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2010issue C1270-83

Constructing a Hull smoother and a Hull-based oscillator gate

A Hull series of length N nests weighted averages of the half lookback, the full lookback, and the integer square root of the lookback. This archive article then applies a relative-strength index to that series, not to raw price, and uses the reading as the oversold and overbought gate in a long-only rule.

  • A Hull series of length N is built from twice the weighted average of the last N/2 observations, minus the weighted average of the last N observations, then a weighted average of that residual over the integer square root of N.
  • The construction prefers a perfect-square-length such as 4, 9, 16, 25, 49, or 81 so the square-root step does not truncate the inner average.
  • The companion oscillator is a nine-bar relative-strength index of a nine-bar Hull series, not of raw price, and that reading gates the long-only entry and exit.
  • An open long closes if the close falls more than 5 percent from the entry close, rises more than 15 percent from the entry close, or if the same nine-bar oscillator exceeds 90.
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Current trackHull moving average6 readingsRelative Strength Index261 readingsRule-based entry292 readings

A Hull moving average is a nested weighted-average smoother that combines a half-length series, a full-length series, and a square-root-length series to reduce lag while keeping a long lookback. Length-n is the lookback in bars used for the outer Hull calculation. Half-N and square-root-N are derived from it.

How the nested averages are built

A Hull series of length N is built by taking twice the weighted average of the last N/2 observations, subtracting the weighted average of the last N observations, then taking a weighted average of that residual over the integer square root of N. A length check rejects N of 1 or less before any nested weighted average is computed.

Perfect-square lookbacks

Because the square-root step truncates non-integers, the construction prefers lookbacks that are perfect squares such as 4, 9, 16, 25, 49, or 81. A perfect-square-length is a lookback chosen so the square-root step is an integer, avoiding truncation of the inner average length.

A short overlay against a simple average

One demonstration overlay uses a four-bar Hull on daily SPY closes against a 50-bar simple average of the close.

Nine-bar Hull average versus nine-bar simple average on SPY

Traders should notice how the nine-bar Hull line turns with price while the nine-bar simple average lags, especially on the July 2009 advance and the May–July 2010 break. Green dots mark buys and red dots mark sells on the source chart. Point values were read from that printed SPY daily plot at each labeled date.
Traders should notice how the nine-bar Hull line turns with price while the nine-bar simple average lags, especially on the July 2009 advance and the May–July 2010 break. Green dots mark buys and red dots mark sells on the source chart. Point values were read from that printed SPY daily plot at each labeled date.SPY · Daily · 2009-05-13T00:00:00.000Z to 2010-07-22T00:00:00.000Z

Both smoothers use a nine-bar window on the close. Coordinates are approximate readings from the raster, not a published table.

Oscillator of the smoother

The companion oscillator is a nine-bar relative-strength index computed on a nine-bar Hull series, not on raw price. The relative-strength index is used here as an overbought or oversold gate on the Hull series rather than on raw closes.

Long-only entry and exit

The rule-based entry is a long-only timing procedure that enters when the Hull-based oscillator is oversold and exits on a percentage move or an overbought oscillator reading. A long is opened when that Hull-based oscillator is below 30 and no position is already open. An open long is closed if the close falls more than 5 percent from the entry close, rises more than 15 percent from the entry close, or if the same nine-bar oscillator exceeds 90. The exit-percent is a fixed percentage gain from the entry close that closes the position without waiting for an oscillator signal.

Packaged strategy inputs

The packaged strategy inputs expose a nine-bar oscillator length, a nine-bar Hull length, close as the Hull price series, and a 15 percent profit exit, with orders generated only on bar close.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
1 of 6 in the Hull moving average track
201526-33 pp.Next on Hull moving averageEvaluating moving-average lag in cycle filtersCompare a simple moving average, an exponential moving average, and a Hull moving average on the same price series so lag and noise can be judged at one lookback.
All readings on this track · 6 readings
  1. 2010Constructing a Hull smoother and a Hull-based oscillator gate
  2. 2015Evaluating moving-average lag in cycle filters
  3. 2017Rebuild a Hull moving average with linear-regression intercepts
  4. 2017A Hull moving-average forecast in a credit-spread trend procedure
  5. 2019Constructing nested regression projections as smoothers
  6. 2020Evaluating countertrend ETF rules against a buy-and-hold control
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Also on Hull moving average5 readings
2017Rebuild a Hull moving average with linear-regression intercepts2015Evaluating moving-average lag in cycle filters2017A Hull moving-average forecast in a credit-spread trend procedure2019Constructing nested regression projections as smoothers2020Evaluating countertrend ETF rules against a buy-and-hold control