observations, the hypothe-ses are said to be empirically identical. The implication is that observationaldata, on its own, cannot tell us which of them is correct.
15. F Schauer, Profiles, Probabilities and Stereotypes (Cambridge, MA: BelknapPress of Harvard, 2003), 59. References to studies on dog breed behavior arefound in footnote 7.
16. In a subsequent chapter it will be shown that in this case the uncertaintywould be 10 times greater with 10 observations than with 1,000 if pricechanges conform to the normal distribution, but may be far more uncertain ifthey do not.
17. http://en.wikipedia.org/wiki/Philosophy_of_science.
18. Shermer, Why People Believe, 24.
19. Richards, Philosophy & Sociology of Science, 45.
20. B.L. Silver, The Ascent of Science (New York: Oxford University Press, 1998),15.
21. Ibid., 14.
22. R.S. Percival, About Karl Popper, adapted from his PhD thesis, available atwww.eeng.dcu.ie/~tkpw/intro_popper/intro_popper.html.
23. This is the frequency definition of probability. There are others.
24. Richards, Philosophy & Sociology of Science, 52.
25. K.R. Popper, The Logic of Scientific Discovery (London: Hutchinson, 1977).
26. K.R. Popper, Conjectures and Refutations (London: Routledge & Kegan 1972).
27. This is known as the underdetermination of theories problem.
of
me eeWN FSD HMNDNotes 493
28.
29.
30.
31.
32.
33.
36.
37.
38.
39.40.
41.
42.
If hypothesis is true, then X should be observed. X is observed (affirms theconsequent). Invalid conclusion: The hypothesis is true (fallacy of affirmingthe consequent).
A necessary condition is one that is required to produce an effect, but italone is not enough to produce the effect. For example if X is a necessarycondition of Y, it means that, if X is lacking, Y will not occur but the fact thatX is present is not enough to insure Y. In shorthand: If not X, then not Y. Asufficient condition is more comprehensive in that its presence is enough (issufficient) to produce the effect. Thus it means: If W, then Y.
Richards, Philosophy & Sociology of Science, 59.
My contention may be challenged in the near future by an objective versionof EWT called EWAVES developed by Robert Prechter’s firm, Elliott Wave In-ternational. Prechter claims it generates definitive signals for which financialperformance can be measured.
Descriptions of EMH-weak vary. Some say only historical price data cannotbe used to generated excess returns. Other versions say any stale marketdata, price, volume, or any other data series that TA practitioners study can-not be used to generated excess returns.
N. Jegadeesh and S. Titman, “Returns to Buying Winners and Selling Losers:Implications for Stock Market Efficiency,” Journal of Finance 56 (1993),699-720.
. R.A. Haugen, The Inefficient Stock Market: What Pays Off and Why (Upper
Saddle River, NJ: Prentice-Hall, 1999), 63.
. F, Nicholson, “Price-Earnings Ratios in Relation to Investment Results,” Fi-
nancial Analysts Journal (January/February 1968), 105-109; J.D. McWilliams,“Prices and Price-Earnings Ratios,” Financial Analysts Journal 22 (May/June,1966), 136-142.
Because there are not infinite time and resources to test all ideas, receptivityis tempered with prior knowledge, experience, and intuition. Some newideas merit more attention than others. However, being overly restrictive atthis phase could have a big cost —missing a breakthrough. In this phase, ifone is to err, it is better to err on the side of openness.
W.A. Wallis and H.V. Roberts, Statistics: A New Approach (New York: TheFree Press of Glencoe, 1964), 5.
P.W. Bridgman, “The Prospect for Intelligence,” Yale Review 34 (1945),444-461, quoted in J.B. Conant,On Understanding Science (New Haven: YaleUniversity Press, 1947), 115.
Wallis and Roberts,.Statistics, 6.
J.E. Russo and P.J.H. Schoemaker, Decision Traps: The Ten Barriers to Bril-liant Decision-Making and How to Overcome Them (New York: Doubleday,1989), 135.
L.R. Goldberg, “Man versus Model of Man: A Rationale, Plus Some Evidencefor a Method of Improving Clinical Inference,” Psychological Bulletin 73(1970), 422-432.
