Application of an ARCH System Equations of 4 Spot Exchange Rates in Terms of CAD/USD, DKK/USD, CHF/USD and JPY/USD to Check the Conditional Variance, Covariance and Correlation

Abstract

In this article, we have tested the volatility of the natural logarithmic monthly returns of the CAD/USD, DKK/USD, CHF/USD and JPY/USD spot exchange rates. We have applied an ARCH – conditional heteroskedasticity method to model the variance, the covariance of the error terms and the correlation of the spot exchange rates. We have used three multivariate ARCH specifications in terms of the diagonal VECH, the conditional constant correlation, and the diagonal BEKK. By applying the system equations 1, which represent a diagonal VECH of a covariance specification multivariate ARCH model, we have found that the log likelihood value of the system equations model is 2935.896. The average log likelihood value is 2.66. The Akaike information criterion is -21.03. The Schwarz criterion is -20.58 and the Hannan – Quinn criterion is -20.85. By applying the system equations 2, which represent a constant conditional correlation of a covariance specification multivariate ARCH model, we have found that the log likelihood value of the system equations model is 2903.890. The average log likelihood value is 2.63. The Akaike information criterion is -20.88. The Schwarz criterion is -20.59 and the Hannan – Quinn criterion is -20.77. By applying the system equations 3, which represent a diagonal BEKK of a covariance specification multivariate ARCH model, we have found that the log likelihood value of the system equations model is 2891.647. The average log likelihood value is 2.62. The Akaike information criterion is -20.84. The Schwarz criterion is -20.63 and the Hannan – Quinn criterion is -20.75. The best fit model to help the arbitrageurs to craft their investment strategy in terms of buying, selling or holding large portfolios of spot exchange rates is the system ARCH equations 1. It has the largest value of the log likelihood and average log likelihood value and the lowest error term value of the Akaike and Hannan - Quinn information criterion. The Schwarz criterion has the highest value. By applying the Wald test of the symmetry restrictions of the natural logarithmic monthly returns of the spot exchange rate returns namely the CAD/USD, DKK/USD, CHF/USD and JPY/USD, we have found that the sample evidence suggests accepting the null hypothesis. In other words, the test has rejected the symmetry restrictions of the mean coefficients. The data that we have used started from 01/01/1990 to 01/01/2013 and represent the CAD/USD, DKK/USD, CHF/USD and JPY/USD spot exchange rates, which total to 277 observations. The natural logarithmic monthly returns total to 276 observations. The data was obtained from the Federal Reserve Statistical Release Department and it is denoted by the symbol, H.10.