Abstract
This paper develops a generalized autoregressive conditional correlation (GARCC) model when the standardized residuals follow a random coefficient vector autoregressive process. As a multivariate generalization of the Tsay (1987, Journal of the American Statistical Association 82, 590–604) random coefficient autoregressive (RCA) model, the GARCC model provides a motivation for the conditional correlations to be time varying. GARCC is also more general than the Engle (2002, Journal of Business & Economic Statistics 20, 339–350) dynamic conditional correlation (DCC) and the Tse and Tsui (2002, Journal of Business & Economic Statistics 20, 351–362) varying conditional correlation (VCC) models and does not impose unduly restrictive conditions on the parameters of the DCC model. The structural properties of the GARCC model, specifically, the analytical forms of the regularity conditions, are derived, and the asymptotic theory is established. The Baba, Engle, Kraft, and Kroner (BEKK) model of Engle and Kroner (1995, Econometric Theory 11, 122–150) is demonstrated to be a special case of a multivariate RCA process. A likelihood ratio test is proposed for several special cases of GARCC. The empirical usefulness of GARCC and the practicality of the likelihood ratio test are demonstrated for the daily returns of the Standard and Poor's 500, Nikkei, and Hang Seng indexes.
Highlights
INTRODUCTIONThe empirical usefulness of the Engle ~1982! autoregressive conditional heteroskedasticity ~ARCH! model and the Bollerslev ~1986! extension of ARCH to the generalized ARCH ~GARCH! model have inspired a new generation of models to capture time-varying conditional volatility in financial time series data+ Such success in modeling conditional volatility has led to several different research developments+ One area of research concerns the flexibility of the models to capture the stylized facts or features that appear in financial time series data+ This direction has led to several useful asymmetric extensions of GARCH, including the Glosten, Jagannathan, and Runkle ~1992! asymmetric ~or threshold! GARCH ~GJR! model, the Nelson ~1991! exponential GARCH ~EGARCH! model, and the Ding, Granger, and Engle ~1993! asymmetric power GARCH ~APGARCH! model+ these univariate models can capture the excessive kurtosis and asymmetric behavior that are often found in financial time series, they do not analyze interdependent ~or spillover! effects in volatility across different markets or assets+
The regularity conditions and asymptotic properties are typically either assumed to hold or sufficient regularity conditions are established to ensure that the asymptotic theory is applicable+ Exceptions to the rule are the VARMAGARCH and VARMA-AGARCH models, for which the structural properties of the models have been developed, the analytical forms of the regularity conditions have been derived, and the asymptotic theory for the QMLE has been established under the second and fourth moments for consistency and asymptotic normality, respectively, and the BEKK model, for which Comte and Lieberman ~2003! showed consistency of the QMLE using the conditions established in Jeantheau ~1998!, and asymptotic normality of the QMLE by assuming the existence of eighth moments+
The plan of the remainder of the paper is as follows+ Section 2 presents the GARCC model and provides a structural justification of the BEKK model+ The structural properties are developed, the analytical forms of the regularity conditions are derived, the asymptotic theory is established, and a likelihood ratio ~LR! test is proposed for several special cases of GARCC in Section 3+ The empirical usefulness of GARCC and the practicality of the LR test are demonstrated for the daily returns in the S&P, Nikkei, and Hang Seng indexes in Section 4+ Section 5 provides some concluding comments+ Proofs of the proposition and theorems are given in the Appendix+
Summary
The empirical usefulness of the Engle ~1982! autoregressive conditional heteroskedasticity ~ARCH! model and the Bollerslev ~1986! extension of ARCH to the generalized ARCH ~GARCH! model have inspired a new generation of models to capture time-varying conditional volatility in financial time series data+ Such success in modeling conditional volatility has led to several different research developments+ One area of research concerns the flexibility of the models to capture the stylized facts or features that appear in financial time series data+ This direction has led to several useful asymmetric extensions of GARCH, including the Glosten, Jagannathan, and Runkle ~1992! asymmetric ~or threshold! GARCH ~GJR! model, the Nelson ~1991! exponential GARCH ~EGARCH! model, and the Ding, Granger, and Engle ~1993! asymmetric power GARCH ~APGARCH! model+ these univariate models can capture the excessive kurtosis and asymmetric behavior that are often found in financial time series, they do not analyze interdependent ~or spillover! effects in volatility across different markets or assets+. Model and provides a motivation for the conditional correlations to be time varying+ GARCC is more general than the DCC and VCC models, as it does not impose unduly restrictive conditions on the parameters of the conditional correlation model+ The structural and statistical properties of GARCC will be derived, which include sufficient conditions for the existence of moments and sufficient conditions for consistency and asymptotic normality of the QMLE+ a formal test is proposed of a variety of cross-equation restrictions that can be imposed on GARCC+ Empirical results using daily returns of the Standard and Poor’s 500 ~S&P!, Nikkei, and Hang Seng indexes from 1 January 1986 to 11 April 2000 show that these crossequation restrictions are violated+ GARCC would seem to be a useful addition to the multivariate GARCH literature for modeling time-varying conditional correlations+. The plan of the remainder of the paper is as follows+ Section 2 presents the GARCC model and provides a structural justification of the BEKK model+ The structural properties are developed, the analytical forms of the regularity conditions are derived, the asymptotic theory is established, and a likelihood ratio ~LR! test is proposed for several special cases of GARCC in Section 3+ The empirical usefulness of GARCC and the practicality of the LR test are demonstrated for the daily returns in the S&P, Nikkei, and Hang Seng indexes in Section 4+ Section 5 provides some concluding comments+ Proofs of the proposition and theorems are given in the Appendix+
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