DSGE Models with Student-t Errors

Abstract

This paper deals with Dynamic Stochastic General Equilibrium (DSGE) models under a multivariate student-t distribution for the structural shocks. Based on the solution algorithm of Klein (2000) and the gamma-normal representation of the t-distribution, the TaRB-MH algorithm of Chib and Ramamurthy (2010) is used to estimate the model. A technique for estimating the marginal likelihood of the DSGE student-t model is also provided. The methodologies are illustrated first with simulated data and then with the DSGE model of Ireland (2004) where the results support the t-error model in relation to the Gaussian model.

Keywords:

• Bayesian inference
• Marginal likelihood
• MCMC
• Metropolis-Hastings
• Particle-filtering
• State-space model
• Multivariate-student-t distribution

JEL Classification:

• C01
• C11
• C15
• C22
• C32
• E27
• E37
• E47

Notes

As defined in Blanchard and Kahn (1980), backward and forward looking refer to ‘predetermined’ and ‘non-predetermined’ variables, respectively. Klein (2000) provides a generalization of these definitions. For instance, a backward-looking variable is one that has an exogenously specified initial value and prediction error.

Notes: 1.) The numerical standard errors are reported in parentheses. 2.) These estimates are based on the following prior specification. Each of g ₁₁ and g ₂₂ are assumed to be apriori normally distributed with a mean of 0.1 and variance of 2.0. For the example without the measurement error, is assumed to follow a lognormal distribution with mean 0.6 and variance 0.95. For the model with the measurement error the prior on both and is lognormal with mean 0.88 and variance 4.46. 3.) The estimates reported for the model without a measurement error are based on u _t ∼ 𝒩(0, 10⁻³). The marginal likelihood estimates for u _t ∼ 𝒩(0, 10⁻²) for models ℳ₁ - ℳ₄ are −119.41, −121.84, −123.68, and −155.52, respectively.

Computed as in Chib (1995).

Alternatively, one could also explicitly derive the posterior distribution of ν as shown in Albert and Chib (1993) at a negligible cost.

Detailed derivations of the solution step and state space representation are available in the notes accompanying the original paper at Peter Ireland's website.

The data is available from the Federal Reserve Bank of St. Louis (FRED) website.

Remark: The results reported in this table are based on prior mean as the starting value. However, the results are insensitive to this choice.