Under the same assumptions, Fuller (1976, Corollary 6.3.5.1) shows that the asymptotic distribution of the vector of autocorrelation coefficient estimators is also multivariate normal:

Уг[р(0)-р(0) p(l)-p(l) p(m)-p(m)] ~ AT(0,G), (2.4.9) where G =

ft - Y, [pWp{t-i+j) + p(l+j)p{l-i)-2p(j)p(l)p(f-i)

t=-oo

- 2p(i)p(t)Pd-j) + 2p(i)p(j)p2(e)}. (2.4.10)

For bnrposes of testing the random walk hypotheses in which all the population autocovariances are zero, these asymptotic approximations reduce to simpler forms and more can be said of their finite-sample means and variances. In particular, if [r,J satisfies RWI and has variance a1 and sixth morrent proportional to a6, then

E[p(A)] =

Cov[p(*), ptf)] =

T-k T(T-l)

+ 0(7-2)

£+0(7-*) if k

г Ф о

0(T~2)

otherwise.

(2.4.11)

(2.4.12)

From (2.4.11) we see that under RWI, where p(k)=0 for all k>0, the sample autocprrelation coefficients p(k) are negatively biased. This negative bias comes from the fact that the autocorrelation coefficient is a scaled sum of cross-products of deviations of r, from its mean, and if the mean is unknown it must be estimated, most commonly by the sample mean (2.4.6). 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.

For smaller samples this effect can be significant: The expected value of p(l) for a sample size of 10 observations is -10%. Under RWI, Fuller (1976) proposes the following bias-corrected estimator p(/c):7

(2.4.13)

7Nol that />(*) is not unbiased, tin- term bias-coi-reeled refers to die Ian thai Е1р(*)1=0(Г-2).

With uniformly hounded sixth moments, he shows that the sample autocorrelation coefficients are asymptotically independent and normally distributed with distribution:

s/Tp(k) ~ JV((>. 1) (2.4.14)

-Д=р(А) ~ MO, I). (2.4.15)

These results yield a variety of autocorrelation-based tests of the random walk hypothesis RWI.

l.o and MacKinlay (1988), Richardson and Smith (1994), and Romano and Thombs (1996) derive asymptotic approximations for sample autocorrelation coefficients under even weaker conditions-iincorrelated weakly tit-pendent observations-and these results may be used to construct tests of RW2 and RW3 (sec Section 2.4.3 below).

2.4.2 Portmanteau Statistics

Since RWI implies that all autocorrelations are zero, a simple test statislic of RWI that has power against many alternative hypotheses is the Q-slatislic due to Чох and Pierce (1970):

Q, = 7][>-(/<). (2.4.16)

Under the RWI null hypothesis, and using (2.4.14), it is easy to see that (l = 7£ = P2(/<) is asymptotically distributed as xf - l.jmvg and Box (1978) provide the following finite-sample correction which yields a better (it to the xi, or small sample sizes:

CZ = П7Ч-2) (2.4.17)

By summing the squared autocorrelations, the Box-Pierce (J-statistic is designed to delect departures from zero autocorrelations in either direction and at all lags. Therefore, it has power against a broad range of alternative hypotheses to the random walk. However, selecting the number of autocorrelations m requires some care-if too few are used, the presence of higher-order autocorrelation may he missed; if loo many arc used, the test may not have much power due to insignificant higher-order autocorrelations. Therefore, while such a portmanteau statistic does have some appeal, better tests of the random walk hypotheses may be available when specific alternative hypotheses can be identified. We shall (urn to such examples in the next sections.

г. I lie rieilictabitily oj Asset Helunis

2.4. 7 Variance Ratios

An important property ol all three random walk, hypotheses is thai the variance of random walk increments must he a linear function of the lime interval.K For example, under RWl for log prices where continuously coin-pounded returns r,= log /,-log /, i are IID, the variance of >,+ >, i must he twice the variance of r,. Therefore, the plausibility of the random walk model may be checked bv comparing the variance of r, + r,-i to twice the variance of r,. Of course, in practice these will not be numerically identical even if RWl were line, but their ratio should be statistically indistinguishable from one. Therefore, lo construct a statistical test of the random walk hypothesis using variance ratios, we require their sampling distribution under the random walk null hypothesis.

