), the phase function will then confirm that the lag between both evolutions has increased, becoming a realtion with a lag of two quarters. Figure 9. Phase between Germany(-1) and Spain. Figure 10. Phase between Germany(+1) and Spain. Even if all is basically about a graphical analysis, these previous movements of one of both compared variables can be combined with contrasts over the phase function to determine roughly the time lag a certain relation among variables dominates. This kind of analysis implemented in all possible paired variables 10 allows the construction of a time lag table as it is shown in Table 5. The results obtained from the frequency domain are not fully coherent. For example, it is deduced from the results that no lag exists neither between the OECD and Spain nor between the OECD and the USA. On the contrary when comparing the USA and Spain a lag of two quarters is obtained. This fact is influenced by various reasons: a) Having independently obtained each lag from the others. 10 From the fourteen variables used in the present research. 9

11 b) That the null values, more than being something evident of the contemporary relation, they enclose the impossibility of rejecting that null hypothesis. c) That the relations among variables are necessarily more complex and the determination of a whole number as lag supposes a large simplification. Consequently, the Table 4 results should be understood as simple explanations of a more complex reality. If all these factors are gathered as estimation errors it is then senseful trying to obtain a slightly different table also allowing the whole results to seem reasonable. Through the average of the results of the initial table of lags the present approach can be then obtained. This research follows this procedure: a) The series referred to the OECD total is used as a reference point. b) All variables related to the OECD are placed in each single row and then a lag related to OECD is obtained. c) As in general a different lag can be obtained according to the information of each row, all these lags are then averaged for each variable. Consequently, a final estimation of each variable lag related to the OECD is obtained. Figure 11. Estimated lag at frequency domain. Figure 12. Estimated lag at time domain. 10

12 d) The value 0 is then assigned to the most leading variable with respect to the OECD (this case being the USA) and the other lags have their importance diminished in relation to this variable. Consequently, an ordering of the whole countries is obtained according to the time lag they go through in relation to the most leading variable (USA) Figures 11 and 12 summarize this ordering and the lag in months of each variable. In one case (Figure 11) the lags obtained in the frequency domain have been used as a reference, whereas in another one (Figure 12) the lags obtained in the time domain. The following conclusions are then deduced from this average: a) Larger lags are obtained when using the time domain results instead of using the frequency domain ones. b) The order obtained by using both methodologies is very similar excepting in the case of Japan. The time domain obtains that Japan finds itself in phase with the USA, whereas the phase function obtains a significative lag between one and two months. This paper intends only to be a first approach to the research tackling which lags produce the different countries economical evolution. That is the reason why, a methodology has been briefly presented allowing the benefit from the tools of the time series spectral analysis in order to establish those lags, even if some aspects of this methodology can be criticized: a) The final result aims to obtain an estimation of the phase function, although this estimation does not have very clear aspects. Among them, for example, the choice of the spectral window to be used and its width. b) The phase function interpretation is not an immediate interpretation despite using the leading and lagging variables. In some variable combinations, it has been impossible to determine by using this analysis which is the prevailing lag between two countries. c) The obtained differences by using the time and frequency domains should catch the 11

13 researcher s eye so that in future researches these may allow to find the causes of those differences. V. Concluding remarks In this paper, one examines stylized facts in the frequency domain via the spectral density function, the Fourier transform of the autocovariance function. The spectral density matrix decomposes variation and covariation among variables by frequency, permitting one to concentrate on the periods of interest (business-cycle, for example, correspond to periods of roughly 6-32 quarters). Transformations of both the real and imaginary parts of the spectral density matrix have immediate interpretation in business-cycle analysis; the coherence between any two economics series effectively charts the strenght of their correlation by frequency, while the phase charts lead/lag relationships by frequency. We conclude that is indeed some empirical support for business cycles periods between six and thirty two quarters, because most of the series have considerable spectral mass in the corresponding frequency band. The commovement of the different countries, as measured by their correlation with OECD, indicating that the contemporaneous correlation coefficients are the highest. Of these, the correlation coefficients for Europe and European Union are highest, followed by those for United States and Canada. The correlation coefficients for Norway are lower. And finally, a methodology has been presented allowing to determine the existing lags in the economical evolution of the analyzed areas. It has been confirmed that the USA is the country having a more leading evolution with respect to the OECD s and Switzerland having a larger lag. References -Baxter, M. and King, R.G. (1995): Measuring business cycles: approximate band-pass filters for economic time series, National Bureau of Economic Research, Working Paper Bowden, R.J. and Martin, V.L. (1992): No, business cycles are not all alike: The United States and Australia compared, Australian Economic Papers, 31,

