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1998issue C041-3

R-squared as a two-state trend filter from a price-time fit

A market filter can classify trending and nontrending states from r-squared on a least-squares fit of closing price to time. Thresholds, optional averaging, and a run-length turn that tightness measure into a two-state gate, while slope, a linear-predicted-value, and percent-error remain secondary outputs.

  • The filter squares the linear association between closing price and a time or bar-index series to obtain r-squared over a fixed lookback.
  • A reading above a 0.70 trend-threshold is treated as trending; a reading below a 0.30 no-trend-threshold is treated as nontrending.
  • Squared r may stay raw or be averaged, and a run-length such as 10 bars can delay a mark or alert until the state persists.
  • The same fit can emit a slope, a linear-predicted-value, and a percent-error without using those series as the state gate.
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Current trackLinear regression112 readingsTrend filter137 readings

How r-squared is formed

The filter begins by measuring the linear association between a closing-price series and a time or bar-index series, then squares that association to obtain r-squared. That quantity is the square of the linear association over a fixed lookback.

Several constructions use a 30-observation lookback for the price-versus-time fit. One construction uses a default correlation period of 10 rather than 30.

Trend and no-trend cutoffs

An r-squared reading above a trend-threshold of 0.70 is treated as a trending state, because price is then aligned with a straight time path. An r-squared reading below a no-trend-threshold of 0.30 is treated as a nontrending state.

SPX r-squared from a 30-bar price-time fit

R-squared on a 30-bar least-squares fit of SPX close to time, June 1997–February 1998, read off the TechniFilter Plus pane. Values above 0.70 mark a tight (trending) fit; values below 0.30 mark a loose (no-trend) fit. Green price bars on the source chart sit where r-squared is above 0.70; red bars sit where it is below 0.30.
R-squared on a 30-bar least-squares fit of SPX close to time, June 1997–February 1998, read off the TechniFilter Plus pane. Values above 0.70 mark a tight (trending) fit; values below 0.30 mark a loose (no-trend) fit. Green price bars on the source chart sit where r-squared is above 0.70; red bars sit where it is below 0.30.SPX · daily · 1997-06-03T00:00:00.000Z to 1998-02-04T00:00:00.000Z

Curve digitized from the published r-squared pane, not from a table. Length is 30 bars (close versus time index). Last printed SPX print is 1006.90 on 4 Feb 1998. Approximate to about 0.02 given raster resolution; do not treat as tick-level data.

Smoothing, persistence, and painted bars

Squared r may be left raw or averaged. One construction defaults the smoother length to 1, which leaves the raw r-squared series in place.

A persistence count can require the reading to stay above or below a threshold for a stated run of bars before a mark or alert is issued. That run-length has one default of 10.

Price bars can be painted as trending or nontrending from the same r-squared series. Equal trend-threshold and no-trend-threshold inputs can force every bar into one of the two states.

Secondary outputs from the same fit

The same least-squares fit can also emit a slope, a linear-predicted-value formed from the prior close plus the prior least-squares slope, and a percent-error that expresses the close as a percentage deviation from that linear-predicted-value.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
28 of 43 in the Linear regression track
20001-4 pp.Next on Linear regressionSecond-order moving-average lag correctionA simple moving average uses only one total of the lookback prices, so it can report the local level but cannot estimate the slope of the same window.
All readings on this track · 43 readings
  1. 1990Constructing dollar baselines from rates, inflation, and residuals
  2. 1990Constructing a nominal index value from forward earnings and fitted yield
  3. 1990Constructing a nominal index price from earnings and a fitted yield
  4. 1990Endpoint-pinned price paths are not forecasts
  5. 1990Constructing least-squares polynomial smoothers
  6. 1991Endpoint growth rates versus linear-regression consistency
  7. 1991Out-of-sample checks for linear growth fits
  8. 1991Trend as persistence, not a straight line
  9. 1991Quadratic trend, residual oscillator, and a secondary cycle calendar
  10. 1991Time-origin offset and residual-price divergence on a quadratic least-squares fit
  11. 1991A least-squares trendline from ordered prices
  12. 1992Constructing log-linear growth and reliability screens
  13. 1992Constructing log-linear growth-rate baselines
  14. 1992Next-session high, low, and close from rolling linear regression
  15. 1993Auditing an index price-earnings multiple with short-rate regression
  16. 1994Regression-seeded nested exponential price filter
  17. 1994Constructing the double exponential average from lag cancellation
  18. 1994Evaluating money supply as a linear leading-index baseline
  19. 1995Constructing least-squares trend channels
  20. 1995Linear baseline holdout checks for annual bill-rate forecasts
  21. 1995Projection bands from high and low regression slopes
  22. 1995Evaluating a least-squares end-point moving average on a known test series
  23. 1996Constructing an endpoint moving average from a least-squares line
  24. 1996Scoring equity path consistency with a k-ratio overlay
  25. 1996Evaluating month-end yield gaps for equity regimes
  26. 1996Constructing session-indexed standard error bands
  27. 1998Evaluating linear regression baselines for index valuation
  28. 1998R-squared as a two-state trend filter from a price-time fit
  29. 2000Second-order moving-average lag correction
  30. 2002Price regression line versus beta for index tracking
  31. 2003Regression slope with an r-squared trend confidence gate
  32. 2003Constructing finite-volume-element divergence with slope comparison
  33. 2004Building a daily score from regression, retracement, and volume
  34. 2004Constructing least-squares trendlines from ordered prices
  35. 2007Rectangle breakout targets beyond height
  36. 2007Confirming a price trend with regression slope and r-squared
  37. 2008A linear-regression angle assembled as one trend filter
  38. 2010A two-state swing machine from four running extremes
  39. 2016Score oil-complex tightness before divergence or regression
  40. 2017Nikkei-yen intermarket divergence as a regime case study
  41. 2017Constructing Calmar ratio and linear regression baselines
  42. 2019Pair-trade layer construction versus average-spread management
  43. 2020A convolution slope built from nested linear regression
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