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Characterizing Moving Average Crossings

Figure 2 illustrated an example of how moving average crossings are used to signal turning points. To formalize these moving average crossings 1 first define

�(3)

The "moving average" rule of technical analysis then uses sign changes in Z, to generate the times {С‚^} sequentially as

Т/ = inf {t: t > т,_ „ Z,Z,_, < 0}, (4)

r

with С‚0 defined as zero.

Now consider what (3) and (4) say in words. I basically calculate two moving averages of the X, process. Assuming that n > m, the first moving average will be smoother than the second one in the sense of having relatively more power at low frequencies. Then, as soon as Z,, С‚,-_| < t changes sign, the rule in (4) will assign the value of t to С‚,-. These latter are signals of major market downturns and upturns according to the moving average method of technical analysis.

1 now show that the {т,} are Markov times, and that they constitute a well-defined method of prediction. Clearly, the product Z,_,Z, is measurable with respect to /,—that is, given /„ the value of Z,_,Z, is known. The т, are then defined as the first entry of Z,_ ,Z, in the interval (-oe, 0) G R. Thus the т, are Markov times according to the theorem above. This makes the moving average method a statistically well-defined procedure. We should, in principle, be able to evaluate the contribution of the {т,} in predicting market turning points using formal tools.

Characterizing Trend Crossings

Methods that use crossings of observed data with trend lines, defined in a variety of ways, constitute the most common form of technical analysis. In contrast to the moving average method, it is not possible to determine a unique definition that would encompass all trend crossing rules. The notion of a moving average immediately suggests a math �ematical formulation, whereas trend crossings appear to be based on arbitrary hand-drawn trends in charts illustrating historical data. Figure 1 displays an example. The main idea behind trend crossing methods is to determine two linear trends, one above, the other below, that would envelop the portion of the data observed since the last turning point. Then, upcrossings (downcrossings) of the upper (lower) enve �lope are taken as signals of market strength (weakness).

It is clear that all trend lines that envelop observed data can be defined by using only two extrema of the portion of the series under consideration. In order to obtain an upper envelope, the two highest local maxima are used. Two lowest local minima define, similarly, a lower envelope. Thus, the theory of local minima (maxima) of time series will play an important role in investigating this type of technical analysis.

I first show that most signals generated using trend lines are not Markov times. Let /, and /0 be the times of onset of the two lowest (highest) local minima (maxima) of X, during the period (С‚,_,, /], where Tj_ i is assumed to be known. Let Xt and X0 be the values of these minima (maxima). Consider the trend line T(t),

TO) = [(ЛГ, - Xt))/(tl - /„)](/) + [(XQtt - *,/<>)/(/, - h)],   (5)

for J, > /0 > Tj_,. This function defines a straight line that goes through the two lowest (highest) local minima (maxima) observed during the interval (С‚,-,, Рі]. As in the previous case, we obtain the times {С‚,} using

z, = x,- no, (6)

т, = inf {f: t> т,_„ Z,Z,_, < 0). (7)

I now show that the times {С‚,} generated by this algorithm will not be Markov times.

It is clear that if the Z, defined by (6) is /,-measurable, and if we adopted the rule in (7) to determine the {С‚,}, then this would be the first entry in the interval [0, by an /,-measurable random variable, and the С‚, would be Markov times. But it turns out that, in general, Z, is not /,-measurable since /, and IQ are never specified as the times of onset of the first two (or the nth) local minima (maxima) during , < t. In practice, the t0 and /, are simply said to be two lowest (highest) local minima (maxima) that occur after some predetermined time С‚,_,. But such an event is not /,-measurable since, before it can be decided whether a local maxima is highest or second highest, one needs to know the levels of subsequent maxima. Figure 4 illustrates this point. None of the trend lines shown here utilize the first two local maxima in determining the '/'(/). There were several local maxima between the selected tt and /0, and these were ignored in obtaining the trend lines of figure 4. Two of these are shown on figure 4 as dotted lines.

Thus we see that trend crossing techniques will not generate Markov times unless one specifies an /,-measurable mechanism for ignoring the local minima (maxima) between /, and /0.