Signal Processing Theory
In signal processing terms, the MACD is a filtered measure of the derivative of the input (price) with respect to time. (The derivative is called "velocity" in technical stock analysis). MACD estimates the derivative as if it were calculated and then filtered by the two low-pass filters in series, multiplied by a "gain" equal to the difference in their time constants. It also can be seen to approximate the derivative as if it were calculated and then filtered by a single low pass exponential filter (EMA) with time constant equal to the sum of time constants of the two filters, multiplied by the same gain. So, for the standard MACD filter time constants of 12 and 26 days, the MACD derivative estimate is filtered approximately by the equivalent of a low-pass EMA filter of 38 days. The time derivative estimate (per day) is the MACD value divided by 14.
The signal line is also a derivative estimate, with an additional low-pass filter in series for further smoothing (and additional lag). The difference between the MACD line and the signal (the "histogram") represents a measure of the second derivative of price with respect to time ("acceleration" in technical stock analysis). This estimate has the additional lag of the signal filter and an additional gain factor equal to the signal filter constant.
A MACD crossover of the signal line indicates that the direction of the acceleration is changing. The MACD line crossing zero suggests that the average velocity is changing direction.
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