i.e., engaging incountertrend strategies).
¢ Information or price discovery premium for promoting marketefficiency: This premium compensates investors for making buy-and-sell decisions that move prices toward rational levels. Because these de-cisions are often based on complex models developed by sophisticatedresearch, this premium might also be called a complexity premium."”For example, a strategy that sells overvalued stocks and buys underval-ued stocks helps move prices back to rational values (price discovery).
The good news about returns that come in the form of a risk premiumis that they are more likely to endure into the future. Of course, it is everyanalyst’s dream to discover true market inefficiencies because the returnsearned from them do not entail additional risk.°’ However, inefficienciesare ephemeral. Sooner or later they tend to be discovered and wiped outby other diligent researchers. However, returns that are justified as a riskpremium are more likely to endure as they represent payment for service.
How do risk premiums explain the existence of systematic pricemovements? Such price movements can provide a mechanism for risk-accepting investors to be compensated. The investor who is willing to pro-vide liquidity to a stockholder with a strong desire to sell winds up buyingstocks that have a systematic tendency to rise over the near term. Thesestocks can be identified with countertrend strategies."*
Therefore, in the context of efficient markets, the profits-earned byTA strategies may be understood as risk premiums; compensation for thebeneficial effect the strategy confers on other investors or the market as awhole. In other words, TA signals may generate profits because they iden-tify opportunities to assume risks that other investors wish to shed.
I should point out that the mere fact that an investor is willing to beara risk does not guarantee that a return will be earned. The risk taker mustbe clever about it. Just as a careless high steel worker or test pilot may notlive long enough to collect a wage that includes the hazardous-duty pre-mium, a careless seeker of risk premiums may not stay in the game longenough to collect, either.
Hedge Risk Premium and the Gains toTrend-Following Commodity Futures
This section describes how the profits to trend followers in the com-modities markets may be explained as a risk transfer premium. Thecommodities futures markets perform an economic function that is fun-Theories of Nonrandom Price Motion 381
damentally different from the stock and bond markets. The stock andbond markets provide companies with a mechanism to obtain equity anddebt financing and provide investors with a way to invest their capital.Because stocks and corporate bond investments expose investors torisks that exceed the risk-free rate (government treasury bills), investorsare compensated with a risk premium—the equity risk premium and thecorporate-bond risk premium.
The economic function of the futures markets has nothing to do withraising capital and everything to do with price risk. Price changes, espe-cially large ones, are a source of risk and uncertainty to businesses thatproduce or use commodities. The futures markets provide a means bywhich these businesses, called commercial hedgers, can transfer pricerisk to investors (speculators).
At first blush, it may seem puzzling that commercial hedgers wouldeven need investors to assume their price risk. Because some hedgersneed to sell, like the farmer who grows wheat, and some need to buy, likethe bread company that uses wheat, why don’t hedgers simply contractwith each other? They do, but often there is an imbalance in their hedgingneeds. Sometimes wheat farmers have more wheat to sell than bakerycompanies and other commercial users of wheat need to buy. At othertimes, the situation is the opposite. Thus there is often a gap between thefarmers’ supply and the bakers’ demand. This creates the need for a thirdgroup of market participants who are willing to buy or sell. That is, theyare willing to fill the supply-demand gap that develops between commer-cial producers and commercial users.
The existence of a supply-demand gap is a predicable consequence ofthe law of supply and demand; as the price of wheat rises, supply goes upwhile demand goes down. As the price falls, supply goes down while de-mand goes up. Data from the Commodity Futures Trading Commission(CFTC), which tracks the buying and selling of commercial hedgers, con-firms this is exactly what occurs in the futures markets. During risingprice trends, commercial hedge selling exceeds hedge buying (i.e.,hedgers are net sellers). During falling trends, commercial hedge buyingexceeds commercial hedge selling (hedgers are net buyers).
In the futures markets, the amount sold and the amount bought mustbe equal just as in real estate transactions the number of homes sold mustequal the number of homes bought. It cannot be otherwise. Given that arising trend motivates more commercial hedge selling than hedge buying,there is a need for additional buyers during rising markets to accommo-date the excess hedge selling. During falling trends, excessive commercialhedge buying creates the need for additional sellers.
Enter the price trend speculator who is willing to buy during uptrendsto meet the unfilled needs of commercial hedge sellers and willing to sellBSS METHODOLOGICAL, PSYCHOLOGICAL, PHILOSOPHICAL, STATISTICAL FOUNDATIONS
short during falling price trends, to meet the unmet needs of the commer-cial hedge buyers. In other words the futures markets need speculators tobe trend followers. And it is the invisible hand of the marketplace that cre-ates the trends to compensate them for filling this need.