PH.K. Chang and C.L. Osler, “Methodical Madness: Technical Analysis andthe Irrationality of Exchange-Rate Forecasts,” Economic Journal,109, no.458 (1999), 636-661; C.L. Osler, “Identifying Noise Traders: The Head-and-494
43.
44.
46.
47.
48.
49.
NOTES
Shoulders Pattern in U.S. Equities,” Publication of Federal Reserve Bank ofNew York, 1998.
R.D. Edwards, J. Magee, and W.H.C. Bassetti (Eds.), Technical Analysis ofStock Trends, 8th ed., (Boca Raton, FL: CRC Press, 2001); M.J. Pring, TechnicalAnalysis Explained, 4th ed. (New York: McGraw-Hill, 2002); J.J. Murphy, Tech-nical Analysis of the Financial Markets: A Comprehensive Guide to TradingMethods and Applications (New York: New York Institute of Finance, 1999).
It may be possible to define chart patterns using fuzzy logic where classmembership is given a class membership function (0 to 1.0) rather than a yesor no. However, evaluating pattern effectiveness will still require a thresholdon the class membership function to decide which patterns belong in a givenclass and which do not.
. §.S. Alexander, “Price Movements in Speculative Markets: Trends or Random
Walks,” Industrial Management Review 2 (1961), 7-26; S.S. Alexander,“Price Movements in Speculative Markets: Trends or Random Walks No. 2,”Industrial Management Review 5 (1964), 25-46.
A.A. Merrill, Filtered Waves: Basic Theory (Chappaqua, NY: Analysis Press,1977).
P.H.K.Chang, and C.L. Osler, “Methodical Madness: Technical Analysis andthe Irrationality of Exchange-Rate Forecasts,” Economic Journal 109, no. 458(1999), 636-661.
C.L. Osler, “Identifying Noise Traders: The Head-and-Shoulders Pattern inU.S. Equities,” Publication of Federal Reserve Bank of New York, 1998.
A.W. Lo, H. Mamaysky, and W. Jiang, “Foundations of Technical Analysis:Computational Algorithms, Statistical Inference, and Empirical Implementa-tion,” Journal of Finance 55, 4 (August 2000), 1705-1765.
. Kernel regression estimates the value of a smoothed curve by averaging
nearby or local observations, where the weight of each observation is in-versely related to its distance from the location for which a smoothed valueis desired. Thus, nearby observations are given the greatest weight. The termkernel refers to the shape of the weighting function used. One commonweight function is based on the shape of the normal probability distribution(Gaussian kernel).
51. The authors of the paper do not come out directly and say the patterns are
not useful for forecasting trends, but the data they present indicates this asdoes the commentary on the paper provided by Narasimahan Jegadeesh (pp.1765-1770). In Table 1 he shows that the post pattern returns (trends) arenot significantly different than when the patterns are not present.
52. T.N. Bulkowski, Encyclopedia of Chart Patterns (New York: John Wiley &
Sons, 2000), 262-304.
CHAPTER 4 Statistical Analysis
1.
2.
David S. Moore, The Basic Practice of Statistics (New York: W.H. Freeman,1999).B. Russell, A History of Western Philosophy (New York: Simon & Schuster,1945).Notes 495
o
10.
11.12.
13.
14.
15.
16.
17.
18.
19.
20.
R.P. Abelson, Statistics as Principled Argument (Mahwah, NJ: LawrenceErlbaum, 1995).
A.W. Wallis and H.V. Roberts, Statistics: A New Approach (New York: TheFree Press of Glencoe, 1956), 101-110.
The method for detrending or zero normalizing the market data series towhich the rules are applied is discussed in Chapter 1.
Probability density is formally defined as the first derivative of the cumula-tive probability of a random variable. For the purposes of this discussion, itcan be thought of as chance or probability that something will happen.Fractional area refers to the proportion of the area of the density function ly-ing to the right of the observed performance relative to the total area cov-ered by the density function.
Instances of dogs missing limbs are excluded from this discussion for pur-poses of simplification.
Wallis and Roberts, Statistics, 101-110.
D.S. Moore, The Basic Practice of Statistics (New York: W.H. Freeman,1999).
S. Kachigan, Statistical Analysis (New York: Radius Press, 1986), 77.
In this example, it is assumed that the underlying phenomenon is stationary;that is, it remains unchanged over time. Financial markets are most likelynonstationary systems. In general, statistical inference is not intended tocope with nonstationary phenomena. Nonstationary is a risk inherent wheninvesting on the basis of any sort of rule, statistically derived or not.
