ward 1>у 73%! Since die mean of V is ч/я/2% 1.25, the mean of die classical roscaled range would be 2.16 for such an AR(1) process.

I.o (НИМ) develops a modification ol the R/S statistic lo account for the effects of short-range dependence, derives an asymptotic sampling theory under several null and alternative hypotheses, and demonstrates via Monte (arlo simulations and empirical examples drawn from recent historical stock market data that the modified R/S statistic is considerably mote act urate, often yielding inferences that contradict those of its classical counterpart. In particular, what the earlier literature had assumed was evidence of long-range dependence in US stock returns may well be the result of quickly decaying short-range dependence instead.

2.7 Unit Root Tests

Л more recent and more specialized class of tests that are often confused with tests of the random walk hypotheses is the collection of unit rant tests in which the null hypothesis is

V, = ц f .V, i + often with the following alternative hypothesis:

.V, - /</ = ф(\, I -/<(/-!))+ <= Ф € (-1, 1). where 6, is any zero-mean stationary process, such that

0 < (7-

lim E

(2.7.1)

(2.7.2)

(2.7.3)

1 leurislic allv, condition (2.7.3) requires that variance of the partial sum 5Z/=i f< increase al approximately the same rate as 7, so that each new e, added to the- partial sum has a nonliivial contribution to the partial sums variance. This condition ensures thai the usual limit theorems are applicable to the c,s, and it is satisfied by virtually all of (he stationary processes that we shall have occasion to study (except for those in Section 2.6).

-- 11 11 tr pai li.tl st n its v.u i.nu < wci с hi gi i iw slower than 7. so that the limit in (2.7.!4 were II. the unc-eitaiiiiv in the seiiieine ol i,\ would he cancelling out over lime and would not he a very tiselul model ol i.uidom price dynamics. An example ol sue It a process is an мл( 1) with a unit root. i.e.. f, = 4, - >1, I. where t is while noise.

II the pat 11.it siuns v.u i.nu e weie lo glow Lister titan 7, so thai the limit ill C .7.:l) weie то. lllis would In- an example ol l j;-wiijr id/niiidiiir. in which the aulocoi relation liuHtioii o( the ,s decay* cciv slowh. Ли example ol sin li a process is a fractionally ilillerenced process 11 - I.)1/1 -- ii,. when- ii, is whin- noise. See Seel inn -.(i and I.о (1991) loi tin i tier discussion.

, IUU i.VltltflU-t 65

The unit root test is designed to reveal whether X( is difference-stationary (the null hypothesis) or trend-stationary (the alternative hypothesis); this distinction rests on whether ф is unity, hence the term unit root hypothesis. The test itself- is formed by comparing the ordinary least squares estimator ф to unity via its (nonstandard) sampling distribution under the null hypothesis (2.7.1), which was first derived by Dickey and Fuller (1979).и Under the null hypothesis, any shock to X, is said to be permanent since E[X,+* I X,] - iik + X, for all k>0, and a shock to X, will appear in the conditional expectation of all future X,+>. In this case X, is often called a stochastic trend since its conditional expectation depends explicitly on the stochastic variable X,. In contrast, under the alternative (2.7.2), a shock to X, is said to be temporary, since Е[Х,+* X,] = n(t+k) + фк(Х,-ц1), and the influence of X, on the conditional expectation of future Xl+k diminishes as k increases.

Because the 6,s are allowed to be an arbitrary zero-mean stationary process under both the unit root null (2.7.1) and alternative hypothesis (2.7.2), the focus of the unit root test is not on the predictability of X as it is under the random walk hypotheses. Even under die null hypothesis (2.7.1), the increments of X, may be predictable. Despite the fact that the random walk hypotheses are contained in the unit root null hypothesis, it is the permanent/temporary nature of shocks to Xi that concerns such tests. Indeed, since there are also nonrandom walk alternatives in the unit root null hypothesis, tests of unit roots are clearly not designed lo detect predictability, but are in fact insensitive to it by construction.

2.8 Recent Empirical Evidence

Predictability in asset returns is a very broad and active research topic, and it is impossible to provide a complete survey of this vast literature in just a few pages. Therefore, in this section we focus exclusively on the recent empirical literature.24 We hope to give readers a sense for the empirical relevance of predictability in recent equity markets by applying the tests developed in the earlier sections to stock indexes and individual stock returns using daily and

weekly data from 1962 to 1994 and monthly data from 1926 to 1994. Despite

wSince then, advances in econometric methods have yielded many extensions and generalizations to this simple framework: tests for multiple unit roots in multivariate ARJMA systems, tests for cointegration, consistent estimation of models with unit roots cointegration, etc. (see Campbell and Perron [ 1991 ] for a thorough survey of this literature).

