Constructing a standard-error envelope around a linear regression

What the envelope is centered on

This article reconstructs a historical workflow for a three-line envelope centered on a linear regression of closes. A second envelope, centered on a moving average, is built beside it so residual width around a fitted line can be compared with volatility around mean price.

Linear regression, in this construction, is a least-squares line through a fixed lookback of ordered closes. It is used as the moving center of the envelope and as the baseline for residual width. The regression line is a least-squares fit drawn through the center of the lookback so observations sit with equal distance above and below the line.

The independent axis for the fit is a sequential bar index rather than calendar dates, so a weekend gap is not counted as multiple periods.

How the outer lines are set

The envelope center is the last point of a 21-period linear regression of closes, and the outer lines sit two standard errors of the estimate above and below that point.

Standard error of the estimate is the residual scatter of closes around the fitted regression line. It is the width term of the envelope. It is zero when every close equals that bar's regression value and rises as noise around the line increases.

Both the regression end-value and the standard-error width are then smoothed with a three-period simple average so a single large close does not reprice the whole envelope.

A mean-centered comparison

Bollinger bands are a three-line envelope whose middle line is a moving average of closes and whose outer lines sit a fixed number of standard deviations away, measuring volatility around mean price.

The comparison construction uses a 20-period moving average of closes plus and minus two standard deviations, so its width tracks volatility around mean price rather than around a fitted line.

A tightness screen divides a 55-period average true range by the width of the 21-period mean-centered bands and treats a reading of 50 percent or more as a compressed-volatility condition. That screen is a reading of mean-centered width, not of scatter around the fitted line.

Companion reads of location and stacked lookbacks

Percent A is the scaled location of the close between the lower and upper standard-error lines, with 100 at the upper line and 0 at the lower line. The companion oscillator equals 100 when the close is on the upper band and 0 when it is on the lower band. It is computed as the close minus the lower band, divided by band width, times 100.

Fibonacci retracement is used here as stacked lookbacks so 8-, 13-, and 21-period calculations can be averaged into one composite path. A Fibonacci composite averages the 8-, 13-, and 21-period versions of the same series so the shortest window leads, the middle window lags, and the longest window mutes one-bar shocks.

The coefficient of determination is the share of squared error removed by using the regression as predictor instead of the sample mean. When that Fibonacci-composite coefficient of determination is below 15, all three lookbacks are read as jointly at extreme lows.