Daily returns refers to returns earned by a rule on zero-centered or de-trended market data as defined in Chapter 1.
S.J. Grossman and J.E. Stiglitz, “On the Impossibility of Informationally Effi-cient Markets,” American Economic Review 70, no. 3 (1980), 393-408.Chapter 7, Theories of Nonrandom Price Motion, discusses the possibilitythat TA rules may be profitable because they signal opportunities to earn riskpremiums. Under this notion, TA rule profits are compensation to the techni-cal trader for providing valuable services to other market participants, suchas liquidity, information in the form of price discovery, or risk transference.Because these are ongoing needs, profits earned by rules that result from ful-filling said needs are likely to persist.
This definition of stationary is not strictly correct but close enough for pur-poses of this discussion.
N.N. Taleb, Fooled by Randomness—The Hidden Role of Chance in theMarkets and in Life (New York: Texere, 2001), 97.
The rigorous definition of a statistic is a function of the cases in the sam-ple that does not depend on any unknown parameter. Thus, if computingthe statistic requires a piece of information that is unknown, then it is nota statistic.
The trimmed mean is computed after removing the largest 5 percent winsand losses. This can provide a more accurate estimate of the populationmean by removing outliers that can skew results.
In fact, the Law of Large Numbers refers to the convergence between asample average and a population average. However, a relative frequency is496
21.22.
23.
24.
27.28.
29.
30.
31.
32.
NOTES
the same as an average if the occurrence of an event is given the value of1 and the nonoccurrence a value of 0. Relative frequency refers to theproportion of event occurrences within an observed sample, whereasthe theoretical idea of probability refers to the relative frequency of theevent in the unlimited number of trials that constitutes the population ofobservations.
Kachigan, Statistical Analysis, 75.
The Law of Large Numbers requires that the number of observations in eachinterval be sufficiently large to reveal the true relative frequency of the val-ues represented by that interval.
Probability density is the slope of the cumulative distribution function. Forany point on the horizontal axis, the height of the probability density func-tion at that point is the first derivative of the probability that an observationwill be less than or equal to the point on the horizontal axis.
For example, the probability of obtaining the value 4.23 is zero. This is not sostrange because a continuous random variable never assumes a value of ex-actly 4.23. The real value is either slightly higher or lower than 4.23, if it weregiven with additional accuracy (e.g., 4.229).
. In fact each individual observation that winds up in the sample is a probabil-
ity experiment.
. A more technical definition of sample statistic: A function of one or more
random variables that does not depend on any unknown parameters. Thisimplies that a statistic is itself a random variable.
Kachigan, Statistical Analysis, 101.
R.S. Witte and J.S. Witte, Statistics, 7th ed. (New York: John Wiley & Sons,2004), 230.
Other statistics include: Z, t, F-ratio, chi-square, and so forth. There aremany.
The Sharpe ratio is defined as the annualized average return in excess of therisk-free rate (e.g., 90-day treasury bills) divided by the standard deviation ofthe returns stated on an annualized basis.
The profit factor is defined as a ratio of the sum of gains of profitable trans-actions to the sum of all losses on unprofitable transactions. The denomina-tor is stated as an absolute value (no sign) so the value of the profit factor isalways positive. A value of 1 would indicate the rule is break even. A supe-rior way to calculate the profit factor to transform it to have a natural zeropoint is to take the log of the ratio.
The Ulcer Index is an alternative and possibly superior measure of risk thatconsiders the magnitude of equity retracements, which are not directly con-sidered by the Sharpe ratio. The standard deviation, the risk measure em-ployed by the Sharpe ratio, does not take into account the sequence ofwinning and losing periods. For a definition of the Ulcer Index, see P.G. Mar-tin and B.B. McCann, The Investor's Guide to Fidelity Funds (New York:John Wiley & Sons, 1989), 75-79. A similar concept, the return to retrace-ment ratio, is described by J.D. Schwager, Schwager on Futures—TechnicalAnalysis (New York: John Wiley & Sons, 1996).Notes 497
33.
34.
36.37.
39.
This figure was inspired by a similar figure in Lawrence Lapin, Statistics forModern Business Decisions, 2nd ed. (New York: Harcourt Brace Jo-vanovich, 1978), Figure 6-10, 186.
