Computes Seasonal (aka. double) moving average (aka. KxM MA).

## Syntax

**NxSMA**(**X**, Order, K, N, EndPoints)

**X**- is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
**Order**- is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).
Order Description 1 Ascending (the first data point corresponds to the earliest date) (default). 0 Descending (the first data point corresponds to the latest date). **K**- is the number of terms (aka points) of the second moving average filter. If missing, K is assumed 1.
**M**- is the number of terms (aka points) in the first moving average filter (e.g., seasonal length). If missing or omitted, N is assumed 1.
**EndPoints**- is a logical value for applying asymmetric weights at the end of the series. If missing or omitted, Endpoints = False, and no treatment for Endpoints is performed.

## Remarks

- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g., #N/A) at either end.
- The NxSMA returns an array-type value to Excel. The user must use CTRL+ALT+ENTER to show returned values.
- The number of terms (i.e., K+N-1) in NxSMA must be odd. In effect, K and N can be both odd or even numbers.
- In practice, the NxSMA is also known as the double-moving average filter, which combines two MA to form a symmetric two-sided moving average filter.
- Why use NxSMA? When working with time series, you often smooth to remove seasonality effects, which requires a period (aka moving window) equal to a seasonal length, which is often an even number, for example, 12 months or 4 quarters. In this case, we use the double moving average (e.g., 2x4 for quarterly data or 2x12 for monthly data, etc.).
- Mathematically, the order in which the MA filters are applied does not matter; they will return the same results. For our application,
- The weights or multiplying factors of the NxSMA filter are derived by convolution of the two moving average filters (K-MA and N-MA).
- The weights or the multiplying factors of the NxSMA are calculated as follows: $$w_j=\left\{\begin{matrix} \frac{j}{K\times M} & j < K\\ \frac{1}{M}& K \leq j \leq M\\ \frac{M+K-j}{K\times M} & M+K > j > M \end{matrix}\right.$$
- By definition, the sum of all weights in the SMA is equal to one(1).
- The NxSMA function is available starting with version 1.66 PARSON.

## Files Examples

## Related Links

## References

- D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906.
- James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
- Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.
- Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848.
- Walter Enders; Applied Econometric Time Series; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568.

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