Which order is too much? An application to a model with staggered price and wage contracts
First-order approximations to the solution to a Dynamic Stochastic General Equilibrium Model (DGSE) are now widely used in the literature. In particular, the solution is usually based on the standard log-linearisation procedure around the steady-state. However, it may be not enough especially when studying welfare across policies. In effect, welfare comparisons cannot be implemented since we need at least second-order effect on the model's deterministic steady-state. This standard method of approximation have been questioned in a series of papers. Except in rather restrictive cases (Obstfeld and Rogoff, 1998, 2002; Devereux and Engel, 2001, and Corsetti and Pesenti, 2001), it is not possible to derive explicitely an exact expression for utility-based welfare functions. Some papers have shown the importance of higher-order approxination. For example, Kim and Kim (2003a) note the importance of second-order approximation in studying welfare effects of trade in a two-country framework. A recent literature have proposed different methods to produce second-order accurate approximation to the solutions to DGSE's from a straightforward second-order approximation of the model. Among others, Judd (2002), Jin and Judd (2002) show how to compute approximation of arbitrary order on discrete-time models. Specifically, they propose a general Taylor series method for computing asymptotically valid approximations to deterministic and stochastic rational expectation models near the deterministic steady-state. Collard and Juillard (2001b), Anderson and Levin (2002), Schmitt-Grohe and Uribe (2004) apply pertubation methods of higher than first order. Sims (2002) generalised the approaches of Judd (1998), Judd and Gaspar (1997) and Judd and Guu (1993) in order to find a second-order accurate solution of discrete-time dynamic equilibrium models. Kim et al. (2003) propose an algorithm in order to compute a second-order approximation in which the error in the approximation is claimed to converge in probability to zero and does not depend on strict boundedness of the support of the distribution of the shocks. Then they apply their method for calculating forecasts and impulse-response functions in DGSE\ models. In this paper, we investigate the accuracy of k-order perturbation method in approximating the solution of a DSGE model. As a benchmark model, we use a version of Erceg, Henderson and Levin (2000) model with staggered price and wage contracts. Using different criteria, we assess to what extent the order of the approximation matters and which order is reliable. Our results show that standard first-order and second-order approximation may lead to misleading interpretations. At the same time, over-approximating the model may also conduct to substantial distortions.
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