Vous consultezVolatility dynamics of the UK business cycle: A multivariate asymmetric garch approach

AuteursKin-Yip Ho du même auteur

1 - Introduction

Two recurrent themes in macroeconomics are the existence of asymmetries or non-linearities in business cycles and the co-movements of macroeconomic variables. Knowing whether business cycles are asymmetric, understanding how the aggregate variables behave and characterizing the conditional variations between the GDP and its components is important for both academics and policymakers. Indeed, as suggested by Diebold and Rudebusch (1996), a successful model for business cycles should not only consider the non-linear behaviour of aggregate variables but also their co-movements.

2 A voluminous literature has developed on output and macroeconomic volatility and their implications for consumption, growth, and welfare. Among others, see Lucas (1987), Schwert (1989), Kose et al. (2003), Ramey and Ramey (1995), Loayza et al. (2005), Calderon and Fuentes (2006), Dopke et al. (2006), Paye (2006), and McKibbin et al. (2008). There are also studies which employ calibrating models to explain the variances and co-variances of macroeconomic variables, such as real output (see Kydland and Prescott, 1982; Canning et al., 1988; Backus et al., 1992). For example, Head (1995) and Canning et al. (1998) find evidence of a negative relationship between the volatility of real output and the country size.

3 Most of the aforesaid papers concentrate on the unconditional volatility of macroeconomic variables and do not capture temporal variations over time. Asymmetries in volatility (unlike business cycle asymmetries) and sectoral variations conditional on the available information are often ignored in these studies. However, it is important to examine the temporal variations of macroeconomic variables for at least two reasons. First, a deeper understanding of the issues of conditional heteroskedasticity and volatility asymmetry of aggregate output is important because of their implications on macroeconomic theory and forecasting. If aggregate output is conditionally heteroskedastic and exhibits volatility asymmetry, then any theory assuming the absence of either of these properties is most likely inadequate (see Valderrama, 2001). Second, economic theory frequently suggests that economic agents respond not only to the mean conditional on the available information, but also to higher moments of economic random variables such as conditional variance (see Engle, 1982). Some articles attempt to capture the property of asymmetric conditional volatility of real output include French and Sichel (1993), Brunner (1993), Henry and Olekalns (2000), Hamori (2000), and Ho and Tsui (2003). In particular, French and Sichel (1993) have detected significant cyclical asymmetry in the volatility of US real GNP. Similarly, Henry and Olekalns (2000) have noted that output volatility is the highest when the economy is contracting. One major drawback of these studies is the failure to examine how the variables are correlated over time, a salient feature emphasised by business-cycle researchers.

4 In this paper, we propose a variety of multivariate asymmetric generalised autoregressive conditional heteroskedasticity (GARCH) models that are capable of capturing not only non-linearities in the conditional variance of macroeconomic variables but also their time-varying correlations. We shall apply these models to three sectoral Index of Industrial Production (IIP) of the United Kingdom. The reason is that, to the best of our knowledge, there are so far no studies on the asymmetric volatility of the UK aggregate output and the correlations among various sectors. Moreover, even though the business cycle has become less volatile in the UK during the past decades compared with previous post-war business cycles, the key unanswered questions remains its sources and how the various sectors are correlated. This study reveals that correlations among the sectors are time-varying, and negative shocks can generate higher volatilities in the future compared to positive shocks of the same magnitude in some sectors, even though this can be sensitive to the model specified. This finding has important implications for macroeconomic policy and forecasting for business cycle.

5 The rest of the paper is organised as follows. Section 2 discusses the methodology employed in this paper, whilst Section 3 focuses on the data sets and the empirical results. Concluding remarks are given in Section 4.

2 - Methodology

6 The problems of using multivariate GARCH models are that they not only increase the number of parameters to be estimated but also complicate the specifications of the conditional variance-covariance matrix. To circumvent these problems, Bollerslev (1990) introduced a multivariate GARCH model assuming constant correlations. Though this model satisfies the positive-definite condition for the variance-covariance matrix, its validity has been rejected in many contexts (Tsui and Yu, 1999). Furthermore, it fails to demonstrate how the correlations of the variables evolve over time. In addition, Bollerslev’s (1990) constant-correlation GARCH (CC-GARCH) model does not capture asymmetries in the conditional variance.

