TAI - R-Squared

Name, Sometimes Called:

r-Squared

Brief Description:

The r-Squared TAI shows the strength of a trend: the more closely prices move in a straight line over time, the stronger the trend.

Definitions, Formulas:

One of the results that can be obtained from Linear Regression is r-squared. The r-squared value shows the “goodness of fit” of a straight line to the price values over a given period. If the r-squared value was 1.0, the fit would be perfect: the straight line would pass through every data point. If the r-squared value over 20 days was 0.70, then 70 percent of the security’s movement over that period can be explained by linear regression. The other 30 percent is unexplained random noise.

The r-Squared TAI and the Linear Regression Slope work together. While the Linear Regression Slope shows the general trend direction (positive or negative), the r-Squared TAI shows the trend’s strength. Hence, high r-Squared values coupled with a large Linear Regression Slope are greatly to be desired.

One formula for r2 is

r-squared = 100 * SS(regression) / SS(total)

where

SS(regression) = the sum of the squared difference between each fitted value of Y and the mean of Y. It describes the variation within the fitted values of Y. The squares are taken to 'remove' the sign (+ or -) from the residual values to make the calculation easier.

and

SS(total) = the sum of the squared difference between each value of Y and the mean of Y. It describes the variation within the values of Y.

Another formula for R squared is

Another formula, which is mathematically equivalent to the previous one, is

where SSregression is the difference between SStotal and SSerror.

r-squared is expressed in terms of a confidence value. That is, when r-squared is above its critical value and increasing, we can say that we are 95 percent confident that the trend is a strong one. This confidence value depends on the number of time periods used in its calculation, as shown in the following table. We use n=14, which corresponds to an r2 critical value of 0.27 for a 95% confidence level.

Positive Development Calculation:

The Linear Regression Slope TAI and the R-squared TAI are used together to compute a single positive development. Both of the following conditions must hold:

(1) The Linear Regression Slope must cross above zero.

and

(2) r squared (r2) must equal or exceed 0.27. (See the table and discussion above.)

Similarly, the Linear Regression Slope and the R-squared TAI are used to determine when a development is no longer positive. If either of the following conditions holds, then a development is no longer positive:

(1) The linear regression slope crosses below zero

or

(2) r squared (r2) is less than 0.27.
To reduce the potential of false positives produced by the traditional specifications, we require that the 3-day price slope also trend upward before a development is considered to be a positive development.

If this TAI is still positive tomorrow, it will no longer be new, but will be a cumulative positive development (CPD).

If this TAI was a new positive development (NPD) yesterday, and is still positive today, then it becomes a cumulative positive development (CPD).

History:

Linear regression is a standard statistical tool. Its outputs are being used here to analyze market prices.

The chart below shows R-squared in use with the Linear Regression Slope indicator. During the period shown there were several new positive developments, but we’ve highlighted only two. First, locate the green-A. It indicates a point where the linear regression slope (LRS) crossed over the zero point. This is, by itself, positive. If you trace down you will see that r-squared was still under 0.27. Both indicators have to be positive before there is a new positive development (NPD). The vertical green bar shows where r-squared crossed over 0.27 and LRS was still positive.

A few days later r-squared dipped below 0.27 and came back up over it. This is a whipsaw. Had you exited your position and gotten back in when r-squared crossed back over 0.27 you’d still be in positive territory but with an increase in your commission expenses.

The red vertical line indicates where these two TAI no longer indicate a positive development (NLPD). Or does it? Find the red-B. Just a little before the day that LRS crossed under zero you can see that r-squared dropped below 0.27 again. Remember, for this technical analysis indicator BOTH have to be positive. That vertical red line should be a bit earlier.