1 equations in 7 + 1 variables are created by summing the 7 + 1 equations developed fromthe general equation by multiplying the second equation by »,, the third by «2, and so on:
an +m > tam t+, > mm = Dya Smita Dat ta, Smt tay > My =D myay S24, Sx t ay SE to ty Ry =D MY
Gay Y Xn $A DKK q FOr Dy Han DE =D wD
The solution to this system of equations can be calculated on most computers using astandard program available for this purpose (see Appendix 2). Those with only a littleexperience in regression analysis should remember that the model is most accurate withinthe range of the data points; when projecting outside the bounds of the sample data, thepredictive qualities of the regression formula decrease with time. It is most likely that agood solution will be found for interest rates and grain prices, which are trading at thesame levels as previous years, but may not be reliable for index markets that have madeerratic new highs over the years.
Many dependent variables (x) may be used to increase the possibility of finding a goodcorrelation. The predictive quality of this solution will depend on the relevance of theindependent variables. It is best to start with the obvious components of a time series, suchas inflation, then add standard economic statistics, as the Consumer Price Index and indus-trial product, as well as supply-and-demand information specific to the market being eval-uated, For both grain and energy markets, the accumulation of inventory, or stocks, is astrong influence on price; these factors also have an expected seasonal variation, which isrepresented in terms of an index of adjustment. Measuring the error of the estimates willhelp determine whether additional factors are necessary. .
.Least-Squares Sinusoidal
A special case of the multiple linear predictor occurs when periodic peaks can be observedin the sample time series data. These peaks and valleys suggest that the time series mayhave a cyclic pattern. One of the more well-known uses of cyclic analysis was performed byHurst in The Profit-Magic of Stock Transaction Timing (Prentice-Hall), in which there is aninteresting example of Fourier analysis applied to the Dow Jones Industrial Averages.
‘The equation for the approximation of a periodic movement is
2nt , 2mt ant , 2ntYF do + at + G2 COS “E+ ay Sin —]~ + at COS“ + st sin ~~ _
which is a special case of the generalized multivariate approximationIY = Ay + Ay, + Az Xz + AzXz + AgXs + AsXs
where P is the number of data points in each cycle and
, the incremental time element
x = cos (2nt/P), a cyclic element -in (2nt/P), a cyclic element
cos (2nt/P), an amplitude-variation element
x, =t sin (2n¢/P), an amplitude-variation element
    
 
 
