Ebooks Econometrics The Econometrics of Financial Markets by Campbell, Ho, Mackinley

By Campbell, Ho, Mackinley

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3 below). 2 Portmanteau Statistics Since RW1 implies that all autocorrelations are zero, a simple test statistic of RWl that has power against many alternative hypotheses is the Q-statistic due to Box and Pierce (1970): k = C # 0 otherwise. 11) we see that under RW1, where p(k)=O for all k>O, the sample autocorrelation coefficients j(k) are negatively biased. 6). But deviations from the sample mean sum to zero by construction; therefore positive deviations must eventually be followed by negative deviations on average and vice versa, and hence the expected value of cross-products of deviations is negative.

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.

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.

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