In this formula, a represents the intercept of the line with the Y axis, and b represents the slope of the line. A straight line can be drawn through a set of data points that visually approximates the best fit.

A detrended version of the original Y data points can be created by subtracting out the line that represents the trend:

Y =Y-YL = Y-(a+£X)

where Y is the transformed (detrended) value of Y.

The trend can be determined in a number of ways. A generally accepted approach is to draw a two-dimensional linear regression line, using the least squares approach. This is designed to minimize the error measured from the squares of the distances of the data points from the trend line.

If there are large errors from this approach, it is necessary to smooth the data using averaging, logarithms, or some other transformation. The data we are studying appears to be very linear, and lends itself, as a first approximation, to representation as a linear trend.

When we proceed to subtract the trend, a new data stream is created. The transformed data stream oscillates around the zero axis, making it easier to pick out other cycles that may exist.

In its transformed version, it is also possible to identify price shocks, which show up as sudden aberrations of the price data. Price shocks may be caused by political, economic, or social events, as well as natural and environmental occurrences. They are worthy of study, and should be considered in any system design.

A determination must be made whether there is any occurrence in the time period studied, which would make it beneficial to divide the data into different parts and analyze each part separately. Several statistical studies, such as ARIMA analysis, are designed for an interrupted time series.

STATISTICA™ is a statistical analysis program that allows for the ready analysis of time series data. Using STATISTICA™, we can statistically evaluate the S&P time series data, using trend subtraction. By eliminating the long-term trend, we can focus in on deviations from the trend.

In Figure 14.2, we transform the data to a time series that oscillates above and below the horizontal "zero" line. This transformation was attained by subtracting price from the trend line, which was determined through linear regression. Using this methodology, it is found that

Y f . = 240.3 + 0.0953 x t

transformed

where Y c , is the transformed value of the S&P, and t (time) is the number of

transformed 7 v

data bars from the starting point.

Different values for the coefficients may be obtained if different starting and ending points are chosen, or if the continuous contract used for study is constructed differently.

The accuracy of this model can be demonstrated by designing and developing a trading system, and then evaluating its financial performance. For a development platform, we use TradeStation®, by Omega Research, Inc. The PowerEditor™ of TradeStation® allowed us to determine whether the transformed S&P prices have predictive value. TradeStation® is a registered trademark and PowerEditor™ is a trademark of Omega Research, Inc.

Figure 14.3 shows the system, called S&P Crossover Buy. It sets Y to be the transformed S&P indicator. If Y crosses above the closing price of the S&P, a buy is signaled. If it crosses below the S&P, the long position is liquidated.

Figure 14.4 shows the performance summary for the S&P Crossover Buy system. In analyzing it, we see that this simple system yields 100 percent profitable trades. This is a significant result and shows that we are working with a promising model. By comparison, a buy-and-hold strategy over the same time period would have yielded similar profit results, but would have experienced much larger drawdowns along the way.

This is not a tradable system. It is based on the slope and intercept of the trend-line through the data points. This information could not have been available at the time the hypothetical trades were entered. This is the extreme example of a totally curve-fit system, where the result is known, and data is manipulated to reach the result.

Figure 14.2 Transformed s&p with trend subtracted.

Figure 14.3 System: S&p crossover buy.

Page I

Type : System

Name : SE Crossover Buy

Notes : Buy when linear regression crosses price.

Last Update . 02/10/97 11 05am Printed on : 02/10/97 11:06am Verified : YES

{Datal is S&P} Var: Y(0);

Y =240 4 + .095 * BarNumber; (Transformed S&P}

If Y crosses above Close of Datal then Buy;

If Y crosses below Close of Datal then ExitLong,

Source: Printed using TradeStation PowerEditor by Omega Research Version 4.02.15-Jul < 09 1996.

Logically, the data points will move back and forth across the trendline in back-testing, since the trend was established by the best fit. Thus, the trendline for the S&P leads the S&P. This has no practical value, but it does provide us with a confirmation that the straight line may be a useful modeling tool. It also provides us with an idea for system development that we will use when we begin to look at intermar-ket relationships.

FIGURE 14.4 PERFORMANCE SUMMARY: S&P CROSSOVER BUY.

S&P Crossover Buy S&P 500 Index - CME-Daily 04/21/82 - 12/19/96

Performance Summary: All Trades

Source: Printed using TradeStation by Omega Research Version 4.02.15-Jul < 09 1996.