impossible to catalog all tests ofllD in any systematic fashion, and we shall mention only a few of the most well-known tests.

Since IID are properties of random variables that are not specific i-i a particular parametric family of distributions, many of these tests fall under the rubric of nonparamelric tests. Some examples are the Spearman rank correlation test, Spearmans footrule test, the Kendall r correlation test, and other tests based on linear combinations of ranks or /{-statistics (see Randies and Wolfe [1979] and Serfling [1980]). By using information contained solely in the ranks of the observations, it is possible to develop tests of dD that are robust across parametric families and invariant to changes in units of measurement. Exact sampling theories for such statistics are generally available but cumbersome, involving transformations of the (discrete) uniform distribution over the set of permutations of the ranks. However, for most оГ these statistics, normal asymptotic approximations to the sampling distributions have been developed (see Serfling [1980]).

jMore recent techniques based on the empirical distribution function of the data have also been used to construct tests of IID. These tests often Require slightly stronger assumptions on the joint and marginal distribution functions of the data-generating-process; hence they fall into the clasj; of semiparametric tests. Typically, such tests form a direct comparison Ibetween the joint and marginal empirical distribution functions or an indirect comparison using the quantjles of the two. For these test statistics,jexact sampling theories are generally unavailable, and we must rely on asynjptolic approximations to perform the tests (see Shorack and Wellner [19Й6]).

Under parametric assumptions, tests of IID are generally easier to construct. For example, to test for independence among к vectors which arc-jointly normally distributed, several statistics may be used: the likelihood ratio statistic, the canonical correlation, eigenvalues of the covariance matrices, etc. (see Muirhead [1983]). Of course, the tractability of such tests must be traded off against their dependence on specific parametric assumptions. Although these tests are often more powerful than their nonparamelric counterparts, even small departures from the hypothesized parametric family can lead to large differences between the actual and nominal sizes of the tests in finite samples.

2.2.2 Sequences and Reversals, and Runs

The early tests of the random walk hypothesis were largely tests of RW1 and RW2. Although they are now primarily of historical interest, nevertheless we can learn a great deal about the properties of the random walk from such tests. Moreover, several recently developed econometric tools rely heavily on RW1 (see, for example, Sections 2.5 and 2.6), hence a discussion of these

tests also provides us with an opportunity to develop some machinery that we shall require later.

Sequences and Reversals

We begin with the logarithmic version of RW1 or geometric Brownian mo- tion in which the log price process p, is assumed lo follow an IID random walk without drift:

Pt = +f e, ~ HDOXV) (2.2.1) 1

and denote by /, the following random variable:

11 if r, = p, - pi-i > 0 1 (2.2.2)

0 if r, = p, - />, , < 0.

Much like the classical Bernoulli coin-toss, /, indicates whether the daie-i continuously compounded return r, is positive or negative. In fact, the coin-tossing analogy is quite appropriate as many of the original tests of RW1 were based on simple coin-iossing probabilities.

One of the first tests of RW1 was proposed by Cowles and Jones (1937) and consists of a comparison of the frequency of sequences and reversals m historical slock returns, where the former are pairs of consecutive returns with the same sign, and the latter are pairs of consecutive returns with opposite signs. Specifically, given a sample of n+l returns rt, ..., r +), the number of sequences Nt and reversals /V, may be expressed as simple functions of the /,s:

N, = £ Y Y, = I, 1 + (1 - /,)(! - /,+,) (2.2.3)

N, = n-Ns. (2.2.4)

If log prices follow a driftless IID random walk (2.2.1), and if we add the further restriction that the distribution of the increments et is symmetric, then whether r, is positive or negative should be equally likely, a fair coin-toss with probability one-half of either outcome. This implies that for any pair of consecutive returns, a sequence and a reversal are equally probable; hence the Cowles-Jones ratio CJ = NJN, should be approximately equal to one. More formally, this ratio may be interpreted as a consistent estimator of the ratio CJ of the probability тг, of a sequence to the probability of a reversal 1 - яг, since:

where denotes convergence in probability. The fact that this ratio exceeded one for many historical slock returns series led Cowlcs and Jones (19.47) lo conclude thai litis represents conclusive evidence of structure in stock prices. 1

However, the assumption of а /его drift is critical in determining the value off;]. In particular, (J will exceed one for an IID random walk with drift, since a drift-either positive or negative-clearly makes sequences more likely than reversals. To see this, suppose that log prices follow a normal random walk with drift:

pi = д+ />,-i+e , ~ ЛЛО.ст2).

Then the indicator variable /, is no longer a fair coin-toss but is biased in the direction of the drift, i.e.,

!l with probability я (2.2.5) 0 with probability 1 - я.

where

PrO, > 0) = 4>()- (2.2.6)

If the drift fi is positive then я > , and if it is negative then л < . Under this more general specification, the ratio of я, to 1 - я, is given by

= я(1-л)2 > 2я(1-я)

As long as the drift is nonzero, it will always be the case that sequences arc more likely than reversals, simply because a nonzero drift induces a trend in the process. It is only for the fair-game case of я = that CJ achieves its lower bound of one.

To see how large an effect a nonzero drift might have on CJ, suppose that Ц - 0.08 and о = 0.21, values which correspond roughly to annual US stock returns indexes over the last halfkentttry. This yields the following estimate of я:

/0.08\

я = Ф I - ) = 0.0484

я, = A + (1 - nf = 0.5440 CJ = 1.19,

Mn .1 lain study, (amies (t№(>) corrects lor biases in time-averaged price data and still lintls C. ratios in est ess ol one. I lowevei. liis conclusion is somewhat more guarded: ... while our various analyses have dist losed a tendency towards persistence in stock price moveinenis, in no rase is llus sullii ienl lo provide more than negligible profits alter payment ol brokei.ige costs.

which is close to the value of 1.17 that Cowlcs and Jones (1937, Table 1) report for the annual returns of an index of railroad stock prices from 1835 to 1935. Is the difference statistically significant?

