Estimation of ergodic agent-based models by simulated minimum distance

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Abstract

Two difficulties arise in the estimation of AB models: (i) the criterion function has no simple analytical expression, (ii) the aggregate properties of the model cannot be analytically understood. In this paper we show how to circumvent these difficulties and under which conditions ergodic models can be consistently estimated by simulated minimum distance techniques, both in a long-run equilibrium and during an adjustment phase.

Introduction

Agent-based (AB) models are sometimes considered as a candidate to replace or at least complement dynamic stochastic general equilibrium (DSGE) as the standard tool for macroeconomic analysis. However, a common critique addressed towards AB models is that they often remain at a theoretical level, and lack a sound empirical grounding (Gallegati and Richiardi, 2009). When present, this is often limited to some ad-hoc calibration of the relevant parameters; this resembles the state of the art in DSGE modeling a few years ago, which has now moved forward toward more formal estimation.1 Estimation is crucial for the empirical validation of a model, for comparing a model with other available models, and for policy analysis. Also, estimation (as opposed to calibration) involves the attainment of clearly specified scientific standards in the way models are confronted with the data, and this sort of “certification” is highly needed to gain confidence and ultimately support from the wider scientific community and the policy circles in the use of a new tool.

The advantage of AB modeling over more conventional techniques ultimately lies in the increased flexibility in model specification, but this flexibility comes at the price of making estimation more difficult. Paradoxically, AB models can provide a more realistic description of how real economies actually work, but this description resists fine tuning to real data. However, fine-tuning is important as small differences in the values of the parameters might be amplified by the nonlinear feedbacks in the system and result in large differences in the effects of the policies under examination. Moreover, AB models are often conceived to investigate what happens “out-of-equilibrium”, during adjustment phases: systemic disruptions, regime shifts, rare events are usual objects of analysis. How to characterize the behavior of a model in such circumstances, beyond a merely qualitative description, is one open issue that we directly address.2 We then turn to some basic questions concerning estimation of AB models: What makes an AB model econometrically tractable? What are the necessary restrictions or assumptions that need to be imposed before an AB model can be estimated? How do they differ from those required for DSGE models?

The answers to these questions depend in part on the structure of the models, and in part on the specific econometric technique employed. We spell out the general conditions under which estimation of an AB model makes sense, and then focus on the application of one specific technique, namely simulated minimum distance (SMD). As this approach to estimation is well known in the literature (see for instance Gouriéroux and Monfort, 1996), our work could be described as a mere transposition of concepts and language to a new area of application. However, the specificities of AB modeling make the exercise far from trivial. As LeBaron and Tesfatsion (2008, p. 249) put it, “the very properties that make ACE [Agent-based Computational Economics] models so interesting to study can cause empirical headaches when estimating them”. Because AB models are composed of many heterogeneous agents interacting together and with the environment, aggregation is not a simple matter and it must be performed numerically, by counting and summing up individual quantities. Therefore, the aggregate properties of an AB model remain hidden in the complexity of the relations among the different elements and the different layers (micro, macro and possibly meso) of the system. The need for a computational solution of the aggregate dynamics of AB models is the same as in DSGE models, but in the latter some aggregate properties can be analytically understood or derived from first principles and primitive assumptions – in particular, whether the system is ergodic and stationary.

We see three merits in our work. First, it provides a formalization of AB models and their behavior that, in our view, is missing and strongly needed in the field. Second, it provides a reference and a guide for empirical work in the field of AB modeling.3 Third, it points out the analogies and differences between AB and DSGE models, not only in terms of methodological assumptions and solution concepts, but also in terms of their econometric consequences.

The rest of the paper is organized as follows. We first put empirically based AB models in retrospective, to highlight their increasing use for quantitative analysis of macro issues (Section 2). We then turn to the DSGE literature, where the problem of estimating many parameters in models that need to be evaluated numerically has already been confronted with, and identify SMD as possibly the most promising method for estimating AB models (Section 3). In Section 4 we present a formal characterization of AB models as recursive systems, and discuss the properties of their implied dynamics. The specific issues concerning estimation of AB models are discussed in Section 5. In Section 6 we give two examples of estimation of ergodic AB models: in the first, estimation is performed in a long-run equilibrium, while in the second estimation is performed on an adjustment phase. Section 7 concludes.

Section snippets

Little AB models grow big

AB models have long been considered as theoretical exercises aimed at investigating the (unexpected) macro-effects arising from the interaction of many individuals – each following possibly simple rules of behavior – or the (unknown) individual routines/strategies underlying some observed macro phenomenon (Richiardi, 2012). As such, the typical AB model is a relatively small “toy” model, which can be used to understand relevant mechanisms of social interaction. Little data is involved with such

Estimation of DSGE models

In DSGE models aggregation is generally not a problem, thanks to a very low level of heterogeneity.8

AB models as recursive systems

AB models are recursive systems (Leombruni and Richiardi, 2005, Epstein, 2006). This is an essential feature as “[t]he elimination of simultaneous equations allows us to get results from a simulation model without having to go through a process of solution” (Bergmann, 1990).20

At each time t an agent i, i1n, is fully described by some state variables xi,tRk. Let

Estimation of AB models

In the following we assume that the model is correctly specified: this is a fundamental hypothesis, which implies that the model perfectly describes the real world, that the structural parameters exist and have a finite value and that all the parameters in the real system are represented in the model.27 Under correct

Examples

In this section we present two examples of estimation of ergodic AB models by SMD: the first model is estimated in an absorbing equilibrium, while the second is estimated in a transient equilibrium. In both cases, because the moments used for estimation are non-linear, we get a small-sample bias (of predictable direction). All estimation strategies are explored by means of Monte Carlo experiments: pseudo-true data are created by running the model with some chosen value of the parameters; then,

Conclusions

In this paper we have identified simulated minimum distance as a natural approach to estimation of AB models. In this approach, the theoretical quantities or statistics used for characterizing the model conditional on the values of the parameters (the moments, for instance), for which no analytical expression is available in AB models, are replaced by their simulated counterparts. This requires that these statistics are appropriately chosen so that their estimates in the simulated data converge

Acknowledgements

Jakob Grazzini received funding from the EU seventh framework collaborative project “Complexity Research Initiative for Systemic InstabilitieS (CRISIS)”, Grant no. 288501, while Matteo Richiardi acknowledges financial support from Collegio Carlo Alberto and Regione Piemonte within the research projects “Causes, Processes and Consequences of Flexsecurity Reform in the EU: Lesson from Bismarckian Countries” (Collegio Carlo Alberto) and “From Work to Health and Back: The Right to a Healthy Working

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