*(t), andthe regression test is performed 1,000 times, providing a distribution of
 
tics in the presence of overlapping observations and devised appropriate correctionsfor hypothesis tests involving regression coefficients. Because I am interested in theexplanatory power, as measured by R? as well as the significance of the regressioncoefficients, I have chosen to use nonoverlapping observations.Patterns of Stock Market Prices 259
regression coefficients, 8*, and R?’:re(t,t +T) = a*(T) + B*(T)r*(t — T, 1) + e* (t,t + T). (2)
The bootstrap not only provides a correction for the autocorrelationbias, but it generates distributions of regression statistics that conformto the null hypothesis that returns are independently and identicallydistributed (i.i.d.). By comparing the values of 8 and R? to their boot-strapped distributions, I may determine how unusual they are, givena null hypothesis that successive annual returns are independent andidentically distributed. In addition, the standard deviation of the boot-strapped distribution of the regression coefficients is a consistent esti-mate of the coefficient standard error (see Efron 1979) and is used toconstruct a f-statistic.
Since the dividend yield may have changed over the course of sev-eral centuries, I also perform a test that allows for long-term variationin the mean return. Instead of the total capital appreciation, I examinethe deviation of the annual capital appreciation return from its 20-yearmoving average:
t-20
r4(t) = r(t) - 5 r(i). (3)
i=t-1
Because the dividend yields are unknown, I cannot distinguish be-tween long-term changes in expected returns and changing dividendpolicies. However, for horizons less than 20 years, I can test whetherdeviations from the long-term mean are reverting or persistent.®
As noted earlier, there is no reason to expect that appreciation re-turns to 3 centuries of stock prices will be homoscedastic. In fact,since the series are spliced from components with varying numbers ofsecurities, I expect the variance of different sections of each index todiffer. Consequently, I perform a stratified-variance bootstrap, usingthe method proposed by Kim, Nelson, and Startz (1988). For both theraw and the demeaned series, I divide the sample into five groups,according to variance, where variance is defined by squared returns.I sample with replacement from each stratum in order to match theapproximate temporal pattern of heteroscedasticity present in the sam-ple.” As with the bootstrap draws under the assumption that returns
6. Given the fact that the mean is allowed to vary with time, the term ‘‘mean rever-sion’’ actually only applies to the deviations from the moving average. Indeed, if themean were allowed to vary each period, a test of ‘‘mean reversion’’ would be absurd.Thus, the test of deviations from the long-term average should be interpreted as condi-tional upon the specification in eq. (3).
7. To the extent that the changing means are components of the squared returns, thiswill tend to make the bootstrap sample resemble the true sample in general. This weak-ens the power of the test, and thus I will report bootstrap quantiles generated by theiid. and stratified variance procedures.260 Journal of Business
are i.i.d., I perform the autocorrelation tests, save the coefficients andR’s, and then report the empirical quantiles exceeded by the statisticsfrom tests performed on the actual series.
B. R/S Tests
Autocorrelation tests detect long-term dependency in stock marketprices if the dependent behavior is periodic and if the periodicity isconsistent over time. Fundamental historical changes may have alteredthe period of market cycles, however. Mandelbrot (1972) proposes astatistic to measure the degree of long-term dependency, in particular,“‘nonperiodic cycles.’’ The rescaled range, or R/S statistic, is formedby measuring the range between the maximum and minimum distancesthat the cumulative sum of a stochastic random variable has strayedfrom its mean and then dividing this by its standard deviation. Anunusually small R/S measure would be consistent with mean reversion,for instance, while an unusually large one would be consistent withreturn persistence.
Mandelbrot (1972) has shown that the R/S statistic is a more generaltest of long-term dependency in time series than either autocorrelationtests or examination of spectral densities. He points out that, in partic-ular, it is robust to changes in periodicity. Lo (1991) points out thatone limitation of the R/S statistic is that it cannot distinguish betweenshort- and long-term dependency, nor is it robust to heteroscedasticity.
