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# Technical Studies Reference

### MACD

The Moving Average Convergence/Divergence (MACD) is a trend-following momentum indicator that shows the relationship between two moving averages of prices. The MACD was developed by Gerald Appel. In Sierra Chart you have a choice of the Moving Average type to use in the calculations. We describe the calculation below.

Let $$X$$ be a random variable denoting the Input Data Input. Let the Inputs Fast Moving Average Length, Slow Moving Average Length, and MACD Moving Average Length be denoted as $$n_F$$, $$n_S$$, and $$n_M$$, respectively. This study calculates and displays three indicators: the MACD, the Moving Average of the MACD, and the MACD Difference. We denote the values of these indicators for the given Inputs at Index $$t$$ as $$MACD_t\left(X,n_F,n_S\right)$$, $$\overline{MACD}_t\left(X,n_F,n_S,n_M\right)$$, and $$\Delta MACD_t\left(X,n_F,n_S,n_M\right)$$, respectively. We describe the methods of calculation of these indicators below.

The MACD is calculated for $$t \geq 0$$ in terms of Exponential Moving Averages as follows.

$$MACD_t\left(X,n_F,n_S\right) = EMA_t\left(X,n_F\right) - EMA_t\left(X,n_S\right)$$

The Moving Average of the MACD is calculated for $$t \geq \max\{n_S,n_F\} + n_M$$ in terms of an Exponential Moving Average as follows.

$$\overline{MACD}_t\left(X,n_F,n_S,n_M\right) = EMA_t\left(MACD\left(X,n_F,n_S\right),n_M\right)$$

In the above formula, $$MACD\left(X,n_F,n_S\right)$$ is a random variable denoting the MACD with Inputs as listed in the parentheses.

Note: Depending on the setting of the Input Moving Average Type, the Exponential Moving Averages in each of the above formulas could be replaced with Linear Regression Moving Averages, Simple Moving Averages, Weighted Moving Averages, Wilders Moving Averages, Simple Moving Averages - Skip Zeros, or Smoothed Moving Averages.

The MACD Difference is calculated for $$t \geq \max\{n_S,n_F\} + n_M$$ in terms of the MACD and the Moving Average of the MACD as follows.

$$\Delta MACD_t\left(X,n_F,n_S,n_M\right) = MACD_t\left(X,n_F,n_S\right) - \overline{MACD}_t\left(X,n_F,n_S,n_M\right)$$