Computational suite of models with heterogeneous agents II: Multi-country real business cycle models

This paper describes the second model considered in the computational suite project that compares the performance of different numerical algorithms. It is a multi-country model in which countries face different productivity shocks. Solving such models is a challenging numerical problem unless the number of countries is small. The solutions are functions of a large set of arguments and the functional forms are unknown. Moreover, the solution procedures have to deal with high-dimensional integration problems.     

Assessing the accuracy of the aggregate law of motion in models with heterogeneous agents

This paper shows that the R² and the standard error have fatal flaws and are inadequate accuracy tests. Using data from a Krusell–Smith economy, I show that approximations for the law of motion of aggregate capital, for which the true standard deviation of aggregate capital is up to 14% (119%) higher than the implied value and which are thus clearly inaccurate, can have an R² as high as 0.9999 (0.99). Key in generating a more powerful test is that predictions of the aggregate law of motion are not updated with the aggregated simulated individual data.     

Comparison of solutions to the incomplete markets model with aggregate uncertainty

This paper compares numerical solutions to the model of Krusell and Smith [1998. Income and wealth heterogeneity in the macroeconomy. Journal of Political Economy 106, 867–896] generated by different algorithms. The algorithms have very similar implications for the correlations between different variables. Larger differences are observed for (i) the unconditional means and standard deviations of individual variables, (ii) the behavior of individual agents during particularly bad times, (iii) the volatility of the per capita capital stock, and (iv) the behavior of the higher-order moments of the cross-sectional distribution. For example, the two algorithms that differ the most from each other generate individual consumption series that have an average (maximum) difference of 1.63% (11.4%).     

Solving the incomplete markets model with aggregate uncertainty using parameterized cross-sectional distributions

This note describes how the incomplete markets model with aggregate uncertainty in Den Haan et al. [Comparison of solutions to the incomplete markets model with aggregate uncertainty. Journal of Economic Dynamics and Control, this issue] is solved using standard quadrature and projection methods. This is made possible by linking the aggregate state variables to a parameterized density that describes the cross-sectional distribution. A simulation procedure is used to find the best shape of the density within the class of approximating densities considered. This note compares several simulation procedures in which there is—as in the model—no cross-sectional sampling variation.     

Solving the incomplete markets model with aggregate uncertainty using the Krusell–Smith algorithm

This paper studies the properties of the solution to the heterogeneous agents model in Den Haan et al. [2009. Computational suite of models with heterogeneous agents: incomplete markets and aggregate uncertainty. Journal of Economic Dynamics and Control, this issue]. To solve for the individual policy rules, we use an Euler-equation method iterating on a grid of pre-specified points. To compute the aggregate law of motion, we use the stochastic-simulation approach of Krusell and Smith [1998. Income and wealth heterogeneity in the macroeconomy. Journal of Political Economy 106, 868–896]. We also compare the stochastic- and non-stochastic-simulation versions of the Krusell–Smith algorithm, and we find that the two versions are similar in terms of their speed and accuracy.     

Solving the incomplete markets model with aggregate uncertainty using explicit aggregation

We propose a method to solve models with heterogeneous agents and aggregate uncertainty. The law of motion describing aggregate behavior is obtained by explicitly aggregating the individual policy rule. The algorithm is simpler and faster than existing algorithms that rely on parameterization of the cross-sectional distribution and/or a computationally intensive simulation step. Explicit aggregation establishes a link between the individual policy rule and the set of necessary aggregate state variables, an insight that can be helpful in determining what state variables to include in other algorithms as well.     

