Asymptotic Distribution of the Cointegrating Vector Estimator in Error Correction Models with Conditional Heteroskedasticity

The notion of cointegration was developed by Engle and Granger (1987), and since then has been considered important in the recent development of time series econometrics. Many statistical methods have been developed for the analysis of the cointegrated systems, and several methods of estimating the cointegrating vector have been proposed. Another development, generalized autoregressive conditional heteroskedasticity (GARCH), was made by Engle (1982) and Bollerslev (1986) to explain the time-varying volatility in the data. This paper explores the asymptotic properties of the maximum likelihood estimator (MLE) of the cointegrating vector in the vector error correction model with conditional heteroskedasticity. This paper is useful and required because the existing estimation methods do not consider conditional heteroskedasticity in the data. This paper considers the asymptotic properties of the maximum likelihood estimator of the cointegrating vector, which estimates the error correction model and the multivariate GARCH process jointly. The existing estimation methods, including the reduced rank regression (RRR) and the regression-based estimators, allow for, but do not treat explicitly conditional heteroskedasticity. Their asymptotic distributions are invariant to conditional heteroskedasticity. However, these estimators ignore the information coming from conditional heteroskedasticity. Many authors, including Bollerslev et al. (1992), show that economic variables such as stock prices and exchange rates have time-varying variances. The clustered volatility and thick tails are typical characteristics of these variables. Although there is vast literature on the cointegrating vector and GARCH, the distribution theory for the cointegrating vector in the error correction model with conditional heteroskedasticity has not been developed. This paper fills this gap in the literature by developing an asymptotic theory for the cointegrating vector estimator in error correction models with conditional heteroskedasticity. This paper finds that the asymptotic distribution of the MLE depends on the conditional heteroskedasticity and the kurtosis of standardized errors. The reduced rank regression (RRR) and the regression-based cointegrating vector estimators do not consider conditional heteroskedasticity in the data, and thus the MLE improves efficiency significantly. Inference on the cointegrating vector also depends on heteroskedasticity. The simulation results show that the efficiency of the MLE increases as the GARCH effect increases.