7 In what follows, we propose a family of multivariate GARCH models by synthesising the methodologies of Tse and Tsui’s (2002) varying-correlation GARCH (VC-GARCH)[2] [2] We are grateful to one referee for pointing out the similarity...
suite, Ding, Engle, and Granger’s (1993) asymmetric power ARCH (APARCH), and Sentana’s (1995) quadratic GARCH (QGARCH). Let yt=(ν1t, ν2t)’ be the bivariate vector of interest with time-varying covariance matrix Ht, and let μt(ξ) be the arbitrary mean functions which depend on ξ, a column vector of parameters. Consider a trivariate GARCH model:

8 Note that Φt –1 is the σ - algebra generated by all the available information up to time (t –1). The random disturbance terms εit and the conditional variance equations hiit are modelled as follows:

9 Denoting the ij -th element (i, j=1,2,3) in Ht by hijt, the conditional correlation coefficients are given by . Tse and Tsui (2002) assumes that the time-varying conditional correlation matrix Γt is generated by the following recursion:

10 To incorporate asymmetric volatility in the VC-GARCH model, we modify the conditional variance equation in (4). The modifications we demonstrate below have the common advantage that they are less restrictive and nest several popular GARCH models in the literature. Details can be found in Sentana (1995), and Ding, Engle, and Granger (1993). One modification advocated by Sentana is the quadratic GARCH (QGARCH) model:

11 Another modification to (4) is based on Ding, Engle, and Granger’s (1993) asymmetric power ARCH (APARCH) model:

3 - Data and results

12 Our data sets comprise the three main sectoral IIP (Index of Industrial Production) series of UK: Intermediate Goods (INT), Investment Goods (INV), and Manufacturing (MFC). These (seasonally adjusted) series are obtained on-line from OECD’s Main Economic Indicators and cover the period from January 1968 to December 2007 (480 observations).[3] [3] We have extended the datasets to December 2007 in order...
suite[4] [4] Thanks to one of the referees for pointing out the issue...
suite To calculate the monthly growth rates of these series on a continuously compounded basis, we take the first difference of the logarithmic IIP:

Table 1 - Summary statistics of the UK IIP growth rates, 1968-2007

13 Before delving into the estimation results, we employ the augmented Dickey-Fuller (ADF) and the Phillips-Perron (PP) tests to check the stationarity of all the IIP growth rate series. Our findings, available upon request, show that all ADF and PP test statistics are significant at the 1% level, thereby indicating that all the series are stationary.

14 Based on a number of factors such as parsimony and residual diagnostics, we adopt the AR(1) model for the conditional mean equation. We have also experimented with longer lags but results are similar to those reported here. Due to space limitations, we only report in table 2 the results for the conditional variance and correlations equations from the VC-GARCH, VC-QGARCH, VC-LOGGARCH, VC-TGARCH, and VC-APARCH models. In addition, table 2 also reports the log-likelihood values and correlation coefficients obtained from the corresponding constant-correlations models to facilitate a comparison between the VC and CC approach (the complete results are available from the authors upon request).

Table 2 - Estimation results of GARCH, QGARCH, LOGGARCH, TGARCH, and APARCH models

15 The results indicate the existence of varying correlations among the different sectors. In most cases, π1 and π2 are individually significant at the 5% level, and the likelihood ratio test suggests that the VC models outperform the restrictive CC models. The exception is the VC-LOGGARCH model, whose π2 is significant only at the 10% level. This finding is consistent with our casual observation that the lower volatility in both price and output and quicker recovery during the recent UK recession are essentially driven by private consumption and investment, in addition to the flexible regulation of labour, product and financial markets. The recent data indicate that the recovery taking place in the UK economy was initially fuelled by a sharp increase in durable consumption from the last quarter of 1992, and then a sharp pick up in residential investment from the second quarter of 1993. Using a vector-autoregression (VAR) approach, Catao and Ramaswamy (1996) also find that the recent recession in the UK was precipitated primarily by shocks to consumption and investment, and consumption shocks have a long lasting impact on the economy. Hence, consumption was expected to drive the recovery in the initial stages once these negative shocks dissipate away.