However, trend followers are exposed to significant risks. For exam-ple, most trend-following systems generate unprofitable signals 50 to 65percent of the time. These occur when markets are trendless, which is thecase a majority of the time. During these times, the trend followers experi-ence the dreaded equity drawdown. During drawdowns it is not unusualfor successful trend followers to lose up to 30 percent of their trading cap-ital. Thus they need a strong motive to tolerate drawdown risk. That mo-tive is the opportunity, though not the guarantee, to profit from pricetrends when they do occur. In other words, large-scale trends provide aprofit opportunity to trend followers who are adequately capitalized andwho manage leverage correctly.
The Mt. Lucas Management Indexof Trend Following Returns
The risk premium earned by commodity trend followers has been quanti-fied with the creation of a benchmark index called the Mt. Lucas Manage-ment Index (MLM). It is a historical record of the returns that can beearned by an extremely simplistic trend-following formula. In otherwords, it assumes no specialized knowledge of a complex forecastingmodel. If it did it could not legitimately be called a benchmark index,whose function is to estimate the returns that can be earned by investingin an asset class with no special skill.
The risk-adjusted returns earned by the MLM index suggest that com-modity futures markets contain systematic price movements that can beexploited with relatively simple TA methods. The MLM index monthly re-turns, which have been computed back to the 1960s, are derived by apply-ing a 12-month moving-average crossover strategy to 25 commoditiesmarkets.!™ At the end of each month, the price of the nearby futures con-tract for each of the markets is compared to its 12-month moving average.If the price is greater than the moving average, a long position is held inthat market for the next month. If the price is below the average, a shortposition is held.
The annualized return’ and risk, as measured by the standard devia-tion in annual returns, of the MLM index is compared with returns andrisks of several other asset-class benchmarks in Figure 7.13.
The risk-adjusted excess return or Sharpe ratio for the MLM index,and the other asset classes are shown Figure 7.14. The trend followerearns a risk premium that is somewhat better than the other asset-classTheories of Nonrandom Price Motion 383
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1816 _|1412Annualized 10} copPercent a Return6 Risk4ol0 =iUS Int. T-Bills = Corp. MLMStocks Stocks Bonds IndexFIGURE 7.13 Returns and standard deviations for five benchmarks.0.70.60.50.4 Risk-Adj. Return0.30.20.1oSUS Int. T-Bills Corp. MLMStocks Stocks Bonds Index
FIGURE 7.14  Risk-adjusted excess returns.
benchmarks.’ The MLM index provides some evidence that the futuresmarkets offer compensation in the form of systematic price movements.If a risk-transfer premium is a valid explanation of trend-follower re-turns, then trend following in stocks should not be as rewarding as it is inthe commodity futures markets. A trend follower in stocks is not provid-ing a risk-transfer service. There are data suggesting that trend followingin stocks is indeed a less rewarding enterprise.’ Lars Kestner comparedthe performance of trend-following systems for a portfolio of futures anda portfolio of stocks. The futures considered were 29 commodities in 8different sectors.'"! The stocks were represented by 31 large-cap stocks in9 different industry sectors,!” and 3 stock indices. Risk-adjusted perfor-mance (Sharpe ratio) was computed for 5 different trend-following sys-tems,' over the period January 1, 1990 through December 31, 2001. TheSharpe ratio averaged over the five trend-following systems in futures was384 METHODOLOGICAL, PSYCHOLOGICAL, PHILOSOPHICAL, STATISTICAL FOUNDATIONS
-604 versus .046 in stocks. These results support the notion that futurestrend followers are earning a risk premium that is not available to trendfollowers in stocks. See Figure 7.15.
Liquidity Premium and the Gainsto Counter Trend Trading in Stocks
The stock market offers risk-taking stock traders a different form of com-pensation. They can earn a premium for providing liquidity to highly moti-vated sellers. In other words, there are systematic price movements instocks that can be exploited with countertrend strategies that buy stocksthat have been very weak over the recent past.
Owners of stock with an urgent need to liquidate their holdings needbuyers. This suggests that the market should offer compensation totraders who are willing to meet unmet needs for liquidity. Evidence pre-sented by Michael Cooper shows that buyers of oversold stocks canearn above-average short-term returns.!"” His study shows that stocksthat have displayed negative price momentum on declining trading vol-ume earn excess returns. In other words, the pattern identifies stockswith distressed sellers in search of buyers. The excess returns appear tobe a liquidity premium. Cooper's study showed that stocks that have de-clined sharply over the prior two weeks on declining volume display asystematic tendency to rise over the following week. The declining vol-ume aspect of this pattern is plausible because it can be interpreted tomean that there are insufficient buyers to meet the acute needs of thesellers. To guard against the possibility of data mining, Cooper used awalk-forward out-of-sample simulation to form portfolios of long andshort positions over the period 1978 to 1993. The long/short portfolios,created from 300 large capitalization stocks, earned an annualized re-turn of 44.95 percent versus a benchmark buy-and-hold return of 17.91
 
