Table 2.2. Kxjiecled runs far n random walk with drift и.

Kxpected total number ol runs in a sample ol n independent Bernoulli (rials representing positive/negative roniinuoiislv < (impounded returns lor a Gaussian geometric Brownian motion with drill /I = 0%......20% and standard deviation a = 21%.

(2.2.10), and the probabilities л, may be estimated directly from the data as the ratios я, = ,/ , where n, is ol the number of runs in the sample of n Q that are the /lb type; thus и = n,.

To develop some sense of the behavior of the total number of runs, consider the Bernoulli case к = 2 corresponding to the indicator variable i /, defined in (2.2.2) of Section 2.2.2 where я denotes the probability that 1 /, = 1. In this case, the expected total number of runs is

! KIAnmsl = 2шг( I -я)+я2 + (1 -я)2. (2.2.17)

j Observe thai lor any n > 1, (2.2.17) is a globally concave quadratic function j in я on [0, 1 ] which attains a maximum value of (n + l)/2 at я = j. There-I fore, a drifiless random walk maximizes the expected total number of runs I for any fixed sample size n or, alternatively, the presence of a drift of either sign will decrease the expected total number of runs.

To see the sensitivity of F. /V ,. with respect to the drift, in Table 2.2 we report the expected total number of runs for a sample of n - 1.000 observations for a geometric random walk with normally distributed increments, drift /< = 0%......20%, and standard deviation rr = 21% (which is

j calibrated lo match annual US stock index returns); hence я = Ф(/</п). From Table 2.2 we see thai as the drift increases, the expected total number of inns declines considerably, from 500.5 for zero-drift to 283.5 for a 20% drift. I lowever, all of these values are still consistent with the random walk hypothesis.

... vj .,и,ишш чаш t.. lnaefienaent increments

To perform a test for the random walk in the Bernoulli case, we may calculate the following statistic:

ли-2шг(1-*) - mi)

2у/пл(1 -я)[1 -Зя(1 - я)]

and perform the usual test of significance. A slight adjustment to this statistic is often made to account for the fact that while the normal approximation yields different probabilities for realizations in the interval [Nmn NronJ 4-1), the exact probabilities are constant over this interval since /Ушт is integer-valued. Therefore, a continuity correction is made in which the z-statistic is evaluated at the midpoint of the interval (see Wallis and Roberts [1956]); thus

Л/П1т-Н-2 я(1-я) a .

2vW(l -л )[1 -Зя(1 -я)]

Other aspects of runs have also been used to test the IID random walk, such as the distribution of runs by length and by sign. Indeed, Moods (1940) seminal paper provides an exhaustive catalog of the properties of runs, including exact marginal and joint distributions, factorial moments, centered moments, and asymptotic approximations. An excellent summary of these results, along with a collection of related combinatorial problems in probability and statistics is contained in David and Barton (1962). Fama (1905) presents an extensive empirical analysis of runs for US daily, four-day, nine-day, and sixteen-day stock returns from 1956 to 1962, and concludes that, ...thereis no evidence of important dependence from either ah investment or a statistical point of view. j

More recent advances in the analysis of Markov chains have generalized the theory of runs to non-IID sequences, and by recasting patterns such as a run as elements of a permutation group, probabilities of very complex patterns may now be evaluated explicitly using the firft-passage or hilling time of a random process defined on the permutation group. For these mor recent results, sec Aldous (1989), Aldous and Diaconis (1986), and Diaconijs (1988).

2.3 Tests of Random Walk 2: Independent Increments

The restriction of identical distributions is clearly implausible, especially when applied to financial data that span several decades. However, testing for independence without assuming identical distributions is quite difficult j . particularly for time series data. If we place no restrictions on how the, marginal distributions of the data can vary through time, it becomes virtually! impossible to conduct statistical inference since the sampling distributions of even the most elementary statistics cannot be derived.

