Skip to main content
Track Linear regression
23 / 43
Search the archiveLibrary
Library

1996issue C021-4

Constructing an endpoint moving average from a least-squares line

The same lookback can be written as a rolling mean or as the last point of a short price-versus-time fit. This archive note reconstructs how the endpoint moving average is entered, how its window advances, and which lookbacks yield integer coefficients.

  • An endpoint moving average is defined for every integer lookback of two or more observations as the last fitted value of a least-squares line of price against time.
  • A rolling window advances one period at a time for an average, a correlation, or a regression, and a paired regression is entered as one shared array formula.
  • Integer-coefficient lookbacks occur when the lookback plus one is a multiple of three, including 2, 5, 8, and 11. With a two-observation lookback the endpoint average equals the last price.
  • Two closed-form expressions for the general endpoint moving average produce identical numerical results because an inner sum is an arithmetic progression.
Entries in this reading2 entries
Current trackLinear regression112 readingsMoving average449 readings

The same lookback, two constructions

An endpoint moving average is defined for every integer lookback of two or more observations. It is a constructed smoother equal to the last fitted value of a least-squares line through a price window of that length, rather than the arithmetic mean of the window.

Endpoint averaging is that least-squares line: ordered prices are fitted against time, and mismatch is absorbed only on the price axis. The last point on the fitted line is the smoother.

How the formula is entered

A 26-week regression written across two spreadsheet cells is entered as one shared array formula rather than as two independently typed cell formulas. The array formula keeps the paired outputs on a single expression so the multi-output regression stays internally consistent.

How the window advances

A 30-week correlation feature is computed on a 30-observation rolling window that advances one period at a time, the same way a moving average advances. The same rolling-window rule recomputes an average, a correlation, or a regression at each new observation.

A 30-week regression feature can be formed as an estimate-to-actual ratio: the fitted estimate for the current interval divided by the contemporaneous actual value of the series being explained.

Which coefficients the algebra produces

Integer endpoint-average coefficients arise when the lookback plus one is a multiple of three. Those integer-coefficient lookbacks include 2, 5, 8, and 11.

With a two-observation lookback, the endpoint average equals the last price because the unique line through two points ends at that last price.

Two closed forms, one numerical result

Two different closed-form expressions for the general endpoint moving average produce identical numerical results. Those expressions match because an inner sum is an arithmetic progression with a closed-form total whose first and last terms depend on the lookback.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
23 of 43 in the Linear regression track
19961-1 pp.Next on Linear regressionScoring equity path consistency with a k-ratio overlayThe evaluation procedure issues no buy or sell signals. It is meant to run at the same time as a separate trade-generating system and write that system’s k-ratio to a log.
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
By fieldIndicators and filters682 readingsBy year199680 readings
All 112 readings tagged Linear regression
Also on Linear regression5 readings
1990Constructing least-squares polynomial smoothers1990Two clocks for copper: a factor regime, a regression baseline, and leftover moving-average timing1991Trend as persistence, not a straight line1993Constructing walk-forward forecasts with linear and moving-average baselines1994Constructing the double exponential average from lag cancellation