and those are discussed in Cochrane (1088), Faust (1992), and I.o and MacKinlay (1988, I9K9).

The variance difference and ratio statistics can he easily generalized to mulliporiod returns. Let our sample consist of nq+\ observations \fa, /i.----/ !. where </ is any integer greater than one and define the estimators:

1 1

/ з - У (ph -/,* ,) = - (/ -fa) (2.4.28)

I tl 7

j i

; = -Yi/k-h-i-ftf (2.4.29)

a~(4) = - ]T(/ty-/>, -qfif (2.4.30)

VO<7) = ,;(</)-a;, VR(<7) = (2.4.31)

Using similar arguments, the asymptotic distributions of VD(f/) and VR((/) under the RWl null hypothesis are

/hT/WXq) ~ Л/ (0,2(<7-1)(т1) (2.4.32)

/Ъ7/(У\Цф - 1) ~ Л/ (0,2(7-1)). (2.4.33)

Two important refinements of these statistics can improve their finite-sample properties substantially. The first is to use overlapping 7-period returns in estimating (he variances by defining the following alternative estimator for a~:

1 Л

<V</> = ~г У (/* - h-4 - ЧЧ)~. (2.4.34)

I *=.,

This esiiinaior contains 110-1/+1 terms, whereas the estimator(У(с/) contains only n terms. Using overlapping (/-period returns yields a more efficient estimator and hence a more powerful test.

The second refinement involves correcting the bias in the variance estimators 0 and fT- before dividing one by the other. Denote (he unbiased estimators as Fff and 07(7), where

j 7

l =--г T (/>. - /> -i - A)2 (2.4.35)

/ - 1 ti

= - Y\ (/*-/*- - о2

hi f-

. ....... Kjiuoiieuuea incremetits

m = ?(n?-9+i)(i Xj, (2.4.36)

and define the statistics:

VD(?) = a2M)-al W(q) = (2.4.37)

This yields an unbiased variance difference estimator, however, the variance ratio estimator is still biased (due to Jensens Inequality). Nevertheless, simulation experiments reported in Lo and MacKinlay (1989) show that the finite-sample properties of VR(q) are closer to their asymptotic limits than VR(<7).

Under the null hypothesis /7, the asymptotic distributions of the variance difference and variance ratio are given by

VD(7) Л n(q.2(2-!>) (2.4.38)

y(VR((?)-l) Л (о,2(2-!)). (2.4.39)

These statistics can then be standardized in the usual way to yield asymptotically standard normal lest statistics. As before, if o4 is estimated by aiin standardizing the variance difference statistic, the result is the same as the standardized variance ratio statistic:

*(,) . №)-1)(2(2? 1>) / (2.4.40)

чМ(?> /2(2?-1)(?-1)у/2 . m })

Sampling Distribution ofVR(q) under RW3 Since there is a growing consensus among financial economists that volatjl-ities change over time (see Section 12.2 in Chapter 12), a rejection of tfje random walk hypothesis because of heteroskedasticity would not be of mucjh interest. Therefore, we seek a test for RW3. As long as returns are uncorrected, even in the presence of heteroskedasticity the variance ratio must stipl approach unity as the number of observations increases without bound, fc-the variance of the sum of uncorrelated increments must still equal the sum of the variances. Howeyer, the asymptotic variance of the variance ratios will clearly depend on the type and degree of heteroskedasticity present.

One approach is to model the heteroskedasticity explicitly as in Section 12.2 of Chapter 12, and then calculate the asymptotic variance ofVR(</) under this specific null hypothesis. However, to allow for more general forms

2. The Predictability of Asset Returns

of heteroskedasticity, we follow the approach taken by l.o and MacKinlay (1988) which relies on the heteroskedasiiciiy-consisient methods of White (1980) and While and Domowitz (1984). This approach applies to a much broader class of log price processes [/>,) than the IID normal increments process of the previous section, a particularly relevant concern for US stock returns as Table 1.1 illustrates.11 Specifically, let r, = ц + <? and define the following compound null hypothesis H*>:

(HI) Tor all t,£[(,} = 0, andV.[e,<r, r] = Oforajiyx ф 0.

