By Oliver de Groot
When a DSGE model features stochastic volatility, is a third-order perturbation approximation sufficient? The answer is often no. A key parameter – the standard deviation of stochastic volatility innovations – does not appear in the coefficients of the decision rules of endogenous variables until a fourth- or sixth-order perturbation approximation (depending on the functional form of the stochastic volatility process). This paper shows analytically this general result and demonstrates, using three models, that important model moments can be imprecisely measured when the order of approximation is too low. i) In the Bansal-Yaron long-run risk model, the equity risk premium rises from 4.5% to 10% by going to sixth-order. ii) In a workhorse real business cycle model, the welfare cost of business cycles also rise when a fourth-order approximation properly accounts for the presence of stochastic volatility. iii) In a canonical New-Keynesian model, the risk-aversion parameter can be lowered while matching the term premium when a fourth-order approximation is used.
It is important to choose an approximation methods that reflects well for the exercise you are trying to perform. That said, there can also be other avenues, in particular with piecewise-linear and/or grid methods that can work better at potentially a lower cost. This paper highlights hat these are important considerations.