ARIMA - 55
The term @¢ will allow for the linear tendencies of the sequence. The term 2m refers toan entire cycle and 2nt/P designates a section (1/P) of a specific cycle ¢; this in turn addsweight to either the sin or cos functions at different points within a cycle.
The solution is calculated in the tabular manner of the other methods, using simulta-neous linear equations derived in the same way as the generalized multivariate equation,substituting from the table of sums and solving the coefficient matrix for do, .. . , as. ACom-plete discussion of curve fitting using trigonometric functions appears in Chapter 7 (“Sea-sonality”).
ARIMA
An Autoregressive Integrated Moving Average (ARIMA) model is created by a process ofrepeated regression analysis over a moving time window, resulting in a forecast value. AnARIMA process automatically applies the most important features of regression analysis ina preset order and continues to reanalyze results until an optimum set of parameters orcoefficients is found. In Chapter 21, the selection and testing of individual parameterswithin a trading strategy are discussed. This involves approximating their initial value andidentifying a testing range. An ARIMA model does all of this as part of its special process.Because it is used to recalculate the best fit each time a new piece of data appears, it maybe thought of as the first adaptive process.
G.E.P Box and G.M. Jenkins refined ARIMA at the University of Wisconsin,* and theirprocedures for solution have become the industry standard. This technique is oftenreferred to as the Box.Jenkins forecast. The two important terms in ARIMA are “autoregres-sion” and “moving average.” Autoregression refers to the use of the same data to selfpredict, for example, using only gold prices to arrive at a gold price forecast. Movingaverage refers to the normal concept of smoothing price fluctuations, using an average ofthe past days. The moving average and popular variations are discussed thoroughly inChapters 4 and 5. This process uses an exponential smoothing technique, which is amongthe most popular methods.
In the ARIMA process, the autocorrelation is used to determine to what extent pastprices will forecast future prices. In a first-order autocorrelation, only the prices on theprevious day are used to determine the forecast. This would be expressed as
P,=axP,,+e
where P, is the price being forecast (dependent variable)P,., is the price being used to forecast (independent variable)a is the coefficient (constant percentage)e is the forecast error
In a second-order autoregression, the previous two prices are used as follows:P,=@,XP,_,+a2X*P,_2+e
where the current forecast P, is based on the two previous prices P,_, and P,-2; there aretwo Unique coefficients and a forecast error. The moving average is used to correct for theforecast error, e. There is also the choice of a first- or second-order moving average process,
First-order: E, = e, -— b x e,-;Second-order: E, =e, — by X @;-1— bz X er-2
where £, is the approximated error terme, is today’s forecast error
*G.EP Box and G.M. Jenkins. Time Series Analysis: Forecasting and Control, 2nd ed, (Holden-Day, San Fran-cisco, 1976). 
56 REGRESSION ANALYSIS
e,-1 and e,_ 2 are.the two previous forecast errors6, and 6; are the two regression coefficients
Because the two constant coefficients, b, and 62, can be considered percentages, the mov-ing average process is similar to exponential smoothing.
‘The success of the ARIMA model is determined by two factors: high correlation in theautoregression and low variance in the final forecast errors. The determination of whetherto use a first- or second-order autoregression is based on a comparison of the correlationcoefficients of the two regressions. If there is little improvement using the second-orderprocess, it is not used due to its time-consuming calculations. The final forecast is con-structed by adding the moving average term, which approximates the errors, back into theautoregressive process
Pl=P,+& +e
where P; is the new forecast, and e’ is the new forecast error. The moving average processis again repeated for the new errors e’, added back into the forecast to get a new value P”and another error e”. When the variance of the errors becomes sufficiently small, theARIMA process is complete. . .
‘The contribution of Box and Jenkins was to stress the simplicity of the solution. Theydetermined that the autoregression and moving average steps could be limited to first- orsecond-ortler processes. To do this, it was first necessary to detrend the data, thereby mak-ing it stationary. Detrending can be accomplished most easily by differencing the data,that is, creating a new series by subtracting each previous term P,_, from the next P,. Ofcourse, the ARIMA program must remember all of these changes, or transformations, torestore the final forecast to the proper price notation by applying these operations inreverse. If a satisfactory solution is not found in the Box-Jenkins process, it is because thedata are still not stationary and further differencing is necessary.
Because of the three features just discussed, the Box-Jenkins forecast is usually shownas ARIMA(p,d,q), where p is the number of autoregressive terms, d is the number of dif-ferences, and q is the number of moving average terms: The expression ARIMA(0, 1,1) isequivalent to simple exponential smoothing, a common technique discussed in the nextchapter. In its normal form, the Box-Jenkins ARIMA process performs the following steps:
1. Specification. Preliminary steps for determining the order of the,autoregressionand moving average to be used.
a. The variance must be stabilized. In many price series, increased volatility is di-rectly related to increased price. In stocks, the common assumption is that thisrelationship is /og-normal. A simple test for variance stability, using the log func-tion, is checked before more complex transformations are used.
b. Prices-are detrended. This uses the technique of first differences; however, a sec-‘ond difference (or more) will be performed if it helps to remove further trendingproperties in the series (this is determined by later steps).
c. Specify the order of the autoregressive and moving average components. Thisfixes the number of prior terms to be used if these approximations (not neces-sarily the same number). In the Box-Jenkins approach, these numbers are usually
smail, often one for both. Large numbers require a rapidly expanding amount of
calculation, even for a computer.
How many terms are necessary? This is a critical part of the ARIMA solutionand may require manual intervention in the computer program. The object ofthis step is.to find the fewest terms necessary to solve the problem. All ARIMAprograms will print a correlogram, a display of the autocorrelation coefficients.‘The correlogram is used to find whether all the trends and well-defined periodic
ARIMA - 87
 
