Department of "Huh!?": This Is All Cosma Shalizi's Fault Department...

So when something comes across my RSS feed stating that it is "2500 words of statisticians quarreling with econometricians about arcane points of statistical theory", how am I supposed to resist getting sucked in?

And so not that many minutes later I find myself reading:

Chris Sims: Time series econometricians long ago got over the idea that frequency domain estimation, which makes only smoothness assumptions on spectral densities, is any more general than time-domain estimation with finite parametrization. In fact, the preferred way to estimate a spectral density is usually to fit an AR or ARMA model…

But surely if you are, to pick an example out of thin air, interested in estimating how much of a time series is made up of its permanent or near-permanent persistent components, you should estimate that directly--not use short-run autocorrelations to fit a low-order ARMA process and then infer the permanent and near-permanent persistent components from the low-order ARMA representation.

In what sense can "fit[ting] an AR or ARMA model" be the "preferred" way to conduct such an esimation?