Constructing MACD signal lines and divergence tests

Two construction stages

The MACD line is the arithmetic difference between two exponential moving averages computed on successive closing prices. It is the signed gap between a faster and a slower exponential smoothing of the close.

The signal line is a further exponential moving average applied to that difference series, not to price. It is the crossover partner for the MACD line.

How each closing-price average is updated

Each EMA update takes the gap between the latest close and the previous EMA, multiplies it by 2/(n+1), and adds the result to the previous EMA.

The factor 2/(n+1) is the smoothing constant that maps a chosen lookback n onto that recursion. A lookback of n = 9 produces the smoothing constant 0.20 under that formula.

One documented lookback set

One documented construction uses a 12-period EMA of the close minus a 26-period EMA of the close, then a 9-period EMA of that difference as the signal line.

Rules applied after both lines exist

A buy-side crossover is defined as the MACD line moving above the signal line. A sell-side crossover is defined as the MACD line moving below it.

One confirmation rule treats an upward crossover as aligned with an uptrend only when both the MACD line and the signal line stand above zero. That requirement is the zero-line filter.

While the MACD line remains above the signal line and both lines remain above zero, the construction is read as an uptrend. The method is trend-following and therefore lags turning points. That delay is the trend-following lag, because both lines react only after the smoothed series have already turned.

Price-indicator divergence is defined as price making a new high or new low while the MACD line fails to better its own prior extreme. That pattern is treated as a reversal warning separate from the crossover rule.