Comparison of solutions to the multi-country Real Business Cycle model

We compare the performance of perturbation, projection, and stochastic simulation algorithms for solving the multi-country RBC model described in Den Haan et al. (this issue). The main challenge of solving this model comes from its large number of continuous-valued state variables, ranging between four and 20 in the specifications we consider. The algorithms differ substantially in terms of speed and accuracy, and a clear trade-off exists between the two. Perturbation methods are very fast but invoke large approximation errors except at points close to the steady state; the projection methods considered are accurate on a large area of the state space but are very slow for specifications with many state variables; stochastic simulation methods have lower accuracy than projection methods, but their computational cost increases only moderately with the state-space dimension. Simulated series generated by different methods can differ noticeably, but only small differences are found in unconditional moments of simulated variables. On the basis of our comparison, we identify the factors that account for differences in accuracy and speed across methods, and we suggest directions for further improvement of some approaches.     

Approximate dynamic programming with post-decision states as a solution method for dynamic economic models

I introduce and evaluate a new stochastic simulation method for dynamic economic models. It is based on recent work in the operations research and engineering literatures (Van Roy et al., 1997, Powell, 2007, Bertsekas, 2011), but also had an early application in economics (Wright and Williams, 1982, Wright and Williams, 1984). The baseline method involves rewriting the household׳s dynamic program in terms of post-decision states. This makes it possible to choose controls optimally without computing an expectation. I add a subroutine to the original algorithm that updates the values of states not visited frequently on the simulation path; and adopt a stochastic stepsize that efficiently weights information. Finally, I modify the algorithm to exploit GPU computing.     

Solving the incomplete markets model with aggregate uncertainty using the Krusell–Smith algorithm and non-stochastic simulations

This article describes the approach to computing the version of the stochastic growth model with idiosyncratic and aggregate risk that relies on collapsing the aggregate state space down to a small number of moments used to forecast future prices. One innovation relative to most of the literature is the use of a non-stochastic simulation routine.     

Solving the incomplete markets model with aggregate uncertainty by backward induction

This paper describes a method to solve models with a continuum of agents, incomplete markets and aggregate uncertainty. I use backward induction on a finite grid of points in the aggregate state space. The aggregate state includes a small number of statistics (moments) of the cross-sectional distribution of capital. For any given set of moments, agents use a specific cross-sectional distribution, called “proxy distribution”, to compute the equilibrium. Information from the steady state distribution as well as from simulations can be used to chose a suitable proxy distribution.     

Optimal monetary policy under incomplete markets and aggregate uncertainty: A long-run perspective

This paper examines the role of monetary policy in an environment with aggregate risk and incomplete markets. In a two-period overlapping-generations model with aggregate uncertainty, optimal monetary policy attains the ex-ante Pareto optimal allocation. This policy aims to stabilize the savings rate in the economy by changing real returns of nominal bonds via variation in expected inflation. Optimal expected inflation is procylical and on average higher than without uncertainty. Simple inflation targeting rules closely approximate the optimal monetary policy.     

Multi-country real business cycle models: Accuracy tests and test bench

This paper describes the methodology used to compare the results of different solution algorithms for a multi-country real business cycle model. It covers in detail the structure of the model, the choice of values for the parameters, the accuracy tests used in the comparison, and the computer program specifically developed for performing the tests.     

The intensity of incentives in firms and markets: Moral hazard with envious agents

While most market transactions are subject to strong incentives, transactions within firms are often not explicitly incentivized. This paper offers an explanation for this observation based on the assumption that agents are envious and suffer utility losses if others receive higher wages. We analyze the impact of envy on optimal incentive contracts in a general moral hazard model and isolate the countervailing effects of envy on the costs of providing incentives. We show that envy creates a tendency towards flat-wage contracts if agents are risk-averse and there is no limited liability. Empirical evidence suggests that social comparisons are more pronounced among employees within firms than among individuals that interact in markets. Flat-wage contracts are then more likely to be optimal in firms.     

Estimating class‐specific parametric models under class uncertainty: local polynomial regression clustering in an hedonic analysis of wine markets

We introduce a method for estimating multiple class regression models when class membership is uncertain. The procedure—local polynomial regression clustering—first estimates a nonparametric model via local polynomial regression, and then identifies the underlying classes by aggregating sample observations into data clusters with similar estimates of the (local) functional relationships between dependent and independent variables. Finally, parametric functions specific to each class are estimated. The technique is applied to the estimation of a multiple‐class hedonic model for wine, resulting in the identification of four distinct wine classes based on differences in implicit prices of the attributes. Copyright © 2009 John Wiley & Sons, Ltd.     

