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I know we can forecast univariate time series using different models of exponential smoothing , but am searching for whether same can be extended to multivariate time series and if yes what are those equations?

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If you have 1 endogenous series and 1 or more exogenous series the approach is to build and identify a Transfer Function Model as described here http://www.autobox.com/pdfs/WHY-WE-FILTER.ppt and here https://autobox.com/pdfs/PREFERRED.pdf and here http://www.autobox.com/pdfs/A.pdf leading to http://www.autobox.com/pdfs/SARMAX.pdf . Analysis will suggest "the form of the appropriate equation" .

EDITED TO BE MORE SPECIFIC AS TO HOW A SIMPLE EXPONENTIAL SMOOTHING MODEL CAN BE GENERALIZED TO INCLUDE PREDICTORS BOTH STOCHASTIC AND DETERMINISTIC

The simple exponential smoothing model is a particular case of an ARIMA model (0,1,1) with coefficient 1-A enter image description here from https://people.duke.edu/~rnau/411arim.htm

Following is the simple exponential smoothing model enter image description here by incorporating X's as in the SARMAX model we are accomplishing the task. Essentially the lag structure in Y is handled by the ARIMA (noise) component in the SARMAX model.

If you have more than 1 endogenous series then either a VARIMA model or VAR model would be appropriate.

AND restrict the arima structure for the NOISE component to a first difference ma(1) model .

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    $\begingroup$ This doesn't seem to answer the question about extending exponential smoothing to multivariate series. If it does, could you so indicate that explicitly? $\endgroup$ – whuber Oct 4 at 17:21

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