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.     

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.     

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.     

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.     

Solving the multi-country real business cycle model using ergodic set methods

We use the stochastic simulation algorithm, described in Judd et al. (2009), and the cluster-grid algorithm, developed in Judd et al. (2010a), to solve a collection of multi-country real business cycle models. The following ingredients help us reduce the cost in high-dimensional problems: an endogenous grid enclosing the ergodic set, linear approximation methods, fixed-point iteration and efficient integration methods, such as non-product monomial rules and Monte Carlo integration combined with regression. We show that high accuracy in intratemporal choice is crucial for the overall accuracy of solutions and offer two approaches, precomputation and iteration-on-allocation, that can solve for intratemporal choice both accurately and quickly. We also implement a hybrid solution algorithm that combines the perturbation and accurate intratemporal-choice methods.     

Behavioural heterogeneity and shift-contagion: Evidence from the Asian crisis

In this paper, we propose an empirical model based on the heterogeneous agents literature. Price changes are induced by fundamental, technical, and international factors. The model is estimated for Hong Kong and Thailand surrounding the Asian crisis. We find that the three sources are relevant and that their relative price impact fluctuates conditional on price impact in the previous period. Results imply that the crisis is triggered in Thailand due to an increased focus on the fundamental price, followed by an increase in chartism and finally aggravated by a focus on foreign developments. Furthermore, the crisis deepens in Hong Kong because of increased attention for foreign markets.     

The market for crash risk

This paper examines the equilibrium when stock market crashes can occur and investors have heterogeneous attitudes towards crash risk. The less crash averse insure the more crash averse through options markets that dynamically complete the economy. The resulting equilibrium is compared with various option pricing anomalies: the tendency of stock index options to overpredict volatility and jump risk, the Jackwerth [Recovering risk aversion from option prices and realized returns. Review of Financial Studies 13, 433–451] implicit pricing kernel puzzle, and the stochastic evolution of option prices. Crash aversion is compatible with some static option pricing puzzles, while heterogeneity partially explains dynamic puzzles. Heterogeneity also magnifies substantially the stock market impact of adverse news about fundamentals.     

Bank failures caused by Large withdrawals: An explanation based purely on liquidity

In a version of the Diamond and Dybvig [Diamond, D., Dybvig, P., 1983. Bank runs, deposit insurance, and liquidity. Journal of Political Economy 91, 401–419.] model with aggregate uncertainty, we show that there exists an equilibrium with the following properties: all consumers deposit at the bank, all patient consumers wait for the last period to withdraw, and the bank fails with strictly positive probability. Furthermore, we show that the probability of a bank failure remains bounded away from zero as the number of consumers increases.This equilibrium explains bank failures driven by extreme withdrawals solely on liquidity since they happen because both banks and depositors are illiquid. Furthermore, it does not require much of the elements typically emphasized, including: consumers well informed about the true state of nature, a non-zero consumption after a crisis, consumers’ panic and sunspots. We therefore think that aggregate risk in Diamond-Dybvig-like environments can be an important element to explain bank crises.     

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.     

MONETARY POLICY IN A NON‐REPRESENTATIVE AGENT ECONOMY: A SURVEY

It is well known that central bank policies affect not only macroeconomic aggregates, but also their distribution across economic agents. Similarly, a number of papers demonstrated that heterogeneity of agents may matter for the transmission of monetary policy to macro variables. Despite this, the mainstream monetary economics literature has so far been dominated by dynamic stochastic general equilibrium models with representative agents. This paper aims to tilt this imbalance towards heterogeneous agents setups by surveying the main positive and normative findings of this line of the literature, and suggesting areas in which these models could be implemented. In particular, we review studies that analyse the heterogeneity of (i) households’ income, (ii) households’ preferences, (iii) consumers’ age, (iv) expectations and (v) firms’ productivity and financial position. We highlight the results on issues that, by construction, cannot be investigated in a representative agent framework and discuss important papers modifying the findings from the representative agent literature.     

Production technologies in stochastic continuous time models

Properties of dynamic stochastic general equilibrium models can be revealed by either using numerical solutions or qualitative analysis. Very precise and intuition-building results are obtained by working with models which provide closed-form solutions. Closed-form solutions are known for a large class of models some of which, however, have some undesirable features such as potentially negative output. This paper offers closed-form solutions for models which are just as tractable but do not suffer from these shortcomings.     

Strategic interaction and aggregate incentives

We consider a model of interdependent efforts, with linear interaction and lower bound on effort. Our setting encompasses asymmetric interaction and heterogeneous agents’ characteristics. We examine the impact of a rise of cross-effects on aggregate efforts. We show that the sign of the comparative static effects is related to a condition of balancedness of the interaction. Moreover, we point out that asymmetry and heterogeneous characteristics are sources of non-monotonic variation of aggregate efforts.