Lo (1991) modifies the R/S statistic so that it is more robust toviolations in the assumption that returns are i.i.d. The modificationconsists of replacing the standard deviation with an estimate that ex-plicitly models short-term temporal dependency using the autocovari-ances up to a finite number of lags, weighted by factors proposed byNewey and West (1987):
1 a ;R/S, = oo ol° ae lms, )— min > »| (4)
 
whereqe =F +2) 0(Q) 4,t=1=,;-—@,(q) = 1 q+Tand
4, = the autocovariance operator.
Lo (1991) points out that the uncorrected R/S statistic is sensitive toheteroscedasticity and cannot distinguish the compounded effects ofPatterns of Stock Market Prices 261
short-horizon patterns from long-term patterns. He derives the distri-bution of the modified (R/S) statistic, allowing it to be used in a hypoth-esis test about long-term dependency in stock market returns.® Whilethe R/S statistic identifies nonperiodic cycles, it is not free of thechoice of return horizon. As with the autocorrelation test, I report thebootstrapped quantile exceeded by the R/S statistic for the raw stockseries and the demeaned stock series, under the i.i.d. and stratifiedvariance procedures. In addition, I report the exceeded quantile of theanalytically derived distribution of the modified R/S reported in Lo(1991).
C. Joint Hypothesis Tests
One problem with examining either the autocorrelation coefficient orthe R/S statistic for a number of different horizons is that a hypothesistest about the significance of a subset of the coefficients or R/S statis-tics is misleading. Thus, as in Goetzmann (1989), Richardson andSmith (1991), and Richardson (in press), I perform a joint significancetest across all 10 autoregression coefficients. To test that all 10 coeffi-cients are zero, I use the Wald test:
W(®) = TQ" 'B) ~ x2, (5)where B denotes the bias-corrected coefficient vector, and the covari-
ance matrix that describes the cross-horizon dependencies is estimatedwith the bootstrap
O45 = F%7,8))- (6)
The x’ distribution is known to be sensitive to deviations from normal-ity in the underlying distribution. Thus the parametric Wald test maybe misspecified. Fortunately, the Wald statistic of equation (5) suggestsa nonparametric test as well. A rejection region for the W-statisticbased on the distribution of the bootstrapped Wald statistic, W*, maybe identified. In other words, I calculate the Wald statistic for eachbootstrapped coefficient vector and use the resulting distribution forhypothesis testing. Thus, while a comparison of the Wald statistic tothe x’ distribution may cause the null to be rejected as a result ofdeviations of the coefficient vector from multivariate normality, a com-parison to the bootstrapped distribution of the Wald statistic will notsince it is based on draws from the empirical coefficient distribution.
As with the autocorrelation test, it is known that the R/S statisticfor each horizon is not independent. Thus, it is necessary to performa joint test of dependency across all 10 lags as before. Lo (1991) dem-onstrates that the distribution of the R/S statistics is defined by the
8. Green and Fielitz (1977) applied the R/S statistic to examine U.S. stock marketbehavior but did not formulate an explicit test of long-term market memory.262 Journal of Business
range of a Brownian bridge process—a distribution that is slightlyright-skew and leptokurtotic in comparison to the normal. This biasesthe Wald test toward rejection; however, as I noted above, a compari-son of the Wald statistic to its bootstrapped distribution, rather thanthe x’ distribution, is robust to departures from normality. It may thusbe used as a measure of whether R/S statistics for each horizon arejointly unusual.
IV. Results
Table 4 and table 5 report the autocorrelation and R/S tests for the rawLSE series and the demeaned LSE series, respectively. The bootstrapt-statistic and the bootstrap percentiles derived from the i.i.d. andstratified variance methods are reported for each autoregression coef-ficient at return horizons from 1 to 10 years.’ I report bootstrap percen-tiles for the R/S statistic, as well as Lo’s (1991) analytically derivedpercentiles. In addition, I report the autoregression R* with boot-strapped percentiles.