movements have been removed from the series by differencing. Figure 3-9 showsthree patterns of correlograms printed by EASI/ARIMA on a PC. The top scaleshows the importance of the coefficients. If they fall within the brackets, theyhave no importance; if they fall outside the brackets, they are highly significant(greater than two standard deviations, or 95%). The left scale gives the /ag rela-tionship, that is, the relationship of the current price to the price n days ago. Anasterisk will appear on line 3 outside the bracket (see Figure 3-96) if a strong pat-tern in the lag 3 relationship to current prices has not yet been removed. To re-move this, additional differencing of the data must be performed. If successful,the resulting correlogram will appear as in Figure 3-9, where only the lag 1 and2 values are outside the bracket. If further differencing results in a pattern shownin Figure 3-9c, all correlation of any importance in the data have been lost. If thepattern in Figure 3-9c were to occur in the first step, it would mean that there isno relationship to be found by autoregression. If it occurs after some differenc-ing has been performed, the data have been differenced too many times.”
2. Estimation: determining the coefficients. The previous step was used to reduce thenumber of autoregressive and moving average terms necessary to the estimationprocess. The ARIMA method of solution is one of minimizing the errors in the fore-cast. In minimization, it will perform a linear or nonlinear regression on price (de-
7 Examples of the correlogram come from Eric Weiss, “Applying ARIMA Forecasts,” Technical Analysis of Stocks &Commodities (May 1983). Refer to Weiss’s two other articles on ARIMA in earlier volumes of the same magazine,especially “ARIMA Forecasting” (January 1983).
 
FIGURE 3-9 ARIMA correlograms. (a) Ideal correlogram results, showing significance in thefirst and seconds lags only. (b) Significance in the first and third lags, indicating that furthertrend elimination is necessary. (c) No significant correlations at any lag values.
   
 
 
“1 ° +1 ( , *2 ( ,*3. ‘ .4 ( *)5 ( ")@1 9 +71 (2 C3 “¢ )4 ( * )5 (oof ) -(6)a 9 +1 +2 ( wy)3 ( * )4 ( . )5 4 7) -(
  
3B REGRESSION ANALYSIS
pending on the number of coefficients selected), determine the errors in the esti-mation, and then approximate those errors using a moving average. It will next lookat the resulting new error series, attempt to estimate and correct the errors in thatone, and repeat the process until it accounts for all price movement.
‘To determine when an ARIMA process is completed, three tests are performedat the end of each estimation pass:
a. Compare the change in the coefficient value. If the last estimation has caused lit-tle or no change in the value of the coefficient(s), the model has successfully con-verged to a solution.
b. Compare the sum of the squares of the error. If the error value is small or if itstays relatively unchanged, the model is completed.
¢. Perform a set number of estimations. Unless a maximum number of estimationsis set, an ARIMA process might continue indefinitely. This safety check is neces-sary in the event the model is not converging to a solution.
Once completed, the errors can be examined using an O-statistic to check for anytrend. If one exists, an additional moving average term may be used to eliminate it.
Forecast Results
Once the coefficients have been determined, they are used to calculate the forecast value.These forecasts are most accurate for the next day and should be used with less confidencefor subsequent days (see Figure 3-10).
What if the forecast does not work? First, the process should be checked to be certainthat it was performed properly. Pay particular attention to the removal of trends using thecorrelogram. Next, check the data used in the process. If the data sample is changing
(which can be observed on a price chart), select either a shorter or longer period that con-
tains more homogeneous data, that is, data similar to the current market period.
Trading Strategies
In the article that originally piqued the interest of traders,® Anon uses a 5-day-ahead fore-cast. If the ARIMA process forecasts an uptrend and if prices fall below the forecast value,\
* Louis J. Anon, “Catch Short-Term Profits with ARIMA,” Commodities Magazine (December 1981).
eyFIGURE 3-10 ARIMA forecast becomes less accurate as it is used farther ahead.
 
