## Sunday, June 26, 2016

### 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.)