VARX Own/Other Group Penalty

The VARX Own/Other Group Penalty model is a variation of the VARX Lag Group Lasso where own lags are shrunk by a smaller factor than other lags, i.e. the autoregressive properties of the included variables are prioritized over the effect of the different variables on each other. This is similar to the Minnesota prior by Litterman which is used in Bayesian analysis. Even in a setting where appropriate indicators are selected, it is common to see that the main variable is highly dependent on its own lags. The VARX Own/Other Group Penalty model only allows lags of indicators to be included in the model if the same lag is included for the main variable.

Mathematically, the penalty can be written as

where ∣∣X∣∣F​ is the Frobenius norm mentioned in the VARX Lag Group Lasso article. The notation A_on(l) and A_off(l) refer to the on and off-diagonal entries of the coefficient matrices for lag l. Since the lags of a variable in its own equation will be represented by the diagonal entries, this shows how the penalty structure is realized mathematically.