1 lowever, we would be remiss if we did not cite the rich empirical tradition on which the; recent literature is built, which includes: Alexander (1961, 1964), Cootner (1964), Cowlei (I9b0), Cowles and Jones (1937), Fama (1965), Fama and Blume (1966) Kendall (1953)? Crangcr and Morgenstem (1963), Mandelbrot (1963), Osborne (1959,1962), Roberts (1959), and Working (1960).

the specificity of these examples, the empirical results illustrate many of the issues that have arisen in the broader search for predictability among asset returns.

iijiiT-

*ЯЩг*-

\ 2. Я. 1 A utocorrelations

Table 2.4 reports the means, standard deviations, autocorrelations, and Box-Pierce Q-statistics for daily, weekly, and monthly CRSP slock returns indexes from July 3, 1962 to December 31, 1994.и During this period, panel Л of Table 2.4 reports that the daily equal-weighted CRSP index has a first-order autocorrelation p(l) of 35.0%. Recall from Section 2.4.1 that under the IID random walk null hypothesis RWI, the asymptotic sampling distribution of /5(1) is normal with mean 0 and standard deviation \/-jT (see (2.4.14)). This implies that a sample si/.е of 8,179 observations yields a standard error of 1.11% for p(l); hence an autocorrelation of 35.0% is clearly statistically significant at all conventional levels of significance. Moreover, the Box-Pierce Q-statistic with five autocorrelations has a value of 263.3 which is significant at all the conventional significance levels (recall that this statistic is distributed asymptotically as a xf, variate for which the 99.5-percentile is 16.7).

Similar calculations for the value-weighted indexes in panel A show that both CRSP daily indexes exhibit statistically significant positive serial correlation at the first lag, although the equal-weighted index has higher autocorrelation which decays more slowly than the value-weighted index. The subsampie autocorrelations demonstrate that the significance of the autocorrelations is not an artifact of any particularly influential subset of the data; both indexes are strongly positively autocorrelated in each subsampie.

To develop a sense of the economic, significance of the autocorrelations in Table 2.4, observe that the R2 of a regression of returns on a constant and its first lag is the square оГ the slope coefficient, which is simply the first-order autocorrelation. Therefore, an autocorrelation of 35.0% implies that 12.3% of the variation in the daily CRSP equal-weighted index return is predictable using the preceding days index return.

?r*Uiijlcss stated otherwise, we lake returns to be continuously compounded. Portfolio returns jrre calculated first from simple returns and then are converted to a continuously compounded return. The weekly return of each security is computed as the return from Tuesdays closing price to the following Tuesdays dosing price. 11 the following Tuesdays price is missing, then Wednesdays price (or Mondays if Wednesdays is also missing) is used. If iHJth Mondays and Wednesdays prices are missing, the return for that week is reported as missing; this occurs only rarely. To compute weekly returns on size-sorted portfolios, for each week all stocks with nonmissing returns that week are assigned to portfolios based on the beginnirtg of year market value. If the l>eginning of year market value is missing, then the end of year value is used. If both market values are missing the stock is not assigned to a portfolio.

2.4. Autocorrelation in daily, weekly, and monthly stock index returns.

Sample- Sample Мс;ш s[) - . -ц Period Size

A. Daily Returns

CRSP Value-Weighted Index

62:07:03-94:12:30 8,179 0.041 0.824 17.6 -0.7 0.1 -0.8 263.3 269.5

1)2:07:03-78:10:27 4,090 0.028 0.738 27.8 1.2 4.6 3.3 329.4 343.5

78:10:30-94:12:30 4,089 0.054 0.901 10.8 -2.2 -2.9 -3.5 69.5 72.1

CRSP Equal-Weighted Index

0> 1)7 03-94:12:30 8,179 0.070 0.764 35.0 9.3 8.5 9.9 1,301.9 1,369.5

6 07:03-78:10:27 4,090 0.063 0.771 4X1 13.0 15.3 15.2 1,062.2 1,110.2

7810:30-94-.12:30 4,080 0.078 0.756 26.2 4.9 2.0 4.9 348.9 379.5

11. Weekly Returns

CRSP Value-Weighted Index

62:07:10-94:12:27 1,695 0.196 2.093 1.5 -2.5 3.5 -0.7

02-07:10-78:10:03 848 0.144 1.994 5.6 -3.7 5.8 1.6

7,4:10:10-94:12:27 847 0.248 2.188 -2.0 -1.5 1.6 -3.3

CRSP Equal-Weighted Index

62:07:10-94:12:27 1,695 0.339 2.321 20.3 6.1 9.1 4.8

(>2 07:10-78:10:03 848 0.324 2.460 21.8 7.5 11.9 6.1

78:10:10-94:12:27 847 0.354 2.174 18.4 4.3 5.5 2.2

8.8 36.7 9.0 21.5 5.3 25.2

94.3 109.3

60.4 68.5 33.7 51.3

C. Monthly Returns

CRSP Value-Weighted Index

62:07:31-94:12:30 390 0.861 4.336 4.3 -5.3 -1.3 6207:31-78-.09:29 195 0.646 4.219 6.4 -3.8. 7.3 78:10:31-94:12:30 195 1.076 4.450 1.3 -6.3 -8.3