 
95%
 
 
ea LS
 
ARIMA - 59
the market can be bought with added confidence (expecting lower risk and more profit).This technique of selecting better entry points may compensate for some of the inaccura-cies latent in any forecasting method. The danger of this approach is that prices may con-tinue to move counter to the forecast, so that caution and a stop-loss are also necessary.
Following the Trend
Use the 1-day-ahead forecast to determine the trend position. If the forecast is for higherprices, a long position should be held; if it is predicting lower prices, shorts are necessary.
Countertrend Indicator
Use the ARIMA confidence bands to determine overbought/oversold levels. Not only can along position be entered when prices penetrate the lowest 95% confidence band, but theycan be closed out when they return to the normal 50% level. A conservative trader willenter the market only in the direction of the ARIMA trend forecast. As shown in Figure 3-10, if the trend is up, only the penetrations of the lower band will be used to enter newJong positions.
Use of Highs and Lows
Both the implied highs and lows as well as the independently forecasted highs and lows canbe the basis for other interesting strategies.” The following two are used with intraday prices.
” 1. Using confidence bands based on the closing prices, buy an intraday penetration ofthe expected high or sell a penetration of the expected low, and liquidate the posi-tion on the close. Use a stop-loss. Consider taking positions only in the direction ofthe ARIMA trend.
2. Using the separate ARIMA models based on the daily high and low prices, buy a pen-etration of the 50% level of the high and sell a penetration of the 50% level of thelows. Liquidate positions on the close. Use a stop-loss.
SlopeThe one-day-ahead forecast suggested in “Following the Trend,” a few paragraphs earlier,is essentially a projection of the slope of the trendline. Because of the frequent erratic pricemovement, also called noise, the purpose of directional analysis, whether regression ormoving averages, is to uncover the true direction of prices. Therefore, the slope of thetrendline, or the direction of the regression forecast, is the logical answer.
‘The popular alternate for triggering a new directional signal is a price penetration ofan envelope or band value. Using regression analysis that band can be replaced by a confi-dence level. While it is true that the number of random, or false, penetrations declines asthe confidence band gets farther away from thé trendline, so does the total number of pen-etrations. At any band distance there are still a large number of erroneous signals. Theslope itself should be viewed as the best approximation of direction.
Kalman Filters
Kalman offers an alternative approach to ARIMA, allowing an underlying forecasting model(message model) to be combined with other timely information (observation model). Themessage model may be any trading strategy, moving average, or regression approach. Theobservation model may be the floor broker’s opening calls, market liquidity, or, in the caseof existing foreign markets, earlier trading activity—all of which have been determined tohave some overriding importance in forecasting.
° John F. Kepka, “Trading With ARIMA Forecasts,” Zechnical Analysis of Stocks & Commodities (August 1985). 
60 REGRESSION ANALYSIS
Assume that the original forecast (message) model can be described asM(B = GPr-1 + me,
and the observation model asOP, = CoP, + 08;
where me and oe are the message and observation model errors, respectively. The com-bined forecast would then use the observation model error to modify the result
Pig = OPi + Kr. 100:
where K is the Kalman gain coefficient,” a factor that adjusts the error term.
LINEAR REGRESSION MODEL
A linear regression, or straight-line fit, could be the basis for a simple trading strategy sim-ilar to a moving average. For example, an n-day linear regression, applied to the closingprices, could be used with the following rules:
1, Buy when the closing price moves above the forecasted value of today's close.2. Sell when the closing price moves below the forecasted value of today’s close."?
There is an important difference between a model based on linear regression and onefounded on a moving average. There is no lag in a regression strategy. If prices continuehigher at the same rate, a moving average system will initially lag behind, then increase atthe same rate. The lag creates a safety zone to absorb some changes in the direction ofPrices, without getting stopped out. (See Chapter 4 for a complete discussion of movingaverages, and Chapter 5 for a comparison of a Linear Regression Slope trading system withfive other popular trending methods.)
A regression model, on the other hand, identifies a change of direction sooner by mea-suring future movement against a straight-line projection in which the current price valuehas little influence. A steady price move, however, will place the fitted line right in the cen-ter of market movement, subject to frequent whipsaws. The area at which a uniform trendchanges from one direction to another is a difficult case for a linear regression system (seeFigure 3-11) and points out the need for using bands. Even with bands, the turning pointof an orderly trend will appear to have much greater variance than during the directionperiod over the same calculation interval.
In Figure 3-11, the three positions of the regression line all show numerous penetra-tions of the price, resulting in many losing trades. This is natural because the regression.line is fitted best when it goes through the center of price movement. For prices to remainon one side of a regression line it must be constantly accelerating or decelerating, anunlikely situation. It is possible that this approach would work better using very few cal-culation periods; however, the use of only a few days is inconsistent with the nature ofsuch a statistical measurement, which gains significance with more data. Figure 3-116shows the more practical use of a confidence band spaced equally around the regressionline. Using a high-confidence band (e.g., 95%), it is possible to turn the erratic perfor-mance in Figure 3-114 into two clearly identifiable trends. Interestingly, the use of a confi-
*° For a more complete discussion, see Andrew D. Seidel and Philip D. Ginsberg’s Commodities Trading (Prentice-
Hall, Englewood Cliffs, NJ, 1983), or R.E, Kalman’s “A New Approach to Linear Filtering and Prediction Problems,”Journal of Basic Engineering (March 1960). .
¥ These rules were used by Frank Hochheimer in Computerized Trading Techniques 1982 (Merrill Lynch Com-
modities, New York, 1982).
 
LINEAR REGRESSION MODEL 61
 
 
FIGURE 3-11 Linear regression model. (a) Simple penetration of the regression lines. (b)Penetration of the channel formed by confidence bands.
 
