We relax the standard assumption in the dynamic stochastic general equilibrium (DSGE) literature that exogenous processes are governed by AR(1) processes and estimate ARMA (p,q) orders and parameters of exogenous processes. Methodologically, we contribute to the Bayesian DSGE literature by using Reversible Jump Markov Chain Monte Carlo (RJMCMC) to sample from the unknown ARMA orders and their associated parameter spaces of varying dimensions. In estimating the technology process in the neoclassical growth model using post war US GDP data, we cast considerable doubt on the standard AR(1) assumption in favor of higher order processes. We find that the posterior concentrates density on hump-shaped impulse responses for all endogenous variables, consistent with alternative empirical estimates and the rigidities behind many richer structural models. Sampling from noninvertibleMA representations, a negative response of hours to a positive technology shock is contained within the posterior credible set. While the posterior contains significant uncertainty regarding the exact order, our results are insensitive to the choice of data filter; this contrasts with our ARMA estimates of GDP itself, which vary significantly depending on the choice of HP or first difference filter.