(H2) ((,) is ф-mixing with coefficients ф{т) ofsiter/{2r-\) or is a-mixing with coefficients a(m) of size r/(r-1), where r > 1, such that for all t and for any r > 0, there exists some & > < oo.

(TO) lim

1 i

- E Etfl =

(114) lor all t, E( , (, с,-/,) = 0 for any nonzero j and к where j ф к.

Condition (HI) is the uncorrelated increments property of the random walk thai we wish to test. Conditions (H2) and (113) are restrictions on the maximum degree of dependence and heterogeneity allowable while still pencilling some form of the Law of Large Numbers and the Central Limit Thedrem lo obtain (see White [1984] for the definitions of ф- and ar-mixiiig random sequences). Condition (114) implies lhat ihe sample autocorrelations pf (, are asymptotically uncorrelated; this condition may be weakened considerably al die expense of computational simplicity (see note 15).

This compound null hypothesis assumes that j>, possesses uncorrelated increments but allows for quite general forms of heteroskedasticity, including deterministic changes in the variance (due, for example, to seasonal factor!?) and Englcs (1982) ARCH processes (in which ihe conditional variance ctepends on past information).

Since VRft?) still approaches one under 11,*, we need only compute its asymptotic variance [call it 0(q)] lo perform the standard inferences. Lo and MacKinlay (1988) do this in two steps. First, recall that the following equality holds asymptotically under quite general conditions:

VR(q) = 1+211

- /5(A).

(2.4.41)

H()f couise, second montcim are Mill assumed lo be finite-; otherwise, the variance ratio is no longer well defined. This rules out distributions with infinite variance, such as those in the stable tareto-levy family (with characteristic exponents that are less than 2) proposed In Mandelbrot (1463) and Fama (1Jlifi). However, many other tonus of lepmkurtosis are allowed, such as that generated by F.ngles (HW21 autoregressive conditionally heteroskedastir (ARC.Ill process (see Section 12.2 in Chapter 12).

2.5. Long-Horizon Returns

Second, И - that under H(* (condition (114)) the autocorrelation coefficient estim (A) are asymptotically uncorrelated.ir If the asymptotic variance b), < ii of the p(A)s can be obtained under 11,*, the asymptotic variance tl{q) ol VR(ty) may be calculated as the weighted sum of the cVs, where the weights are simply die weights in relation (2.4.41) squared. Denote by 8k and 0(q) the asymptotic variances of p(k) and VR(<y), respectively. Then under die null hypothesis 11,* Lo and MacKinlay (1988) show that

1. The statistics VD(r/), and VR(ty)-1 converge almost surely to zero for all q ;ts n increases without bound.

2. The following is a heteroskedasticity-consistent estimator of c5*:

Sk = ; -----~-. (2.4.42)

[EjLVft-ft-.-A)*]

3. The following is a heteroskedaslicily-consistetu estimator of (?(</):

9(q) = 4 E ~ ~j (2.4.43)

Despite the presence of general heteroskedasticity, the standardized test

statistic fi*(q)

can be used lo test HJ in the usual way.

2.5 Long-Horizon Returns

Several recent studies have focused on the properties of long-horizon returns to test the random walk hypotheses, in some cases using 5- to 10-year returns over a 65-year sample. There are fewer nonoverlapping long-horizon returns for a given time span, so sampling errors are generally

