the market can be bought with added confidence (especting lower risk and more profit). This technique of selecting better entry points may compensate for some of the inaccuracies latent in any forecasting method. The danger of this approach is tiiat prices may continue to move counter to the forecast so that caution and a stop-loss are also necessarj..

Following tiie Trend

Use the 1-day-ahead forecast to determine the trend position, if the forecast is for higher prices, a long position should be held; if it is predicting lower prices, shorts are necessarj.

pending on the number of coefficients selected), determine the errors in the estimation, and then approximate those errors using a moving average, it will nest look at the resulting new error series, attempt to estimate and correct the errors in that one, and repeat the process until it accounts for all price movement.

To determine when an ARIMA process is completed, three tests are performed at the end of each estimation pass:

a. Compare the change in the coefficient value. If the last estimation has caused litUe or no change in the value of the coefficienH s), the model has successfully converged to a solution.

b. Compare the sum of the squares of the error If the error value is small or if it stajs relatively unchanged, the model is completed.

c. Perform a set number of estimations. Unless a maximum number of estimations is set, an ARIMA process might continue indefinitely This safe"- check is necessarj in the event the model is not converging to a solution.

Once completed, the errors can be examined using an 0-statistic to check for any trend. 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 nest day, and should be used with less confidence for subsequent days (see Figure 3-10).

What if the forecast does not work? First, the process should be checked to be certain that it was performed properly. Pay particular attention to the removal of trends using the correlogram. Nest, 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 contains more homogeneous data, that is. data similar to the current martlet period.

Trading Strategies

In the article that ongmally piqued the interest of traders. Anon uses a 5-day-ahead forecast. If the ARIMA process forecasts an uptrend and if prices fall below the forecast value.

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FIGURE3-10 ARIMA forecast becomes less accurate as it is used farther ahead.

Countertrend Indicator

Use the ARIMA confidence bands to determine overbought/Oversold levels. Not only, can a long position be entered when prices penetrate the lowest 95>o confidence band, but they can be closed out when they retum to the normal SOo level. A conservative trader will enter the market only in the direction of the ARIMA trend forecast. As shown in Figure 310, if the trend is up, only the penetrations of the lower band will be used to enter new long positions.

Use of Highs and Lows

Both the implied highs and lows as well as the independently forecasted highs and lows can be the basis for other interesting shategies. The following two are used with intraday prices.

1 . Using confidence bands based on the closing prices, buy an intraday penetration of the expected high or sell a penetration of the expected low, and liquidate the position on the close. Use a stop-loss. Consider taking positions only in the direction of the ARIMA trend.

2. Using the separate ARIMA models based on the daily high and low prices, buy a penetration of the 50°o level of the high and sell a penetration of the 50°o level of the lows. Liquidate positions on the close. Use a stoploss.

Slope

The 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 price movement, also called noise, the purpose of directional analjsis, whether regression or moving averages, is to uncover the true direction of prices. Therefore, the slope of the frendline, or the direction of the regression forecast is the logical answer.

The popular alternate for friggering a new directional signal is a price penetration of an envelope or band value. Using regression analjsis that band can be replaced by a confidence level. While it is true that the number of random, or false, penetrations declines as the confidence band gets farther away from the frendline, so does the total number of penetrations. At any band distance there are still a large number of erroneous signals. The slope itself should be viewed as the best approximation of direction.

Kalman Filters

Kalman offers an alternative approach to ARIMA, allowing an underljTng forecasting model (message model) to be combined with other timely information (observation model). The message model may be any frading strategj, moving average, or regression approach. The observation model may be the floor brokers opening calls, market liquidity, or. in the case of existing foreign maitets, earlier trading activity-all of which have been dd;emiined to have some overriding importance in forecasting.

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Assume that the original forecast (message) model can be described as

MiP.) = CjP..,*me, and the observation model as

where me and oe are the message and observation model errors, respectively. The combined forecast would then use the observation model error to modifj the result

where 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 strategj similar to a movmg average. For example, an n-day linear regression, applied to the closing prices, could be used with the following rules:

1. Buy when the closing price moves above the forecasted value of todays close.

2. Sell when the closing price moves below the forecasted value of todays close.

There is an important difference between a model based on linear regression and one founded on a moving average. There is no lag in a regression strategj. If prices continue higher at the same rate, a moving average sjstem will initially lag behind, then increase at the same rate. The lag creates a safety zone to absorb some changes in the direction of prices, without getting stopped out. (See Chapter 4 for a complete discussion of moving averages, and Chapter 5 for a comparison of a Linear Regression Slope trading sjstem with five other popular trending methods.)

A regression model, on the other hand, identifies a change of direction sooner by measuring future movement against a straight-line projection in which the current price value has litUe influence. A steadj price move, however, will place the fitted line right in the center of market movement, subject to frequent whipsaws. The area at which a uniform trend changes from one direction to another is a difficult case for a linear regression sjstem (see Figure 3-11) and points out the need for using bands. Even with bands, the fuming point of an orderly frend will appear to have much greater variance than during the direction period over the same calculation interval.

In Figure 3-1 la, the three positions of the regression line all show numerous penetrations of the price, resulting in many losing frades. This is natural because the regression line is fitted best when it goes through the center of price movement. For prices to remain on one side of a regression line it must be constantly accelerating or decelerating, an unlikely situation, it is possible that this approach would work better using very few calculation periods; however, the use of only a few dajs is inconsistent with the nature of such a statistical measurement, which gains significance with more data. Figure 3-llb shows the more practical use of a confidence band spaced equally around the regression line Using a high-confidence band (e.g., 95" o), it is possible to tum the erratic performance in Figure 3-1 la into two clearly identifiable frends. Interestingly, the use of a confi

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FIGURE 3-11 Linear regression model, (a) Simple penetration of the regression lines, (b) Pendration of the channel formed by confidence bands