I Sonic of (he nonparamelric methods mentioned in Section 2.2.1 such as rank correlations do test for independence without also requiring identical distributions, but the number of distinct marginal distributions is typically

fa litjtite and small number. For example, a test of independence between IQ scores ami academic performance involves two distinct marginal dis-tribiitions: one for 1Q scores and the other for academic performance. Multiple observations are drawn from each marginal distribution and vari-ous iiionparametric tests can be designed to check whether the product of Ш the marginal distributions equals the joint distribution of the paired ot>

щ. scrvjitions. Such an approach obviously cannot succeed if we hypothesize a unique marginal distribution for each observation of IQ and academic щр. performance.

Щ Nevertheless, there are two lines of empirical research that can be

viewed as a kind of economic test of RW2: filter rides, and technical analysis. Although neither of these approaches makes much use of formal statistical Шр- inference, both have captured the interest of the financial community for

practical reasons. This is not to say that statistical inference cannot be applied to these modes of analysis, but rather that the standards of evidence in this literature have evolved along veiy different paths. Therefore, we shall present only a cursory review of these techniques.

2.3.1 Filter Rules

To test RW2, Alexander (1961, 1964) applied a filter rule in which an asset is purchased when its price increases by x%, and (short)sold when iLs price drops by x%. Such a rule is said to be an x% filter, and was proposed by Alexander (1961) for the following reasons:

Suppose we tentatively assume the existence of trends in stock market prices but believe them to be masked by the jiggling of the market. We might fitter out all movements smaller than a specified size and examine the remaining movements.

The total return of this dynamic portfolio strategy is then taken to be a measure of the predictability in asset returns. A comparison of the total return to the return from a buy-and-hold strategy lor the Dow Jones and Standard and Poors industrial averages led Alexander to conclude that

... there <ire trends in stock market prices

Fama (1965) and Fama and Blume (1966) present a more detailed empiric: ] analysis of filter rules, correcting for dividends and trading costs, and conclude that such rules do not perform as well as the buy-and-hold strategy. In the absence of transactions costs, very small filters (1% in Alexander [ 1964] and between 0.5% and 1.5% in Fama and Blume [ 1966]) do yield superior returns, but because small filters generate considerably

more freement trading, Fama and Blume (1966) show that even a 0.1% roundtrip transaction cost is enough to eliminate the profits from such filler rules.

2.3.2 Technical Analysis

As a measure of predictability, the filler rule has the advantage of practical relevance-it is a specific and readily implemenlable trading strategy, and the metric of its success is total return. The filter rule is just one example ol a much larger class of trading rules arising from technical analysis or charting. Technical analysis is an approach lo investment management based on the belief that historical price series, trading volume, and other market statistics exhibit regularities-often (but not always) in the form of geometric patterns such as double bottoms, head-and-shoulders, and support and resistance levels-that can be profitably exploited lo extrapolate future price movements (see, for example, Edwards and Magee [1966] and Murphy [1986]). hi the words of Edwards and Magee (1966):

Technical analysis is the science of recording, usually in graphic form, die actual history of trading (price changes, volume of transactions, etc;) in a certain stock or in the averages and then deducing from ihat pictured history the probable future trend.

Historically, technical analysis has been the black sheep of die academic finance community. Regarded by many academics as a pursuit that lies somewhere between astrology and voodoo, technical analysis has never enjoyed the same degree of acceptance that, for example, fundamental analysis has received. This state of affairs persists today, even though the distinction between technical and fundamental analysis is becoming progressively fuzzier.5 Perhaps some of the prejudice against technical analysis can be attributed to semantics. Because fundamental analysis is based on quantities familiar to most financial economists-for example, earnings, dividends, and other balance-sheet and income-statement items-it possesses a natural bridge to the academic literature. In contrast, the vocabulary of the technical analyst is completely foreign to the academic and often mystifying to the general public. Consider, for example, the following, which might be found in any recent academic finance journal:

The magnitudes and decay pattern of the first twelve autocorrelations and the statistical significance of the Box-Pierce Q-statistic suggest the presence of a high-frequency predictable component in stock returns.