(a)
 
dence band with the regression analysis looks remarkably similar to a channel on a stan-dard bar chart.
Hochheimer’s 1982 study, performed without a channel or confidence band, shows thebest selection of the regression interval at 60 to 70 days (the maximum tested was 70 days).Silver had a remarkable 1,615 trades—oné each day. Only two markets showed averagetrade duration greater than 2 days. We can draw the conclusion that successful regression-based systems seek out longer time periods. 
 
Trend Calculations
                                   
the market, those erratic moves that seem to be meaningless, and find the currentrection of prices. Because there may be more than one trend at any one time,caused by short-term events and long-term policy, it is possible to search for the strongest,or most dominant trend, or a minor trend that corresponds to your expected time frame.The technique that is used to uncover the right trend depends upon whether any of thetrend characteristics are known. Does it have a seasonal or cyclic component, is it based onlong-term monetary policy, or is it an overnight effect? The more you know about the rea-sons why prices trend, the better you will be able to find the most reliable calculation forseparating the price direction from market noise.
Chapter 3, “Regression Analysis,” produced forecasts by finding relationships bétween.two or more price series. Once you know that there is a fundamental relationship betweendata, based on measuring the properties of dependence and correlation, a formula can befound that expresses one price movement in terms of the other prices and data. The pre-dictive qualities of these methods are best when applied to data that has been seen before,that is, prices that are within the range of historic data. Forecasting reliability decreasessharply when values are based on extrapolation outside the previous occurrences. Thisphenomenon will also be true of other trending methods. Because of the way we test anddefine the final trend calculation, it is based on the movement of historical data; whenprices move to new levels, the results of the modei will often deteriorate.
T° purpose of all trend identification methods is to see past the underlying noise in
 
 
FORECASTING AND FOLLOWING
\There is a clear distinction between forecasting the trend and determining the currenttrend. Forecasting, predicting the future price, is much more desirable but very complex.As shown in the previous chapter, it may involve combining many pieces of data and resultin a forecasted price with a confidence level. The further you try to forecast into the future,the lower the confidence. _
The techniques most commonly used for evaluating the direction or tendency of-prices both within prior ranges or at new levels are called autoregressive functions. Unlikeforecasting models, they are only concerned with evaluating the current price direction.This analysis is normally used to conclude that prices are moving in an upward, down-ward, or sideways direction. From this simple building block, it will be possible to formrules of action and develop complex strategies of anticipation. From a practical viewpoint,these trending methods are more flexible than the traditional regression models, but to “3achieve success they introduce a lag. A great effort has been spent trying to reduce this lag. *
In an autoregressive model, one or more of the previous day’s prices determine thenext sequential price. If P, represents today’s price, P;_,; yesterday’s, and so on, thentomorrow’s expected price will be
   
  
Py) = Got QP, + @2P,_, ++ +aPit+e
ee
LEAST-SQUARES MODEL 63
 
where each price is given a corresponding weighting a, and combined to give the resultantprice for tomorrow P,,, = e (where e represents an error factor). The simplest example isthe use of yesterday's price alone to generate tomorrow's price:
Prav= ot aP,te
which you may also recognize as the formula for a straight line, y = @ + bx.
The autoregressive model does not have to be linear; each prior day can have a non-linear predictive quality. Thus each expected price P,,, could be represented by a curvilin-ear expression, P,,1 = ao + dP; + @,P?_, + e, or by an exponential or logarithmic farmula,In P,.=4o+@In P, + azIn P,_, +e, which is commonly used in equity analysis. All of theseexpressions could then be combined to form an autoregressive forecasting model for P,, 1.In going from the simple to the complex, it is natural to want to know which of thesechoices will perform best. The answer can only be found by application to a specific prob-lem, validation, and experience. Various methods, based on a sound premise, must beattempted and applied to actual data in real-time or extrapolated situations to determinethe predictive quality of any model. Chapters 5 and 15 will discuss which choices havebeen most popular and Chapter 21 will show testing methods that are most likely to leadto robust results.
 