CRSP Equal-Weighted Index

62:07:31-94:12:30 390 1.077 5.749 17.1 -3.4 -3.3 02:07:31-78:09:29 195 1.049 6.148 18.4 -2.5 4.4 78:10:31-94:12:30 195 1.105 5.336 15.0 -1.6-12.4

Auiororrelaiiiin coefficients (in percent) and tlox-Iierce -statistics lor CRSP daily, weekly, anil monthly value- and equal-weighted return indexes for the sample period from July 3, 1962 lo December 30, 111У4 and subperiods.

I. I lie. Predictability oj Asset Returns

The weekly and monthly reinrn autocorrelations reported in panels В and С of Table 2.4, respectively, exhibit patterns similar to those of the daily autocorrelations: positive and statistically significant at the first lag over the entire sample and for all subsamples, with smaller and sometimes negative higher-order autocorrelations.

2.tV. 2 Variance Ratios

The fact that the autocorrelations of daily, weekly, and monthly index returns in Table 2.4 are positive and often significantly different from zero has implications for the behavior of the variance ratios of Section 2.4 and we explore these implications in this section for the returns of indexes, portfolios, and individual securities.

CRSP Indexes

The autocorrelations in Table 2.4 suggest variance ratios greater than one, and this is confirmed in Table 2.5 which reports variance ratios VR defined in (2.4.37) and, in parentheses, hcteroskedasticity-consistenl asymptotically standard normal test statistics defined in (2.4.44), for weekly CRSP

equal-and value-weighted market return indexes.215 Panel Л contains results for the equal-weighted index and panel В contains results for the value-weighted index. Within each panel, the first row presents the variance ratios and test statistics for the entire 1,095-weck sample and the next two rows present similar results lor the two subsamples of 848 and 847 weeks.

Panel A shows that the random walk null hypothesis RW3 is rejected at all the usual significance levels for the entire time period and all subperiods for the equal-weighted index. Moreover, the rejections are not due to changing variances since the t (r/)s are hcteroskedasticity-consistenl. The estimates of the variance ratio are larger than one for all cases. For example, the entries in the first column of panel A correspond lo variance ratios with an aggregation value q of 2. In view of (2.4.18), ratios with q=2 arc approximately equal to 1 plus the first-order autocorrelation coefficient estimator of weekly returns; hence, the entry in the first row, 1.20, implies that the first-order autocorrelation for weekly returns is approximately 20%, which is consistent with the value reported in Table 2.4. With a corresponding ф*(>/) statistic ol 4.53, the random walk hypothesis is resoundingly rejected.

The subsample results show that although RW3 is easily rejected over both halves of the sample period, the variance ratios arc slightly larger and the rejections slightly stronger over the first half. This pattern is repealed in Table 2.0 and in other empirical studies of predictability in US slock

-Since in our sample the values ol /*(i/)-torn puled under the null hypothesis RW3-arc always sialism ally less significant than the values ol tytq) calculated under KWI, lo conserve space we report only the mote < onservalive statistics.

♦

2.8. Recent Empirical Evidence

Table 2.5. Variance ratios for weekly slock index returns.

Sample period

Number nq of base observations

Number q of base observations aggregated to form variance ratio

A. CRSP Equal-Weighted Index 62:07:10-94:12:27 1,695

62:07:10-78:10:03 78:10:10-94:12:27

848 847

B. CRSP Value-Weighted Index

Variance-ratio test of the random walk hypothesis for CRSP equal-and value-weighted indexes, for the sample period from July 10, 1962 to December 27, 1994 and subperiods. The variance ratios VR(<,) are reported in the main rows, with heteroskedasiiciry-consistent test statistics tytq) given in parentheses immediately below each main row. Under the random walk null hypothesis, the value of 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.

returns: the degree of predictability seems to be declining through time. To the extent that such predictability has been a source of excess profits, its decline is consistent with the fact that financial markets have become increasingly competitive over the sample period.

The variance ratios for the equal-weighted index generally increase with q: the variance ratio climbs from 1.20 (for q=1) to 1.74 (for q = 16), and the subsample results show a similar pattern. To interpret this pattern, observe that an analog of (2.4.18) can be derived for ratios of variance ratios:

where p,(l) is the first-order autocorrelation coefficient for (j-period returns

r,+r, 4-----l-r, ?+. Therefore, the fact that the variance ratios in panel A

of Table 2.5 are increasing implies positive serial correlation in multiperiod