Serguei Maliar

Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models

We develop numerically stable and accurate stochastic simulation approaches for solving dynamic economic models. First, instead of standard least-squares methods, we examine a variety of alternatives, including least-squares methods using singular value decomposition and Tikhonov regularization, least-absolute deviations methods, and principal component regression method, all of which are numerically stable and can handle ill-conditioned problems. Second, instead of conventional Monte Carlo integration, we use accurate quadrature and monomial integration. We test our generalized stochastic simulation algorithm (GSSA) in three applications: the standard representative agent neoclassical growth model, a model with rare disasters and a multi-country models with hundreds of state variables. GSSA is simple to program, and MATLAB codes are provided.

The representative consumer in the neoclassical growth model with idiosyncratic shocks

This paper studies a complete-market version of the neoclassical growth model, where agents face idiosyncratic shocks to earnings. We show that if agents possess identical preferences of either the CRRA or the addilog type, then the heterogeneous-agent economy behaves as if there was a representative consumer who faces three kinds of shocks, to preferences, to technology and to labor. We calibrate and simulate the constructed representative-consumer models. We find that idiosyncratic uncertainty can have a non-negligible effect on aggregate labor-market fluctuations.

HETEROGENEITY IN CAPITAL AND SKILLS IN A NEOCLASSICAL STOCHASTIC GROWTH MODEL

Does a heterogeneous agents version of a neoclassical model with labor-leisure choice replicatethe distributions of consumption and working hours observed in the cross-sectional data? Doesincorporating heterogeneity enhance the aggregate performance of the representative agentmodel? We address these questions in a complete market model economy with two sources ofheterogeneity: initial endowments and non-acquired skills. We find positive answers to bothquestions.

Parameterized Expectations Algorithm and the Moving Bounds

The Parameterized Expectations Algorithm (PEA) is a powerful tool for solving nonlinear stochastic dynamic models. However, it has an important shortcoming: it is not a contraction mapping technique and thus does not guarantee a solution will be found. We suggest a simple modification that enhances the convergence property of the algorithm. The idea is to rule out the possibility of (ex)implosive behavior by artificially restricting the simulated series within certain bounds. As the solution is refined along the iterations, the bounds are gradually removed. The modified PEA can systematically converge to the stationary solution starting from the nonstochastic steady state.

Merging simulation and projection approaches to solve high-dimensional problems with an application to a new Keynesian model

We introduce a numerical algorithm for solving dynamic economic models that merges stochastic simulation and projection approaches: we use simulation to approximate the ergodic measure of the solution, we cover the support of the constructed ergodic measure with a fixed grid, and we use projection techniques to accurately solve the model on that grid. The construction of the grid is the key novel piece of our analysis: we replace a large cloud of simulated points with a small set of “representative” points. We present three alternative techniques for constructing representative points: a clustering method, an e-distinguishable set method, and a locally-adaptive variant of the e-distinguishable set method. As an illustration, we solve one- and multi-agent neoclassical growth models and a large-scale new Keynesian model with a zero lower bound on nominal interest rates. The proposed solution algorithm is tractable in problems with high dimensionality (hundreds of state variables) on a desktop computer.

Numerical Methods for Large-Scale Dynamic Economic Models

Abstract We survey numerical methods that are tractable in dynamic economic models with a finite, large number of continuous state variables. (Examples of such models are new Keynesian models, life-cycle models, heterogeneous-agents models, asset-pricing models, multisector models, multicountry models, and climate change models.) First, we describe the ingredients that help us to reduce the cost of global solution methods. These are efficient nonproduct techniques for interpolating and approximating functions (Smolyak, stochastic simulation, and e -distinguishable set grids), accurate low-cost monomial integration formulas, derivative-free solvers, and numerically stable regression methods. Second, we discuss endogenous grid and envelope condition methods that reduce the cost and increase accuracy of value function iteration. Third, we show precomputation techniques that construct solution manifolds for some models’ variables outside the main iterative cycle. Fourth, we review techniques that increase the accuracy of perturbation methods: a change of variables and a hybrid of local and global solutions. Finally, we show examples of parallel computation using multiple CPUs and GPUs including applications on a supercomputer. We illustrate the performance of the surveyed methods using a multiagent model. Many codes are publicly available.

The Neoclassical Growth Model with Heterogeneous Quasi-Geometric Consumers

This paper investigates how the assumption of quasi-geometric (hyperbolic) discounting affects the distributional implications of the standard one-sector neoclassical growth model with infinitely lived heterogeneous agents. The agents are subject to idiosyncratic shocks and face borrowing constraints. We confine attention to an interior Markov recursive equilibrium. The consequence of quasi-geometric discounting is that the effective discount factor of an agent is not a constant, but an endogenous variable which depends on the agents current state. We show, both analytically and by simulation, that this new feature can significantly affect the distributional implications of the neoclassical growth model.

INCOME AND WEALTH DISTRIBUTIONS ALONG THE BUSINESS CYCLE: IMPLICATIONS FROM THE NEOCLASSICAL GROWTH MODEL

This paper studies the business cycle dynamics of income and wealth distributions in the context of the neoclassical growth model where agents are heterogeneous in initial wealth and non-acquired skills. Our economy admits a representative consumer which enables us to characterize distributive dynamics by the evolution of aggregate quantities. We show that inequality in both wealth and income follow a countercyclical pattern: the former is countercyclical because labor income is more sensitive to the business cycle than capital income, while the latter is countercyclical due to the wealth-distribution effect. We find that the predictions of the model about the income distribution dynamics accord well with the U.S. data.

Capital-Skill Complementarity and Balanced Growth

We construct a general equilibrium version of the Krusell et al. Econometrica 68, 1029, 2000 model with capital-skill complementarity. We assume several sources of growth simultaneously: exogenous growth of skilled and unskilled labour, equipment-specific technological progress, skilled and unskilled labour-augmenting technological progress and Hicks-neutral technological progress. We derive restrictions that make our model consistent with balanced growth. A calibrated version of our model can account for the key growth patterns in the US data, including those for capital equipment and structures, skilled and unskilled labour and output, but it fails to explain the long-run behaviour of skilled-labour wages and, consequently, the skill premium.

A Tractable Framework for Analyzing a Class of Nonstationary Markov Models

We study a class of infinite-horizon nonlinear dynamic economic models in which preferences, technology and laws of motion for exogenous variables can change over time either deterministically or stochastically, according to a Markov process with time-varying transition probabilities, or both. The studied models are nonstationary in the sense that the decision and value functions are time-dependent, and they cannot be generally solved by conventional solution methods. We introduce a quantitative framework, called extended function path (EFP), for calibrating, solving, simulating and estimating such models. We apply EFP to analyze a collection of challenging applications that do not admit stationary Markov equilibria, including growth models with anticipated parameters shifts and drifts, unbalanced growth under capital augmenting technological progress, anticipated regime switches, deterministically time-varying volatility and seasonal fluctuations. Also, we show an example of estimation and calibration of parameters in an unbalanced growth model using data on the U.S. economy. Examples of MATLAB code are provided.