Least-squares construction of channel trendlines

Where the trendline can sit

A trendline is a straight reference placed through highs, lows, or the interior of a plotted price series to describe direction. It may be placed through successive highs, successive lows, or through the interior of the price series rather than only at the extremes.

A price channel is the band of prices around a central path, whose midline can be read as the channel's growth axis.

Scale and the growth axis

A straight line through the middle of a price channel on a logarithmic scale reports the series' percentage rate of advance or decline.

Least-squares instead of freehand

A freehand center line is open to disagreement. A least-squares fit removes that ambiguity by minimizing the sum of squared deviations of the plotted points from the line. It is the unique straight line that minimizes the sum of squared vertical distances from the observed points.

The constructed regression line is specified by two constants in the form Y equals A plus B times X, where A is the intercept, the fitted vertical value when the independent-axis reading is zero, and B is the slope, the constant change in the fitted vertical value for each one-unit move along the independent axis.

Worksheet columns and a worked line

Intercept and slope can be computed from four worksheet columns together with the observation count: elapsed periods, the observed series which may be logged, the squared period index, and the product of period and series.

In the worked numerical example the fitted constants are a slope of 0.50 and an intercept of 1.10, so the line advances one-half vertical unit for each unit of the independent axis.

Stock price against a market index

The same least-squares construction, with a stock price on the vertical axis and a market index on the horizontal axis, yields intercept and slope constants used as alpha and beta.