"Frontmatter". In: Analysis of Financial Time Series

CROSS-CORRELATION 303where ̂D is the k × k diagonal matrix of the sample standard deviations of the componentseries.Similar to the univariate case, asymptotic properties of the sample cross-correlationmatrix ˆρ l have been investigated under various assumptions; see, for instance,Fuller (1976, Chapter 6). The estimate is consistent, but is biased in a finite sample.For asset return series, the finite sample distribution of ˆρ l is rather complicated partlybecause of the presence of conditional heteroscedasticity and high kurtosis. If thefinite-sample distribution of cross-correlations is needed, we recommend that properbootstrap resampling methods be used to obtain an approximate estimate of the distribution.For many applications, a crude approximation of the variance of ˆρ ij (l) issufficient.Example 8.1. Consider the monthly log returns of IBM stock and the S&P500 index from January 1926 to December 1999 with 888 observations. The returnsinclude dividend payments and are in percentages. Denote the returns of IBM stockand the S&P 500 index by r 1t and r 2t , respectively. These two returns form a bivariatetime series r t = (r 1t , r 2t ) ′ . Figure 8.1 shows the time plots of r t using the same scale.Figure 8.2 shows some scatterplots of the two series. The plots show that the tworeturn series are concurrently correlated. Indeed, the sample concurrent correlation(a) IBM monthly log returns: 1/1926-12/1999ibm-20 0 201940 1960 1980 2000year(b) Monthly log returns of SP500 index: 1/1926-12/1999sp5-20 0 201940 1960 1980 2000yearFigure 8.1. Time plot of monthly log returns in percentages for IBM stock and the S&P 500index from January 1926 to December 1999.