To perform a formal comparison of the two values 1.19 and 1.17 require a sampling theory for the estimator CJ. Such a theory may be obtained by noting from (2.2.3) that the estimator N, is a binomial random variable, i.e., the sum of n Bernoulli random variables V, where

I with probability я, = я2 + (1 - я)2; 0 with probability 1 - я,

hence we may approximate the distribution of Ns for large n by a normal distribution with mean E[iV,] = пя1 and variance Var[N,]. Because each pair of adjacent Y,s will be dependent,4 the variance of N, is not пя,{\ -я,) j-the usual expression for the variance of a binomial random variable-but is instead

Var[N,] = пя,(1 -я,) + 2пСоу[К K(+1]

= пя,(1 - я,) + 2 (яя + (1 -я)*-я2). (2.2.7)

Applying a first-order Taylor approximation or the delta method (see Section A.4 of the Appendix) to CJ = N,/(n - N,) using the normal asymptotic approximation for the distribution of Ns then yields

qA( iEL-, .a-J + + a-)B-?) ), (2.2.8)

where ~ indicates that the distributional relation is asymptotic. Since the Cowlesandjones (1937) estimate of 1.17 yields я, = 0.5392 andji = 0.6399, with a sample size n of 99 returns, (2.2.8) implies that the approximate standard error of the 1.17estimate is0.2537. Therefore, the estimate 1.17 is not statistically significantly different from 1.19. Moreover, under the null hypothesis я = , CJ has a mean of one and a standard deviation of 0.2010; hence neither 1.17 or 1.19 is statistically distinguishable from one. This provides little evidence against the random walk hypothesis.

On the other hand, suppose the random walk hypothesis were false- would this be detectable by the CJ statistic? To see how departures from the random walk might affect the ratio CJ, let the indicator /, be the following

,ln (act, I, is a two-state Markov chain with probabilities PrfK, = 1 Y,-\ = I) = (p1 + (1 - )V/, and Vr(Y, = 0 I = 0) = 1/2.

(wo-statc Markov chain:

2. The Iledit lability of Awl Retains

- и ft

(2.2.9)

wlici.c a denotes the conditional probability that r,+ \ is negative, conditional on a Positive r,. and fi denotes the conditional probability that r,+ \ is positive, conditional on a negative II a = 1-/5, this reduces to the case examined above (set л = \-a): the III) random walk with drill. As long as а ф I -fi, I, (hj-nce r,) will be serially correlated, violating RW1. In this case, the thcoicliral value of die ratio CJ is given by

<3 =

(I - a)ft + (I -fi)a 2afi

(2.2.10)

whicli can take on anv nonnegative i eal value, as illustrated by the lollowiiiL table.

As ct and ft both approach one, the likelihood ol reversals increases and hence CJ approaches 0. As cither a or fi approaches /его, the likelihood of sequences increases and CJ increases without bound. In such cases, GJ is clearly a reasonable indicator of departures from RW1. However, note that there exist combinations of (a. fi) for which афХ-fi and CJ=1, e.g., (ct, fi)=(-v д); hence the CJ statistic cannot distinguish these cases from RWI (see Problem 2.3 for further discussion).

Huns

Another common test for RWI is the rims lest, in which the number of sequences of consecutive positive and negative returns, or runs, is tabulated and compared against its sampling distribution under ihe random walk hypothesis. For example, using the indicator variable /, defined in (2.2.2). a particular sequence of 10 returns may be represented by 1001 1 I (1100. containing three runs of Is (of length 1. 3. and 1. respectively and three runs

2.2. Tests of Random Walk 1: III) Increments

of Os (of length 2, 1, and 2, respectively), thus six runs in total. In contrast, the sequence 0000011111 contains the same number of 0s and Is, bul only 2 runs. Ну comparing the number of runs in the data with the expected number of runs under RWI, a test of the IID random walk hypothesis may be constructed. To perform the lest, we require the sampling distribution of the total number of runs N(um in a sample of n. Mood (1940) was the fust lo provide a comprehensive analysis of runs, and we shall provide a brief summary of his most general results here.

Suppose that each of n IID observations lakes on one of q possible

values with probability n i - 1.....q (hence - ) me case of

the indicator variable /, defined in (2.2.2), q is equal to 2; we shall return to this special case below. Denote by Nmm(i) the total number of runs of type i (of any length), i = \,...,q; hence the total number of runs Nnms = IIi Mims()- Using combinatorial arguments and the properties of the multinomial distribution, Mood (1940) derives die discrete distribution of A/nln.s( ) from which he calculates the following moments:

F[iV,4ms(i)] = 7от,(1-я,) + я,- (2.2.11)

Var[/Vums(<)] = 1 - An-, + (wf - Зл;)

+ 7г;(3-87г, + Гт;) (2.2.12)

Cov[/v rom(<). КхтЛ])] = -М7Г,7Г,(1 - 2тГ, - 2л, + ЗтГ,Л})

- л, л,(2л, + 2kj - 5л, л,). (2.2.13)

Moreover, Mood (1940) shows thai the distribution of the number of runs converges lo a normal distribution asymptotically when properly normalized, ln particular, we have

<Чшъ(<) - iTi( 1 - л,) - л} Xj = -

~ /V(0,*i(l-л,)-3л;(1 (2.2.14)

CovU-.jc,! = -7г,л-,(1 - 2л, - 2л, + Зя,я;) (2.2.15)

А/,. - н(1 - Е,71 , ) х и - -

where = indicates that die equality holds asymptotically. Tests of RWI may then be performed using the asymptotic approximations (2.2.14) or