Marginal Data Density Research Articles

Forecast evaluation of DSGE models: Linear and nonlinear likelihood

We apply a Bayesian approach to estimate a small-scale New Keynesian Dynamic Stochastic General Equilibrium (DSGE) model, solved by linear and nonlinear solution techniques. In particular, the likelihood function for the linearized and nonlinearized models is approximated via the linear and two types of nonlinear filters, respectively. We evaluate the performance of the linearized/nonlinearized model in terms of the log marginal data density and out-of-sample forecasting exercise over the transition periods of “Great Inflation” to “Great Moderation” (GI-GM) and “Great Moderation” to “Great Recession” (GM-GR). Several results are found as follows. First, it is not clear whether the nonlinearized DSGE model has a better fit than a linearized one to the full sample, and the result is also applied over the transition period of GI-GM when we compute the time-varying marginal data density. Second, under the point forecast evaluation, we do not find any statistical evidence that the forecasting performance of DSGE models, either the linearized or nonlinearized one, is better than the VAR models on predicting GDP growth rate, inflation, and interest rates over two transition periods. Moreover, the forecast errors for three data series calculated from either the individual model or the case of “Aggregation across models” in “ Normal” period are significantly lower than those in “ Crisis” period. Third, under the density forecast evaluation across DSGE models, we do not find any significant difference on the sum of log predictive likelihood across models at all forecast horizons for three individual data series. In particular, the above results are also applied to two transition periods and real-time data sets.

New Estimators of the Bayes Factor for Models with High-Dimensional Parameter and/or Latent Variable Spaces.

Formal Bayesian comparison of two competing models, based on the posterior odds ratio, amounts to estimation of the Bayes factor, which is equal to the ratio of respective two marginal data density values. In models with a large number of parameters and/or latent variables, they are expressed by high-dimensional integrals, which are often computationally infeasible. Therefore, other methods of evaluation of the Bayes factor are needed. In this paper, a new method of estimation of the Bayes factor is proposed. Simulation examples confirm good performance of the proposed estimators. Finally, these new estimators are used to formally compare different hybrid Multivariate Stochastic Volatility–Multivariate Generalized Autoregressive Conditional Heteroskedasticity (MSV-MGARCH) models which have a large number of latent variables. The empirical results show, among other things, that the validity of reduction of the hybrid MSV-MGARCH model to the MGARCH specification depends on the analyzed data set as well as on prior assumptions about model parameters.

Bayesian Model Averaging and Prior Sensitivity in Stochastic Frontier Analysis

This paper discusses Bayesian model averaging (BMA) in Stochastic Frontier Analysis and investigates inference sensitivity to prior assumptions made about the scale parameter of (in)efficiency. We turn our attention to the “standard” prior specifications for the popular normal-half-normal and normal-exponential models. To facilitate formal model comparison, we propose a model that nests both sampling models and generalizes the symmetric term of the compound error. Within this setup it is possible to develop coherent priors for model parameters in an explicit way. We analyze sensitivity of different prior specifications on the aforementioned scale parameter with respect to posterior characteristics of technology, stochastic parameters, latent variables and—especially—the models’ posterior probabilities, which are crucial for adequate inference pooling. We find that using incoherent priors on the scale parameter of inefficiency has (i) virtually no impact on the technology parameters; (ii) some impact on inference about the stochastic parameters and latent variables and (iii) substantial impact on marginal data densities, which are crucial in BMA.

Fiscal Stimulus on Bayesian DSGE Models

I take a Bayesian approach to estimate and forecast the effects of fiscal stimulus in various versions of the model by Smets and Wouters (2007) for the US economy. Specifically, I proxy various simpler DSGE sub-models by imposing a tight prior on a single parameter or a combination of tight priors on multiple parameters in the Smets-Wouters model. I find that the present-value government spending multipliers obtained are all in a reasonable range. Moreover, I forecast the effect of fiscal stimulus in a scenario similar to the 2008/2009 recession in the US, where the public expects a large and temporary increase in government spending to stimulate a fragile economy. The forecasts, generated individually by a group of representative models, are weighted averaging by means of the posterior model probabilities that are computed on the basis of their corresponding marginal data densities. According to the Diebold-Mariano test, I find that the forecast error of the combination forecast, computed via Bayesian model averaging (BMA), is statistically larger than the individual forecast, obtained only from the one that has the best fit among those DSGE models.

