Should-Read: What Noah Smith does not say is: this is a horrible research program.
Taking your residual, putting it on the right hand side, and calling it a "productivity" shock may allow you to fit some things, but it doesn't allow you to explain or _understand. And there is no theory and no interest in developing any theory of where these "productivity shocks" come from.
Compare this to medieval Ptolemaic astronomy—well, Judah al-Barceloni had a theory that it was Sammael, the Angel of Mars, who guided the planet around its epicycles.
Ljungqvist and Sargent haven't even reached that level of Popperian potential falsification: their productivity shocks do not emerge from any economic process whereby businesses learn and forget production technologies, but they simply crystallize out of the air:
Noah Smith: Realism in macroeconomic modeling: "Ljungqvist and Sargent are trying to solve the Shimer Puzzle... http://noahpinionblog.blogspot.com/2017/09/realism-in-macroeconomic-modeling.html
...the fact that in classic labor search models of the business cycle, productivity shocks aren't big enough to generate the kind of employment fluctuations we see in actual business cycles. A number of theorists have proposed resolutions to this puzzle-i.e., ways to get realistic-sized productivity shocks to generate realistic-sized unemployment cycles. Ljungqvist and Sargent look at these and realize that they're basically all doing the same thing-reducing the value of a job match to the employer, so that small productivity shocks are more easily able to stop the matches from happening..."