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1995issue C071-10

Constructing projection bands from least-squares slopes

Projection bands are assembled by fitting a least-squares slope on a rolling lookback, moving each past price to the current bar along that slope, and reading the close inside the projected range. Adaptive statistical envelopes such as Bollinger bands are the contrast for how those pieces are wired.

  • A projection envelope is built in three steps: fit a least-squares slope on a lookback period, project each observation in that window to the current bar, and take the highest and lowest projected values as the band edges.
  • Bollinger bands remain the contrast case. They treat a moving average as typical movement and treat variation around that average as the source of relatively extreme prices.
  • A projection oscillator reports the current close as a percentage of the span between the projected bands. The construction is presented as a stochastic oscillator with a trend correction.
  • The same stack can use closes only or separate high and low slopes, and it also yields a projection-bandwidth reading of relative envelope width.
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Current trackBollinger Bands84 readingsLinear regression112 readingsStochastic oscillator170 readings

Statistical envelopes as a contrast

Trading-band construction progressed from hand-drawn envelopes and percentage bands around moving averages to adaptive statistical envelopes such as Bollinger bands.

Those bands treat a moving average as typical movement and treat variation around that average as the source of relatively extreme prices.

A shared slope for channel boundaries

Channel envelopes connect peaks and troughs with straight lines that can be extended as boundaries.

A least-squares fit can supply a quantitative slope that those boundaries often share, though the analyst still chooses the start and end of the window.

Projecting lookback prices to the current bar

Projection bands are upper and lower envelopes formed from the extreme values of lookback prices after each price is moved forward along a fitted slope.

Each observation in the lookback period is projected forward along the fitted trend of that window. The current upper and lower bands are the maximum and minimum of those projected values. The lookback is a rolling window of trading days used both to fit the slope and to generate the projected extrema, and other lengths can be used.

For a close-only series, the projected value of a past close is that close moved forward by the least-squares close slope of the current lookback, according to the number of bars from that past bar to the current bar.

A projection oscillator with a trend correction

A close-based projection oscillator reports the current close as a percentage of the span between the lower and upper projected-close bands.

A slower version is an exponential moving average of that oscillator. The construction is presented as a stochastic oscillator with a trend correction: a placement of the current close between recent extremes after the lookback prices have been shifted along the fitted slope.

Separate high and low slopes

When highs and lows are used, a least-squares slope is fit separately to highs and to lows. Each high and low in the lookback is projected on its own slope, and the band edges are the maximum projected high and the minimum projected low.

The high-low projection oscillator places the current close between those projected high and low bands as a percentage. A typical slow version applies an exponential moving average.

Bandwidth and condition of the series

Projection bandwidth reports the relative width of the envelope as a percentage. It is the distance between the two bands scaled by their midpoint.

The same least-squares projection stack can be applied to different series, but readings are described as needing different interpretation in ranging versus trending conditions.

A lag of price change behind the fitted trend can appear in the bands before a stochastic oscillator would register a comparable change.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
3 of 45 in the Bollinger Bands track
19951-6 pp.Next on Bollinger BandsConstructing regression projection bands and range oscillatorsSeparate least-squares slopes from a 14-bar window of highs and a 14-bar window of lows drive every later projection.
All readings on this track · 45 readings
  1. 1992Constructing volatility-scaled bands with relative strength index confirmation
  2. 1994Implied volatility as a band-defined regime filter for index options
  3. 1995Constructing projection bands from least-squares slopes
  4. 1995Constructing regression projection bands and range oscillators
  5. 1996Constructing Bollinger bands, percent-b, and stochastics
  6. 1996Constructing mechanical rules from Bollinger Bands and stochastics
  7. 1996Constructing a standard-error envelope around a linear regression
  8. 1996Dual-horizon ratio envelopes and regression error channels
  9. 1997Rational group structure with a trend screen, RSI, and bands
  10. 1997Asymmetric volatility band construction
  11. 1998Constructing three-state filters from Bollinger band envelopes
  12. 1999Combination filters with Bollinger Bands and the relative strength index
  13. 1999Constructing stochastic timed exits and band-RSI reversals
  14. 1999Evaluating Bollinger Bands against fixed-width and range-based envelopes
  15. 2000Constructing a Bollinger Band target as a forward price
  16. 2001Numeric candlestick encoding with local size bands
  17. 2001Ranked candlestick sentiment to band-cross entries
  18. 2002Combining Bollinger Bands, RSI, and a stop-loss
  19. 2002Bollinger Bands remain filters, not forecasts
  20. 2002Constructing a stochastic RSI with Bollinger bands
  21. 2002Constructing a StochRSI and Bollinger mechanical system
  22. 2003Constructing volatility-scaled Bollinger envelopes
  23. 2003Why tick breadth fails as a market personality
  24. 2005Constructing Bollinger bands versus fixed trading bands
  25. 2006Squared versus absolute deviation in envelope construction
  26. 2006Confirming yen crossovers with implied volatility and bands
  27. 2006A daily candle reversal is a hypothesis until shorter sessions fail at the same zone
  28. 2008Rebuild the Relative Strength Index as price-scale bands
  29. 2008Reading Relative Strength Index extremes on one price axis with Bollinger Bands and moving averages
  30. 2011Three-filter confirmation for short-swing futures
  31. 2011Constructing an inverse Fisher stochastic with bands and averages
  32. 2012Constructing a Bollinger Band indicator suite
  33. 2012Stacking price extremes, crossovers, bands, and MACD
  34. 2012Adaptive Bollinger band impulse, trend, and momentum filters
  35. 2013Rescaling stochastic, percent-B, and wave-count parameters
  36. 2014Industry-group quartile pivots as a Bollinger Bands case study
  37. 2014Bollinger Bands as adaptive price envelopes: a 2014 classroom case
  38. 2016Trend-channel entry rules from stacked moving averages
  39. 2016A permission stack for Bollinger, RSI, and the 50-period average
  40. 2017Constructing weighted Bollinger bands and volume averages
  41. 2017Four swing-entry rules that share a timed exit
  42. 2017Two-wave monthly cycles as a regime filter
  43. 2019Constructing exponential-deviation-bands from a midline-average
  44. 2020Critiquing exponential variants of Bollinger Bands
  45. 2020Constructing selectable volatility and moving-average bands
By fieldIndicators and filters682 readingsBy year199575 readings
All 84 readings tagged Bollinger Bands
Also on Bollinger Bands5 readings
1995Constructing regression projection bands and range oscillators1995Constructing price envelopes from averages, volatility, and regression1996Dual-horizon ratio envelopes and regression error channels1996Constructing a standard-error envelope around a linear regression1996Standard-error bands, width gates, and weekday counts