Bringing an elementary agent-based model to the data: Estimation via GMM and an application to forecasting of asset price volatility*

https://doi.org/10.1016/j.jempfin.2016.02.002Get rights and content

  • We explore the issue of estimating a simple agent-based model of asset price formation.

  • We derive analytic moment conditions and apply GMM estimation technique.

  • Efficient estimates can only be obtained with particular estimation designs.

  • We use the behavioral model to compute forecasts for future volatility.

  • The model's forecasts often are not encompassed by those of GARCH(1,1).

Abstract

We explore the issue of estimating a simple agent-based model of price formation in an asset market using the approach of Alfarano et al. (2008) as an example. Since we are able to derive various moment conditions for this model, we can apply generalized method of moments (GMM) estimation. We find that we can get relatively accurate parameter estimates with an appropriate design of the GMM estimator that reduces the biases arising from strong correlations of the estimates of certain parameters. We apply our estimator to a sample of long records of returns of various stock and foreign exchange markets as well as the price of gold. Using the estimated parameters to form the best linear forecasts for future volatility we find that the behavioral model generates sensible forecasts that get close to those of a standard GARCH(1,1) model in their overall performance, and often provide useful information on top of the information incorporated in the GARCH forecasts.

Introduction

Asset markets have been known for a long time to be characterized by a set of ubiquitous ‘stylized facts' that are hard to explain by any traditional approach to asset pricing. The best known of these are the fat tails of the unconditional distribution of returns and the volatility clustering characterizing their conditional distribution. Known since the 60s, but largely mysterious in terms of their behavioral origins, these salient features have also occasionally been classified as ‘anomalies'. However, the latter term appears odd in view of the fact that these are really the constants in the statistical analysis of time series of financial markets across time, countries and asset types, i.e. the imprint of their apparently ‘normal’ mode of operation.

Under the traditional ‘efficient market paradigm’ the time series (a)noma-lies are interpreted as the mere reflection of the same (a)nomalies of the ‘fundamental factors' of asset prices. However, the fundamentals consist of a conglomerate of diverse factors (macroeconomic, firm-specific, etc.) and as an ensemble they are not observable so that this aspect of the efficient market paradigm cannot be tested directly. While there do not even exist examples of fundamental factors whose time-variation would share the phenomenology of fat tails and clustered volatility, these features could also suggest a more behavioral explanation rather than the mere transmission of information from fundamental factors into changing prices. For instance, ‘fat tails' are represented by an unusually high number of extremely large observations which resonates with the notion of excessive volatility of financial markets. Indeed, one of the most convincing components of the body of empirical evidence against the efficient market hypothesis is evidence for excessive volatility according to the test strategy developed first by Shiller (1981). In a similar vein, clustering of volatility could originate from waves of speculative behavior or overoptimism of market participants occasionally switching to overpessimism, ‘risk appetite’ changing over time, and similar descriptions of financial market turmoil.

The first attempts at explaining the stylized facts with behavioral models have come forth since about the beginning of the 1990s. Examples include Kirman (Kirman, 1991, Kirman, 1993), De Grauwe et al. (1993), and Lux (1995), among others. While early contributions were targeting phenomena like excessive volatility and endogenous emergence of bubbles and crashes, the subsequent literature has also concentrated on reproducing time series from behavioral market models that could reproduce those of empirical records. Most of this research is conducted via simulation studies, since ‘stylized facts' are statistically characterized by higher-order conditional and unconditional moments that for complex models of the market process with heterogeneous agents are hard to derive analytically. Meanwhile, a broad range of models exists that all get close to empirical market behavior or even generate synthetic data that are hard to distinguish with statistical tests from ‘real’ ones, cf. the surveys by Hommes, 2006, LeBaron, 2006, Samanidou et al., 2007 and Lux (2008). It appears that the combination of stabilizing and destabilizing (centripedal and centrifugal) forces as represented by, e.g., trend following and chartist strategies, on the one hand, and arbitrage activities based on some perception of an underlying fundamental value by some agents, on the other hand, is generally sufficient to generate a process that with a bit of stochasticity added, is able to provide very realistic data for a broad class of models varying in their exact details. Models in this vein range from simple “zero intelligence” settings (Kirman, 1991, Cont and Bouchaud, 2000) over chartist-fundamentalist models (Lux and Marchesi, 1999, Brock and Hommes, 1998) to models in which agents continuously develop their strategies with some kind of artificial learning algorithm (LeBaron et al., 1999, Lux and Schornstein, 2005).

With this literature having reached a status of consolidation, one natural further research direction is the empirical estimation and validation of such models. This is an endeavor economists are not accustomed to as empirical implementation in an economic context has typically been concerned with (sets of) reduced form equations of behavioral characterizations of representative agent models (expressed, for instance, via the Euler equations characterizing the optimal path of economic activity of such a representative agent). In the present context one would rather have to estimate models with an ensemble of agents with more or less complex interactions. However, there is no fundamental problem involved in such an undertaking. Typical agent-based models can be represented as Markovian stochastic processes and, thus, often generically fulfill a number of ‘regularity conditions' that are needed for the application of certain estimation strategies. Relatively simple models might also be amenable to some kind of reduced-form condensation which, in fact, will be the case for the model studied in the following chapters.