The qualifying condition for the mean of a sample to approach the mean ofthe parent population is that the mean in the parent population be finite.
5. This is true when the test statistic is the sample mean, but it is not true in
general. A critical assumption is that the observations comprising the sampleare independent. If the observations are serially correlated, as might be thecase in a time series, the variance reduction is slower than the square rootrule suggests.
The population has a finite mean and a finite standard deviation.
The general principle that bigger samples are better applies to stationaryprocesses. In the context of financial market time series, which are mostlikely not stationary, old data may be irrelevant, or even misleading. Forexample, indicators based on short selling volume by NYSE specialistsseem to have suffered a decline in predictive power. Larger samples maynot be better.
. The Central Limit Theorem applies to a sample for which observations are
drawn from the same parent population, which are independent, and inwhich the mean and standard deviation of the parent population are finite.This figure was inspired by a similar figure in Lawrence Lapin, Statistics forModern Business Decisions, 2nd ed. (New York: Harcourt Brace Jo-vanovich, 1978), 216-217.
CHAPTER 5 Hypothesis Tests and
o
Confidence Intervals
J.E. Burt and G.M. Barber, Elementary Statistics for Geographers, 2nd ed.(New York: The Guilford Press, 1996), 5.
We saw in Chapter 2 that because of the self-attribution bias, people canmaintain a delusion by rationalizing a loss: headache, shoelaces were untied,argued with wife before game, the other player put an evil eye on me, and soon. Distasteful evidence can always be explained away if loose evidencestandards are applied.
As discussed in Chapter 3, according to Popper and Hume, no amount ofconfirmatory evidence can provide certitude for a claim. However, evidencecan be used to falsify a claim. For this reason, a science test claims by focus-ing on evidence that has the potential to show that the claim is false ratherthan on evidence that might confirm the claim.
R.S. Witte and J.S. Witte, Statistics, 7th ed. (Hoboken, NJ: John Wiley &Sons, 2004), 264.
Ibid.
Eric W. Noreen, Computer Intensive Methods for Testing Hypotheses: AnIntroduction (New York: John Wiley & Sons, 1989), 2.498 NOTES
7. David S. Moore, The Basic Practice of Statistics, 2nd ed. (New York: W. H.Freeman, 2000), 314.
8. As described in Chapter 1, detrending transforms the market data such that.it has a mean daily change of zero. This eliminates the distortion in a rulesperformance due to the conjoint effects of position bias and the market'strend during the back-test period.
9. Randomly paired means that the historical daily price changes of the marketare scrambled, and then each is randomly paired with a rule output value (+1or -1).
10. Quantmetrics, 2214 El Amigo Road, Del Mar, CA 92014.
11. B. Efron, “Bootstrap Methods: Another Look at the Jackknife,” Annals ofStatistics 7(1979), 1-26.
12. E.W. Noreen in Computer Intensive Methods for Testing Hypotheses citesthe following references to bootstrapping: B. Efron and G. Gong, “A LeisurlyLook at the Bootstrap, the Jacknife and Cross-Validation,” American Staisti-can 37 (February 1984), 36-48; B. Efron, “Better Bootstrap Confidence Inter-vals,” LCS Technical Report No. 14, Department of Statistics and StanfordLinear Accelerator; B. Efron and R. Tibshirani, “Bootstrap Methods for Stan-dard Errors, Confidence Intervals, and Other Methods of Statistical Accu-racy,” Statistical Science 1 (1984), 54-77.
13. Here, ordered means the values occur in their proper temporal order.
14. A channel breakout rule, under traditional TA interpretation, registers a buy(sell) signal when the rule’s input series is at a maximum (minimum) valuefor the current day and all prior days in the rule’s look-back window.
15. LJ. Kazmier, Statistical Analysis for Business and Economics (New York:McGraw-Hill, 1978), 217.
16. Standard error refers to the standard deviation of sample means around thepopulation mean when many independent samples are taken from the samepopulation.
17. Kazmier, Statistical Analysis, 216.
18. Ibid.
CHAPTER 6_Data-Mining Bias: The Fool’s Goldof Objective TA
1. H. White, “A Reality Check for Data Snooping,” Econometrica 68, no. 5(September 2000), 1102, Proposition 2.1. The proof shows that as the sam-ple size goes to infinity, the probability that the rule with the highest ex-pected return will also have the highest observed performanceapproaches 1.0.
