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**A test for constant correlations in a multivariate GARCH model.**
*(English)*
Zbl 0968.62066

Summary: We introduce a Lagrange Multiplier (LM) test for the constant-correlation hypothesis in a multivariate GARCH model. The test examines the restrictions imposed on a model which encompasses the constant-correlation multivariate GARCH model. It requires the estimates of the constant-correlation model only and is computationally convenient. We report some Monte Carlo results on the finite-sample properties of the LM statistic. The LM test is compared against the Information Matrix (IM) test due to A.K. Bera and S. Kim [Testing constancy of correlation with an application to international equity returns. Mimeo (1996)]. The LM test appears to have good power against the alternatives considered and is more robust to nonnormality. We apply the test to three data sets, namely, spot-futures prices, foreign exchange rates and stock market returns. The results show that the spot futures and foreign exchange data have constant correlations, while the correlations across national stock market returns are time varying.

### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

91B28 | Finance etc. (MSC2000) |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

62P20 | Applications of statistics to economics |

91B84 | Economic time series analysis |

### Keywords:

information matrix test; Lagrange multiplier test; Monte Carlo experiment; multivariate conditional heteroscedasticity; constant-correlation hypothesis
Full Text:
DOI

### References:

[1] | Baillie, R. T.; Myers, R. J.: Bivariate GARCH estimation of the optimal commodity futures hedge. Journal of applied econometrics 6, 109-124 (1991) |

[2] | Bera, A. K.; Higgins, M. L.: ARCH models: properties, estimation and testing. Journal of economic survey 7, 305-366 (1993) |

[3] | Bera, A.K., Kim, S., 1996. Testing Constancy of Correlation with an Application to International Equity Returns. Mimeo. Center for International Business Education and Research (CIBER) working paper 96-107, University of Illinois, Urbana-Champaign. |

[4] | Bollerslev, T.: Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model. Review of economics and statistics 72, 498-505 (1990) |

[5] | Bollerslev, T.; Engle, R. F.; Wooldridge, J. M.: A capital asset pricing model with time-varying covariances. Journal of political economy 96, 116-131 (1988) |

[6] | Diebold, F. X.; Nerlove, M.: The dynamics of exchange rate volatility: a multivariate latent factor ARCH model. Journal of applied econometrics 4, 1-21 (1989) · Zbl 1126.91365 |

[7] | Engel, C.; Rodrigues, A. P.: Tests of international CAPM with time-varying covariances. Journal of applied econometrics 4, 119-138 (1989) |

[8] | Engle, R. F.; Kroner, K. F.: Multivariate simultaneous generalized ARCH. Econometric theory 11, 122-150 (1995) |

[9] | Engle, R. F.; Ng, V. K.; Rothschild, M.: Asset pricing with a factor-ARCH covariance structure: empirical estimates for treasury bills. Journal of econometrics 45, 213-237 (1990) |

[10] | Godfrey, L. G.: Misspecification tests in econometrics. (1988) · Zbl 0849.62063 |

[11] | Karolyi, G. A.: A multivariate GARCH model of international transmissions of stock returns and volatility: the case of the united states and Canada. Journal of business and economic statistics 13, 11-25 (1995) |

[12] | Kroner, K. F.; Claessens, S.: Optimal dynamic hedging portfolios and the currency composition of external debt. Journal of international money and finance 10, 131-148 (1991) |

[13] | Kroner, K.F., Sultan, J., 1991. Exchange rate volatility and time varying hedge ratios, In: Rhee, S.G., Chang, R.P. (Eds.), Pacific-Basin Capital Market Research, Vol. 2, pp. 397–412. |

[14] | Kroner, K. F.; Sultan, J.: Time varying distribution and dynamic hedging with foreign currency futures. Journal of financial and quantitative analysis 28, 535-551 (1993) |

[15] | Li, W. K.; Mak, T. K.: On the squared residual autocorrelations in non-linear time series with conditional heteroscedasticity. Journal of time series analysis 15, 627-636 (1994) · Zbl 0807.62070 |

[16] | Lien, D.; Luo, X.: Multiperiod hedging in the presence of conditional heteroscedasticity. Journal of futures markets 14, 927-955 (1994) |

[17] | Lien, D.; Tse, Y. K.: Hedging time-varying downside risk. Journal of futures markets 18, 705-722 (1998) |

[18] | Park, T. H.; Switzer, L. N.: Bivariate GARCH estimation of the optimal hedge ratios for stock index futures: a note. Journal of futures markets 15, 61-67 (1995) |

[19] | Tse, Y. K.: The conditional heteroscedasticity of the yen-dollar exchange rate. Journal of applied econometrics 13, 49-55 (1998) |

[20] | Tse, Y. K.; Tsui, A. K. C.: Conditional volatility in foreign exchange rates: evidence from the malaysian ringgit and Singapore dollar. Pacific-basin finance journal 5, 345-356 (1997) |

[21] | Tse, Y. K.; Zuo, X. L.: Stock volatility and impact of news: the case of four Asia-Pacific markets. Advances in international banking and finance 2, 115-137 (1996) |

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