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PURPOSE OF A MOVING AVERAGE A moving average is used to reduce unwanted noise in a time series so that the underlying behavior, unmasked by interference, can be more clearly perceived; it serves as a data smoother As a smoothing agent, a moving average is a rudimentary low-passJ%er, ie, a filter that permits low frequency activity to pass through unimpeded while blocking higher frequency activity In the time domain, high frequency activity appears as rapid up-and-down jiggles, ie, noise, and low frequency activity appears as more gradual trends or undulations Ehlers (1989) discusses the relationship between moving averages and low-pass filters He provides equations and compares several formal low-pass filters with various moving averages for their usefulness Moving averages may be used to smooth any time series, not just prices.

This departure from ordinary split pea soup has subtle undertones of the exotic spices from Thailand. I especially like the taste of freshly ground spices from whole seeds. Freshly ground mustard, cardamom, and allspice seeds taste powerfully different from the stale preground and packaged spices available in most stores.

THE ISSUE OF LAG Besides their ability to decrease the amount of noise in a time series, moving averages are versatile, easy to understand, and readily calculated However, as with any well-damped low-pass filter or real-time data smoothing procedure, reduced noise comes at a cost: lag Although smoothed data may contain less noise and, therefore, be easier to analyze, there will be a delay, or lag, before events in the original time series appear in the smoothed series Such delay can be a problem when a speedy response to events is essential, as is the case for traders Sometimes lag is not an issue, eg, when a moving average of one time series is predictive of another series This occurs when the predictor series leads the series to be predicted enough to compensate for the lag engendered by the moving average.

This kind of Web response can be very pro table. Overall, the direct mail produced a response rate of 1.42 percent and a registration rate of Figure 14-10 Results of the Quiz

It is then possible to benefit from noise reduction without the cost of delay Such a scenario occurs when analyzing solar phenomena and seasonal tendencies Also, lag may not be a serious problem in models that enter when prices cross a moving average line: In fact, the price must lead the moving average for such models to work Lag is more problematic with models that use the slope or turning points in the average to make trading decisions In such cases, lag means a delayed response, which, in turn, will probably lead to unprofitable trades A variety of adaptive moving averages and other sophisticated smoothing techniques have been developed in an effort to minimize lag without giving up much noise reduction One such technique is based on standard time series forecasting methods to improve moving averages To eliminate lag, Mulloy (1994) implements a linear, recursive scheme involving multiple moving averages.

Figure 14-11

When the rate of movement in the market is appropriate to the filter, lag is eliminated; however, the filters tend to overshoot (an example of insufficient damping) and deteriorate when market behavior deviates from filter design specifications Chande ( 1992) took a nonlinear approach, and developed a moving average that adapts to the market on the basis of volatility Sometimes lag can be controlled or.

1.01 percent. These registrants provided their names and company information and took the survey. The email promotion produced a response rate of 3.49 percent with a registration rate of 1.95 percent. It is signi cant that the email promotion was so much more productive. Figuring in the cost of the mailing, the results are shown in Table 14-5. Email promotion proved to be 11 times as cost-effective as direct mail. Why would that be There seem to be several reasons: The product was a computer product, so potential users all had computers. The audience was IT directors, who were used to email and were on the Web every day. The click-through process from an email takes only a second. Trying to enter a URL from a direct-mail piece takes at

eliminated by combining several moving averages to create a band-pass filter. Band-pass filters can have effectively zero lag for signals with periodicities near the center of the pass-band; the smoothed signal can be coincident with the original, noisy signal when there is cyclic activity and when the frequency (or periodicity) of the cyclic activity is close to the frequency maximally passed by the filter.

Direct mail Unit cost Cost for 50,000 Response rate Number of responses Registration rate Number registered Cost per lead $0.71 $35,500 1.42% 710 1.01% 505 $70.30 Email $0.12 $6,000 3.49% 1,745 1.95% 975 $6.15

All moving averages, from the simple to the complex, smooth time series data by some kind of averaging process. They differ in how they weigh the sample points that are averaged and in how well they adapt to changing conditions. The differences between moving averages arose from efforts to reduce lag and increase responsiveness. The most popular moving averages (equations below) are the simple moving average, the exponential moving average, and the front-weighted triangular moving average. Less popular is Chande s adaptive moving average (1992).