VARReduce

Generalized VAR forecaster using tabularized regression.

As special cases, can be used to construct classical L1 (Lasso) or elastic VAR forecasting models.

VARReduce is constructed with a tabular scikit-learn regressor (e.g., Lasso, Ridge, etc.) and is designed to be used with multivariate time series data.

The input data Y_in is a multivariate time series data containing n time series. An example with n = 2:

Fitting proceeds in two steps:

1. Tabularization:

   For each time step and each time series within Y_in, lagged values X are generated. The number of lagged values are determined by the lags parameters.

   Below is the X for the sample Y_in with lags = 2. Note the absence of the earliest 2 timesteps as no corresponding lag value is available.

   index	ts1_lag1	ts2_lag1	ts1_lag2	ts2_lag2
   3	12	7	11	6
   4	13	8	12	7
   5	14	9	13	8

2. Regression:

   The chosen regressor is fitted with `Y_in` as a target and X as predictors. Care is taken to first remove the first lags data points in `Y_in` as they do not have corresponding indices in X (i.e. the first two data points in the above example).

For forecasting, the last lags observations in Y_in are reframed as lagged predictors X_forecast and passed to the trained regressor to obtain the forecasts. X_forecast is shown below.

By default, LinearRegression is used, yielding results equivalent to a traditional VAR model. Alternatively, any scikit-learn compatible regressor can be used to introduce regularization and/or non-linearity.

For example:

• VARReduce(regressor = Ridge()) is equivalent to VAR with L2 regularization.

• VARReduce(regressor = Lasso()) is equivalent to VAR with L1 regularization.

• VARReduce(regressor = ElasticNet()) is equivalent to elastic VAR.

These specific models are well-known classical generalizations of VAR. They can be used to incorporate regularization and prevent overfitting when the input data contain a large number of individual time series relative to data points.