By Campbell, Ho, Mackinley
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Since the idiosyncratic noise is largely attenuated 'X~id-sample market values are used as the size measure. 74 2. The Predictability of Asset Returns by forming portfolios, we would expect to uncover the predictable systematic component more readily when securities are combined. Nevertheless, the weak negative autocorrelations of the individual securities are an interesting contrast to the stronger positive autocorrelation of the portfolio returns. 8. 8. Cross-autocorrelation matrices fm size-sorted portfolio returns.
Nevertheless, the weak negative autocorrelations of the individual securities are an interesting contrast to the stronger positive autocorrelation of the portfolio returns. 8. 8. Cross-autocorrelation matrices fm size-sorted portfolio returns. 3 Cross-Autocorrelations and Lead-Lag Relations Despite the fact that individual security returns are weakly negatively autocorrelated, portfolio returns-which are essentially averages of individual security returns-are strongly positively autocorrelated.
Denote by and B(q) the asymptotic variances of 6(k) and m ( q ) , respectively. Then under the null hypothesis Hg Lo and MacKinlay (1988) show that 1. The statistics m ( q ) , and m ( q ) - 1 converge almost surely to zero for all q as n increases without bound. 2. The following is a heteroskedasticity-consistentestimator of 6k: 3. The following is a heteroskedasticityconsistent estimator of 8(q): (H4) Forallt, E [ t t r t - , r t e t - k ] = 0 for any nonzeroj and k wherej # k. Condition ( H l ) is the uncorrelated increments property of the random walk that we wish to test.