returns. For example, VR(4)/VR(2)= 1.42/1.20= 1.18, which implies that 2-week returns have a first-order autocorrelation coefficient of approximate!* 18%.

Panel В of Table 2.5 shows that the value-weighted index behaves differently. Over the entire sample period, the variance ratios are all grc.ui-i than one, but not by much, ranging from 1.02 for <y=2 to 1.04 for q=H. Moreover, the test statistics (q) are all statistically insignificant, hence RW3 cannot be rejected for any q. The subsampie results show that during the first half of the sample period, the variance ratios for the value-weighted index do increase with q (implying positive serial correlation for multiperiod returns), but during the second half of the sample, the variance ratios decline with q (implying negative serial correlation for multiperiod returns). These two opposing patterns arc responsible for the relatively stable behavior of the variance ratios over the entire sample period.

Although the test statistics in Table 2.5 are based on nominal stock returns, it is apparent that virtually the same results would obtain with real or excess returns. Since the volatility of weekly nominal returns is so much larger than that of the inflation and Treasury-bill rates, the use of nominal, real, or excess returns in volatility-based tests will yield practically identical inferences.

Size-Sorted Portfolios

The fact that RW3 is rejected by the equal-weighted index but not by the value-weighted index suggests that market capitalization or size may play a role in the behavior of the variance ratios. To obtain a better sense of this intuition, Table 2.6 presents variance ratios for the returns of size-sorted portfolios. We compute weekly returns for five size-sorted portfolios from the CkSP NYSF.-AMFX daily returns file. Slocks with returns lor any given week arc assigned to portfolios based on which quimile their beginning-of-yeatmarket capitalization belongs to. The portfolios are equal-weighted and halve a changing composition.-7 Panel A of Table 2.6 reports the results for the portfolio of small firms (first quimile), panel В reports the results for thcvportfolio of medium-size firms (third qtiintile), and panel С repot ts the results for the portfolio of large firms (fifth quintile).

Evidence against the random walk hypothesis for the portfolio of companies Jn the smallest quintile is strong for the entire sample and for both subsaiijples: in panel A all the i*(q) statistics are well above the 5% critical value of 1.96, ranging from 4.67 to 10.74. The variance ratios, are all greater

-We also performed our tests using value-weighted portfolios and obtained essential!* the same, results. The only dilTerence appears in die largest quintile of the value-weighted portfolio, lor which the random walk hypothesis was generally not rejected, this, of course, is not surprising, given that the largest value-weighted quintile is quite similar lo the value-weiglitedjinarket index.

Number Number ц of base observations aggregated

, , of base <> < rm variance nitio

observations 2 4 8 Hi

Л Portfolio of linns with market values in smallest CRSP quimile

Turn- к liod

Variance-ratio test of the random walk hypothesis for size-sorted portfolios, lor the sample period Iroin uly К), 1У02 to December 27, HUM, and subperiods. The variance ratios VR(i;) are reported in the main rows, with heteroskedasticiiy-consistent test statistics <ji(q) given in parentheses immediately below each main row. Under the random walk null hypothesis, the value ol the variance ratio is one and the test statistics have a standard normal distribution asymptotically. Test statistics marked with asterisks indicate that the corresponding variance ratios are statistically different from one at the 5% level of significance.

than one, implying a first-order autocorrelation of 35% for weekly returns over the entire sample period.

For the portfolios of medium-size companies, the ii(q) statistics in panel В shows that there is also strong evidence against RW3, although the variance ratios are smaller now, implying lower serial correlation. For the portfolio of the largest funis, panel 0 shows that evidence against RW3 is sparse, limited only to the first half of the sample period.

Table 2.6. Variance ratios for weekly size-sorted portfolio returns.

The results lor size-based portfolios are generally consistent with those lor the market indexes: variance ratios are generally greater than one and increasing in , implying positive serial correlation in mitltiperiod returns, statistically significant for portfolios of all but the largest companies, and more significant during the first half of the sample period than the second half.

Individual Securities

1 laving shown that the random walk hypothesis is inconsistent with the behavior of the equal-weighted index and portfolios of small- and medium-size companies, we now turn to the case of individual security returns. Tabic 2.7 reports the cross-sectional average of the variance ratios of individual slocks thai have complete return histories in the CRSP database for our entire I ,()9.r>-weok sample period, a sample of 411 companies. Panel Л contains the cross-sectional average of the variance ratios of the 411 slocks, as well as of the 100 smallest, 100 intermediate, and 100 largest stocks.w Cross-sectional standard deviations are given in parentheses below the main rows. Since the variance ratios are clearly not cross-seclionally independent, these standard deviations cannot be used lo form the usual tests of significance-thev are reported only to provide some indication of the cross-sectional dispersion of the variance ratios.

