The Linear Regression Of Time and Price

By Ryan Campbell AAA

Technical and quantitative analysts have applied statistical principles to the financial market since its inception. Some attempts have been very successful while some have been anything but. The key is to find a way to identify price trends without the fallibility and bias of the human mind. One approach that can be successful for investors and is available in most charting tools is linear regression.

Linear regression analyzes two separate variables in order to define a single relationship. In chart analysis, this refers to the variables of price and time. Investors and traders who use charts recognize the ups and downs of price printed horizontally from day-to-day, minute-to-minute or week-to-week, depending on the evaluated time frame. The different market approaches are what make linear regression analysis so attractive. (Learn more about quantitative analysis in Quantitative Analysis Of Hedge Funds.)

Bell Curve Basics
Statisticians have used the bell curve method, also known as a normal distribution, to evaluate a particular set of data points. Figure 1 is an example of a bell curve, which denoted by the dark blue line. The bell curve represents the form of the various data point occurrences. The bulk of the points normally take place toward the middle of the bell curve, but over time, the points stray or deviate from the population. Unusual or rare points are sometimes well outside of the "normal" population.


Figure 1: A bell curve, normal distribution.
Source: ProphetCharts

As a reference point, it is common to average the values to create a mean score. The mean doesn't necessarily represent the middle of the data, and instead represents the average score including all outlying data points. After a mean is established, analysts determine how often price deviates from the mean. A standard deviation to one side of the average is usually 34% of the data, or 68% of the data points if we look at one positive and one negative standard deviation, which is represented by the orange arrow section. Two standard deviations include approximately 95% of the data points and are the orange and pink sections added together. The very rare occurrences, represented by purple arrows, occur at the tails of the bell curve. Because any data point that appears outside of two standard deviations is very rare, it is often assumed that the data points will move back toward the average or regress. (For further reading, see Modern Portfolio Theory Stats Primer.)

Stock Price as a Data Set

Imagine if we took the bell curve, flipped it on its side and applied it to a stock chart. This would allow us to see when a security is overbought or oversold and ready to revert to the mean. In Figure 2, the linear regression study is added to the chart, giving investors the blue outside channel and the linear regression line through the middle of our price points. This channel shows investors the current price trend and provides a mean value. Using a variable linear regression, we can set a narrow channel at one standard deviation, or 68%, to create green channels. While there isn't a bell curve, we can see that price now reflects the bell curve's divisions, noted in Figure 1.


Figure 2: Illustration of trading the mean reversion using four points
Source: ProphetCharts

Trading the Mean Reversion
This setup is easily traded by using four points on the chart, as outlined in Figure 2. No.1 is the entry point. This only becomes an entry point when the price has traded out to the outer blue channel and has moved back inside the one standard deviation line. We don't simply rely on having the price as an outlier because it may get another further out. Instead, we want the outlying event to have taken place and the price to revert to the mean. A move back within the first standard deviation confirms the regression. (Check out how the assumptions of theoretical risk models compare to actual market performance, read The Uses And Limits Of Volatility.)

No.2 provides a stop-loss point in case the cause of the outliers continues to negatively affect the price. Setting the stop-loss order easily defines the trade's risk amount.

Two price targets at No.3 and No.4 will be set for profitable exits. Our first expectation with the trade was to revert to the mean line, and in Figure 2, the plan is to exit half of the position near $26.50 or the current mean value. The second target works under the assumption of a continuing trend, so another target will be set at the opposite end of the channel for the other standard deviation line, or $31.50. This method defines an investor's possible reward.


Figure 3: Filling the mean price
Source: ProphetCharts

Over time, price will move up and down and the linear regression channel will experience changes as old prices fall off and new prices appear. However, targets and stops should remain the same until the mean price target fills (see Figure 3). At this point, a profit has been locked in and the stop-loss should be moved up to the original entry price. Assuming it is an efficient and liquid market, the remainder of the trade should be without risk. (Learn more in Working Through The Efficient Market Hypothesis.)


Figure 4: Filling the mean price.
Source: ProphetCharts

Remember, a security doesn't have to close at a particular price for your order to fill; it only needs to reach the price intraday. You may have been filled on the second target during any of the three areas in Figure 4.

Truly Universal
Technicians and quant traders often work one system for a particular security or stock and find that the same parameters won't work on other securities or stocks. The beauty of linear regression is that the security's price and time period determine the system parameters. Use these tools and the rules defined within this article on various securities and time frames and you will be surprised at its universal nature. (For further reading, see Bettering Your Portfolio With Alpha And Beta and Style Matters In Financial Modeling.)

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