14 -Bowden, R.J. and Martin, V.L. (1994): International business cycles and financial integration, The Review of Economics and Statistics, 77, Burns, A.F. and Mitchell, W.C. (1946): Measuring business cycles, National Bureau of Economic Research, New York. -Diebold, F.X. and Rudebusch, G.D. (1996): Measuring business cycles: a modern perspective, The Review of Economics and Statistics, 78, Granger, C.W.J. and Hatanaka, M. (1964): Spectral analysis of economic time series, Princeton University Press, Princeton, New York. -Hodrick, R.J. and Prescott, E.C. (1980): Postwar U.S. business cycles: an empirical investigation, Carnegie-Mellon University, Discussion Paper, Kydland, F.E. and Prescott, E.C. (1982): Time to build and aggregate fluctuations, Econometrica, 50, Kydland, F.E. and Prescott, E.C. (1990): Business cycles: real facts and a monetary myth, Federal Reserve Bank of Minneapolis Quarterly Review, 14, Layton, A.P. (1986): A causality analysis of Australia s growth cycle and the composite index of leading indicators, Australian Economic Papers, 25, Long, J.B. and Plosser, C.I. (1983): Real business cycles, Journal of Political Economy, 91, Lucas, R.E. (1977): Understanding business cycles. In Brunner, K. and Meltzer, A.H. (1977) (eds.): Stabilisation of the domestic and international economy, Carnegie-Rochester Conference Series on Public Policy, 5, North Holland, Amsterdam. -Martin, V.L. (1987): Leads and lags in the Australian business cycle: a canonical approach in the frequency domain, Australian Economic Papers, 26, Martin, V.L. (1990): Derivation of a leading index for the United States using Kalman filters, The Review of Economics and Statistics, 72, Niemira, M.P. and Klein, P.A. (1994): Forecasting financial and economic cycles, Wiley, New York. -Priestley, M.B. (1981): Spectral analysis and time series, Academic Press, New York. -Sargent, T.J. (1987): Macroeconomic theory, 2nd edition, Academic Press, Boston. -Zarnowitz, V. (1992): Business cycles: Theory, history, indicators and forecasting, National Bureau of Economic Research, Ballinger Publishing Company, Cambridge. 13

15 Table 1. Variance attributed to different frequencies Country >32 quarters 6-32 quarters <6 quarters OECD Europe European Union Australia Canada France Germany Italy Japan Norway Spain Switzerland United Kingdom United States Table 2. Coherence estimated with OECD at different frequencies Country >32 quarters 6-32 quarters <6 quarters Europe European Union Australia Canada France , Germany Italy Japan Norway Spain Switzerland United Kingdom United States

16 Table 3. Cross-correlation with OECD GDP t-5 GDP t-4 GDP t-3 GDP t-2 GDP t-1 GDP t GDP t+1 GDP t+2 GDP t+3 GDP t+4 GDP t+5 Europe European Union Australia Canada France Germany Italy Japan Norway Spain Switzerland United Kingdom United States

17 Table 4. Countries in which a contemporary relation can be rejected. OECD Europe E. Union Australia Canada France Germany Italy Japan Norway Spain Switzerland U. Kingdom U. States OECD * * Europe * * * European Union * * Australia Canada * * * * * * France * * * * Germany * * * * * Italy * * * * * * * Japan * * * * * Norway * * * * Spain * * * * Switzerland * * * * * United Kingdom * * * * * * * * United States * * * * * * * * *

18 Table 5. Estimated lags and leads from GDP phase OECD Europe E. Union Australia Canada France Germany Italy Japan Norway Spain Switzerland U. Kingdom U. States OECD Europe European Union Australia Canada France Germany Italy Japan Norway Spain Switzerland United Kingdom United States

19 Figure 1. Estimated spectrum of quarterly GDP series Figure 1.1 Spectrum of OECD. Figure 1.2 Espectrum of Europe. Figure 1.3 Spectrum of European Union. Figure 1.4 Spectrum of Germany. Figure 1.5 Spectrum of France. Figure 1.6 Spectrum of Italy. 18

20 Figure 1 (continued) Figure 1.7 Spectrum of United Kingdom. Figure 1.8 Spectrum of Spain. Figure 1.9 Spectrum of United States. Figure 1.10 Spectrum of Canda. Figure 1.11 Spectrum of Japan. Figure 1.12 Spectrum of Australia. Figure 1.13 Spectrum of Norway. Figure 1.14 Spectrum of Switzerland. 19

21 Figure 2. Estimated coherences with OECD Figure 2.1 Coherence of OECD and Europe. Figure 2.2 Coherence of OECD and E. Union. Figure 2.3 Coherence of OECD and Germany. Figure 2.4 Coherence of OECD and France. Figure 2.5 Coherence of OECD and Italy. Figure 2.6 Coher. of OECD and Un. Kingdom. 20

22 Figure 2 (continued) Figure 2.7 Coherence of OECD and Spain. Figure 2.8 Coherence of OECD and USA. Figure 2.9 Coherence of OECD and Canada. Figure 2.10 Coherence of OECD and Japan. Figure 2.11 Coherence of OECD and Australia. Figure 2.12 Coherence of OECD and Norway. Figure 2.13 Coh. of OECD and Switzerland. 21