Previous attempts of estimation of behavioral or agent-based models of financial markets are sparse. Examples include: Amilon (2008) who estimated the model of Brock and Hommes (1998) by maximum likelihood and efficient method of moments approaches, Gilli and Winker (2003) who attempted to estimate the ‘ant’ model of Kirman (1993) via nonlinear optimization techniques, Alfarano et al. (2005) who estimated an extended version of the same model via an approximate likelihood approach, and a recent series of papers by Franke and Westerhoff (Franke and Westerhoff, 2011, Franke and Westerhoff, 2012, Franke and Westerhoff, 2014) in which a variety of extremely simplified versions of agent-based models are estimated via moment matching approaches. Jang (2013) also uses a simulated method of moments approach for a closely related model, and reports certain principal difficulties in estimating even a very basic agent-based model: He finds the surface of the objective function to be very flat over certain regions so that the chosen moments provide little scope in differentiating between different parameter sets, and he highlights that a very rugged surface of the objective function also makes the search for a global optimum computationally difficult as the danger to get trapped in one out of many local minima of the objective function is hard to assess.

In the present study, we attempt to contribute to this literature by exploring more systematically the issues surrounding the estimation of a prototype agent-based behavioral market model. Our model is of the same class as investigated already by Gilli and Winker, 2003, Alfarano et al., 2005 and Jang (2013). Among a number of closely related varieties we choose the specification of Alfarano et al. (2008). The latter has the advantage that the authors have derived already a set of moments that with a bit more of effort can be expanded into moment conditions of an estimable version of their model. These are the basic moments that characterize the stochastic dimension of the stylized facts, i.e. higher moments of returns and autocorrelations of such higher moments. We use these moments in a generalized method of moments (GMM) setting which, due to the absence of simulation in the estimation process, is more transparent than an SMM approach without any analytical input. By and large, we will see that the problems highlighted in previous literature can be overcome with a judicious choice of the moment conditions and the estimation strategy. An empirical application indicates that the present simple model already gets so close to the key moments characterizing fat tails and clustered volatility that it can mostly not be rejected as the underlying data-generating process for these moments even for relatively long data sets extending over several decades. A forecasting exercise demonstrates that the behavioral model possesses significant forecast capacity for short- and medium-term volatility. While its forecast performance remains generally somewhat inferior to that of a baseline GARCH model, it is often not ‘encompassed’ by the GARCH model, i.e. it adds valuable information on top of that extracted by the GARCH model.

The rest of this paper proceeds as follows. Section 2 introduces the agent-based model and its ‘reduced form’ representation in the form of a stochastic differential or Langevin equation. Section 3 provides an exposition of the GMM estimation and the moment conditions used. Section 4 provides Monte Carlo results for different specifications of the estimator. Section 5, then, contains the empirical application, and Section 6 concludes. In an Appendix we provide the details of the derivation of our moment conditions as well as results from an alternative simulation algorithm for the underlying model.

Section snippets

An elementary agent-based model of sentiment formation and asset price dynamics

The model investigated in Alfarano et al. (2008) basically extends Kirman's (1993) seminal herding model into a simple asset-pricing model. The overall market consists of fundamental traders as well as those driven by sentiment and the time-variation of sentiment is formalized via the herding dynamics. Referring to the second group of agents as noise traders, any one of these at any point in time might be labeled as an optimist or pessimist. The number of agents in an optimistic mood will be

Estimation

We will assume throughout that we are given equidistant observations of market prices, pti, ti = Δti, i = 0,…,t, making a sample of size (T + 1). We calculate from Eqs. (6), (7) and (8)rt=σfεt+NTcTfxt+1xt,where εt is a standard normal variate that is independent from xt at all lags. Thus, we observe a sequence of the log-returns rt of size T. The parameter vector θ to be estimated is θ : =(a,b,σf). Note that we do not estimate the parameters N, Tc, and Tf. We assume that these are given by N = 100,

Monte Carlo results

In order to assess the quality of the proposed GMM estimator, we have conducted a series of Monte Carlo experiments.

In the baseline setting, we assume that the log returns follow Wiener Brownian motion without drift. The fundamental dynamics is, thus, simply characterized by its standard deviation σf. We choose the parameter set θ = (a,b,σf) = (0.0003,0.014,0.03) together with the number of agents N = 100. This is a parameter set in the vicinity of typical empirical estimates, and it satisfies a

An empirical application

Given the acceptable performance of our modified GMM estimates, we turn to an empirical application. We have selected a number of important financial indices and other assets to explore the performance of our model. Table 4 exhibits estimation results for three stock market indices, three exchange rates and the price of gold. The stock market indices are: the S & P 500, the German DAX and the Japanese Nikkei. For these series and for the gold price, we have used daily data from the start of

Conclusions

In this paper, we have explored the issues evolving around the estimation of agent-based asset pricing models as they have mushroomed over the last two decades. While we have concentrated on the particular example of the model by Alfarano et al. (2008), we believe that some of our findings would also be relevant for other models. Since we were able to derive analytical moment conditions, we believe that important features showed up more clearly than would have been under the additional

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    *

    We gratefully acknowledge financial support from the European Union's 7th Framework Programme under the grant agreement no. 612955. We are also thankful for helpful comments from Zhenzi Chen, Reiner Franke and from the audience of various seminar and conference presentations.

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