Heteroscedasticity and Multivariate Volatility

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.