Some International Evidence for Keynesian Economics Without the Phillips Curve

Farmer and Nicolo (2018)… show that the Farmer Monetary (FM)‐model outperforms the three‐equation New Keynesian (NK)‐model in post‐war U.S. data. In this paper, we compare the marginal data density of the FM‐model with marginal data densities for determinate and indeterminate versions of the NK‐model for three separate samples using U.S., U.K. and Canadian data. We estimate versions of both models that restrict the parameters of the private sector equations to be the same for all three countries. Our preferred specification is the constrained version of the FM‐model which has a marginal data density that is more than 40 log points higher than the NK alternative. Our findings also demonstrate that cross‐country macroeconomic differences are well explained by the different shocks that hit each economy and by differences in the ways in which national central banks reacted to those shocks.

Some International Evidence for Keynesian Economics Without the Phillips Curve

Farmer and Nicolò (2018) show that the Farmer Monetary (FM)-model outperforms the three-equation New-Keynesian (NK)-model in post war U.S. data. In this paper, we compare the marginal data density of the FM-model with marginal data densities for determinate and indeterminate versions of the NK-model for three separate samples using U.S., U.K. and Canadian data. We estimate versions of both models that restrict the parameters of the private sector equations to be the same for all three countries. Our preferred specification is the constrained version of the FM-model which has a marginal data density that is more than 30 log points higher than the NK alternative. Our findings also demonstrate that cross-country macroeconomic differences are well explained by the different shocks that hit each economy and by differences in the ways in which national central banks reacted to those shocks.

Checking if the Straitjacket Fits

Pesaran and Smith (2011) concluded that DSGE models were sometimes a straitjacket which hampered the ability to match certain features of the data. In this paper we look at how one might assess the fit of these models using a variety of measures, rather than what seems to be an increasingly common device - the Marginal Data Density. We apply these in the context of models by Christiano et.al (2014) and Ireland (2004), finding they fail to make a match by a large margin. Against this, there is a strong argument for having a straitjacket as it enforces some desirable behaviour on models and makes researchers think about how to account for any non-stationarity in the data. We illustrate this with examples drawn from the SVAR literature and also more eclectic models such as Holston et al (2017) for extracting an estimate of the real natural rate.

A Comparison Analysis of Monetary Policy Effect Under an Open Economy Model

The paper analyzes and compares the effects of domestic monetary policy using DSGE, DSGE-VAR, and VAR based on a two-country open economy model of Korea and the U.S. According to impulse response analysis, a domestic interest rate hike raises won value in the case of DSGE and DSGE-VAR models, while in the case of the unrestricted VAR model, it lowers won value. In the marginal data density standard, DSGE-VAR (μ=1) is superior to DSGE or Bayesian VAR over the sample period. Conversely, in the in-sample RMSE criterion, especially for the won/dollar exchange rate, VARs are superior to DSGE or DSGE-VAR. It is necessary to study further if these differences are caused by model misspecification or omitted variable bias.

A Comparison Analysis of Monetary Policy Effect Under an Open Economy Model

The paper analyzes and compares the effects of domestic monetary policy using DSGE, DSGE-VAR, and VAR based on a two-country open economy model of Korea and the U.S. According to impulse response analysis, a domestic interest rate hike raises won value in the case of DSGE and DSGE-VAR models, while in the case of the unrestricted VAR model, it lowers won value. In the marginal data density standard, DSGE-VAR (μ=1) is superior to DSGE or Bayesian VAR over the sample period. Conversely, in the in-sample RMSE criterion, especially for the won/dollar exchange rate, VARs are superior to DSGE or DSGE-VAR. It is necessary to study further if these differences are caused by model misspecification or omitted variable bias.