2. R.N. Kahn, “Data Mining Made Easy: Seven Quantitative Insights into ActiveManagement: Part 5,” www.barra.com/Newsletter/NL165/SevIns5NL165.asp.
3. Bible code scholars claim that a prediction that Israeli Prime Minster YitzhakRabin would be assassinated was made in advance of the event. They alsoclaim that efforts were made to warn Rabin, but to no avail. I have been un-able to authenticate the veracity of this claim.Notes 499
4.
no
11.
12.
13.
14.
15.
16.
17.
18.
The code scholars do not define in advance what constitutes a significantpattern. It can be any group of words that are deemed to be in close proxim-ity in the text and to be related, in the opinion of the code researcher, to thepurportedly predicted event. They are free to use any spacing interval theywish to define the pattern.
See “The Case Against the Codes,” Professor Barry Simon, www.wopr.con/biblecodes/.
Ibid.
Kahn, “Data Mining Made Easy.”
Here, the term prediction does not necessarily mean foretelling the future,though forecasting is certainly the ultimate intent of TA. Rather, the termprediction refers to observations not yet made, regardless of whether theylie in the future. It is crucial that the observations have not yet been made be-cause prior observations can always be explained after the fact by an infinitenumber of hypotheses or rules. The only way to determine which of thesenumerous hypotheses might be correct is for predictions to be made that arethen compared with new observations. Thus the term new observation canrefer to the back test of a newly proposed rule. (See Chapter 3 for a fuller ex-position of this point.)
Out-of-sample data refers to data not used for back testing the rule.
R. Pardo, Design, Testing and Optimization of Trading Systems (NewYork: John Wiley & Sons, 1992).
A.W. Lo, “The Adaptive Markets Hypothesis: Market Efficiency from an Evo-lutionary Perspective,” Journal of Portfolio Management 30 (2004), 15-29;A. Timmermann, “Structural Breaks: Incomplete Information, and StockPrices,” Journal of Business & Economic Statistics 19, no. 3 (2001),299-314.
W.V. Kidd and W.B. Borsen, “Why Have Returns to Technical Analysis De-creased?” Journal of Economics and Business 56, no. 3 (May/June 2004),159-176. The article examines the deterioration of technical system perfor-mance over the course of the out-of-sample data. Thus, it could not be due todata-mining bias.
D. Meyers, “Optimization: The World of Illusion,” Active Trader (March2004), 52; D. Meyers, “The Optimization Trap,” Active Trader (November2001), 68; D. Meyers, “The Siren Call of Optimized Trading Systems, 1996,available on the author's web site, www.MeyersAnalytics.com.
T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learn-ing: Data Mining, Inference and Prediction, Springer Series in Statistics(New York: Springer, 2001).
S.H. Weiss and N. Indurkhya, Predictive Data Mining: A Practical Guide(San Francisco: Morgan Kaufmann, 1998).
LH. Witten and E. Frank, Data Mining: Practical Machine Learning Toolsand Techniques, 2nd ed. (San Francisco: Morgan Kaufmann, 2005).
D.D. Jensen and PR. Cohen, “Multiple Comparisons in Induction Algo-rithms,” Machine Learning 38, no. 3 (March 2000), 309-338.
The Ulcer Index was discussed by P.G. Martin and B.B. McCann, The In-vestor’s Guide to Fidelity Funds (New York: John Wiley & Sons, 1989). A500
19.
al.
22.
23.
24.25.26.27.28.29.30.
31.
32.33.34.
35.36.
37.38.39.40.
Al.
NOTES
similar measure of risk based on the average magnitude of equity retrace-ment was previously proposed by J.D. Schwager, A Complete Guide to theFutures Markets (New York: John Wiley & Sons, 1984), 471-472.
R. Pardo, Design, Testing and Optimization of Trading Systems (NewYork: John Wiley & Sons, 1992).
. J.O. Katz and D.L. McCormick, The Encyclopedia of Trading Strategies
(New York: McGraw-Hill, 2000).
PJ. Kaufman, New Trading Systems and Methods, 4th ed. (Hoboken, NJ:John Wiley & Sons, 2005).
