Regularization for Long Memory

Two earlier regularization posts focused on panel data and generic time series contexts. Now consider a specific time-series context: long memory. For exposition consider the simplest case of a pure long memory DGP,  \( (1-L)^d y_t = \varepsilon_t \) with  \( |d| < 1/2  \).  This \( ARFIMA(0,d,0) \) process is  is \( AR(\infty) \) with very slowly decaying coefficients due to the long memory. If you KNEW the world was was \(ARFIMA(0,d,0)\) you'd just fit \(d\) using GPH or Whittle or whatever, but you're not sure, so you'd like to stay flexible and fit a very long \(AR\) (an \(AR(100) \), say). But such a profligate parameterization is infeasible or at least very wasteful. A solution is to fit the \(AR(100) \) but regularize by estimating with ridge or a LASSO variant, say.

Related, recall the Corsi "HAR" approximation to long memory. It's just a long autoregression subject to coefficient restrictions. So you could do a LASSO estimation, as in Audrino and Knaus (2013). Related analysis and references are in a Humboldt University 2015 master's thesis.)

Finally, note that in all of the above it might be desirable to change the LASSO centering point for shrinage/selection to match the long-memory restriction. (In standard LASSO it's just 0.)