16 The log-likelihood values also indicate that the VC-TGARCH model outperforms the rest of the models. Even though the VC-APARCH model has a slightly higher log-likelihood value than VC-TGARCH, the difference is statistically insignificant, which can be shown using the likelihood ratio test. As noted by Nelson and Foster (1994), the TGARCH model is a consistent estimator of the conditional variance of near diffusion processes, and, in the presence of leptokurtic error distributions, the TGARCH model is a more efficient filter of the conditional variance than Bollerslev’s (1986) GARCH model.

17 As regards the existence of asymmetric volatility, the coefficient γ is significant at the 5% level for the Intermediate Goods (INT) series in the VC-QGARCH, VC-TGARCH, VC-LOGGARCH and VC-APARCH models. However, for the Manufacturing (MFC) sector, significant asymmetric volatility is detected only in the VC-LOGGARCH model.

18 On the diagnostics front, we employ a battery of diagnostic tests, such as the Ljung-Box Q-statistic and the runs test, to check for model adequacy (the results are available upon request). It is found that most Ljung-Box Q-statistics based on the cross product of the standardised residuals suggest the absence of serial correlation. However, the diagnostic results for the constant-correlation models are less favourable, as there is some evidence of serial correlation in the cross product of the standardised residuals.

19 Finally, the upper panels of figures 1 and 2 show that volatility is usually higher around the recessionary phase of the UK business cycle, such as the periods of 1974-1976 and 1979-1981. This accords with the view that economic downturns are periods of increased uncertainty and volatility. Furthermore, output volatility has declined in the 1980s and 1990s compared with the early 1970s, and this is consistent with the recent finding of Stock and Watson (2002). Undoubtedly the structural flexibility and the stabilization and growth policy implemented in the recent years have contributed to this low volatility of business cycle. On the other hand, as shown in the lower panels of figures 1 and 2, correlations are usually stronger when output volatility is higher. In passing, we note that this observation further reinforces the advantage of a multivariate GARCH model with time-varying correlations, because it would not be obvious in a constant- correlation framework.

Figure 1Conditional standard deviation and conditional correlations: VC-APARCH

Figure 2Conditional standard deviation and conditional correlations: VC-QGARCH

4 - Conclusion

20 We have employed a variety of multivariate asymmetric GARCH models with time-varying correlations to characterise the output volatility dynamics of three UK sectors. The results indicate the existence of asymmetric volatility, and it is sensitive to the structure of the conditional variance. The correlations and volatility of the variables are found usually higher around the recession phase of the UK economy. This finding is consistent with our casual observation that the lower volatility in both price and output and quicker recovery during the recent UK recession are essentially driven by private consumption and investment, in addition to the flexible regulation of labour, product and financial markets. It is indicated in the recent data that the recovery was initially fuelled by a sharp increase in durable consumption from the last quarter of 1992, followed up by a sharp pick up in residential investment from the second quarter of 1993. The findings of Catao and Ramaswamy (1996) also confirm that the recent recession in the UK was precipitated primarily by shocks to consumption and investment, and hence consumption was expected to drive the recovery in the initial stages once these negative shocks dissipate away.

21 The estimated conditional-correlation path provides an interesting time history that would otherwise be lost in a constant-correlation framework.

22 The finding of asymmetric volatility has important policy implications. When negative output shocks induce greater future volatilities, it may vindicate the government’s role in stabilising the macroeconomic environment during recessions. This is because the adverse impact of negative shocks could be mitigated through effective counter-cyclical measures. Furthermore, negative economic disturbances arising from one sector may spill over to another sector through strong sectoral linkages. As such, economic policy co-ordination becomes imperative to ameliorate the effect of shocks originating from one sector.