LEAST-SQUARES MODEL
The least-squares regression model is the same technique that was used in the previouschapter to find the relationship between two dependent markets, corn and soybeans, or tofind how prices moved when driven by known related factors such as supply and demand.Here, the least-squares model will be used to find the relationship between time and price,rather than between two prices, where the price forecast that we are seeking is dependentupon time. The regression model will also be applied in an autoregressive way by recalcu-lating the expected price daily and using the slope of the resulting straight line or curvilin-ear fit to determine the direction of the trend.
A simple error analysis can be used to evaluate the predictive qualities of this method.Assume that there is a lengthy price series for a market and that we would like to knowhow many prior days are optimum for predicting the next day’s price. The answer is found
- by looking at the average error in the predictions. If the number of days in the calculation
increases and if the predictive error decreases, the answer is improving; if the error stopsdecreasing, the accuracy limit has been reached. Error analysis can improve most trend cal-culations, and it is also covered in the section on exponential smoothing later in this chap-ter. As an example, start by using only one prior day to find the price forecast
Prii= aot aP,P,= A+ aPy-1
Pri =o + ayPr-2
“P= yt @P;
and work up to a large number of days:
Py = gt GP, + G2P,- 4 + AnP inedP,
Pry = Gq + QyPy2 + G2Pyeg t+ AnPrin
Sot GPp + GzPr-2t 0 + AnPron
  
> 64 TREND CALCULATIONS
 
In the last case, it takes days of prior prices to generate each new prediction. In all cases,n + 2 equations can be written to solve the n + 1 coefficients ao, 4), @2,. .- , Gn USINE thematrix elimination method found in Appendix 3. The result is a predicted price P;’ for eachactual price P,. We will use the notation P’(7), to mean the predicted price for day ¢ usingan n-day linear regression; therefore, P’(3)2s = 58.00 means that the predicted value of Ponday 25 was 58.00 using a 3-day linear regression analysis (a straight-line fit of the 3 priordays). The error occurring in each prediction is defined as
 
E(),=P-P(n),
the difference between the actual and predicted values for that day using the 7-day linearregression. As an example of error analysis, a copper sequence was selected for a period of20 trading days that included a slight upward and a slight downward move, with someintermediate changes of direction. Table 4-1 shows the actual predictions using linearregressions for 2 through 7 prior prices and a prediction of one day forward. Table 4-2shows the relative error in these predictions and an analysis of the errors. The mean,shown at the bottom of Table 4-2, is a simple average of all the points and gives the nettrend bias during the sample period; the small range from negative to positive shows verylittle bias. The standard deviation measures the accuracy of the forecast. When the stan-dard deviation is small, the prédicted values are closer to the actual values and the forecasterror is smaller. ‘
The standard deviation and variance are both measurements of the distribution ofpoints around the fitted line. As discussed in Chapter 2, the standard deviation measuresthe occurrence of points near the predictions; the smaller the value, the closer the group-ing and the better the estimation. The variance is another way of looking at the deviations;the smaller the variance, the better the forecast. The copper error analysis shows the small-est values for E(4),, the 4-day linear regression. This is an interesting result because wewould expect the shortest forecasting interval to produce the smallest error simplybecause prices cannot move as far from the forecast in 2 days as they can in 7 days. There-fore, we should expect the absolute size of the error, as measured by the standard devia-tion or variance, to get larger as the forecasting period gets larger. The smailer errorproduced by the 4-day regression shows that there was a pattern during this short test thatcould be fit better by the 4-day model than by any of the others.
Determination of the best predictive model using error analysis can be applied to anyforecasting technique. This works particularly well when comparing the errors of two dif-ferent forecasting methods evaluated over the same number of periods, eliminating thebias caused by longer and shorter intervals. It is also practical to carry the error analysisone step further and include the results of the prediction error on day t+ 1, f + 2, and soon. This gives a measure of out-of-sample forecast accuracy and lends confidence to thepredictive qualities of the technique- ~
Having selected the most accurate forecast model, the size of the prior day predictiveerror canbe used to resolve trading decisions. Consider the following situations:
1. The prediction and the actual price are very clase (high confidence level). For ex-ample, the long-term copper error may have I standard deviation = .25.
2. Today’s forecast error is within 1 standard deviation of expectations; therefore, wecontinue to follow the trend strategy.
3. Today’s forecast error is between 1 and 3 standard deviations of expectations; there-fore, we are cautious, yet understand that this is normal but less frequent.
4. Today’s forecast error is greater than 3 standard deviations. This is unusual, indi-cates high risk, and may identify a price shock. Alternately, it could indicate a trendturning point.
 
THE MOVING AVERAGE 65
TABLE 4-1 Analysis of Predictive Error for Copper
 
Sequence Price Price Predictions for n-Day Linear Regression
 
 
) P P(2), P(3). PA) PS). P(6). PUT).
I 60.30
2 59.30
3 58.70
4 57.80
5 5