0.70.6 a
0.5SharpeRatio 0.4
0.30.20.1
 
 
 
 
 
 
   
 
 
 
Futures Stocks
FIGURE 7.15 Risk-adjusted returns to trend-following futures versus stocks.Theories of Nonrandom Price Motion 385
percent. Again, we see evidence of systematic price movements that canbe explained as a risk premium, in this case a liquidity premium, whichcan be exploited with TA.
In the context of efficient equity and futures markets, signals given byTA methods can be viewed as identifying opportunities to fill a marketneed. In effect, they are want ads, “risk adopter for hire by hedgers,” or“desperately seeking liquidity provider—will pay.” TA traders profitingfrom these signals are not getting a free lunch. They are simply readingthe market’s Help Wanted advertisements.
CONCLUSION
 
In this chapter, I have made a case for the existence of nonrandom pricemovements in financial markets. Without them, there can be no justifica-tion for technical analysis. With them, TA has an opportunity to capturesome portion of that nonrandom behavior. Ultimately, it is up to eachmethod of TA to prove that it can. 
 
Case Study:Signal Rulesfor theS&P 500 Index
 
 Case Studyof Rule Data
Mining forthe S&P 500
 
 
of which are reported in Chapter 9. The study evaluates the statis-
tical significance of 6,402 individual TA rules back tested on theS&P 500 Index over the period from November 1, 1980 through July 1,2005.
ke chapter describes a case study in rule data mining, the results
DATA MINING BIAS AND RULE EVALUATION
 
The primary purpose of the case study is to illustrate the application ofstatistical methods that take into account the effects of data-mining bias.To recap, data mining is a process in which the profitability of many rulesis compared so that one or more superior rules can be selected. Aspointed out in Chapter 6, this selection process causes an upward bias inthe performance of the selected rule(s). In other words, the observed per-formance of the best rule(s) in the back test overstates its (their) ex-pected performance in the future. This bias complicates the evaluation ofstatistical significance and may lead a data miner to select a rule with nopredictive power (i.e., its past performance was pure luck). This is thefool's gold of the objective technician.
This problem can be minimized by using specialized statistical-inference tests. The case study illustrates the application of two suchmethods: an enhanced version of White's reality check and Masters’s Monte-Carlo permutation method. Both take advantage of a recent improvement,!
389390 CASE STUDY: SIGNAL RULES FOR THE S&P 500 INDEX
which reduces the probability that a good rule will be overlooked (Type IIerror). That is to say, the improvement increases the power of the tests.
A secondary purpose of the case study was the possible discovery ofone or more rules that generate statistically significant profits. However,when the case study’s rules were proposed, it was unknown if any wouldbe found to be statistically significant.
AVOIDANCE OF DATA SNOOPING BIAS
 