Kcir example, many technical analyses no longer base their forecasts solely on past prices and volume but also use earnings and dividend information and other fundamental data, and as many fundamental analysis now look at past price and volume patterns in addition lo more inidiiional variables.

Given a covariance-stationary time series jr,}, the ftth order autocovariance and autocorrelation coefficients, y(k) and p{k), respectively, are definedis6

Y(k) s Cov[r r,+*l (2.4.2)

p{k) s Cov[r = Cov[r r+] = Ш (943)

v/VirTrJ~v/Var[r(+*] Var[r,] y(0) V

where the second equality in (2.4.3) follows from the covariance-stalionarity of (r,]. For a given sample (r,[ J= autocovariance and autocorrelation coefficients may be estimated in the natural way by replacing population moments with sample counterparts:

УМ = ~У>-7-)(г,+*-гт). 0 < к < T (2.4.4)

У (A)

P(k) = (2.4.5)

K(0)

1 T

fr s -Vr(. (2.4,6)

The sampling theory for y(k) and p(k) depends, of course, on the data-generating process for \r,}. For example, if r, is a finite-order moving average,

м *=o

where (c,) is an independent sequence with mean 0, variance аг, fourth moment ца*, and finite sixth moment, then Fuller (1976, Theorem 6.3.5) shows that the vector of autocovariance coefficient estimators is asymptotically multivariate normal:

[P(0)-K(0) i>(l)-y(D )>Ы)-у(т)} ~ JV(0.V). (2.4.7J) where

V = [Vii]

Vij s (r,-?>)y{i)y(j)+ £ [y(t)Yil-i+j)

t=-oo

+ Y(l+f)Y(e-i)]. (2.4.8)

The requirement of rt>variance-siationariry ь primarily for notational convenience- odierwise y(k) and p{k) maybe functions of I as well as k, and may not even be well-defined il second moments are not finite.

Contrast this with the statement:

The presence of clearly identified support and resistance levels, coupled with a one-third retracement parameter when prices lie between them, suggests die presence of strong buying and selling opportunities in the near-lerm.

Both statements have ihe same meaning: Using historical prices, one can predict future prices to some extent in the short run. But because the two statements are so laden with jargon, the type of response they elicit depends very much on the individual reading them.

Despite the differences in jargon, recent empirical evidence suggests that technical analysis and more traditional financial analysis may have much in common (see, in particular, Section 2.8). Recentstudiesby Blume, Easley, and OHara (1994), Brock, Lakonishok, and LeBaron (1992), Brown and Jennings (1989), LeBaron (1996), Neftci (1991), Pan (1991), Taylor and Allen (1992), and Treynor and Ferguson (1985) signal a growing interest in lechnical analysis among financial academics, and so it may become a more active research area in the near future.

2.4 Tests of Random Walk 3: Uncorrelated Increments

One of the most direct and intuitive tests of the random walk and martingale hypotheses for an individual time series is lo check for serial correlation, correlation between two observations of the same series at different dates. Under the weakest version of the random walk, RW3, the increments or first-differences of the level of the random walk are uncorrelated at all leads and lags. Therefore, we may test RW3 by testing the null hypothesis that die autocorrelation coefficients of the first-differences at various lags are all zero.

This seemingly simple approach is the basis for a surprisingly large variety of tests of the random walk, and we shall develop these tests in this chapter. For example, tests of the random walk may be based on the autocorrelation coefficients themselves (Section 2.4.1). More powerful tests may be constructed from the sum of squared autocorrelations (Section 2.4.2). Lineal combinations of the autocorrelations may also have certain advantages in delecting particular departures from the random walk (Sections 2.4.3 and 2.5). Therefore, we shall devote considerable attention to the properties of autocorrelation coellit ieiils in the coming sections.

2.4.1 Autocorrelation Coefficients

The autocorrelation coefficient is a natural liinc-series extension of the well-known correlation coefficient between two random variables .vand y: