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Foundations of Technical Analysis:Computational Algorithms, Statistical

Inference, and Empirical Implementation

ANDREW W. LO, HARRY MAMAYSKY, AND JIANG WANG*

ABSTRACT

Technical analysis, also known as charting, has been a part of financial practicefor many decades, but this discipline has not received the same level of academicscrutiny and acceptance as more traditional approaches such as fundamental analy-sis. One of the main obstacles is the highly subjective nature of technical analy-sisthe presence of geometric shapes in historical price charts is often in the eyesof the beholder. In this paper, we propose a systematic and automatic approach totechnical pattern recognition using nonparametric kernel regression, and we applythis method to a large number of U.S. stocks from 1962 to 1996 to evaluate theeffectiveness of technical analysis. By comparing the unconditional empirical dis-tribution of daily stock returns to the conditional distributionconditioned on spe-cific technical indicators such as head-and-shoulders or double-bottomswe findthat over the 31-year sample period, several technical indicators do provide incre-mental information and may have some practical value.

ONE OF THE GREATEST GULFS between academic finance and industry practiceis the separation that exists between technical analysts and their academiccritics. In contrast to fundamental analysis, which was quick to be adoptedby the scholars of modern quantitative finance, technical analysis has beenan orphan from the very start. It has been argued that the difference be-tween fundamental analysis and technical analysis is not unlike the differ-ence between astronomy and astrology. Among some circles, technical analysisis known as voodoo finance. And in his inf luential book A Random Walkdown Wall Street, Burton Malkiel ~1996! concludes that @u#nder scientificscrutiny, chart-reading must share a pedestal with alchemy.

However, several academic studies suggest that despite its jargon and meth-ods, technical analysis may well be an effective means for extracting usefulinformation from market prices. For example, in rejecting the Random Walk

* MIT Sloan School of Management and Yale School of Management. Corresponding author:Andrew W. Lo [email protected]!. This research was partially supported by the MIT Laboratory forFinancial Engineering, Merrill Lynch, and the National Science Foundation ~Grant SBR9709976!. We thank Ralph Acampora, Franklin Allen, Susan Berger, Mike Epstein, Narasim-han Jegadeesh, Ed Kao, Doug Sanzone, Jeff Simonoff, Tom Stoker, and seminar participants atthe Federal Reserve Bank of New York, NYU, and conference participants at the Columbia-JAFEE conference, the 1999 Joint Statistical Meetings, RISK 99, the 1999 Annual Meeting ofthe Society for Computational Economics, and the 2000 Annual Meeting of the American Fi-nance Association for valuable comments and discussion.

THE JOURNAL OF FINANCE VOL. LV, NO. 4 AUGUST 2000

1705

Hypothesis for weekly U.S. stock indexes, Lo and MacKinlay ~1988, 1999!have shown that past prices may be used to forecast future returns to somedegree, a fact that all technical analysts take for granted. Studies by Tabelland Tabell ~1964!, Treynor and Ferguson ~1985!, Brown and Jennings ~1989!,Jegadeesh and Titman ~1993!, Blume, Easley, and OHara ~1994!, Chan,Jegadeesh, and Lakonishok ~1996!, Lo and MacKinlay ~1997!, Grundy andMartin ~1998!, and Rouwenhorst ~1998! have also provided indirect supportfor technical analysis, and more direct support has been given by Pruitt andWhite ~1988!, Neftci ~1991!, Brock, Lakonishok, and LeBaron ~1992!, Neely,Weller, and Dittmar ~1997!, Neely and Weller ~1998!, Chang and Osler ~1994!,Osler and Chang ~1995!, and Allen and Karjalainen ~1999!.

One explanation for this state of controversy and confusion is the uniqueand sometimes impenetrable jargon used by technical analysts, some of whichhas developed into a standard lexicon that can be translated. But there aremany homegrown variations, each with its own patois, which can oftenfrustrate the uninitiated. Campbell, Lo, and MacKinlay ~1997, 4344! pro-vide a striking example of the linguistic barriers between technical analystsand academic finance by contrasting this statement:

The presence of clearly identified support and resistance levels, coupledwith a one-third retracement parameter when prices lie between them,suggests the presence of strong buying and selling opportunities in thenear-term.

with this one:

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

Despite the fact that both statements have the same meaningthat pastprices contain information for predicting future returnsmost readers findone statement plausible and the other puzzling or, worse, offensive.

These linguistic barriers underscore an important difference between tech-nical analysis and quantitative finance: technical analysis is primarily vi-sual, whereas quantitative finance is primarily algebraic and numerical.Therefore, technical analysis employs the tools of geometry and pattern rec-ognition, and quantitative finance employs the tools of mathematical analy-sis and probability and statistics. In the wake of recent breakthroughs infinancial engineering, computer technology, and numerical algorithms, it isno wonder that quantitative finance has overtaken technical analysis inpopularitythe principles of portfolio optimization are far easier to pro-gram into a computer than the basic tenets of technical analysis. Neverthe-less, technical analysis has survived through the years, perhaps because itsvisual mode of analysis is more conducive to human cognition, and becausepattern recognition is one of the few repetitive activities for which comput-ers do not have an absolute advantage ~yet!.

1706 The Journal of Finance

Indeed, it is difficult to dispute the potential value of price0volume chartswhen confronted with the visual evidence. For example, compare the twohypothetical price charts given in Figure 1. Despite the fact that the twoprice series are identical over the first half of the sample, the volume pat-terns differ, and this seems to be informative. In particular, the lower chart,which shows high volume accompanying a positive price trend, suggests thatthere may be more information content in the trend, e.g., broader partici-pation among investors. The fact that the joint distribution of prices andvolume contains important information is hardly controversial among aca-demics. Why, then, is the value of a visual depiction of that joint distributionso hotly contested?

Figure 1. Two hypothetical price/volume charts.

Foundations of Technical Analysis 1707

In this paper, we hope to bridge this gulf between technical analysis andquantitative finance by developing a systematic and scientific approach tothe practice of technical analysis and by employing the now-standard meth-ods of empirical analysis to gauge the efficacy of technical indicators overtime and across securities. In doing so, our goal is not only to develop alingua franca with which disciples of both disciplines can engage in produc-tive dialogue but also to extend the reach of technical analysis by augment-ing its tool kit with some modern techniques in pattern recognition.

The general goal of technical analysis is to identify regularities in the timeseries of prices by extracting nonlinear patterns from noisy data. Implicit inthis goal is the recognition that some price movements are significanttheycontribute to the formation of a specific patternand others are merely ran-dom fluctuations to be ignored. In many cases, the human eye can perform thissignal extraction quickly and accurately, and until recently, computer algo-rithms could not. However, a class of statistical estimators, called smoothingestimators, is ideally suited to this task because they extract nonlinear rela-tions [m~{! by averaging out the noise. Therefore, we propose using these es-timators to mimic and, in some cases, sharpen the skills of a trained technicalanalyst in identifying certain patterns in historical price series.

In Section I, we provide a brief review of smoothing estimators and de-scribe in detail the specific smoothing estimator we use in our analysis:kernel regression. Our algorithm for automating technical analysis is de-scribed in Section II. We apply this algorithm to the daily returns of severalhundred U.S. stocks from 1962 to 1996 and report the results in Section III.To check the accuracy of our statistical inferences, we perform several MonteCarlo simulation experiments and the results are given in Section IV. Weconclude in Section V.

I. Smoothing Estimators and Kernel Regression

The starting point for any study of technical analysis is the recognition thatprices evolve in a nonlinear fashion over time and that the nonlinearities con-tain certain regularities or patterns. To capture such regularities quantita-tively, we begin by asserting that prices $Pt % satisfy the following expression:

Pt 5 m~Xt ! 1 et , t 5 1, . . . ,T, ~1!

where m~Xt ! is an arbitrary fixed but unknown nonlinear function of a statevariable Xt and $et % is white noise.

For the purposes of pattern recognition in which our goal is to construct asmooth function [m~{! to approximate the time series of prices $ pt % , we setthe state variable equal to time, Xt 5 t. However, to keep our notation con-sistent with that of the kernel regression literature, we will continue to useXt in our exposition.

When prices are expressed as equation ~1!, it is apparent that geometricpatterns can emerge from a visual inspection of historical price seriesprices are the sum of the nonlinear pattern m~Xt ! and white noiseand


that such patterns may provide useful information about the unknown func-tion m~{! to be estimated. But just how useful is this information?

To answer this question empirically and systematically, we must first de-velop a method for automating the identification of technical indicators; thatis, we require a pattern-recognition algorithm. Once such an algorithm isdeveloped, it can be applied to a large number of securities over many timeperiods to determine the efficacy of various technical indicators. Moreover,quantitative comparisons of the performance of several indicators can beconducted, and the statistical significance of such performance can be as-sessed through Monte Carlo simulation and bootstrap techniques.1

In Section I.A, we provide a brief review of a general class of pattern-recognitiontechniques known as smoothing estimators, and in Section I.B we describe insome detail a particular method called nonparametric kernel regression on whichour algorithm is based. Kernel regression estimators are calibrated by a band-width parameter, and we discuss how the bandwidth is selected in Section I.C.

A. Smoothing Estimators

One of the most common methods for estimating nonlinear relations suchas equation ~1! is smoothing, in which observational errors are reduced byaveraging the data in sophisticated ways. Kernel regression, orthogonal se-ries expansion, projection pursuit, nearest-neighbor estimators, average de-rivative estimators, splines, and neural networks are all examples of smoothingestimators. In addition to possessing certain statistical optimality proper-ties, smoothing estimators are motivated by their close correspondence tothe way human cognition extracts regularities from noisy data.2 Therefore,they are ideal for our purposes.

To provide some intuition for how averaging can recover nonlinear rela-tions such as the function m~{! in equation ~1!, suppose we wish to estimatem~{! at a particular date t0 when Xt0 5 x0. Now suppose that for this oneobservation, Xt0, we can obtain repeated independent observations of theprice Pt0, say Pt0

1 5 p1, . . . , Pt0n 5 pn ~note that these are n independent real-

izations of the price at the same date t0, clearly an impossibility in practice,but let us continue this thought experiment for a few more steps!. Then anatural estimator of the function m~{! at the point x0 is

[m~x0! 51

n (i51n

pi 51

n (i51n

@m~x0! 1 eti# ~2!

5 m~x0! 11

n (i51n

eti , ~3!

1A similar approach has been proposed by Chang and Osler ~1994! and Osler and Chang~1995! for the case of foreign-currency trading rules based on a head-and-shoulders pattern.They develop an algorithm for automatically detecting geometric patterns in price or exchangedata by looking at properly defined local extrema.

2 See, for example, Beymer and Poggio ~1996!, Poggio and Beymer ~1996!, and Riesenhuberand Poggio ~1997!.


and by the Law of Large Numbers, the second term in equation ~3! becomesnegligible for large n.

Of course, if $Pt % is a time series, we do not have the luxury of repeatedobservations for a given Xt . However, if we assume that the function m~{! issufficiently smooth, then for time-series observations Xt near the value x0,the corresponding values of Pt should be close to m~x0!. In other words, ifm~{! is sufficiently smooth, then in a small neighborhood around x0, m~x0!will be nearly constant and may be estimated by taking an average of the Ptsthat correspond to those Xts near x0. The closer the Xts are to the value x0,the closer an average of corresponding Pts will be to m~x0!. This argues fora weighted average of the Pts, where the weights decline as the Xts getfarther away from x0. This weighted-average or local averaging procedureof estimating m~x! is the essence of smoothing.

More formally, for any arbitrary x, a smoothing estimator of m~x! may beexpressed as

[m~x! [1

T (t51T

vt ~x!Pt , ~4!

where the weights $vt ~x!% are large for those Pts paired with Xts near x, andsmall for those Pts with Xts far from x. To implement such a procedure, wemust define what we mean by near and far. If we choose too large aneighborhood around x to compute the average, the weighted average will betoo smooth and will not exhibit the genuine nonlinearities of m~{!. If wechoose too small a neighborhood around x, the weighted average will be toovariable, ref lecting noise as well as the variations in m~{!. Therefore, theweights $vt ~x!% must be chosen carefully to balance these two considerations.

B. Kernel Regression

For the kernel regression estimator, the weight function vt ~x! is con-structed from a probability density function K~x!, also called a kernel:3

K~x! $ 0, EK~u!du 5 1. ~5!By rescaling the kernel with respect to a parameter h . 0, we can change itsspread; that is, let

Kh~u! [1

hK~u0h!, EKh~u!du 5 1 ~6!

3 Despite the fact that K~x! is a probability density function, it plays no probabilistic role inthe subsequent analysisit is merely a convenient method for computing a weighted averageand does not imply, for example, that X is distributed according to K~x! ~which would be aparametric assumption!.


and define the weight function to be used in the weighted average ~equation~4!! as

vt, h~x! [ Kh~x 2 Xt !0gh~x!, ~7!

gh~x! [1

T (t51T

Kh~x 2 Xt !. ~8!

If h is very small, the averaging will be done with respect to a rather smallneighborhood around each of the Xts. If h is very large, the averaging will beover larger neighborhoods of the Xts. Therefore, controlling the degree ofaveraging amounts to adjusting the smoothing parameter h, also known asthe bandwidth. Choosing the appropriate bandwidth is an important aspectof any local-averaging technique and is discussed more fully in Section II.C.

Substituting equation ~8! into equation ~4! yields the NadarayaWatsonkernel estimator [mh~x! of m~x!:

[mh~x! 51

T (t51T

vt, h~x!Yt 5(t51

T

Kh~x 2 Xt !Yt

(t51

T

Kh~x 2 Xt !

. ~9!

Under certain regularity conditions on the shape of the kernel K and themagnitudes and behavior of the weights as the sample size grows, it may beshown that [mh~x! converges to m~x! asymptotically in several ways ~seeHrdle ~1990! for further details!. This convergence property holds for awide class of kernels, but for the remainder of this paper we shall use themost popular choice of kernel, the Gaussian kernel:

Kh~x! 51

h%2pe2x

202h2 ~10!

C. Selecting the Bandwidth

Selecting the appropriate bandwidth h in equation ~9! is clearly central tothe success of [mh~{! in approximating m~{!too little averaging yields afunction that is too choppy, and too much averaging yields a function that istoo smooth. To illustrate these two extremes, Figure 2 displays the NadarayaWatson kernel estimator applied to 500 data points generated from the relation:

Yt 5 Sin~Xt ! 1 0.5eZt , eZt ; N ~0,1!, ~11!

where Xt is evenly spaced in the interval @0,2p# . Panel 2 ~a! plots the rawdata and the function to be approximated.


(a)

(b)

Figure 2. Illustration of bandwidth selection for kernel regression.


(c)

(d)

Figure 2. Continued


Kernel estimators for three different bandwidths are plotted as solid lines inPanels 2~b!~c!. The bandwidth in 2~b! is clearly too small; the function is toovariable, fitting the noise 0.5eZt and also the signal Sin~{!. Increasing thebandwidth slightly yields a much more accurate approximation to Sin ~{! asPanel 2~c! illustrates. However, Panel 2~d! shows that if the bandwidth is in-creased beyond some point, there is too much averaging and information is lost.

There are several methods for automating the choice of bandwidth h inequation ~9!, but the most popular is the cross-validation method in which his chosen to minimize the cross-validation function

CV~h! 51

T (t51T

~Pt 2 [mh, t !2, ~12!

where

[mh, t [1

T (ttT

vt, hYt . ~13!

The estimator [mh, t is the kernel regression estimator applied to the pricehistory $Pt% with the tth observation omitted, and the summands in equa-tion ~12! are the squared errors of the [mh, ts, each evaluated at the omittedobservation. For a given bandwidth parameter h, the cross-validation func-tion is a measure of the ability of the kernel regression estimator to fit eachobservation Pt when that observation is not used to construct the kernelestimator. By selecting the bandwidth that minimizes this function, we ob-tain a kernel estimator that satisfies certain optimality properties, for ex-ample, minimum asymptotic mean-squared error.4

Interestingly, the bandwidths obtained from minimizing the cross-validationfunction are generally too large for our application to technical analysiswhen we presented several professional technical analysts with plots of cross-validation-fitted functions [mh~{!, they all concluded that the fitted functionswere too smooth. In other words, the cross-validation-determined bandwidthplaces too much weight on prices far away from any given time t, inducingtoo much averaging and discarding valuable information in local price move-ments. Through trial and error, and by polling professional technical ana-lysts, we have found that an acceptable solution to this problem is to use abandwidth of 0.3 3 h*, where h* minimizes CV~h!.5 Admittedly, this is an adhoc approach, and it remains an important challenge for future research todevelop a more rigorous procedure.

4 However, there are other bandwidth-selection methods that yield the same asymptotic op-timality properties but that have different implications for the finite-sample properties of ker-nel estimators. See Hrdle ~1990! for further discussion.

5 Specifically, we produced fitted curves for various bandwidths and compared their extremato the original price series visually to see if we were fitting more noise than signal, and weasked several professional technical analysts to do the same. Through this informal process, wesettled on the bandwidth of 0.3 3 h* and used it for the remainder of our analysis. This pro-cedure was followed before we performed the statistical analysis of Section III, and we made norevision to the choice of bandwidth afterward.


Another promising direction for future research is to consider alternativesto kernel regression. Although kernel regression is useful for its simplicityand intuitive appeal, kernel estimators suffer from a number of well-knowndeficiencies, for instance, boundary bias, lack of local variability in the de-gree of smoothing, and so on. A popular alternative that overcomes theseparticular deficiencies is local polynomial regression in which local averag-ing of polynomials is performed to obtain an estimator of m~x!.6 Such alter-natives may yield important improvements in the pattern-recognitionalgorithm described in Section II.

II. Automating Technical Analysis

Armed with a mathematical representation [m~{! of $Pt % with which geo-metric properties can be characterized in an objective manner, we can nowconstruct an algorithm for automating the detection of technical patterns.Specifically, our algorithm contains three steps:

1. Define each technical pattern in terms of its geometric properties, forexample, local extrema ~maxima and minima!.

2. Construct a kernel estimator [m~{! of a given time series of prices sothat its extrema can be determined numerically.

3. Analyze [m~{! for occurrences of each technical pattern.

The last two steps are rather straightforward applications of kernel regres-sion. The first step is likely to be the most controversial because it is herethat the skills and judgment of a professional technical analyst come intoplay. Although we will argue in Section II.A that most technical indicatorscan be characterized by specific sequences of local extrema, technical ana-lysts may argue that these are poor approximations to the kinds of patternsthat trained human analysts can identify.

While pattern-recognition techniques have been successful in automatinga number of tasks previously considered to be uniquely human endeavorsfingerprint identification, handwriting analysis, face recognition, and so onnevertheless it is possible that no algorithm can completely capture the skillsof an experienced technical analyst. We acknowledge that any automatedprocedure for pattern recognition may miss some of the more subtle nuancesthat human cognition is capable of discerning, but whether an algorithm isa poor approximation to human judgment can only be determined by inves-tigating the approximation errors empirically. As long as an algorithm canprovide a reasonable approximation to some of the cognitive abilities of ahuman analyst, we can use such an algorithm to investigate the empiricalperformance of those aspects of technical analysis for which the algorithm isa good approximation. Moreover, if technical analysis is an art form that can

6 See Simonoff ~1996! for a discussion of the problems with kernel estimators and alterna-tives such as local polynomial regression.


be taught, then surely its basic precepts can be quantified and automated tosome degree. And as increasingly sophisticated pattern-recognition tech-niques are developed, a larger fraction of the art will become a science.

More important, from a practical perspective, there may be significantbenefits to developing an algorithmic approach to technical analysis becauseof the leverage that technology can provide. As with many other successfultechnologies, the automation of technical pattern recognition may not re-place the skills of a technical analyst but can amplify them considerably.

In Section II.A, we propose definitions of 10 technical patterns based ontheir extrema. In Section II.B, we describe a specific algorithm to identifytechnical patterns based on the local extrema of price series using kernelregression estimators, and we provide specific examples of the algorithm atwork in Section II.C.

A. Definitions of Technical Patterns

We focus on five pairs of technical patterns that are among the most popularpatterns of traditional technical analysis ~see, e.g., Edwards and Magee ~1966,Chaps. VIIX!!: head-and-shoulders ~HS! and inverse head-and-shoulders ~IHS!,broadening tops ~BTOP! and bottoms ~BBOT!, triangle tops ~TTOP! and bot-toms ~TBOT!, rectangle tops ~RTOP! and bottoms ~RBOT!, and double tops~DTOP! and bottoms ~DBOT!. There are many other technical indicators thatmay be easier to detect algorithmicallymoving averages, support and resis-tance levels, and oscillators, for examplebut because we wish to illustratethe power of smoothing techniques in automating technical analysis, we focuson precisely those patterns that are most difficult to quantify analytically.

Consider the systematic component m~{! of a price history $Pt % and sup-pose we have identified n local extrema, that is, the local maxima andminima, of $Pt %. Denote by E1, E2, . . . , En the n extrema and t1

* , t2* , . . . , tn

* thedates on which these extrema occur. Then we have the following definitions.

Definition 1 (Head-and-Shoulders) Head-and-shoulders ~HS! and in-verted head-and-shoulders ~IHS! patterns are characterized by a sequence offive consecutive local extrema E1, . . . , E5 such that

HS [ 5E1 is a maximum

E3 . E1, E3 . E5

E1 and E5 are within 1.5 percent of their average

E2 and E4 are within 1.5 percent of their average,

IHS [ 5E1 is a minimum

E3 , E1, E3 , E5

E1 and E5 are within 1.5 percent of their average

E2 and E4 are within 1.5 percent of their average.


Observe that only five consecutive extrema are required to identify a head-and-shoulders pattern. This follows from the formalization of the geometryof a head-and-shoulders pattern: three peaks, with the middle peak higherthan the other two. Because consecutive extrema must alternate betweenmaxima and minima for smooth functions,7 the three-peaks pattern corre-sponds to a sequence of five local extrema: maximum, minimum, highestmaximum, minimum, and maximum. The inverse head-and-shoulders is sim-ply the mirror image of the head-and-shoulders, with the initial local ex-trema a minimum.

Because broadening, rectangle, and triangle patterns can begin on eithera local maximum or minimum, we allow for both of these possibilities in ourdefinitions by distinguishing between broadening tops and bottoms.

Definition 2 (Broadening) Broadening tops ~BTOP! and bottoms ~BBOT!are characterized by a sequence of five consecutive local extrema E1, . . . , E5such that

BTOP [ 5E1 is a maximum

E1 , E3 , E5

E2 . E4

, BBOT [ 5E1 is a minimum

E1 . E3 . E5

E2 , E4

.

Definitions for triangle and rectangle patterns follow naturally.

Definition 3 (Triangle) Triangle tops ~TTOP! and bottoms ~TBOT! are char-acterized by a sequence of five consecutive local extrema E1, . . . , E5 such that

TTOP [ 5E1 is a maximum

E1 . E3 . E5

E2 , E4

, TBOT [ 5E1 is a minimum

E1 , E3 , E5

E2 . E4

.

Definition 4 (Rectangle) Rectangle tops ~RTOP! and bottoms ~RBOT! arecharacterized by a sequence of five consecutive local extrema E1, . . . , E5 suchthat

RTOP [ 5E1 is a maximum

tops are within 0.75 percent of their average

bottoms are within 0.75 percent of their average

lowest top . highest bottom,

7 After all, for two consecutive maxima to be local maxima, there must be a local minimumin between and vice versa for two consecutive minima.


RBOT [ 5E1 is a minimum

tops are within 0.75 percent of their average

bottoms are within 0.75 percent of their average

lowest top . highest bottom.

The definition for double tops and bottoms is slightly more involved. Con-sider first the double top. Starting at a local maximum E1, we locate thehighest local maximum Ea occurring after E1 in the set of all local extremain the sample. We require that the two tops, E1 and Ea, be within 1.5 percentof their average. Finally, following Edwards and Magee ~1966!, we requirethat the two tops occur at least a month, or 22 trading days, apart. There-fore, we have the following definition.

Definition 5 (Double Top and Bottom) Double tops ~DTOP! and bottoms~DBOT! are characterized by an initial local extremum E1 and subsequentlocal extrema Ea and Eb such that

Ea [ sup $Ptk* : tk

* . t1* , k 5 2, . . . , n%

Eb [ inf $Ptk* : tk

* . t1* , k 5 2, . . . , n%

and

DTOP [ 5E1 is a maximum

E1 and Ea are within 1.5 percent of their average

ta*2 t1

* . 22

DBOT [ 5E1 is a minimum

E1 and Eb are within 1.5 percent of their average

ta*2 t1

* . 22

B. The Identification Algorithm

Our algorithm begins with a sample of prices $P1, . . . , PT % for which we fitkernel regressions, one for each subsample or window from t to t 1 l 1 d 2 1,where t varies from 1 to T 2 l 2 d 1 1, and l and d are fixed parameterswhose purpose is explained below. In the empirical analysis of Section III,we set l 5 35 and d 5 3; hence each window consists of 38 trading days.

The motivation for fitting kernel regressions to rolling windows of data isto narrow our focus to patterns that are completed within the span of thewindowl 1 d trading days in our case. If we fit a single kernel regressionto the entire dataset, many patterns of various durations may emerge, andwithout imposing some additional structure on the nature of the patterns, it


is virtually impossible to distinguish signal from noise in this case. There-fore, our algorithm fixes the length of the window at l 1 d, but kernel re-gressions are estimated on a rolling basis and we search for patterns in eachwindow.

Of course, for any fixed window, we can only find patterns that are com-pleted within l 1 d trading days. Without further structure on the system-atic component of prices m~{!, this is a restriction that any empirical analysismust contend with.8 We choose a shorter window length of l 5 35 tradingdays to focus on short-horizon patterns that may be more relevant for activeequity traders, and we leave the analysis of longer-horizon patterns to fu-ture research.

The parameter d controls for the fact that in practice we do not observe arealization of a given pattern as soon as it has completed. Instead, we as-sume that there may be a lag between the pattern completion and the timeof pattern detection. To account for this lag, we require that the final extre-mum that completes a pattern occurs on day t 1 l 2 1; hence d is the numberof days following the completion of a pattern that must pass before the pat-tern is detected. This will become more important in Section III when wecompute conditional returns, conditioned on the realization of each pattern.In particular, we compute postpattern returns starting from the end of trad-ing day t 1 l 1 d, that is, one day after the pattern has completed. Forexample, if we determine that a head-and-shoulder pattern has completedon day t 1 l 2 1 ~having used prices from time t through time t 1 l 1 d 2 1!,we compute the conditional one-day gross return as Z1 [ Yt1l1d110Yt1l1d .Hence we do not use any forward information in computing returns condi-tional on pattern completion. In other words, the lag d ensures that we arecomputing our conditional returns completely out-of-sample and without anylook-ahead bias.

Within each window, we estimate a kernel regression using the prices inthat window, hence:

[mh~t! 5(s5t

t1l1d21

Kh~t 2 s!Ps

(s5t

t1l1d21

Kh~t 2 s!

, t 5 1, . . . ,T 2 l 2 d 1 1, ~14!

where Kh~z! is given in equation ~10! and h is the bandwidth parameter ~seeSec. II.C!. It is clear that [mh~t! is a differentiable function of t.

Once the function [mh~t! has been computed, its local extrema can be readilyidentified by finding times t such that Sgn~ [mh' ~t!! 5 2Sgn~ [mh' ~t 1 1!!, where[mh' denotes the derivative of [mh with respect to t and Sgn~{! is the signum

function. If the signs of [mh' ~t! and [mh

' ~t 1 1! are 11 and 21, respectively, then

8 If we are willing to place additional restrictions on m~{!, for example, linearity, we canobtain considerably more accurate inferences even for partially completed patterns in any fixedwindow.


we have found a local maximum, and if they are 21 and 11, respectively, thenwe have found a local minimum. Once such a time t has been identified, weproceed to identify a maximum or minimum in the original price series $Pt % inthe range @t 2 1, t 1 1# , and the extrema in the original price series are usedto determine whether or not a pattern has occurred according to the defini-tions of Section II.A.

If [mh' ~t! 5 0 for a given t, which occurs if closing prices stay the same for

several consecutive days, we need to check whether the price we have foundis a local minimum or maximum. We look for the date s such that s 5 inf $s .t : [mh

' ~s! 0% . We then apply the same method as discussed above, excepthere we compare Sgn~ [mh' ~t 2 1!! and Sgn~ [mh' ~s!!.

One useful consequence of this algorithm is that the series of extrema thatit identifies contains alternating minima and maxima. That is, if the kthextremum is a maximum, then it is always the case that the ~k 1 1!th ex-tremum is a minimum and vice versa.

An important advantage of using this kernel regression approach to iden-tify patterns is the fact that it ignores extrema that are too local. For exam-ple, a simpler alternative is to identify local extrema from the raw price datadirectly, that is, identify a price Pt as a local maximum if Pt21 , Pt and Pt . Pt11and vice versa for a local minimum. The problem with this approach is that itidentifies too many extrema and also yields patterns that are not visually con-sistent with the kind of patterns that technical analysts find compelling.

Once we have identified all of the local extrema in the window @t, t 1 l 1d 2 1# , we can proceed to check for the presence of the various technicalpatterns using the definitions of Section II.A. This procedure is then re-peated for the next window @t 1 1, t 1 l 1 d # and continues until the end ofthe sample is reached at the window @T 2 l 2 d 1 1,T # .

C. Empirical Examples

To see how our algorithm performs in practice, we apply it to the dailyreturns of a single security, CTX, during the five-year period from 1992 to1996. Figures 37 plot occurrences of the five pairs of patterns defined inSection II.A that were identified by our algorithm. Note that there were norectangle bottoms detected for CTX during this period, so for completenesswe substituted a rectangle bottom for CDO stock that occurred during thesame period.

In each of these graphs, the solid lines are the raw prices, the dashed linesare the kernel estimators [mh~{!, the circles indicate the local extrema, andthe vertical line marks date t 1 l 2 1, the day that the final extremumoccurs to complete the pattern.

Casual inspection by several professional technical analysts seems to con-firm the ability of our automated procedure to match human judgment inidentifying the five pairs of patterns in Section II.A. Of course, this is merelyanecdotal evidence and not meant to be conclusivewe provide these fig-ures simply to illustrate the output of a technical pattern-recognition algo-rithm based on kernel regression.


(a) Head-and-Shoulders

(b) Inverse Head-and-Shoulders

Figure 3. Head-and-shoulders and inverse head-and-shoulders.


(a) Broadening Top

(b) Broadening Bottom

Figure 4. Broadening tops and bottoms.


(a) Triangle Top

(b) Triangle Bottom

Figure 5. Triangle tops and bottoms.


(a) Rectangle Top

(b) Rectangle Bottom

Figure 6. Rectangle tops and bottoms.


(a) Double Top

(b) Double Bottom

Figure 7. Double tops and bottoms.


III. Is Technical Analysis Informative?

Although there have been many tests of technical analysis over the years,most of these tests have focused on the profitability of technical tradingrules.9 Although some of these studies do find that technical indicators cangenerate statistically significant trading profits, but they beg the questionof whether or not such profits are merely the equilibrium rents that accrueto investors willing to bear the risks associated with such strategies. With-out specifying a fully articulated dynamic general equilibrium asset-pricingmodel, it is impossible to determine the economic source of trading profits.

Instead, we propose a more fundamental test in this section, one thatattempts to gauge the information content in the technical patterns of Sec-tion II.A by comparing the unconditional empirical distribution of returnswith the corresponding conditional empirical distribution, conditioned on theoccurrence of a technical pattern. If technical patterns are informative, con-ditioning on them should alter the empirical distribution of returns; if theinformation contained in such patterns has already been incorporated intoreturns, the conditional and unconditional distribution of returns should beclose. Although this is a weaker test of the effectiveness of technical analy-sisinformativeness does not guarantee a profitable trading strategyit is,nevertheless, a natural first step in a quantitative assessment of technicalanalysis.

To measure the distance between the two distributions, we propose twogoodness-of-fit measures in Section III.A. We apply these diagnostics to thedaily returns of individual stocks from 1962 to 1996 using a procedure de-scribed in Sections III.B to III.D, and the results are reported in Sec-tions III.E and III.F.

A. Goodness-of-Fit Tests

A simple diagnostic to test the informativeness of the 10 technical pat-terns is to compare the quantiles of the conditional returns with their un-conditional counterparts. If conditioning on these technical patterns providesno incremental information, the quantiles of the conditional returns shouldbe similar to those of unconditional returns. In particular, we compute the

9 For example, Chang and Osler ~1994! and Osler and Chang ~1995! propose an algorithm forautomatically detecting head-and-shoulders patterns in foreign exchange data by looking atproperly defined local extrema. To assess the efficacy of a head-and-shoulders trading rule, theytake a stand on a class of trading strategies and compute the profitability of these across asample of exchange rates against the U.S. dollar. The null return distribution is computed by abootstrap that samples returns randomly from the original data so as to induce temporal in-dependence in the bootstrapped time series. By comparing the actual returns from tradingstrategies to the bootstrapped distribution, the authors find that for two of the six currenciesin their sample ~the yen and the Deutsche mark!, trading strategies based on a head-and-shoulders pattern can lead to statistically significant profits. See, also, Neftci and Policano~1984!, Pruitt and White ~1988!, and Brock et al. ~1992!.


deciles of unconditional returns and tabulate the relative frequency Zdj ofconditional returns falling into decile j of the unconditional returns, j 51, . . . ,10:

Zdj [number of conditional returns in decile j

total number of conditional returns. ~15!

Under the null hypothesis that the returns are independently and identi-cally distributed ~IID! and the conditional and unconditional distributionsare identical, the asymptotic distributions of Zdj and the corresponding goodness-of-fit test statistic Q are given by

!n~ Zdj 2 0.10! ;a N ~0,0.10~1 2 0.10!!, ~16!

Q [ (j51

10 ~nj 2 0.10n!2

0.10n;a

x92, ~17!

where nj is the number of observations that fall in decile j and n is the totalnumber of observations ~see, e.g., DeGroot ~1986!!.

Another comparison of the conditional and unconditional distributions ofreturns is provided by the KolmogorovSmirnov test. Denote by $Z1t %t51

n1 and$Z2t %t51

n2 two samples that are each IID with cumulative distribution func-tions F1~z! and F2~z!, respectively. The KolmogorovSmirnov statistic is de-signed to test the null hypothesis that F1 5 F2 and is based on the empiricalcumulative distribution functions ZFi of both samples:

ZFi ~z! [1

ni(k51

ni

1~Zik # z!, i 5 1,2, ~18!

where 1~{! is the indicator function. The statistic is given by the expression

gn1, n2 5 S n1 n2n1 1 n2D102

sup2`,z,`

6 ZF1~z! 2 ZF2~z!6. ~19!

Under the null hypothesis F1 5 F2, the statistic gn1, n2 should be small. More-over, Smirnov ~1939a, 1939b! derives the limiting distribution of the statisticto be

limmin~n1, n2!r`

Prob~gn1, n2 # x! 5 (k52`

`

~21!k exp~22k2x 2 !, x . 0. ~20!


An approximate a-level test of the null hypothesis can be performed by com-puting the statistic and rejecting the null if it exceeds the upper 100athpercentile for the null distribution given by equation ~20! ~see Hollander andWolfe ~1973, Table A.23!, Cski ~1984!, and Press et al. ~1986, Chap. 13.5!!.

Note that the sampling distributions of both the goodness-of-f it andKolmogorovSmirnov statistics are derived under the assumption that re-turns are IID, which is not plausible for financial data. We attempt to ad-dress this problem by normalizing the returns of each security, that is, bysubtracting its mean and dividing by its standard deviation ~see Sec. III.C!,but this does not eliminate the dependence or heterogeneity. We hope toextend our analysis to the more general non-IID case in future research.

B. The Data and Sampling Procedure

We apply the goodness-of-fit and KolmogorovSmirnov tests to the dailyreturns of individual NYSE0AMEX and Nasdaq stocks from 1962 to 1996using data from the Center for Research in Securities Prices ~CRSP!. Toameliorate the effects of nonstationarities induced by changing market struc-ture and institutions, we split the data into NYSE0AMEX stocks and Nas-daq stocks and into seven five-year periods: 1962 to 1966, 1967 to 1971,and so on. To obtain a broad cross section of securities, in each five-yearsubperiod, we randomly select 10 stocks from each of f ive market-capitalization quintiles ~using mean market capitalization over the subperi-od!, with the further restriction that at least 75 percent of the priceobservations must be nonmissing during the subperiod.10 This procedureyields a sample of 50 stocks for each subperiod across seven subperiods~note that we sample with replacement; hence there may be names incommon across subperiods!.

As a check on the robustness of our inferences, we perform this samplingprocedure twice to construct two samples, and we apply our empirical analy-sis to both. Although we report results only from the first sample to con-serve space, the results of the second sample are qualitatively consistentwith the first and are available upon request.

C. Computing Conditional Returns

For each stock in each subperiod, we apply the procedure outlined in Sec-tion II to identify all occurrences of the 10 patterns defined in Section II.A.For each pattern detected, we compute the one-day continuously com-pounded return d days after the pattern has completed. Specifically, con-sider a window of prices $Pt % from t to t 1 l 1 d 2 1 and suppose that the

10 If the first price observation of a stock is missing, we set it equal to the first nonmissingprice in the series. If the tth price observation is missing, we set it equal to the first nonmissingprice prior to t.


identified pattern p is completed at t 1 l 2 1. Then we take the conditionalreturn R p as log~1 1 Rt1l1d11!. Therefore, for each stock, we have 10 sets ofsuch conditional returns, each conditioned on one of the 10 patterns ofSection II.A.

For each stock, we construct a sample of unconditional continuously com-pounded returns using nonoverlapping intervals of length t, and we comparethe empirical distribution functions of these returns with those of the con-ditional returns. To facilitate such comparisons, we standardize all returnsboth conditional and unconditionalby subtracting means and dividing bystandard deviations, hence:

Xit 5Rit 2 Mean@Rit #

SD@Rit #, ~21!

where the means and standard deviations are computed for each individualstock within each subperiod. Therefore, by construction, each normalizedreturn series has zero mean and unit variance.

Finally, to increase the power of our goodness-of-fit tests, we combine thenormalized returns of all 50 stocks within each subperiod; hence for eachsubperiod we have two samplesunconditional and conditional returnsand from these we compute two empirical distribution functions that wecompare using our diagnostic test statistics.