The Accuracy of Linear and Nonlinear Estimation in the Presence of the Zero Lower Bound

This paper evaluates the accuracy of linear and nonlinear estimation methods for dynamic stochastic general equilibrium models. We generate a large sample of artificial datasets using a global solution to a nonlinear New Keynesian model with an occasionally binding zero lower bound (ZLB) constraint on the nominal interest rate. For each dataset, we estimate the nonlinear model—solved globally, accounting for the ZLB—and the linear analogue of the nonlinear model—solved locally, ignoring the ZLB—with a Metropolis-Hastings algorithm where the likelihood function is evaluated with a Kalman filter, unscented Kalman filter, or particle filter. In datasets that resemble the U.S. experience, the nonlinear model estimated with a particle filter is more accurate and has a higher marginal data density than the linear model estimated with a Kalman filter, as long as the measurement error variances in the particle filter are not too big.

Estimating the Marginal Likelihood Using the Arithmetic Mean Identity

In this paper we propose a conceptually straightforward method to estimate the marginal data density value (also called the marginal likelihood). We show that the marginal likelihood is equal to the prior mean of the conditional density of the data given the vector of parameters restricted to a certain subset of the parameter space, A, times the reciprocal of the posterior probability of the subset A. This identity motivates one to use Arithmetic Mean estimator based on simulation from the prior distribution restricted to any (but reasonable) subset of the space of parameters. By trimming this space, regions of relatively low likelihood are removed, and thereby the efficiency of the Arithmetic Mean estimator is improved. We show that the adjusted Arithmetic Mean estimator is unbiased and consistent.

Is Rotemberg pricing justified by macro data?

Structural models used to study monetary policy often include sticky prices. Calvo pricing is more common but Rotemberg pricing has become popular due to its computational advantage. To determine whether the data supports that change, we estimate a nonlinear New Keynesian model with a zero lower bound (ZLB) constraint and each type of sticky prices. The models produce similar parameter estimates and the filtered shocks are nearly identical when the Fed was not constrained, but the Rotemberg model has a higher marginal data density and it endogenously generates more volatility at the ZLB, which helps explain data from 2008–2011.

Plastic to fuel and wax

Structural models used to study monetary policy often include sticky prices. Calvo pricing is more common but Rotemberg pricing has become popular due to its computational advantage. To determine whether the data supports that change, we estimate a nonlinear New Keynesian model with a zero lower bound (ZLB) constraint and each type of sticky prices. The models produce similar parameter estimates and the filtered shocks are nearly identical when the Fed was not constrained, but the Rotemberg model has a higher marginal data density and it endogenously generates more volatility at the ZLB, which helps explain data from 2008–2011.

Real-Time Forecasting With a Mixed-Frequency VAR

This article develops a vector autoregression (VAR) for time series which are observed at mixed frequencies—quarterly and monthly. The model is cast in state-space form and estimated with Bayesian methods under a Minnesota-style prior. We show how to evaluate the marginal data density to implement a data-driven hyperparameter selection. Using a real-time dataset, we evaluate forecasts from the mixed-frequency VAR and compare them to standard quarterly frequency VAR and to forecasts from MIDAS regressions. We document the extent to which information that becomes available within the quarter improves the forecasts in real time. This article has online supplementary materials.