Population Propel ties of Variance Ratios

Before deriving such sampling distributions, we develop some intuition for the population values of the variance ratio statistic under various scenarios. Consider again the ratio of the variance of a two-period continuously compounded return ,( ) = r, + > i lo twice the variance of a one-period return i and for the moment let us assume nothing about the time series of returns other than slationai ity. Then this variance ratio, which we write as VR(2), reduces to:

Varr,(2) = Уаг[г,+ ,) 2Va.r, 2Varr,

2Varr,l +2Cov[r,.r,.l1 2Var[r,]

l+p(l), (2.4.18)

where p{\) is the fust-order autocorrelation coefficient of returns (>,). For any stationary lime series, the population value of the variance ratio statistic VR(2) is simply one plus the fust-order autocorrelation coefficient. In particular, under RWl all the autocorrelations are zero, hence VR(2)=1 in this case, as expected.

In the presence of positive lirsi-order autocorrelation, VR(2) will exceed one. If returns are posilivclv autocorrelated, the variance of (he sum of two

This linearity property is шоп- diHicnlt lo stale in ilie ease of RW2 and KYV:i because ilie variances ol increments inav vary through time. However, even in these cases the variance ol lite sum must c<u.tl iIn- sum ol the variances, and this is the linearity property which the variant с ratio lest exploits. We shall ciinstruct tests ol all three hypotheses below.

Mam studies have exploited ibis piopel IV ol the random walk hypothesis in devising empirical tests ol piedn lability; tei ent examples include (lanipbell and Mankiw (I9H7). < ioi lii.ine (I.IKH), l-aust (IlUJh In .mil M.u-Kintav (IlXX). Ioteiba and Suuimeis (19XX). Iti. haiiUou (199:1). and Kit b.inlson anil Sun к ( l-ISO).

VR(2) =

VR(2) =

2.4. Tests of Random Walk 3: Uncorrelated Increments

one-period returns will be larger than the sum of (he one-period returns variances; hence variances will grow faster than linearly. Alternatively, in the presence of negative first-order autocorrelation, the variance of the sum ol two one-period returns will be smaller than (he sum of the one-period returns variances; hence variances will grow slower than linearly.

For comparisons beyond one- and two-period returns, higher-order autocorrelations come into play. In particular, a similar calculation shows tha( the general 17-pcriod variance ratio statistic VR(q) satisfies the relation:

Var[r,(?)] ,-1

<7-Var[r t=

to-;)-

VR(<?) = -Г7-7= 1 +2 у 1 - - )p(k), (2.4.19)

where r,(k) = r, + r, i H-----rr, *+i and/0(A) is the kih order autocorrelation

coefficient of (r,. This shows that VR(q) is a particular linear combination of (he first fc-1 autocorrelation coefficients of r,j, with linearly declinir weights.

Under RWl, (2.4.19) shows that for all q, VR(<7)=1 since in this casje p(A)=0 for all k> 1. Moreover, even under RW2 and RW3, VR(g) must still equal one as long as the variances of r, are finite and the average variance Y.L1 Var[r,]/7converges to a finite positive number. But (2.4.19) is even more informative for alternatives to the random walk because it relates the behavior of VR(q) to the autocorrelation coefficients of (r,) under such alternatives. For example, under an AR(1) alternative, r, = 0r, + (2.4.19) implies that

izl / k

VR(9) = I+2W1-

k= i v

= 1 +

фч ф-4

q qO-ф)

Relations such as this are critical for constructing alternative hypotheses for which the variance ratio test has high and low power, and we shall return to this issue below.

Sampling Distribution of VD(q) andVR(q) under RWl

To construct a statistical test for RWl we follow the exposition of Lo and MacKinlay (1988) and begin by stating the null hypothesis H0 under which the sampling distribution of the test statistics will be derived.10 I.el p, denote (lie log price process and r, = p,-p,-\ continuously compounded returns.

Im ahernaiive expositions see Campbell and Mankiw (19X7), Cochrane (19XX), Faust (199.). Ioteiba and Summers (I9HX), Richardson (1993). and Richardson and Stock (19X9).

Then the null hypothesis we consider in this section is11

Ж H0 : r, = /<+e e, UDAf(0,a2).

our data consist of 2/i-f-1 observations of log prices p\.....fan), and

р. consider the following estimators for ц and cr2:

i i

= 2 E (/* - /*-t> = (A* - A)) (2.4.20) 1 2

a s E<A*-/*- -/ О (2.4.21)

< s it; E (Ь - А -* - 2£)2. (2.4.22)

Equations (2.4.20) and (2.4.21) are the usual sample mean and variance estimators. They are also the maximum-likelihood estimators of ц and a1 (see Section 9.3.2 in Chapter 9). The second estimator a2 of a2 makes use of the random walk nature of/),: Under RWI the mean and variance of increments are linear in the increment interval, hence the a2 can be estimated by one-half the sample variance of the increments of even-numbered observations ...../blunder standard asymptotic theory, all three estimators are strongly consistent: Holding all other parameters constant, as the total number of observations 2n increases without bound the estimators converge almost surely to their population values. In addition, it is well known that a/and a2 possess the following normal limiting distributions (see, for example, Stuart and Ord [1987]): i

j. i Vbi(a2~a2) ~ Я(()Лстл) (2.4.23)

g I Vbi(d2 -a2) ~ Af(0,4a4). (2.4.24)

However, we seek the limiting distribution of the ratio of the variances. Щ Although it may readily be shown that the ratio is also asymptotically normal ** with uliit mean under RWI, the variance of the limiting distribution is not Ш apparent since the two variance estimators arc clearly not asymptotically

uncorrected.

Ш But since the estimator a2 is asymptotically efficient under the null

hypothesis RWI, we may use llausmans (1978) insight that the asymptotic

fe We assume normality only loi exiiosiiioii.il convenience-the results in this section apply

Ш much more generally to log price processes with III) increments that possess linite tomtit

inoments.

variance of the difference of a consistent estimator and an asymptotically cfliciciii estimator is simply the difference of llicjisymplolic variances.12 If we deline the variance different с estimator as VD(2) s rr,; - <72, then (2.4.23), (2.4.24), and llausmans result implies:

s/2~i7Vf)(2) ~ A/ (0.2a). (2.4.25)

The null hypothesis can then be tested using (2.4.25) and any consistent estimator 2a 1 of 2a1 (for example, 2(<т2)2): Construct the standardized statistic- Vl)(2)/s/2cJ which has a limiting standard normal distribution under RWI, and reject the null hypothesis at the 5% level if it lies outside the interval [-1.9(i, 1.90].

The asymptotic distribution of the two-period variance ratio statistic VR(2) = afjcr2 now follows directly from (2.4.25) using a first-order Taylor approximation or the delta method (see Section A.4 of the Appendix):

VR(2) s % , s/27,(VR(2) - 1) ~ ЛА >,2). (2.4.2C.)

Tlit null hypothesis Hn can be tested by computing the standardized statistic s/2h(VR(2)-1)/\/2 which is asymptotically standard normal-if it lies outside the interval [-1.90, 1.9(i, RWI may be rejected al the 5% level of significance.

Although the variance ratio is often preferred lo the variance dilfcrcncc because the ratio is scale-free, observe that if 2(<т2)- is used to estimate 2r> , then the standard significance test of V1)=0 for the difference will yield the same inferences as the corresponding test of VR-1=0 for the ratio since:

2 V1)(2) fUal - a2) (VR(2) - I) J}

vj Ло* У2

Therefore, in this simple context the two test statistics are equivalent. However, there are other reasons that make the variance ratio more appealing

-Iriclh. llausiuan (HI7H) exploits the fact that any asymptotic ally ellicienl estimator ol a parameter I), sav t l mrrst possess the prupeity that it is asymptotically utrcoi related with the ililTi-reni с i> - ( /,., where (5 is any other estimator ol II. II not, then there exists a linear comhiuation ol II,. audi,- r that is more ellicienl than I) < ontr.idiciiiig the assumed elliciency ol и,. The result lollows directly, then, since:

aV.rri, = aVar(),-H< - <i = aVai r aV.n - ,

=> .iVar(7 -fi,. = aVarlil - aVarlil.

where a\.u I 1 denotes the asymptotic variance operator.

1 in particular, apply the delta method to [\ll\. l\,)=U\fth, where So/-c;,;. Ih= !,. ami obsei vt- I bar nj-п;, ami n; ar e asymptolir ally rtirr or r r-l.tled l)r< arise cit] is an el lit ient estimator.