Practical differences of first and second order (or higher) approximation of the solution

Dear all. I’m aware that some use second order approximation e.g. analyzing welfare under different policies, since an optimal policy needs to for example maximize welfare, which is measured as a second order approximation.

My question is, in practice what are other differences that provide second, or even higher order approximation, in terms of interpretation, irfs, theoretical moments, etc.? Since I see authors often use first-order approximation and second if using some welfare analysis.

Also, why when computing higher order approximation of the solution, e.g. 5, only Var-Cov matrix of shocks are printed and not the rest of usual output?


My 2 cents on this: in terms of absolute quantities/levels, 1st-order solutions and 2nd-order ones shouldn’t make much difference. If they do, it means that your 1st-order solution (essentially, an approximation to the original DSGE system) is utterly wrong, or it means that the variance of the shocks – the uncertainties – are too much, so you have to use second-order approximation.
But if you care about statistics or moments that are 2nd-order or above (for example, welfare analysis, uncertainties, risk-premiums, etc), then 1st-order approximation is for sure insufficient. The other case that you have to use a higher-order approximation is when your system deviates from the steady state a lot (for example, there’s a large shock to your system) and 1st-order approximation will underestimate the effects.

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With perturbation solutions, your are doing a local polynomial approximation to a typically nonlinear problem. Thus, the more nonlinear the problem is and the further you move away from the approximation point, the worse the approximation will get.
This tells you that there are two reasons for higher order approximations:

  1. You are interested in some inherently nonlinear properties like welfare or the effect of uncertainty shocks.
  2. The model is so nonlinear and/or the shocks so big and persistent that the accuracy of a first order solution is not sufficient anymore.

The reason you are not getting moments at higher order is that these moments cannot be analytically computed anymore (unless you use pruning). You would need to set the periods-option.

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