In addition to data mining bias, rule studies can also suffer from an evenmore serious problem, the data-snooping bias. Data snooping refers tousing the results of prior rule studies reported by other researchers. Be-cause these studies typically do not disclose the amount of data miningthat led to the discovery of whatever it was that was discovered, there isno way to take its effects into account and hence no way to properly eval-uate the statistical significance of the results. As pointed out in Chapter 6,depending on which method is being used—White’s Reality Check or theMonte Carlo permutation method—information about each rule testedmust be available to construct the appropriate sampling distribution.
For example, suppose the case study in this book had included a ruledeveloped by Dr. Martin Zweig known as the double 9:1 upside/downsidevolume rule. This rule signals long positions when the daily ratio of upsideto downside volume on the NYSE exceeds a threshold value of 9 on twoinstances within a three-month time window. Note, this rule has three freeparameters: the threshold value on the ratio (9), the number of instanceson which the ratio exceeds the threshold (2), and the maximum time sep-aration between the threshold violations (3 months). According to tests ofthis rule conducted by students in my technical analysis class, the signaldoes have predictive power over the following 3, 6, 9, and 12 months. Inother words, the rule is statistically significant, but only under the as-sumption that Zweig did not engage in data mining to discover the para-meter values he recommends: 9, 2, and 3. Zweig has not reported if hetested other versions of the rule using different parameter combinationsor, if he did, how many combinations were tried to find the specific setthat defines his rule. If Zweig’s rule were to be included in the case studyand it was selected as the best rule, it would be impossible to take into ac-count the true amount of data mining that led to its discovery.
In an effort to avoid data-snooping bias, the case study did not explic-itly include any rules discussed by other researchers. Even though it ispossible, perhaps even likely, that some of the study’s rules were similarto those tested in prior rule studies, these similarities were by accident,Case Study of Rule Data Mining for the S&P 500 391
not by design. This precaution mitigated the data-snooping bias but couldnot eliminate it entirely because, in proposing the 6,402 rules tested, Icould not help but be affected by rule studies I have previously read.
ANALYZED DATA SERIES
 
Although all rules were tested for their profitability on the S&P 500 Index,the vast majority of rules utilized data series other than the S&P 500 togenerate buy-and-sell signals. These other data series included: other mar-ket indices (e.g., transportation stocks), market breadth (e.g., upside anddownside volume), indicators that combine price and volume (e.g., on-balance volume), prices of debt instruments (e.g., BAA bonds), and interest-rate spreads (duration spread between 10-year treasury notes and 90-daytreasury bills). This approach is in the spirit of intermarket analysis as dis-cussed by Murphy.’ All series and rules are detailed below.
TECHNICAL ANALYSIS THEMES
 
The 6,402 rules tested were derived from three broad themes of TA: (1)trends, (2) extremes and transitions, and (3) divergences. Trend rules gen-erate long and short market positions based on the current trend of thedata series analyzed by the rule. Extreme and transition rules generatelong and short positions when the analyzed series reaches an extremehigh or low value or as it makes a transition between extreme values. Di-vergence rules generate signals when the S&P 500 Index trends in one di-rection while a companion data series trends in the other. The specificdata transformations and signal logic for each rule are described later inthis chapter.
The rules employ several common analysis methods including movingaverages, channel breakouts, and stochastics, which I refer to as channelnormalization.
PERFORMANCE STATISTIC: AVERAGE RETURN
 
The performance statistic used to evaluate each rule was its average re-turn over the period (1980 to 2005) when back tested on detrended S&P500 data. As pointed out in Chapter 1, detrending involves subtracting theS&P 500’s average daily price change over the back-test period from each392 CASE STUDY: SIGNAL RULES FOR THE S&P 500 INDEX
day’s actual price change. This results in a new data series for which theaverage daily change is equal to zero.
As discussed in Chapter 1, detrending eliminates any benefit or detri-ment that would accrue to a rule’s performance as a result of either a longor short position bias. A binary rule can be position biased if one of itsconditions (long or short) is restrictive relative to the other condition(long or short). For example, if the long-position condition is more diffi-cult to satisfy than the short-position condition, the rule will have a short-position bias and tend to spend a majority of its time in short positions. Ifsuch a rule were to be applied to market data with a positive trend (aver-age daily price change > 0), its average return would be penalized. This re-duction in the rule’s performance would have nothing to do with itspredictive power and thus cloud its evaluation.
NO COMPLEX RULES WERE EVALUATED
 
To keep the scope of the case study manageable, it was restricted to testsof individual rules. Complex rules, which are derived by combining two ormore individual rules with mathematical and/or logical operators, werenot considered. The combining method can be as simple an unweightedaverage or as complex as an arbitrarily nonlinear function derived withsophisticated data-modeling software.
Limiting the case study to individual rules was detrimental for tworeasons. First, few practitioners rely on a single rule to make decisions.Second, complex rules exploit informational synergies between individualrules. It is not surprising, therefore, that complex rules have demonstratedhigher levels of performance. At least one study’ has shown that complexrules can produce good performance, even when the simple rules com-bined to form the complex rule are individually unprofitable. Combiningrules intuitively is extremely difficult, but there are now effective auto-mated methods? for synthesizing complex rules.
THE CASE STUDY DEFINED IN STATISTICAL TERMS
 
