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1990issue C081-12

Constructing a lag-aligned triangular MACD

Before cycles can be inspected, trend is removed so an ordered market series is stationary. This construction subtracts two even-length triangular moving averages only after each average is shifted by its midpoint lag, placing a reduced-amplitude residual back in time with the parent series.

  • Trend is removed first so remaining variation can be inspected for cyclic structure. Listed constructions include first differences, residuals from a fitted trend, departures from a centered average, and the difference of two moving averages of unequal length.
  • The oscillator is the difference of two even-length triangular moving averages, with the shorter window recommended as half the longer. A worked pair uses lengths 20 and 10.
  • Each triangular average lags the parent series by half its period minus one, so the length-10 average lags by 4 observations and the length-20 average lags by 9. The oscillator at time t uses each average evaluated at t plus its own lag.
  • End-of-sample oscillator points are unavailable until the longer window is complete. A current index of 54 leaves the last computed value at 45, a gap of 9 equal to half the longer window.
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Current trackTriangular moving average8 readingsMACD115 readingsMoving average449 readings

Remove trend before inspecting cycles

Before cycles can be inspected in ordered market series, trend is removed. The aim is stationarity: a series whose remaining variation can be inspected for cyclic structure. Listed constructions include first differences, residuals from a fitted trend, departures from a centered average, and the difference of two moving averages of unequal length.

MACD, in this setting, is the difference of two unequal-length moving averages, used to cancel shared trend and leave a stationary residual. The moving average inside the construction is a lookback-window smoother whose window length is constrained to an even count. The triangular moving average supplies the weights: they rise linearly to a midpoint peak and fall back to zero.

Specify even-length triangular windows

The constructed oscillator is the difference of two triangular moving averages whose window lengths are even, with the shorter window recommended as half the longer. A worked pair uses lengths 20 and 10. The even-window constraint keeps the triangular peak on a midpoint observation.

Under the stated weighting convention, the peak triangular weight is 2.0 at the window midpoint, the step size is 2 divided by half the period, the count of effective weights is one less than the period, and the weights sum to one plus that count. The weight-sum is the divisor of the triangular average. A triangular average value equals the weight-times-observation sum inside the window, divided by the sum of the weights.

Shift each average by its midpoint lag

Because the triangular window is symmetric, each average has a lag equal to half the even window length minus one. The length-10 average lags by 4 observations and the length-20 average lags by 9.

The oscillator at time t is the shorter triangular average evaluated at t plus its lag, minus the longer triangular average evaluated at t plus its lag. That placement puts the difference back in time with the parent series at a reduced amplitude.

Leave the sample end incomplete

End-of-sample oscillator points are unavailable until the longer window is complete. In the spreadsheet illustration, a current index of 54 leaves the last computed value at 45, a gap of 9 equal to half the longer window.

Inspect the residual on test and market series

In a sine-wave test series of period 40, the first length-10 triangular average needs nine prior points and the first length-20 average needs nineteen prior points. The lag of the longest average is half its period minus one.

Chart overlays on an individual equity series, a broad stock index, and a volume-adjusted advance/decline series were used to inspect whether the triangular-average difference recovered smooth peaks and troughs that were faint in the raw series.

A short program accepts operator-chosen periods, builds the triangular weights, and plots the input beside the oscillator. The bundled test input is a sine series whose period is twice the longer triangular window.

Lag-aligned triangular MACD on the 40-period test sine

Once the 10- and 20-period triangular averages are slid back by their midpoint lags, the residual turns with the parent 40-period sine instead of trailing it, at a much smaller swing. Every point is a cell from Johnson’s Figure 2 worksheet.
Once the 10- and 20-period triangular averages are slid back by their midpoint lags, the residual turns with the parent 40-period sine instead of trailing it, at a much smaller swing. Every point is a cell from Johnson’s Figure 2 worksheet.Hypothetical 40-period sine

Windows are even-length 10 and 20. Each average is shifted by lag (P/2)−1 before the subtract, so a live series would leave the last nine MACD bars unknown without extrapolation.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
4 of 8 in the Triangular moving average track
19981-3 pp.Next on Triangular moving averagePaired tests for moving-average lag and smoothnessLinear-input-lag can treat a longer single-pass exponential average and a multi-pass exponential average as equally delayed even when a peaked path shows extra phase-lag for the stacked version.
All readings on this track · 8 readings
  1. 1989Constructing lag-matched triangular moving averages
  2. 1990Lag-aligned MACD from triangular moving averages
  3. 1990Constructing triangular moving average weights
  4. 1990Constructing a lag-aligned triangular MACD
  5. 1998Paired tests for moving-average lag and smoothness
  6. 1999Fixed-lag construction of moving-average smoothers
  7. 2003Weighting recipes inside one moving-average lookback
  8. 2010Assembling a smoothed percent-b oscillator from Heikin-Ashi and stacked averages
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Also on Triangular moving average5 readings
1990Lag-aligned MACD from triangular moving averages1989Constructing lag-matched triangular moving averages1998Paired tests for moving-average lag and smoothness1999Fixed-lag construction of moving-average smoothers2003Weighting recipes inside one moving-average lookback