What exactly is certainty equivalence in the context of DSGE models?

Question

I keep reading about certainty equivalence in the context of DSGE models.

I understand that it has something to do with "Getting rid of the expectation operator", but I'm not entirely sure? What exactly is "certainty equivalence" in the context of DSGE models?

Do we have certainty equivalence if the solution to a stochastic model $M_S$ is the same as the solution to the non-stochastic model $M_N$, if $M_N$ is obtained by taking the equations of $M_S$ and simply removing the expectation operator and the random shocks?

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EDIT: I just read in the Dynare user manual, that a first order taylor approximation of a DSGE model has "certainty equivalence", because future shocks drop out when you take the expectation operator. Yet current shocks do not drop out, as we can see in the following equation:

Note the $u$ at the very end. This denotes the current-time shock.

So if this is a "certainty equivalent model", but we still have an exogenous shock, then what does certainty equivalence mean? Apparently it doesn't mean "the absence of uncertainty" since there is still an exogenous shock.

Answer 1

Intuitively, it means that the model has such characteristics that "the best we can say" about remaining uncertainty, is that it will be zero. From general experience, we know that it won't be zero, but the information we possess does not permit us to say anything else than that it will be zero. The even deeper assumption here is that the information and knowledge we possess make all things that remain unpredictable to average to zero.

In such a case, of course the actual realized path of the model is affected by these totally unpredictable shocks, but the decisions we take are based on a rule "as if" the future shocks will be zero (it is implied that we have to take decisions prior to observe the realization of the shock).

More technically, quoting from Lungqvist & Sargent's "Recursive Macroeconomic Theory" 2004, 2nd ed:

a) p.113: "CERTAINTY EQUIVALENCE PRINCIPLE: The decision rule that solves the stochastic optimal linear regulator problem is identical with the decision rule for the corresponding nonstochastic linear optimal regulator problem.

b) p.115: The certainty equivalence principle is a special property of the optimal linear regulator problem, and comes from the quadratic objective function. the linear transition equation, and the property that (future disturbances have zero mean conditional on the current state). Certainty equivalence does not characterize stochastic control problems generally.

Please read also this post, so that if you do further search, to make sure that you look for "certainty equivalence in macroeconomics".