D. Conditioning on Volume

Given the prominent role that volume plays in technical analysis, we alsoconstruct returns conditioned on increasing or decreasing volume. Specifi-cally, for each stock in each subperiod, we compute its average share turn-over during the first and second halves of each subperiod, t1 and t2,respectively. If t1 . 1.2 3 t2, we categorize this as a decreasing volumeevent; if t2 . 1.2 3 t1, we categorize this as an increasing volume event. Ifneither of these conditions holds, then neither event is considered to haveoccurred.

Using these events, we can construct conditional returns conditioned ontwo pieces of information: the occurrence of a technical pattern and the oc-currence of increasing or decreasing volume. Therefore, we shall comparethe empirical distribution of unconditional returns with three conditional-return distributions: the distribution of returns conditioned on technical pat-terns, the distribution conditioned on technical patterns and increasing volume,and the distribution conditioned on technical patterns and decreasing volume.

Of course, other conditioning variables can easily be incorporated into thisprocedure, though the curse of dimensionality imposes certain practicallimits on the ability to estimate multivariate conditional distributionsnonparametrically.


E. Summary Statistics

In Tables I and II, we report frequency counts for the number of patternsdetected over the entire 1962 to 1996 sample, and within each subperiod andeach market-capitalization quintile, for the 10 patterns defined in Sec-tion II.A. Table I contains results for the NYSE0AMEX stocks, and Table IIcontains corresponding results for Nasdaq stocks.

Table I shows that the most common patterns across all stocks and overthe entire sample period are double tops and bottoms ~see the row labeledEntire!, with over 2,000 occurrences of each. The second most commonpatterns are the head-and-shoulders and inverted head-and-shoulders, withover 1,600 occurrences of each. These total counts correspond roughly to fourto six occurrences of each of these patterns for each stock during each five-year subperiod ~divide the total number of occurrences by 7 3 50!, not anunreasonable frequency from the point of view of professional technical an-alysts. Table I shows that most of the 10 patterns are more frequent forlarger stocks than for smaller ones and that they are relatively evenly dis-tributed over the five-year subperiods. When volume trend is consideredjointly with the occurrences of the 10 patterns, Table I shows that the fre-quency of patterns is not evenly distributed between increasing ~the rowlabeled t~;!! and decreasing ~the row labeled t~'!! volume-trend cases.For example, for the entire sample of stocks over the 1962 to 1996 sampleperiod, there are 143 occurrences of a broadening top with decreasing vol-ume trend but 409 occurrences of a broadening top with increasing volumetrend.

For purposes of comparison, Table I also reports frequency counts for thenumber of patterns detected in a sample of simulated geometric Brownianmotion, calibrated to match the mean and standard deviation of each stockin each five-year subperiod.11 The entries in the row labeled Sim. GBMshow that the random walk model yields very different implications for thefrequency counts of several technical patterns. For example, the simulatedsample has only 577 head-and-shoulders and 578 inverted-head-and-shoulders patterns, whereas the actual data have considerably more, 1,611and 1,654, respectively. On the other hand, for broadening tops and bottoms,the simulated sample contains many more occurrences than the actual data,1,227 and 1,028, compared to 725 and 748, respectively. The number of tri-angles is roughly comparable across the two samples, but for rectangles and

11 In particular, let the price process satisfy

dP~t! 5 mP~t!dt 1 sP~t!dW~t!,

where W~t! is a standard Brownian motion. To generate simulated prices for a single securityin a given period, we estimate the securitys drift and diffusion coefficients by maximum like-lihood and then simulate prices using the estimated parameter values. An independent priceseries is simulated for each of the 350 securities in both the NYSE0AMEX and the Nasdaqsamples. Finally, we use our pattern-recognition algorithm to detect the occurrence of each ofthe 10 patterns in the simulated price series.


double tops and bottoms, the differences are dramatic. Of course, the simu-lated sample is only one realization of geometric Brownian motion, so it isdifficult to draw general conclusions about the relative frequencies. Never-theless, these simulations point to important differences between the dataand IID lognormal returns.

To develop further intuition for these patterns, Figures 8 and 9 display thecross-sectional and time-series distribution of each of the 10 patterns for theNYSE0AMEX and Nasdaq samples, respectively. Each symbol represents apattern detected by our algorithm, the vertical axis is divided into the fivesize quintiles, the horizontal axis is calendar time, and alternating symbols~diamonds and asterisks! represent distinct subperiods. These graphs showthat the distribution of patterns is not clustered in time or among a subsetof securities.

Table II provides the same frequency counts for Nasdaq stocks, and de-spite the fact that we have the same number of stocks in this sample ~50 persubperiod over seven subperiods!, there are considerably fewer patterns de-tected than in the NYSE0AMEX case. For example, the Nasdaq sample yieldsonly 919 head-and-shoulders patterns, whereas the NYSE0AMEX samplecontains 1,611. Not surprisingly, the frequency counts for the sample of sim-ulated geometric Brownian motion are similar to those in Table I.

Tables III and IV report summary statisticsmeans, standard deviations,skewness, and excess kurtosisof unconditional and conditional normalizedreturns of NYSE0AMEX and Nasdaq stocks, respectively. These statisticsshow considerable variation in the different return populations. For exam-ple, in Table III the first four moments of normalized raw returns are 0.000,1.000, 0.345, and 8.122, respectively. The same four moments of post-BTOPreturns are 20.005, 1.035, 21.151, and 16.701, respectively, and those ofpost-DTOP returns are 0.017, 0.910, 0.206, and 3.386, respectively. The dif-ferences in these statistics among the 10 conditional return populations, andthe differences between the conditional and unconditional return popula-tions, suggest that conditioning on the 10 technical indicators does havesome effect on the distribution of returns.

F. Empirical Results

Tables V and VI report the results of the goodness-of-fit test ~equations~16! and ~17!! for our sample of NYSE and AMEX ~Table V! and Nasdaq~Table VI! stocks, respectively, from 1962 to 1996 for each of the 10 technicalpatterns. Table V shows that in the NYSE0AMEX sample, the relative fre-quencies of the conditional returns are significantly different from those ofthe unconditional returns for seven of the 10 patterns considered. The threeexceptions are the conditional returns from the BBOT, TTOP, and DBOTpatterns, for which the p-values of the test statistics Q are 5.1 percent, 21.2percent, and 16.6 percent, respectively. These results yield mixed supportfor the overall efficacy of technical indicators. However, the results of Table VItell a different story: there is overwhelming significance for all 10 indicators


Tab

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1962

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574

812

9411

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1620

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2710

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65

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314

322

066

671

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263

769

197

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55

361

440

933

730

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252

355

277

653

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84,3

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218

178

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8299

279

256

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295

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119

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7816

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120

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64

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1962

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309

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146

150

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228

299

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83,9

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1962

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262

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256

106

108

182

190

182

207

313

299

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7368

116

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1987

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310

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513

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3673

8710

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1992

to19

96E

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299

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102

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173

194

292

315

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368

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9188

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123

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105

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9296

143

104


Tab

leII

Fre

quen

cyco

un

tsfo

r10

tech

nic

alin

dica

tors

dete

cted

amon

gN

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qst

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from

1962

to19

96,

infi

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ear

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ds,

insi

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inti

les,

and

ina

sam

ple

ofsi

mu

late

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Bro

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mot

ion

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each

five

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rsu

bper

iod,

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ocks

per

quin

tile

are

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cted

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ndo

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ong

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ksw

ith

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ast

80%

non

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sin

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ices

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chst

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tory

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ann

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ran

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ence

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TO

P!,

tria

ngl

ebo

ttom

~TB

OT

!,re

ctan

gle

top

~RT

OP

!,re

ctan

gle

bott

om~R

BO

T!,

dou

ble

top

~DT

OP

!,an

ddo

ubl

ebo

ttom

~DB

OT

!.T

he

Sam

ple

colu

mn

indi

cate

sw

het

her

the

freq

uen

cyco

un

tsar

eco

ndi

tion

edon

decr

easi

ng

volu

me

tren

d~

t~ '

!!,

incr

easi

ng

volu

me

tren

d~

t~ ;

!!,

un

con

diti

onal

~E

nti

re!

