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.)

Indicator Saturation Estimation

In an earlier post, "Fixed Effects Without Panel Data",  I argued that you could allow for (and indeed estimate) fixed effects in pure cross sections (i.e., no need for panel data) by using regularization estimators like LASSO. The idea is to fit a profligately-parameterized model but then to recover d.f. by regularization.

Note that you can use the same idea in time-series contexts.  Even in a pure time series, you can allow for period-by-period time effects, broken polynomial trend with an arbitrary number of breakpoints, etc., via regularization.  

It turns out that a fascinating small literature on so-called "indicator saturation estimation" pursues this idea.  The "indicators" are things like period-by-period time dummies, break-date location dummies, etc., and "saturation" refers to the profligate parameterization.  Prominent contributors include David Hendry and Soren Johanssen; see this new paper and those that it cites.  (Very cool application, by the way, to detecting historical volcanic eruptions.)