A normalized-slope trend filter from linear regression

A slope is not a rank

Linear regression is a least-squares fit through every observation in a chosen window. It produces a signed slope that depends on the whole path rather than only the first and last prints.

That raw slope inherits the price scale of the series, so a higher-priced series can post a larger numerical slope than a lower-priced series with the same relative path. The raw number therefore cannot be used to rank securities against one another.

Why correlation is not the filter

A Pearson correlation is a dimensionless two-series association between minus one and plus one. It is a two-series statistic and does not, by itself, classify the trend of a single ordered price path.

The slope equals the correlation of the paired series multiplied by the ratio of the population standard deviation of the dependent series to that of the independent series. Equal volatilities make the slope coincide with the correlation and stay inside plus or minus 1. That identity scales the slope. It does not turn correlation into a single-series trend classifier.

The whole path, not the end points

Because a fitted line uses every point in the window, two 10-observation paths that both finish with a 10% end-to-end change can still produce opposite slopes, shown as 0.0032 versus -0.0483.

A rate of change is an end-point slope that uses only the first and last observations and can therefore disagree with a path-sensitive regression slope.

From raw slope to normalized slope

The standard error of the regression is the root-mean residual distance of the observations from the fitted line, measured in the units of the dependent series. The standard error of the slope is the residual standard error of the fit divided by the product of the square root of sample size minus one and the sample standard deviation of the independent variable.

Dividing the raw slope by that standard error yields a normalized slope identical to the t-statistic of the slope.

How large a reading counts as on

Absolute readings beyond about 2 correspond to a conventional two-sided significance level near 0.025. Degrees of freedom equal sample size minus one, so windows smaller than about 30 need a higher absolute t-threshold than the large-sample normal values 1.96, 1.65, and 1.28 for 95%, 90%, and 80% coverage. A 16-observation window needs about 2.13 for 95%.

A trend filter is a gate on whether a fitted slope is strong enough, relative to the uncertainty of the fit, to treat the window as directionally on rather than as noise.

What the finished reading reports

The finished series is a trend filter that reports signed direction, strength, and a probability-style quality mark in one reading. Multiple securities can be ranked on that common scale.

Unlike an average directional index construction, which summarizes unsigned trend strength, the normalized slope keeps the sign of the move and attaches an explicit significance threshold. An average directional index does not report the sign of the slope or a t-style significance mark.

A 26-observation construction check

In a 26-observation construction check, the normalized-slope series and a MACD built with lengths 12, 26, and 1 shared the same general shape and crossed zero at the same observations. The normalized series could move against price when the MACD continued with price.