Mean-reversion rules are based on the concept that when a data seriesmoves to extreme high or low values relative to recent values, it should re-vert back to its mean value.
Divergence rules are based on the concept that if two data series typicallymove up and down together, it is important information when they divergefrom each other. Thus, divergence rules give signals when data series departfrom this normal pattern of activity.
Hastie, Tibshirani, and Friedman, Elements of Statistical Learning.
White, “Reality Check,” proof of Proposition 2.1
This diagram is conceptual and is not intended to accurately convey the pre-dictive power of scientific laws or of TA rules except in a general sense.Jensen and Cohen, “Multiple Comparisons in Induction Algorithms.”
White, “Reality Check.”
Jensen and Cohen, “Multiple Comparisons in Induction Algorithms.”Although this is not an example of data mining, it is an example of using amultiple comparison procedure to find a best solution to a problem.Actually, randomness has been put to good use in the design of some of themost powerful data-mining algorithms used for complex rule synthesis. Anexample of this is Random Forests, developed by the late Leo Brieman, pro-fessor of statistics, at the University of California, Berkeley.
E. Peters, Fractal Market Analysis: Applying Chaos Theory to Investment& Economics (New York: John Wiley & Son., 1994), 21-38.
A rule selected because of its superior performance in the mined data.
This figure was inspired by a similar figure in Lawrence Lapin, Statistics forModern Business Decisions, 2nd ed. (New York: Harcourt Brace Jo-vanovich, 1978), Figure 6-10, 186.
Pardo, Design, Testing and Optimization, 108.
M. De La Maza, “Backtesting,” Chapter 8 in Computerized Trading: Maai-mizing Day Trading and Overnight Profit, M. Jurik (Ed.) (Paramus, NJ:Prentice-Hall, 1999).
Katz and McCormick, Encyclopedia of Trading Strategies.
Kaufman, New Trading Systems.
P-H. Hsu and C.-M. Kuan, “Rexamining the Profitability of Technical Analysiswith Data Snooping Checks,” Journal of Financial Econometrics 3, no. 4(2005), 606-628.
H.M. Markowitz and G.L. Xu, “Data Mining Corrections: Simple and Plausi-ble,” Journal of Portfolio Management (Fall 1994), 60-69.
De La Maza, “Backtesting.”Notes 501
42.
43.
44.
45.
46.
B. Efron, “Bootstrap Methods: Another Look at the Jackknife,” Annals ofStatistics 7 (1979), 1-26; B. Efron and G. Gong, “A Leisurely Look at theBootstrap, the Jackknife, and Cross-Validation,” American Statistician 37(February 1983), 36-48.
Quantmetrics, 2214 El Amigo Road, Del Mar, CA 92014; contact ProfessorHalbert White, PhD.
Note that the subtraction of the average daily return from each daily re-turn can also be done prior to bootstrapping. In this illustration it wasdone afterward.
PR. Hansen, “The Reality Check for Data Snooping: A Comment on White,”Brown University Department of Economics; P.R. Hansen, “A Test for Supe-rior Predictive Ability,” Journal of Business and Economic Statistics 23(2005), 365-380.
J.P. Romano and M. Wolf, “Stepwise Multiple Testing as Formalized DataSnooping,” Econometrica 73, no. 4 (July 2005), 1237-1282. Note, this paperand many others use the term data snooping for what I refer to as the data-mining bias.
CHAPTER 7 Theories of Nonrandom
a
10.
Price Motion
C.R. Lightner, “A Rationale for Managed Futures,” Technical Analysis ofStocks & Commodities (March 1999).
Kepler actually proposed three distinct laws.
R.D. Edwards and J. Magee, Technical Analysis of Stock Trends, 4th ed.(Springfield, MA: John Magee, 1958).
J.J. Murphy, Technical Analysis of the Futures Markets: A ComprehensiveGuide to Trading Methods and Applications (New York: New York Instituteof Finance, 1999), 2.
E. Fama, “Efficient Capital Markets: A Review of Theory and Empirical Work”Journal of Finance 25 (1970), 383-417.
R.J. Shiller, Irrational Exuberance (Princeton, NJ: Princeton UniversityPress, 2000), 135.
Ibid.
In this chapter in section “Nonrandom Price Motion in the Context of Effi-cient Markets,” I state the case of those who say EMH allows for price pre-dictability, namely, that random walks are not a