,or

for

asa

mpl

eof

sim

ula

ted

geom

etri

cB

row

nia

nm

otio

nw

ith

para

met

ers

cali

brat

edto

mat

chth

eda

ta~

Sim

.G

BM

!.

Sam

ple

Raw

HS

IHS

BT

OP

BB

OT

TT

OP

TB

OT

RT

OP

RB

OT

DT

OP

DB

OT

All

Sto

cks,

1962

to19

96E

nti

re41

1,01

091

981

741

450

885

078

911

3413

2012

0811

47S

im.

GB

M41

1,01

043

444

712

9711

3911

6913

0996

9156

757

9t

~ '!

40

826

869

133

429

460

488

550

339

580

t~ ;

!

284

325

234

209

185

125

391

461

474

229

Sm

alle

stQ

uin

tile

,19

62to

1996

En

tire

81,7

5484

6441

7311

193

165

218

113

125

Sim

.G

BM

81,7

5485

8434

128

933

436

711

1214

012

5t

~ '!

36

256

2056

5977

102

3181

t~ ;

!

3123

3130

2415

5985

4617

2nd

Qu

inti

le,

1962

to19

96E

nti

re81

,336

191

138

6888

161

148

242

305

219

176

Sim

.G

BM

81,3

3667

8424

322

521

922

924

1299

124

t~ '

!

9451

1128

8610

911

113

169

101

t~ ;

!

6657

4638

4522

8512

090

42

3rd

Qu

inti

le,

1962

to19

96E

nti

re81

,772

224

186

105

121

183

155

235

244

279

267

Sim

.G

BM

81,7

7269

8622

721

021

423

915

1410

510

0t

~ '!

10

866

2335

8791

9084

7814

5t

~ ;!

71

7956

4939

2984

8612

258

4th

Qu

inti

le,

1962

to19

96E

nti

re82

,727

212

214

9211

618

717

929

630

328

929

7S

im.

GB

M82

,727

104

9224

221

920

925

523

2611

597

t~ '

!

8868

1226

101

101

127

141

7714

3t

~ ;!

62

8357

5634

2210

493

118

66

Lar

gest

Qu

inti

le,

1962

to19

96E

nti

re83

,421

208

215

108

110

208

214

196

250

308

282

Sim

.G

BM

83,4

2110

910

124

419

619

321

923

2710

813

3t

~ '!

82

5817

2499

100

8392

8411

0t

~ ;!

54

8344

3643

3759

7798

46


All

Sto

cks,

1962

to19

66E

nti

re55

,969

274

268

7299

182

144

288

329

326

342

Sim

.G

BM

55,9

6969

6316

312

313

714

924

2277

90t

~ '!

12

999

1023

104

9811

513

696

210

t~ ;

!

8310

348

5137

2310

111

614

464

All

Sto

cks,

1967

to19

71E

nti

re60

,563

115

120

104

123

227

171

6583

196

200

Sim

.G

BM

60,5

6358

6119

418

418

118

89

890

83t

~ '!

61

2915

4012

712

326

3949

137

t~ ;

!

2457

7151

4519

2516

8617

All

Sto

cks,

1972

to19

76E

nti

re51

,446

3430

1430

2928

5155

5558

Sim

.G

BM

51,4

4632

3711

511

310

711

05

646

46t

~ '!

5

40

45

712

83

8t

~ ;!

8

71

22

05

128

3

All

Sto

cks,

1977

to19

81E

nti

re61

,972

5653

4136

5273

5765

8996

Sim

.G

BM

61,9

7290

8423

616

517

621

219

1911

098

t~ '

!

77

12

48

1212

79

t~ ;

!

66

51

40

58

76

All

Sto

cks,

1982

to19

86E

nti

re61

,110

7164

4644

9710

710

911

512

097

Sim

.G

BM

61,1

1086

9016

216

814

717

423

2197

98t

~ '!

37

198

1446

5845

5240

48t

~ ;!

21

2524

1826

2242

4238

24

All

Sto

cks,

1987

to19

91E

nti

re60

,862

158

120

5061

120

109

265

312

177

155

Sim

.G

BM

60,8

6259

5722

918

720

524

47

779

88t

~ '!

79

4611

1973

6913

014

050

69t

~ ;!

58

5633

3026

2810

012

289

55

All

Sto

cks,

1992

to19

96E

nti

re59

,088

211

162

8711

514

315

729

936

124

519

9S

im.

GB

M59

,088

4055

198

199

216

232

98

6876

t~ '

!

9064

2431

7097

148

163

9499

t~ ;

!

8471

5256

4533

113

145

102

60


(a)

(b)

Figure 8. Distribution of patterns in NYSE/AMEX sample.


(c)

(d)

Figure 8. Continued


~e!

(f)

Figure 8. Continued


(g)

(h)

Figure 8. Continued


(i)

(j)

Figure 8. Continued


(a)

(b)

Figure 9. Distribution of patterns in Nasdaq sample.


(c)

(d)

Figure 9. Continued


(e)

(f)

Figure 9. Continued


(g)

(h)

Figure 9. Continued


(i)

(j)

Figure 9. Continued


Tab

leII

IS

um

mar

yst

atis

tics

~mea

n,

stan

dard

devi

atio

n,

skew

nes

s,an

dex

cess

kurt

osis

!of

raw

and

con

diti

onal

one-

day

nor

mal

ized

retu

rns

ofN

YS

E0

AM

EX

stoc

ksfr

om19

62to

1996

,in

five

-yea

rsu

bper

iods

,an

din

size

quin

tile

s.C

ondi

tion

alre

turn

sar

ede

fin

edas

the

dail

yre

turn

thre

eda

ysfo

llow

ing

the

con

clu

sion

ofan

occu

rren

ceof

one

of10

tech

nic

alin

dica

tors

:h

ead-

and-

shou

lder

s~H

S!,

inve

rted

hea

d-an

d-sh

ould

ers

~IH

S!,

broa

d-en

ing

top

~BT

OP

!,br

oade

nin

gbo

ttom

~BB

OT

!,tr

ian

gle

top

~TT

OP

!,tr

ian

gle

bott

om~T

BO

T!,

rect

angl

eto

p~R

TO

P!,

rect

angl

ebo

ttom

~RB

OT

!,do

ubl

eto

p~D

TO

P!,

and

dou

ble

bott

om~D

BO

T!.

All

retu

rns

hav

ebe

enn

orm

aliz

edby

subt

ract

ion

ofth

eir

mea

ns

and

divi

sion

byth

eir

stan

dard

devi

atio

ns.

Mom

ent

Raw

HS

IHS

BT

OP

BB

OT

TT

OP

TB

OT

RT

OP

RB

OT

DT

OP

DB

OT

All

Sto

cks,

1962

to19

96M

ean

20.

000

20.

038

0.04

02

0.00

52

0.06

20.

021

20.

009

0.00

90.

014

0.01

72

0.00

1S

.D.

1.00

00.

867

0.93

71.

035

0.97

90.

955

0.95

90.

865

0.88

30.

910

0.99

9S

kew

.0.

345

0.13

50.

660

21.

151

0.09

00.

137

0.64

32

0.42

00.

110

0.20

60.

460

Ku

rt.

8.12

22.

428

4.52

716

.701

3.16

93.

293

7.06

17.

360

4.19

43.

386

7.37

4

Sm

alle

stQ

uin

tile

,19

62to

1996

Mea

n2

0.00

02

0.01

40.

036

20.

093

20.

188

0.03

62

0.02

00.

037

20.

093

0.04

32

0.05

5S

.D.

1.00

00.

854

1.00

20.

940

0.85

00.

937

1.15

70.

833

0.98

60.

950

0.96

2S

kew

.0.

697

0.80

21.

337

21.

771

20.

367

0.86

12.

592

20.

187

0.44

50.

511

0.00

2K

urt

.10

.873

3.87

07.