The average variance ratio with <z=2 is 0.90 for the 41 1 individual securities, implying that (here is negative serial correlation on average. For all stocks, the average serial correlation is -4%, and -5% for the smallest 100 slocks. However, the serial correlation is both statistically and economically insignificant and provides little evidence against the random walk hypothesis. For example, the largest average i/*(q) statistic over all stocks occurs for i/=4 and is -0.90 (with a cross-sectional standard deviation of 1.19); the largest average i/*(q) for the 100 smallest stocks is -1.07 (for q=2, with a cross-sectional standard deviation of 1.75). These results are consistent with French and Rolls (1980) finding that daily returns of individual securities are slightly negatively autocorrelated.

For comparison, panel В reports the variance ratio of equal- and value-weighted portfolios of the 411 securities. The results are consistent with those in Tables 2.5 and 2(>: significant positive autocorrelation for the equal-weighted portfolio, and autocorrelation (lose 10 zero lor the value-weiglued portfolio.

That the ret urns of individual securities have statistically insignificant autocorrelation is not surprising. Individual returns contain much company-specific or idiosyncratic noise thai makes it difficult to detect the presence of predictable components. Since the idiosyncratic noise is largely atleuuaied

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by forming portfolios, wc 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.

2.8.3 Cross-Autocorrelations and Lead-Lag Relations

Despite the fact that individual security returns are weakly negatively att-tocorrelated, portfolio returns-which are essentially averages of individual security returns-are strongly positively autocorrelated. This somewhat paradoxical result can mean only one thing: large positive cross-autocorrelations across individual securities across time.

To see this, consider a collection of N securities and denote by R, the (Nxl) vector of their period-/ simple returns [ Ru RNl }. We switch to simple returns here because the locus of our analysis is on the interaction of returns within portfolios, and continuously compounded returns do not aggregate across securities (see Section 1.4.1 in Chapter 1 for further discussion). For convenience, we maintain the following assumption throughout this stYclion:29

(Al) К, is a jointly covariance-slalionary stochastic process with expectation F.[R,] = fJi l= [ Hi Hi Mw ] and autocovariance matrices E[(R, * - /x)(R, -P-)] = Г(к) where, with no loss of generality, we take k>0 sinceT(k) = Г(-к).

If l is defined to be a vector of ones [1 ... 1 ], we can express the equal-weighted market index as /{ , = iR,/N. The first-order autocovariance of Ля,may then be decomposed into the sum of the first-order own-autocovariances and cross-autocovariances of the component securities:

Cov[/{, , R

tR,-i

(.тан

(2.8.2)

and therefore the first-order autocorrelation of R, , can be expressed as Cov!rtm, i, ii ,] lT{\)L lT(l)l

Var[ Rml]

iT(0) t

п(Г(1)) (г(Г(1))

(-T(O)t

(2.8.3)

where (r() is the trace operator which sums the diagonal entries of its squarc-inalrix argument. The first term of (he right side of (2.8.3) contains only

Assumption (Al) is nude lot uotatioiial simplicity, sin<e joint covariaure-staiionaiity allows ns to eliminate time-indexes liom population moments such as/; and Г(А); the qualitative features of our results will not change under the weaker assumptions of weakly dependent hetcrogeneonsly distributed vectors /(,. Ibis would merely require replacing expectations with corresponding probability limits ol suitably defined time-averages. See l.o and MacKinlay (IJ.XIc) for further details.

Table 2.8.

T =

T, =

T, =

Cmw-uuloi uncial ion matrices jor uievnted fmrtfolio returns.

T, =

Autocorrelation matrices of the vector X, 2 /( Нг, Ни Ни /<г ) where li is the week-

t return on the equal-weighted portfolio ol stocks in (be ,th quintile, i=l.....5 (quiniile

1 contains the smallest slocks), for the sample of NYSt-AMKX stocks from July 10. 1902 to December27, 1994 (I ,(i9f>observations). Note that T(A) = D /2El(X,- - )(X,-))D-12

where D s diaglcrj.....a*); thus the (i, /)tli element is the correlation between K and

К . Asymptotic standard errors for the autocorrelations under an 111) null hypothesis are given by l/s/7 = 0.024.

cross-autocovariances and the second term only the own-autocovariances. If the own-autocovariances are generally negative, and index autocovariance is positive, then the cross-autocovariances must be positive. Moreover, the cross-autocovariances must be large, so large as to exceed the sum of the negative own-autocovariances. ,