23 K.-Y. H., A. K. T. & Z. Z.[5] [5] The authors wish to thank two anonymous referees for their...
suite

Bibliographie

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Notes

[ 1] Corresponding author: Zhaoyong Zhang, Associate Professor of economics at Edith Cowan University, Australia ( zhaoyong.zhang@ecu.edu.au).
Kin-Yip Ho, Lecturer, Swinburne University of Technology, Australia; Albert K. Tsui, Associate Professor of econometrics, National University of Singapore.

[ 2] We are grateful to one referee for pointing out the similarity between the Dynamic Conditional Correlations (DCC) model proposed by Engle (2002) and the Time Varying-Correlations (VC) model proposed by Tse and Tsui (2002), both published in the same issue of Journal of Business and Economic Statistics. In terms of parameter estimation, VC models jointly estimate the conditional variance and correlation equations whereas DCC models separate the estimation of the conditional mean and variance equations by using a two-stage process. We have also attempted to estimate the 2-step DCC and noted that the parameter estimates are somewhat similar to those reported in our paper. This is unsurprising, since we are already able to obtain convergence for the 1-step procedure of the VCC. In the case of the DCC, we have to go through the extra step of modifying the standard errors of the conditional correlation parameters in order to ensure the reliability of our statistical inference (see Engle and Sheppard, 2001).

[ 3] We have extended the datasets to December 2007 in order to determine the robustness of our results. We did not include the more recent observations as they are subject to revision by OECD. In general, our original results still remain relatively robust.

[ 4] Thanks to one of the referees for pointing out the issue of deterministic and stochastic seasonality. Most of the OECD countries (including UK) employ X11 ARIMA or X12 ARIMA model to calculate seasonally adjusted series (see Hong and Chavoix and Mannato, 2000). In particular, the UK IIP series in our paper are seasonally adjusted according to the X11 ARIMA method (UK version), not the method of deterministic seasonality. As such, stochastic seasonality has already been taken care of. The possible presence of deterministic seasonality in the series can be removed by including dummy variables in the conditional mean equation (see Pierce, 1978). This procedure has been implemented separately and we notice that the parameter estimates are generally similar to those we have reported in our paper, which is also consistent with the asymptotic result of Nelson and Foster (1994) that in general the conditional mean specification does not have significant impact on the specification of the conditional variance. Studies such as French and Sichel (1993), Hamori (1998), and Ho and Tsui (2003) also use seasonally adjusted data from sources like OECD and IMF to model business cycle fluctuations. As noted by Sims (1990), it is usually true that rational expectations modelling in macroeconomics with seasonally adjusted data (treating the adjusted data as if it were actual data) gives approximately correct results, and naïve extensions of standard modelling techniques to seasonally unadjusted data may give worse results than naïve use of adjusted data.

[ 5] The authors wish to thank two anonymous referees for their very helpful comments and suggestions. The third author wishes to acknowledge the financial support of a strategic research grant from ECU.

Résumé

Cet article étudie la dynamique de la volatilité du cycle au Royaume-Uni. À cette fin, l’étude s’appuie sur quatre nouveaux modèles GARCH multivariés qui prennent en compte non seulement le caractère asymétrique de la volatilité, mais aussi les corrélations variant dans le temps. Les résultats montrent que la volatilité est asymétrique, mais sensible à la structure de la variance conditionnelle. Les corrélations et la volatilité sont en général plus élevées au cours de la phase de récession. Ces différents points ont des conséquences importantes en matière de politique macroéconomique et de prévision faites sur le cycle.
Classification JEL : E32 ; E37.

PLAN DE L'ARTICLE

• 1 - Introduction
• 2 - Methodology
• 3 - Data and results
• 4 - Conclusion

Kin-Yip Ho et al. « Volatility dynamics of the UK business cycle: A multivariate asymmetric garch approach », Economie internationale 1/2009 (n° 117), p. 31-46.
URL : www.cairn.info/revue-economie-internationale-2009-1-page-31.htm.