Table 4 suggests that returns with horizons greater than 5 years arestrongly persistent. The bootstrapped percentiles indicate that the 6-,8-, 9-, and 10-year coefficients are strongly positive and that the ex-planatory power of the regression, as measured by R?, is around 10%-20%. This result may be partially due to the variance structure of thetime series, however. Only the 8-year coefficient exceeds the ninety-fifth percentile of the stratified variance coefficient distribution. TheR/S statistics are unusual at the first three horizons but not at longerintervals. Since they exceed one for horizons up to 3 years, they sug-gest persistence rather than reversion.'®
Table 5 reports the results from the demeaned LSE series. It showsthat much of the persistence in raw LSE returns may in fact be dueto long-term changes in the mean. Once the 20-year moving averageis subtracted, all of the coefficients become negative, and for the 4-,5-, 6-, and 7-year horizons they are significantly so. In fact, the coeffi-cients display the U-shaped pattern that Fama and French (1988) pre-dict for returns generated by a process having both permanent andtemporary components. The evidence from the R/S statistics is lessclear. Few are unusual when compared to Lo’s analytically deriveddistribution or to the bootstrapped distributions.
Table 6 and table 7 report the autocorrelation and R/S tests for the
9. The bootstrap t-statistic is formed by dividing the bias-adjusted coefficient vectorby the standard deviation of the bootstrapped coefficient samples, in the manner pro-posed by Efron (1979).
10. The fact that the R/S statistic does not exceed one for longer horizons does notimply a contradiction between the autocorrelation test and the R/S test since Lo’s (1991)adjustment of the R/S statistic includes multiple lags.263
Patterns of Stock Market Prices
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266Patterns of Stock Market Prices 267
raw NYSE and demeaned NYSE series, respectively. The coefficientsfor both series are negative at each horizon, although they do notfollow the U shape hypothesized by Fama and French (1988). Thet-statistics and the bootstrap probability levels suggest that some bias-adjusted coefficients may differ significantly from zero. Table 7 reportsthe results for the demeaned NYSE series. As with the demeaned LSEseries, long-horizon returns appear significantly negatively autocorre-lated. In addition, the R/S shows evidence of reversion at horizonsless that 4 years.
While the statistics about the individual horizons are suggestive ofmean-reverting behavior, the joint tests reported in table 8 indicate
TABLE 8 Joint Hypothesis Test for 1-10 Horizons Based on the Wald StatisticDescribed in Equation (5)
 
TestStatistic Quantile 90% 95% 99%
x distribution 15.99 18.31 23.21LSE raw capital appreciation seriesautocorrelation coefficients:
 
iid. bootstrap 100.43 1.00 16.76 19.65 29.80Stratified variance 38.41 1.00 16.20 19.12 26.13LSE demeaned capital appreciationseries autocorrelation coeffi-cients:iid. bootstrap 13.20 79 16.32 19.45 28.07Stratified variance 16.57 90 16.02 18.57 24.07NYSE raw capital appreciationseries autocorrelation coeffi-cients:iid. bootstrap 14.37 84 16.99 18.15 24.73Stratified variance 8.73 44 16.06 18.13 21.42NYSE demeaned capital apprecia-tion series autocorrelationcoefficients:iid. bootstrap 16.11 -90 15.48 18.26 24.09Stratified variance 7.48 31 15.97 18.21 23.54LSE raw capital appreciation seriesR/S statistics:iid. bootstrap 35.08 98 19.02 23.83 41.18Stratified variance 6.93 32 16.90 21.32 28.10LSE demeaned capital appreciationseries R/S statistics:iid. bootstrap 34.57 .98 16.22 22.66 57.56Stratified variance 45.06 1.00 17.84 24.30 38.89NYSE raw capital appreciationseries R/S statistics:iid. bootstrap 7.65 68 14.76 21.65 61.59Stratified variance 6.20 34 16.00 19.18 38.47
NYSE demeaned capital apprecia-tion series R/S statistics:iid. bootstrap 32.84 96 16.78 26.39 72.59Stratified variance 34.73 -98 16.73 22.82 56.47268 Journal of Business
how unusual the pattern of 10