VARX Own/Other Sparse Group Penalty

The VARX Own/Other Sparse Group Penalty model is a variation of the VARX Lag Group Lasso which is very similar to the VARX Own/Other Group Penalty model 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 main difference compared to the VARX Own/Other Group Penalty model is that it allows sparsity within groups, i.e. a lag of the indicators may be included for the main variable equation for only some of the indicators.

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. Looking at the last term and the α parameter, it is also possible to see similarities to the elastic net penalty. Here a mixture between group lasso and regular lasso is achieved similar to how elastic net creates a mixture of lasso and ridge regression.