SEQUENTIAL MONTE CARLO SAMPLING FOR DSGE MODELS

SUMMARYWe develop a sequential Monte Carlo (SMC) algorithm for estimating Bayesian dynamic stochastic general equilibrium (DSGE) models; wherein a particle approximation to the posterior is built iteratively through tempering the likelihood. Using two empirical illustrations consisting of the Smets and Wouters model and a larger news shock model we show that the SMC algorithm is better suited for multimodal and irregular posterior distributions than the widely used random walk Metropolis–Hastings algorithm. We find that a more diffuse prior for the Smets and Wouters model improves its marginal data density and that a slight modification of the prior for the news shock model leads to drastic changes in the posterior inference about the importance of news shocks for fluctuations in hours worked. Unlike standard Markov chain Monte Carlo (MCMC) techniques; the SMC algorithm is well suited for parallel computing. Copyright © 2014 John Wiley & Sons, Ltd.

Sequential Monte Carlo Sampling for DSGE Models

SUMMARY We develop a sequential Monte Carlo (SMC) algorithm for estimating Bayesian dynamic stochastic general equilibrium (DSGE) models; wherein a particle approximation to the posterior is built iteratively through tempering the likelihood. Using two empirical illustrations consisting of the Smets and Wouters model and a larger news shock model we show that the SMC algorithm is better suited for multimodal and irregular posterior distributions than the widely used random walk Metropolis–Hastings algorithm. We find that a more diffuse prior for the Smets and Wouters model improves its marginal data density and that a slight modification of the prior for the news shock model leads to drastic changes in the posterior inference about the importance of news shocks for fluctuations in hours worked. Unlike standard Markov chain Monte Carlo (MCMC) techniques; the SMC algorithm is well suited for parallel computing. Copyright © 2014 John Wiley & Sons, Ltd.

Ocena stopnia zgodności wybranego modelu równowagi ogólnej z danymi empirycznymi

The paper presents the derivation of structural equations of an estimated general equilibrium model, illustrates techniques of its connection with the data observed and discusses a method of assessing the degree of acceptance by the data. The results suggest that the model is not confirmed by the data. Logarithms of the marginal data density tend to systematically decrease when the weight of the structural model is being increased in hybrid vector autoregression. This defines an econometric tool which can capture a general equilibrium model on the one hand and almost unrestricted vector autoregression on the other. As a consequence of high structural model misspecification, economic conclusions it leads one to draw may be highly inaccurate.

Estymowane modele równowagi ogólnej i autoregresja wektorowa. Aspekty praktyczne

The DSGE-VAR model consists of two models of vector autoregressions: the first one approximates the linearised solution of the dynamic stochastic general equilibrium model and is used as a tool for construction of a prior distribution for the second one, estimated with the observed data. The main purpose of the paper is to present practical aspects of DSGE-VAR estimation, verification and comparison, based on the marginal data density. It can be obtained after considering conditional models or by estimation of fully specified models, after assuming uniform, generalised gamma and modified beta distributions. The conditional models lead to serious variability of the Bayes factors that has little economic interpretation. Posterior inference for the weighting parameter from fully estimated models is less sensitive to its prior specification. In the second part of the paper author discusses convergence diagnostics used for checking stability of MCMC algorithms.

Estymowane modele równowagi ogólnej i autoregresja wektorowa. Aspekty teoretyczne

The DSGE-VAR model consists of two models of vector autoregressions: the first one approximates linearised solution of the dynamic stochastic general equilibrium model and is used as a tool for construction of a prior distribution for the second one, estimated with the observed data. Combined inference is possible on the basis on probability distributions with the Bayesian techniques. The key role in the hybrid model is played by the weighting parameter that defines the relative proportions of the structural and autoregressive models. It has crucial impact for the marginal data density that allows to compare the power of different models. The main purpose of the paper is to present in details model assumptions and estimation.

Methods for computing marginal data densities from the Gibbs output

We introduce two estimators for estimating the Marginal Data Density (MDD) from the Gibbs output. Our methods are based on exploiting the analytical tractability condition, which requires that some parameter blocks can be analytically integrated out from the conditional posterior densities. This condition is satisfied by several widely used time series models. An empirical application to six-variate VAR models shows that the bias of a fully computational estimator is sufficiently large to distort the implied model rankings. One of the estimators is fast enough to make multiple computations of MDDs in densely parameterized models feasible.