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Moving Average - Exponential Regression

The Moving Average - Exponential Regression (ERMA) study calculates and displays an exponential regression moving average of the data specified by the Input Data Input.

Let \(X\) be a random variable denoting the Input Data, and let \(X_t\) denote the value of the Input Data at Index \(t\). Let the Length Input be denoted as \(n\). Then we denote the Moving Average - Exponential Regression at Index \(t\) as \(ERMA_t(X,n)\).

The regression model is of the form \(\hat{X} = k_t(X,n)e^{r_t(X,n) \cdot T}\), where \(\hat{X}\) is an estimator for \(X\). We call \(k_t(X,n)\) the coefficient, and we call \(r_t(X,n)\) the growth constant.

Taking the logarithm of both sides of this model, we obtain the following linear regression model for the logarithm of the Input Data.

\(\displaystyle{\ln(\hat{X}) = r_t(X,n) \cdot T + \ln(k_t(X,n))}\)

We then compute linear regression statistics for \(\ln(X_t)\). That is, we compute \(b_t(\ln(X),n)\) and \(a_t(\ln(X),n)\).

This yields the following.

  • \(b_t(\ln(X),n) = r_t(X,n)\)
  • \(a_t(\ln(X),n) = \ln(k_t(X,n))\), or \(k_t(X,n) = e^{a_t(\ln(X),n)}\).

Finally, we compute \(ERMA_t(X,n)\) for \(t \geq n - 1\) as follows.

\(\displaystyle{ERMA_t(X,n) = k_t(X,n)e^{r_t(X,n) \cdot n}}\)

The value of \(ERMA_t(X,n)\) is the vertical coordinate of the right endpoint of the exponential regression trendline of Length \(n\).

Inputs

Spreadsheet

The spreadsheet below contains the formulas for this study in Spreadsheet format. Save this Spreadsheet to the Data Files Folder.

Open it through File >> Open Spreadsheet.

Moving_Average_-_Exponential_Regression.485.scss


*Last modified Wednesday, 07th July, 2021.