ь Ahhough [his restriction on (lie fourth cross-moments ol may seem somewhat uniiilu-iuvc, и i.s satisfied lor .my process with independent increments (regardless of heterogeneity) and л1м> lor linear (laussian ARCH processes. This assumption may be relaxed entirely, requiring the estimation ollhe asymptotic covariances ol the autocorrelation estimators in order to estimate die limiting variance в o( VR() via {1AAI). Although (he resulting estimator ol в uould be more сеян plicated than equation (2.4.-4H), it is conceptually straightforward and may readily be iormed along the lines olNewey and West (HJH7). An even more genet al (and possibly more exact) sampling theory tor the variance ratios may he obtained using the results of Did our (HWI) and DuJour and Roy (1УК5). Again, this would sacrilice much of l lie simplicity tA our asymptotic results.

larger for statistics based on long-horizon returns. But for some alternatives to the random walk, long-horizon returns can be more informative than their shorter-horizon counterparts (sec Section 7.2.1 in Chapter 7 and I.о and MacKinlay {1989]).

One motivation for using long-horizon returns is the permanent/transitory components alternative hypothesis, first proposed by Muth (1960) in a macroeconomic context. In this model, log prices arc composed of two components: a random walk and a stationary process,

= щ+у, (2.5.1)

w, = /( + + f e, ~ IID(0,cr) y, = any zero-mean stationary process,

and (n>,l and \y,) are mutually independent. The common interpretation lor (2.5.1) as a model of stock prices is that wt is the fundamental component that reflects the efficient markets price, and y, is a zero-mean stationary component that reflects a short-term or transitory deviation from the efficient-markets price > implying the presence of fads or other market inefficiencies. Since v, is stationary, it is mean-reverting by definition and reverts to its mean of zero in the long run. Although there are several difficulties with such an interpretation of (2.5.1)-not the least of which is the fact that market efficiency is tautological without additional economic structure-nevertheless, such an alternative provides a good laboratory for studying the variance ratios performance.

While VR(</) can behave in many ways under (2.5.1) for small q (depending on the correlation structure оГу(), as q gets larger the behavior of VR(q) becomes less arbitrary. In particular, observe that

>, = h - /V i = il + <i + y, - y,-\ (2.5.2)

Va. I r,(q) I = I 1 + 2к,(0) - 2y,(q). (2.5.4)

where yv(iy)= Covy yn (/ is the autocovariance function of y,. Therefore, in tliis case the population value of the variance ratio becomes

VR(7, = = рт + 2Ут(0)-2)

4\:\r\r,\ y(a-r-2yv(0)-2yv(l))

- -- as я - 00 (2.5.6)

ff + 2yT(0)-2y,(l)

2)/v(0)-2)/,(l)

a* + 2y,(0)-2ft(l)

Var[Ay] Var[Ay] + Var[Ai ]

where (2.5.6) requires the additional assumption that y,(q)-*0 as q-юо, an asymptotic independence condition that is a plausible assumption for most economic time series. This shows that for a sufficiently long horizon q, the permanent/transitory components model must yield a variance ratio less than one. Moreover, the magnitude of the difference between the long-horizon variance ratio and one is the ratio of the variance of Ду, to the variance of Ap a kind of signal/ (signal+noise) ratio, where the signal is the transitory component and the noise is the permanent markets component. In fact, one might consider extracting the signal/noise ratio from VR(r/) in the obvious way:

1 Var[Ay]

VR( ) Var[Aw]

2.5.1 Problems with Ij>ng-Horizon Inferences

There are, however, several difficulties with long-horizon returns that stem from the fact that when the horizon q is large relative to the total time span T=nq, the asymptotic approximations that are typically used to perform inferences break down.

For example, consider the test statistic VR(9)-1 which is asymptotically normal with mean 0 and variance:

у = 2<2?-l)(7-l) = A

?>nql 3n

*2 J

(2.5.8)

under the RWl null hypothesis. Observe that for all q>2, the bracketed term in (2.5.8) is bounded between j and 1 and is monotonically increasing in q. Therefore, for fixed n, this implies upper and lower bounds for V are ~ and r, respectively. Now since variances cannot be negative, the lower

litis is implied by ergodirity, and even the long-range4lependent time series discussed Section 2.b satisfy this condition.