143

6.70

10.

575

4.18

512

.532

1.79

34.

384

2.58

13.

989

2nd

Qu

inti

le,

1962

to19

96M

ean

20.

000

20.

069

0.14

40.

061

20.

113

0.00

30.

035

0.01

80.

019

0.06

72

0.01

1S

.D.

1.00

00.

772

1.03

11.

278

1.00

40.

913

0.96

50.

979

0.86

80.

776

1.06

9S

kew

.0.

392

0.22

31.

128

23.

296

0.48

52

0.52

90.

166

21.

375

0.45

20.

392

1.72

8K

urt

.7.

836

0.65

76.

734

32.7

503.

779

3.02

44.

987

17.0

403.

914

2.15

115

.544

3rd

Qu

inti

le,

1962

to19

96M

ean

20.

000

20.

048

20.

043

20.

076

20.

056

0.03

60.

012

0.07

50.

028

20.

039

20.

034

S.D

.1.

000

0.88

80.

856

0.89

40.

925

0.97

30.

796

0.79

80.

892

0.95

61.

026

Ske

w.

0.24

62

0.46

50.

107

20.

023

0.23

30.

538

0.16

60.

678

20.

618

0.01

32

0.24

2K

urt

.7.

466

3.23

91.

612

1.02

40.

611

2.99

50.

586

3.01

04.

769

4.51

73.

663

4th

Qu

inti

le,

1962

to19

96M

ean

20.

000

20.

012

0.02

20.

115

0.02

80.

022

20.

014

20.

113

0.06

50.

015

20.

006

S.D

.1.

000

0.96

40.

903

0.99

01.

093

0.98

60.

959

0.85

40.

821

0.85

80.

992

Ske

w.

0.22

20.

055

0.59

20.

458

0.53

72

0.21

72

0.45

62

0.41

50.

820

0.55

02

0.06

2K

urt

.6.

452

1.44

41.

745

1.25

12.

168

4.23

78.

324

4.31

13.

632

1.71

94.

691

Lar

gest

Qu

inti

le,

1962

to19

96M

ean

20.

000

20.

038

0.05

42

0.08

12

0.04

20.

010

20.

049

0.00

90.

060

0.01

80.

067

S.D

.1.

000

0.84

30.

927

0.99

70.

951

0.96

40.

965

0.85

00.

820

0.97

10.

941

Ske

w.

0.17

40.

438

0.18

20.

470

21.

099

0.08

90.

357

20.

167

20.

140

0.01

10.

511

Ku

rt.

7.99

22.

621

3.46

53.

275

6.60

32.

107

2.50

90.

816

3.17

93.

498

5.03

5


All

Sto

cks,

1962

to19

66M

ean

20.

000

0.07

00.

090

0.15

90.

079

20.

033

20.

039

20.

041

0.01

92

0.07

12

0.10

0S

.D.

1.00

00.

797

0.92

50.

825

1.08

51.

068

1.01

10.

961

0.81

40.

859

0.96

2S

kew

.0.

563

0.15

90.

462

0.36

31.

151

20.

158

1.26

42

1.33

72

0.34

12

0.42

72

0.87

6K

urt

.9.

161

0.61

21.

728

0.65

75.

063

2.67

44.

826

17.1

611.

400

3.41

65.

622

All

Sto

cks,

1967

to19

71M

ean

20.

000

20.

044

0.07

92

0.03

52

0.05

60.

025

0.05

72

0.10

10.

110

0.09

30.

079

S.D

.1.

000

0.80

90.

944

0.79

30.

850

0.88

50.

886

0.83

10.

863

1.08

30.

835

Ske

w.

0.34

20.

754

0.66

60.

304

0.08

50.

650

0.69

72

1.39

30.

395

1.36

00.

701

Ku

rt.

5.81

03.

684

2.72

50.

706

0.14

13.

099

1.65

98.

596

3.25

44.

487

1.85

3

All

Sto

cks,

1972

to19

76M

ean

20.

000

20.

035

0.04

30.

101

20.

138

20.

045

20.

010

20.

025

20.

003

20.

051

20.

108

S.D

.1.

000

1.01

50.

810

0.98

50.

918

0.94

50.

922

0.87

00.

754

0.91

40.

903

Ske

w.

0.31

62

0.33

40.

717

20.

699

0.27

22

1.01

40.

676

0.23

40.

199

0.05

62

0.36

6K

urt

.6.

520

2.28

61.

565

6.56

21.

453

5.26

14.

912

3.62

72.

337

3.52

05.

047

All

Sto

cks,

1977

to19

81M

ean

20.

000

20.

138

20.

040

0.07

62

0.11

40.

135

20.

050

20.

004

0.02

60.

042

0.17

8S

.D.

1.00

00.

786

0.86

31.

015

0.98

91.

041

1.01

10.

755

0.95

60.

827

1.09

5S

kew

.0.

466

20.

304

0.05

21.

599

20.

033

0.77

60.

110

20.

084

0.53

40.

761

2.21

4K

urt

.6.

419

1.13

21.

048

4.96

12

0.12

52.

964

0.98

91.

870

2.18

42.

369

15.2

90

All

Sto

cks,

1982

to19

86M

ean

20.

000

20.

099

20.

007

0.01

10.

095

20.

114

20.

067

0.05

00.

005

0.01

12

0.01

3S

.D.

1.00

00.

883

1.00

21.

109

0.95

60.

924

0.80

10.

826

0.93

40.

850

1.02

6S

kew

.0.

460

0.46

40.

441

0.37

22

0.16

50.

473

21.

249

0.23

10.

467

0.52

80.

867

Ku

rt.

6.79

92.

280

6.12

82.

566

2.73

53.

208

5.27

81.

108

4.23

41.

515

7.40

0

All

Sto

cks,

1987

to19

91M

ean

20.

000

20.

037

0.03

32

0.09

12

0.04

00.

053

0.00

30.

040

20.

020

20.

022

20.

017

S.D

.1.

000

0.84

80.

895

0.95

50.

818

0.85

70.

981

0.89

40.

833

0.87

31.

052

Ske

w.

20.

018

20.

526

0.27

20.

108

0.23

10.

165

21.

216

0.29

30.

124

21.

184

20.

368

Ku

rt.

13.4

783.

835

4.39

52.

247

1.46

94.

422

9.58

61.

646

3.97

34.

808

4.29

7

All

Sto

cks,

1992

to19

96M

ean

20.

000

20.

014

0.06

92

0.23

12

0.27

20.

122

0.04

10.

082

0.01

10.

102

20.

016

S.D

.1.

000

0.93

51.

021

1.40

61.

187

0.95

31.

078

0.81

40.

996

0.96

01.

035

Ske

w.

0.30

80.

545

1.30

52

3.98

82

0.50

22

0.19

02.

460

20.

167

20.

129

20.

091

0.37

9K

urt

.8.

683

2.24

96.

684

27.0

223.

947

1.23

512

.883

0.50

66.

399

1.50

73.

358


Tab

leIV

Su

mm

ary

stat

isti

cs~m

ean

,st

anda

rdde

viat

ion

,sk

ewn

ess,

and

exce

ssku

rtos

is!

ofra

wan

dco

ndi

tion

alon

e-da

yn

orm

aliz

edre

turn

sof

Nas

daq

stoc

ksfr

om19

62to

1996

,in

five

-yea

rsu

bper

iods

,an

din

size

quin

tile

s.C

ondi

tion

alre

turn

sar

ede

fin

edas

the

dail

yre

turn

thre

eda

ysfo

llow

ing

the

con

clu

sion

ofan

occu

rren

ceof

one

of10

tech

nic

alin

dica

tors

:h

ead-

and-

shou

lder

s~H

S!,

inve

rted

hea

d-an

d-sh

ould

ers

~IH

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Documents

Foundations of Technical Analysis: Computational ...mkearns/teaching/cis700/lo.pdf · Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, ... Technical