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What are predictor variables in time series forecasting?
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Predictor variables are often useful in time series forecasting. For example, suppose we wish to forecast the hourly electricity demand (ED) of a hot region during the summer period. A model with predictor variables might be of the form

$\mathrm{ED}=f$ (current temperature, strength of economy, population, time of day, day of week, error).

The relationship is not exact $-$ there will always be changes in electricity demand that cannot be accounted for by the predictor variables. The "error" term on the right allows for random variation and the effects of relevant variables that are not included in the model. We call this an explanatory model because it helps explain what causes the variation in electricity demand.
Because the electricity demand data form a time series, we could also use a time series model for forecasting. In this case, a suitable time series forecasting equation is of the form
$$\mathrm{ED}_{t+1}=f\left(\mathrm{ED}_{t}, \mathrm{ED}_{t-1}, \mathrm{ED}_{t-2}, \mathrm{ED}_{t-3}, \ldots, \text { error }\right)$$
where $t$ is the present hour, $t+1$ is the next hour, $t-1$ is the previous hour, $t-2$ is two hours ago, and so on. Here, prediction of the future is based on past values of a variable, but not on external variables which may affect the system. Again, the "error" term on the right allows for random variation and the effects of relevant variables that are not included in the model.

There is also a third type of model which combines the features of the above two models. For example, it might be given by
$\mathrm{ED}_{t+1}=f\left(\mathrm{ED}_{t}\right.$, current temperature, time of day, day of week, error).
These types of "mixed models" have been given various names in different disciplines. They are known as dynamic regression models, panel data models, longitudinal models, transfer function models, and linear system models (assuming that $f$ is linear).

An explanatory model is useful because it incorporates information about other variables, rather than only historical values of the variable to be forecast. However, there are several reasons a forecaster might select a time series model rather than an explanatory or mixed model.

First, the system may not be understood, and even if it was understood it may be extremely difficult to measure the relationships that are assumed to govern its behaviour.

Second, it is necessary to know or forecast the future values of the various predictors in order to be able to forecast the variable of interest, and this may be too difficult.

Third, the main concern may be only to predict what will happen, not to know why it happens.

Finally, the time series model may give more accurate forecasts than an explanatory or mixed model.

The model to be used in forecasting depends on the resources and data available, the accuracy of the competing models, and the way in which the forecasting model is to be used.

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