The following section defines the case study in terms of the key elementsof a statistical study: the population at issue, the parameter of interest,sample data used, the statistic of interest, the null and alternative hy-potheses considered, and the designated significance level.Case Study of Rule Data Mining for the S&P 500 393
The Population
The population at issue is the set of daily returns that would be earned byarule if its signals were to be applied to the S&P 500 over all possible real-izations of the immediate practical future.’ This is an abstract popula-tion, in the sense that its observations have not yet occurred and it isinfinite in size.
Population Parameter
The population parameter is the rule’s expected average annualized returnin the immediate practical future.
The Sample
The sample consists of the daily returns earned by a rule applied to de-trended S&P 500 Index prices over the back-test period from November 1,1980 until July 1, 2005.
Sample Statistic (Test Statistic)
The sample statistic is the average annualized return earned by a rulewhen applied to the detrended S&P 500 price data from November 1, 1980until July 1, 2005.
The Null Hypothesis (H,)
The null hypothesis states that all 6,402 rules tested are without predictivepower. This implies that any observed profits in a back test were due tochance (sampling variability).
In reality, there are two version of the null hypothesis in the casestudy. This stems from the fact that two different methods were used toassess statistical significance: White’s reality check and Masters’s MonteCarlo permutation method. Although both methods assert that all rulesconsidered were devoid of predictive power, they implement this assump-tion differently. The null hypothesis tested by White’s reality check is thatall rules tested have expected returns equal to zero (or less). The nulltested by Masters’s Monte Carlo permutation is that all rules generatedtheir long and short positions in a random fashion. In other words, longand short were randomly permuted with the market’s one-day-forwardprice change.394 CASE STUDY: SIGNAL RULES FOR THE S&P 500 INDEX
The Alternative Hypothesis
This study’s alternative hypothesis asserts that a rule’s back-tested prof-itability stems from genuine predictive power. Again, the two methodsassert this in slightly different ways. In the case of White’s reality check,the alternative hypothesis says that there is at least one rule within thetested universe that has an expected return greater than zero. It shouldbe noted that White’s reality check does not assert that the rule with thehighest observed performance is necessarily that best rule (i.e., the rulewith the highest expected return). However, under fairly reasonableconditions, they are one and the same.° The alternative hypothesis de-clared by Masters’s method is that a rule’s back-tested profitability is theresult of an informative pairing of its long and short positions with themarket’s one-day-forward price change. In other words, the rule has pre-dictive power.
The Statistical Significance Level
A5 percent level of significance was chosen as a threshold for rejection ofthe null hypothesis. This means there was a 0.05 probability of rejectingthe H, hypothesis when the H, was, in fact, true.
Practical Significance
The practical significance of a result is different from its statistical signifi-cance. The latter is the probability that a rule that has no predictive power(i.e., H, is true) would earn a return as high as or higher than the returnproduced by the rule in a back test by luck. In contrast, practical signifi-cance relates to the economic value of the observed rule return. Whensample sizes are large, as they are in the case study (i.e., over 6,000 days),the H, can be rejected even if a rule’s return is positive by a very minoramount. In other words, a rule can be statistically significant even thoughits practical value is negligible. Practical significance depends upon atrader's objectives. One trader may be satisfied with a rule for which ex-pected return is 5 percent whereas another may reject any rule for whichexpected return is less than 20 percent.
RULES: TRANSFORMING DATA SERIESINTO MARKET POSITIONS
 
