Abstract
In this chapter we develop a number of approaches to handle cointegration under variance that changes over the sample, in the first instance caused by volatility and or time varying heteroscedasticity. There is some debate in the literature about the influence of volatility on the Johansen estimator, but it is quite clear that the distribution towards which the conventional Johansen test tends is different under volatility (Cavaliere and Taylor 2008). And, although it is suggested that the test is asymptotically invariant to the presence of volatility, in the presence of volatility the tests on the cointegrating relations might suffer from significant size distortion in relatively large samples (Cavaliere and Taylor 2008), while Rahbek et al. (2002) are often construed to be suggesting that the impact of volatility on the Johansen estimator and test is innocuous. However, as the volatility becomes persistent, this may not be the case. More specifically the rate at which the test converges to the asymptotic distribution is sensitive to the spectral radius of the ARCH/GARCH polynomial matrices or the dimension of the largest eigenvalue. It has been observed that for a spectral radius in excess of 0.85 the simulated series exhibit quite extreme behaviour and for these types of numbers Rahbek et al. (2002) suggest that inference might only be appropriate when the sample lies in the range 600–1000. Furthermore, inference on the long-run parameters and loadings may also be affected by both inefficiency and inaccurate computation of the conventional Johansen estimator. It has also been shown by Seo (2007) that there can be significant distortion in the estimates of the long-run parameters caused by time varying variance structures.
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Notes
- 1.
Where ϱ{D, B} defines the largest eigenvalue of the ARCH and GARCH matrices represented by D and B respectively.
- 2.
Returns on the world stock index for all countries in the sample except the USA are represented by returns on the S&P 500 index. In the case of the USA, the world stock index is represented by the Morgan Stanley Capital International (MSCI) world (excluding the USA) index. The world oil price is represented by the West Texas Intermediate Cushing Crude Oil Spot Index, in US dollars per barrel. The data have been obtained from DataStream.
- 3.
The procedure was implemented with a convergence criterion of 0.00001, using the quasi-Newton method of Broyden, Fletcher, Goldfarb and Shanno, which does not require exact estimates of the matrix of second derivatives in contrast to the approach of Berndt, Hall, Hall and Hausman (see Sargan 1988).
- 4.
In this section the notation is intended to follow that of Johansen, which, when compared, is in contrast to the use of z t to define the exogenous variables in the conventional VECM in Chap. 5.
References
Al-Sadoon, M. M. (2015). A general theory of rank testing. Paper presented at the Econometrics Study Group Conference, Bristol.
Amano, R. A., & van Norden, S. (1998). Oil prices and the rise and fall of the US real exchange rate. Journal of International Money and Finance, 17, 299–316.
Bauwens, L., Deprins, D., & Vandeuren, J.-P. (1997). Bivariate Modelling of Interest Rates with a Cointegrated VAR-GARCH Model. Discussion Paper CORE, The Catholic University, Louvain La Nueve DP 9780.
Bauwens, L., Laurent, S., & Rombouts, J. V. K. (2006). Multivariate GARCH models: A survey. Journal of Applied Econometrics, 21, 79–109.
Bollerslev, T. P., & Wooldridge, J. M. (1992). Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews, 11, 143–172.
Boswijk, H. P. (1996). Cointegration, identification and exogeneity: Inference in structural error correction models. Journal of Business and Economics and Statistics, 14, 153–160.
Brooks, C. Burke, S.P. & and Persand, G. (2003). Multivariate GARCH models: software choice and estimation issues. Journal of Applied Econometrics 18, 725–734.
Burke, S. P., & Hunter, J. (2012). Arbitrage, market definition and monitoring a time series approach. SSRN Electronic Journal. Available at SSRN 2108398.
Caporale, G.-M., Hunter, J. & Menla-Ali, F. (2013). On the Linkages Between Stock Prices and Exchange Rates: Evidence from the Banking Crisis of 2007–2010. German Institute for Economic Research Discussion Paper Series 1289. DIW Berlin.
Cavaliere, G., Rahbek, A., & Taylor, A. M. R. (2010). Testing for co-integration in vector autoregressions with non-stationary volatility. Journal of Econometrics, 158, 7–24.