Rational expectations and the econometric modeling of markets subject to uncertainty: A Bayesian approach

Rational expectations theory is synthesized with Bayesian econometric theory to yield econometrically relevant models of competitive markets subject to uncertainty. The theory is used to derive both optimal estimators of the parameters of a Cobb-Douglas production function from time series data, and the equilibrium predictor of a future price. It is shown that a rational expectations price predictor is always an unbiased predictor, but that the converse is not true. It is also shown that the rational expectations equilibrium is a natural extension of the usual notion of a competitive equilibrium.     

Estimation and inference for spatial models with heterogeneous coefficients: An application to US house prices

This paper considers the estimation and inference of spatial panel data models with heterogeneous spatial lag coefficients, with and without weakly exogenous regressors, and subject to heteroskedastic errors. A quasi maximum likelihood (QML) estimation procedure is developed and the conditions for identification of the spatial coefficients are derived. The QML estimators of individual spatial coefficients, as well as their mean group estimators, are shown to be consistent and asymptotically normal. Small‐sample properties of the proposed estimators are investigated by Monte Carlo simulations and results are shown to be in line with the paper's key theoretical findings, even for panels with moderate time dimensions and irrespective of the number of cross‐section units. A detailed empirical application to US house price changes during the 1975–2014 period shows a significant degree of heterogeneity in spatiotemporal dynamics over the 338 Metropolitan Statistical Areas considered.     

Simple agent-based dynamical system models for efficient financial markets: Theory and examples

We propose an agent-based framework, based on simple piecewise linear time-invariant continuous-time dynamical systems models, as a means for describing efficient financial markets. We show by examples that many of the common agent-specific trading strategies occurring in the academic literature, including chartists and fundamentalists of various kinds, can be described in the proposed framework. We present definitions for weak and strong market efficiency and provide necessary and sufficient conditions for them to hold. We present minimal examples of strongly and weakly efficient markets to show that these concepts are natural and easy to satisfy in agent-based models, and that the models can reproduce both statistical and behavioral stylized facts of real markets. We provide examples to demonstrate that the framework can be extended for agents with delays in information processing, as well as for agents with time-varying strategies and for nonlinear market impact functions. We also provide a counterexample to show that the proposed market efficiency concepts may require modification in generalizations for nonlinear trading strategies.     

The transmission of US economic policy uncertainty shocks to Asian and global financial markets

This paper proposes a novel approach to investigating the spillover effects of US economic policy uncertainty shocks on the global financial markets. Employing a factor-augmented vector autoregression (FAVAR), we model US economic policy uncertainty jointly with the latent factors extracted from equity prices, exchange rates, and commodity prices. We find that US economic policy uncertainty affects these factors significantly. A country-level analysis shows heterogeneous responses to an increase in US economic policy uncertainty. With regard to equities, US economic policy uncertainty adversely affects equity prices. However, its impact on the Chinese equity market is relatively small. As for foreign exchange markets, while many currencies depreciate in response to an increase in US economic policy uncertainty, the US dollar and the Japanese yen appreciate, reflecting their safe-haven status. The Chinese yuan, whose nominal exchange rate is closely linked to the US dollar, also appreciates in response to uncertainty shocks.     

The relation between sunspot effects and multiplicity in incomplete markets models with numeraire assets

This paper considers the necessity and sufficiency of multiple certainty equilibria for sunspot effects, and shows that neither implication is valid. This claim is made for models with incomplete markets and numeraire assets. First, I prove that a multiplicity of certainty equilibria is neither necessary nor sufficient for sunspot effects by way of two counter-examples. Second, I verify over an entire subset of economies that equilibrium with sunspot effects can never be characterized as a randomization over multiple certainty equilibria.