Eric Eisenstat

Marginal Likelihood Estimation with the Cross-Entropy Method

We consider an adaptive importance sampling approach to estimating the marginal likelihood, a quantity that is fundamental in Bayesian model comparison and Bayesian model averaging. This approach is motivated by the difficulty of obtaining an accurate estimate through existing algorithms that use Markov chain Monte Carlo (MCMC) draws, where the draws are typically costly to obtain and highly correlated in high-dimensional settings. In contrast, we use the cross-entropy (CE) method, a versatile adaptive Monte Carlo algorithm originally developed for rare-event simulation. The main advantage of the importance sampling approach is that random samples can be obtained from some convenient density with little additional costs. As we are generating independent draws instead of correlated MCMC draws, the increase in simulation effort is much smaller should one wish to reduce the numerical standard error of the estimator. Moreover, the importance density derived via the CE method is grounded in information theory, and therefore, is in a well-defined sense optimal. We demonstrate the utility of the proposed approach by two empirical applications involving womens labor market participation and U.S. macroeconomic time series. In both applications, the proposed CE method compares favorably to existing estimators.

Stochastic Model Specification Search for Time-Varying Parameter VARs

This article develops a new econometric methodology for performing stochastic model specification search (SMSS) in the vast model space of time-varying parameter vector autoregressions (VARs) with stochastic volatility and correlated state transitions. This is motivated by the concern of overfitting and the typically imprecise inference in these highly parameterized models. For each VAR coefficient, this new method automatically decides whether it is constant or time-varying. Moreover, it can be used to shrink an otherwise unrestricted time-varying parameter VAR to a stationary VAR, thus providing an easy way to (probabilistically) impose stationarity in time-varying parameter models. We demonstrate the effectiveness of the approach with a topical application, where we investigate the dynamic effects of structural shocks in government spending on U.S. taxes and gross domestic product (GDP) during a period of very low interest rates.

Modelling Inflation Volatility

This paper discusses estimation of US inflation volatility using time varying parameter models, in particular whether it should be modelled as a stationary or random walk stochastic process. Specifying inflation volatility as an unbounded process, as implied by the random walk, conflicts with priors beliefs, yet a stationary process cannot capture the low frequency behaviour commonly observed in estimates of volatility. We therefore propose an alternative model with a change-point process in the volatility that allows for switches between stationary models to capture changes in the level and dynamics over the past forty years. To accommodate the stationarity restriction, we develop a new representation that is equivalent to our model but is computationally more efficient. All models produce effectively identical estimates of volatility, but the change-point model provides more information on the level and persistence of volatility and the probabilities of changes. For example, we find a few well defined switches in the volatility process and, interestingly, these switches line up well with economic slowdowns or changes of the Federal Reserve Chair. Moreover, a decomposition of inflation shocks into permanent and transitory components shows that a spike in volatility in the late 2000s was entirely on the transitory side and a characterized by a rise above its long run mean level during a period of higher persistence.

Bayesian Model Comparison for Time-Varying Parameter VARs with Stochastic Volatility

We develop importance sampling methods for computing two popular Bayesian model comparison criteria, namely, the marginal likelihood and deviance information criterion (DIC) for TVP-VARs with stochastic volatility. The proposed estimators are based on the integrated likelihood, which are substantially more reliable than alternatives. Specifically, integrated likelihood evaluation is achieved by integrating out the time-varying parameters analytically, while the log-volatilities are integrated out numerically via importance sampling. Using US and Australian data, we find overwhelming support for the TVPVAR with stochastic volatility compared to a conventional constant coefficients VAR with homoscedastic innovations. Most of the gains, however, appear to have come from allowing for stochastic volatility rather than time variation in the VAR coefficients or contemporaneous relationships. Indeed, according to both criteria, a constant coefficients VAR with stochastic volatility receives similar support as the more general model with time-varying parameters.

Behavioural model uncertainty in estimation of structural oligopoly models

The focus of this paper is on developing a methodology for dealing with behavioural model uncertainty in structural oligopoly models. It is well recognised that being an essential part of the identification strategy, the particular choice of a behavioural model embodies a highly influential, yet largely arbitrary, set of assumptions in the structural framework. The methods developed here are founded in Bayesian model averaging techniques and provide a practically and conceptually desirable way of accommodating behavioural model uncertainty in structural estimation. Moreover, a substantial feature of this approach is that it yields straightforward model comparison through the model posterior distribution. These methods are applied to estimate the parameters of the industry demand curve and firms cost functions in oligopoly markets (e.g., marginal costs, markups, etc.). Three models of oligopoly behaviour are considered: one non-cooperative and two variations of cooperative with unobserved demand shocks. The specific industry analysed is the 1800s railroad cartel, commonly known as the Joint Executive Committee, which is widely familiar to industrial organisations economists. The results indicate that the algorithm performs quite well in correctly identifying cooperative behaviour, in additional to offering a clear view of the way in which model averaging resolves conflicts in inference arising from competing behavioural models.