A rule is an input/output process. That is to say, it transforms input(s),consisting of one or more time series, into an output, a new time series 
Case Study of Rule Data Mining for the S&P 500
consisting of +1’s and -1’s that indicate long and short positions in themarket being traded (i.e., S&P 500). This transformation is defined by theone or more mathematical,” logical,’ or time series® operators that are ap-plied to the input time series. In other words, a rule is defined by a set ofoperations. See Figure 8.1.
Some rules utilize inputs that are raw time series. That is to say, theone or more time series used as inputs to the rule are not transformed inany way prior to the rule’s position logic being applied, for example theS&P 500 closing price. This is illustrated in Figure 8.2.
Other rules utilize input series that have been derived from one ormore raw market series by applying various transformations to the mar-ket data. These preprocessed inputs are referred to as constructed dataseries or indicators. An example of an indicator is the negative volumeindex. It is derived from transformations of two raw data series; S&P 500closing price and total NYSE daily volume. This is illustrated in Figure8.3. The transformations used in the creation of the negative volume in-dex and other indicators used in the case study are described in the fol-lowing section.
INPUT OUTPUTMarket Time SeriesTime ofSeries Market Position{________»Time
Mathematical,
 
 
Logical,& .Time-SeriesTi me OperatorseeeEtc.
FIGURE 8.1 TA rule transforms input into output.396 CASE STUDY: SIGNAL RULES FOR THE S&P 500 INDEX
Input Output
V/
RawS&P 500Series
 
 
Rule
 
 
 
 
 
 
FIGURE 8.2 Raw market time series as rule input.
Raw RawTime Series Time SeriesS&P 500 NYSEP Volume
Negative Volume IndexFormula
Indicator(Negative Volume Index)
 
 
 
 
 
 
 
 
 
Rule Output
 
FIGURE 8.3 Transformed market time series as rule input.
TIME-SERIES OPERATORS
This section describes the time-series operators used in the case study totransform raw data series into indicators. Time-series operators are mathe-matical functions that transform a time series into a new time series. Thecase study utilized several common time-series operators: channel break-out, moving average, and channel normalization. The channel breakout op-erator is used to identify trends in a time series for rules that are based onCase Study of Rule Data Mining for the S&P 500 397
trends. The moving-average operator, which can also identify trends, is usedin the case study for smoothing. The channel-normalization operator, orstochastics, is used to eliminate the trend of a time series (i.e., detrending).
Channel Breakout Operator (CBO)
“The purpose of all trend identification methods is to see past the underly-ing noise in a time series, those erratic moves that seem to be meaning-less, and find the current direction of the series.”!” One such method is then-period channel-breakout operator.'! Here, n refers to the number oftime periods (days, weeks, and such) into the past that are examined todefine the upper and lower channel boundaries that enclose the time se-ries. The lower boundary is defined by the minimum value of the time se-ries over the past n-periods, not including the current period. The upperboundary is defined by the time series’ maximum value over the pastn-periods, not including the current period. Despite its extreme simplicity,the channel-breakout operator has proven to be as effective as more com-plex trend-following methods.”
As conventionally interpreted, the channel breakout operator signalslong positions if the analyzed series exceeds its maximum value estab-lished over the past n-periods. Conversely, short positions are signaledwhen the time series falls below its n-period minimum. This is illustratedin Figure 8.4. In practice, the channel is redrawn as each new data point
Look-Back
SpanTime eee = == 4Series
  
      
      
ChannelBreakout
     
ChannelBreakout
  
+1 prttte crore nnn n rec r enews ,: Rule :
 
1 \ ; Output ; a
FIGURE 8.4 Channel-breakout operator.398 CASE STUDY: SIGNAL RULES FOR THE S&P 500 INDEX
becomes available, but in Figure 8.4 only several channels are shown to il-lustrate its function. Although the channel has a constant look-back spanwith respect to the time axis, its vertical width with respect to the axisthat represents the value of the time series adjusts dynamically to therange of the series over the past n-periods. This feature may explain theeffectiveness of the method. This is to say, the channel breakout’s dy-namic range may reduce the likelihood of false signals caused by an in-crease in the series’ volatility rather than an actual change in its trend.
The n-period breakout operator has a single free parameter, its look-back span, the number of time periods into the past used to establish thechannel's boundaries. In general, the larger the look-back span, the widerthe channel, and hence, the less sensitive the indicator. Thus, larger valuesof n are used to identify larger trends. The case study tested 11 differentlook-back spans for which lengths were separated by a factor of approxi-mately 1.5. The specific spans used were: 3, 5, 8, 12, 18, 27, 41, 61, 91, 137,and 205 days.
Moving-Average Operator (MA)
The moving average is one of the most widely used times-series operatorsin TA. It clarifies trends by filtering out high frequency (short-period) fluc-tuations while passing through low frequency (long-period) components.Thus, the moving-average operator is said to