Constructing r-squared trend filters with dual lookbacks

Editorial: treat trend as a two-stage construction problem rather than a chart slogan. The first stage is a least-squares linear regression of price on time. That fit returns slope, a fitted forecast, the standard error of the estimate, and r-squared. The second stage is a directional overlay, and it is allowed to speak only after r-squared has been published as a trend filter.

What the fit publishes

A linear regression of price on time is estimated by least squares. Besides the slope of the fitted time line, the estimate returns a fitted forecast, the standard error of the estimate, and r-squared.

R-squared is a zero-to-one goodness-of-fit reading. It states how much of the observed price path is explained by the fitted time line.

What a moving average leaves out

A simple moving average smooths noise but does not report slope, goodness of fit, or forecast error. A 20-session average is conceptually centered 10 sessions back, and the lookback is chosen arbitrarily.

R-squared as a published trend filter

R-squared is bounded between 0 and 1. High readings were treated as a linear-trend state and low readings as a nontrend or range state. The statistic can gate whether trend-following or range methods apply.

On a 30-observation sideways close series the fitted r-squared was 0.00014. On a steadily rising series it was 0.95.

Lookback, thresholds, and runs

A 30-session lookback was used as the shortest window that does not require a small-sample correction. Shorter windows need that correction. Longer windows lose short-horizon responsiveness.

In a 30-day linear-regression sample of a broad equity index from August 1988 to summer 1997, r-squared above 0.70 was labeled a good fit and r-squared below 0.30 a poor fit. Readings in the trending band occurred about 35 percent of the time.

Consecutive sessions with r-squared above 0.70 were counted as a trend run. The longest such run in that sample lasted 65 sessions. The longest stretch below 0.70 lasted 204 sessions.

Direction after the filter

Because r-squared does not sign direction, bars were colored by regression slope once r-squared reached 0.70.

Signed percentage residuals were plotted as an early-warning overlay on the completed lookback. Each residual is the signed gap between actual and predicted price, expressed as a percent of the prediction: actual minus predicted, divided by predicted, times 100.

Two lookbacks stacked like averages

Two r-squared series with 21-session and 63-session lookbacks were stacked the way moving-average crossovers stack two averages. The shorter fit cycles inside the longer fit.

Limits of a straight-line fit

A straight-line regression cannot describe an exponential or parabolic path. Neither regression nor a moving average supplies a uniquely correct lookback length.