Cavaliere, G., Rahbek, A., & Taylor, A. M. R. (2012). Bootstrap determination of the co-integration rank in VAR models. Econometrica, 80, 1721–1740.
Cavaliere, G., & Taylor, A. M. R. (2008). Bootstrap unit root tests for time series with nonstationary volatility. Econometric Theory, 24, 43–71.
Davidson, R., & MacKinnon, J. G. (2004). Econometric theory methods. New York: Oxford University Press.
Engle, R. F., & Kroner, K. F. (1995). Multivariate simultaneous generalized ARCH. Econometric Theory, 11, 122–150.
Gregory, A. W., & Hansen, B. E. (1996). Residual-based tests for cointegration in models with regime shifts. Journal of Econometrics, 70, 99–126. Gregory, A. W. & Hansen, B. E. (1996). Residual-based tests for cointegration in models with regime shifts. Journal of Econometrics, 70, 99–126.
Hosking, J. R. M. (1981). Equivalent forms of the multivariate portmanteau statistic. Journal of the Royal Statistical Society, 43, 261–262.
Hunter, J. (1990). Cointegrating exogeneity. Economics Letters, 34, 33–35.
Hunter, J., & Burke, S. P. (2007). Common trends, cointegration and competitive price behaviour. In Royal Economic Society conference, Warwick.
Johansen, S. (1991). Estimation and hypothesis testing of cointegrating vectors in gaussian vector autoregressive models. Econometrica, 59, 1551–1580.
Johansen, S. (1992) Testing weak exogeneity and the order of cointegration in UK money demand data. Journal of Policy Modelling, 14, 313–334. Special Issue: Cointegration, Exogeneity and Policy Analysis.
Johansen, S. (1995). Likelihood-inference in cointegrated vector auto-regressive models. Oxford: Oxford University Press.
Kurita, T. (2008). Common stochastic trends and long run price leadership in the US gasoline market. Fukuoka University, Faculty of Economics WP-2008-003.
Li, W. K., Ling, S., & Wong, H. (2001). Estimation for partially nonstationary multivariate autoregressive models with conditional heteroscedasticity. Biometrika, 88, 1135–1152.
Ling, S. & Li, D. (2008). Asymptotic Inference for a Nonstationary Double AR(1) Model. Biometrika, 95, 257-263.
Ling, S., & McAleer, M. (2003). Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory, 19, 280–310.
Lütkepohl, H. (1996). Handbook of matrices. Wiley: Chichester.
Mosconi, R., & Giannini, C. (1992). Non-causality in cointegrated systems: Representation, estimation and testing. Oxford Bulletin of Economics and Statistics, 54, 399–417.
Phylaktis, K., & Ravazzolo, F. (2005). Stock prices and exchange rate dynamics. Journal of International Money and Finance, 24, 1031–1053.
Rahbek, A., Hansen, E., & Dennis, J. G. (2002). ARCH Innovations and Their Impact on Cointegration Rank Testing. Working Paper No. 22, Centre for Analytical Finance, University of Copenhagen.
Sargan, J. D. (1988). Lectures on advanced econometric theory. Oxford: Basil Blackwell.
Seo, B. (2007). Asymptotic distribution of the cointegrating vector estimator in error correction models with conditional heteroskedasticity. Journal of Econometrics, 137, 68–111.
Sims, C. (1980). Macroeconomics and reality. Econometrica, 48, 11–48.
Snell, A. (1999). A test of r versus r-1 cointegrating vectors. Journal of Econometrics, 88, 151–191.
Spliid, H. (1983). A fast estimation method for the vector auto-regressive moving average model with exogenous variables. Journal of the American Statistical Association, 78, 843–849.
Ülkü, N., & Demirci, E. (2012). Joint dynamics of foreign exchange and stock markets in emerging Europe. Journal of International Financial Markets, Institutions & Money, 22, 55–86.
Wong, H., Li, W. K., & Ling, S. (2005). Joint modeling of cointegration and conditional heteroscedasticity with applications. Annals of the Institute of Statistical Mathematics, 57, 83–103.
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Hunter, J., Burke, S.P., Canepa, A. (2017). Heteroscedasticity and Multivariate Volatility. In: Multivariate Modelling of Non-Stationary Economic Time Series. Palgrave Texts in Econometrics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-137-31303-4_7
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