توضیح تئوری بازی تکاملی اثرات ARCH

While ARCH/GARCH equations have been widely used to model financial market data, formal explanations for the sources of conditional volatility are scarce. This paper presents a model with the property that standard econometric tests detect ARCH/GARCH effects similar to those found in asset returns. We use evolutionary game theory to describe how agents endogenously switch among different forecasting strategies. The agents evaluate past forecast errors in the context of an optimizing model of asset pricing given heterogeneous agents. We show that the prospects for divergent expectations depend on the relative variances of fundamental and extraneous variables and on how aggressively agents are pursuing the optimal forecast. Divergent expectations are the driving force leading to the appearance of ARCH/GARCH in the data.

ARCH/GARCH models have been used to describe the behavior of inflation, interest rates and exchange rates,1 and they have become the standard tool for analyzing returns in financial markets.2 Despite the widespread empirical successes of ARCH/GARCH models, discovering underlying mechanisms that lead to time-varying volatility has proved to be an elusive goal. We propose that time-varying volatility is a natural feature of models with forward-looking agents. Our key condition is that agents are not constrained by assumption to agree on a single expectation. Instead, we apply recent developments in evolutionary game theory to explain how forward-looking agents might choose among differing forecasts. We assume that information arrives uniformly over time so that changes in volatility are due entirely to agents’ behavior. Furthermore, our agents do not set out simply to invent ARCH. They use ideas drawn from the literature on rational expectations to choose among forecasts based on what they perceive to be fundamentals. We establish conditions under which ARCH effects will be a normal feature of the resulting data. The mechanism generating time-varying volatility has a general formulation. Evolutionary game theory describes how fractions xt=(x1,t,…,xk,t)xt=(x1,t,…,xk,t) of the population using forecasting strategies st=(s1,t,…,sk,t)st=(s1,t,…,sk,t) evolve according to the performances of the strategies. An asset pricing model then shows how the price ytyt depends on the fractions xtxt and other information ΘtΘt so that yt=y(xt,Θt)yt=y(xt,Θt). The fractions xtxt are taken to be known and fixed before ytyt is realized. We divide ΘtΘt into information ΩtΩt available to agents before ytyt is realized and all other information ɛtɛt. The model generating ytyt can then be written in the form equation(1) yt=y(xt,Ωt,ɛt).yt=y(xt,Ωt,ɛt). Turn MathJax on After agents observe ytyt, they choose strategies for period t+1t+1, updating xtxt using a procedure of the form equation(2) xt+1=g(xt,yt,Ωt).xt+1=g(xt,yt,Ωt). Turn MathJax on Given this structure, the variance of the asset price can generally be written as equation(3) V(yt|xt,Ωt,Σt)=h(xt,Ωt,Σt),V(yt|xt,Ωt,Σt)=h(xt,Ωt,Σt), Turn MathJax on where ΣtΣt is a measure of the volatility of ɛtɛt. To clarify the source of conditional heteroskedasticity, we assume that ΣtΣt is constant. The variance of ytyt will not, however, be constant if it depends on the mix xtxt of agents’ strategies. We demonstrate that ARCH effects can appear to be important in an empirical model for ytyt if the econometrician does not take into account heterogeneity. A typical representative agent model for ytyt, for example, would omit xtxt from (1) leaving yt=y′(Ωt,ɛt).yt=y′(Ωt,ɛt). Turn MathJax on The corresponding representative agent model of the conditional variance (3) would reduce to equation(4) V′(yt|Ωt,Σt)=h′(Ωt,Σt).V′(yt|Ωt,Σt)=h′(Ωt,Σt). Turn MathJax on Standard econometric tests may well diagnose that the representative agent model (4) leaving out the fractions xtxt has ARCH, making it appear to be necessary to account for changes over time in ΣtΣt. We provide specific simulations that illustrate how apparent ARCH can be an artifact of ignoring heterogeneous expectations. These ARCH effects take place in a standard mean–variance optimization model of asset prices extended to an environment with heterogeneous agents. Brock and Hommes (1998) develop the theoretical basis for this model. They apply the model in an environment with multiple trader types who use an assortment of linear forecasting rules. Brock et al. (2005) and Gaunersdorfer et al. (2003) extend these results.3 Other studies with heterogeneous expectations include Chiarella and He (2002), DeGrauwe (1993) and DeLong et al. (1990). Our study differs in that we focus on agents pursuing goals set forth in the literature on rational expectations. Our specific example features agents choosing among three forecasting strategies. A fundamentalist uses only expected future dividends to form his forecast. A mystic uses fundamentals, but may also experiment with other extraneous information because, perhaps, there is some uncertainty about what belongs in the fundamentals. A reflectivist incorporates all available information about agents’ expectations, calculating the average expectation using population share weights. We implement the procedure for updating agents’ choices of forecasting strategies (2) by having agents switch to strategies that have exhibited lower squared forecast errors. The switching probabilities are determined by the evolutionary dynamic of Hofbauer and Weibull (1996). That dynamic allows for a nonlinear weighting function in the forecast evaluation. We add an important dimension to our analysis by using the nonlinearity to parameterize how aggressively agents switch forecasts. Standard econometric tests applied to simulated data confirm that the extent of ARCH effects depends on agent aggressiveness and on the variance of the potential extraneous element that might enter the mystical forecast. If the latter variance is small relative to the variance of the fundamentals or if agents are not very aggressive, then the asset price tends to follow fundamentals nearly all the time. If the variance of the extraneous element is larger and agents are more aggressive, then asset prices show occasional bubble behavior and both Engle's (1982) test for ARCH and estimates of a GARCH(1,1) model support the conclusion that the data can be described as ARCH/GARCH for many of the simulations. The role of heterogeneity has been noted in other related contexts with boundedly rational forecasting strategies. Lux and Marchesi (2000) construct an asset market with fundamentalists and two different types of chartists, who respond to trends in the data. The switching probabilities between the two strategies are determined by a modified discrete choice model that allows for sluggish adjustment. Their model shows that switching between strategies can produce ARCH effects for certain parameter values. Föllmer et al. (2005) show the existence of bubbles, but also show the existence of limiting distributions of asset prices in a discrete choice model with forecasting strategies put forward by ‘gurus’ that could include chartists and fundamentalists. In a similar framework, Gaunersdorfer and Hommes (2006) derive theoretical results indicating the presence of bubbles and volatility clustering. LeBaron et al. (1999) study the time series features of a simulated asset market and show the existence of ARCH effects and many other features of financial market data. They use a computational approach with many trader types introduced throughout the simulation according to a genetic algorithm. They find that the evidence of ARCH effects is much stronger in an environment they term fast learning than it is given slow learning. The focus in this paper on the possibility of heterogeneous forecasts stands in contrast to those who argue that martingale solutions should be ruled out according to criteria such as transversality (see Cochrane, 2001, p. 27), minimum state variables (McCallum, 1983 and McCallum, 1997), and expectational stability (Evans and Honkapohja, 2001).4 In our context, the importance of the martingale solutions depends on the parameter values and, while convergence to a single expectation is possible under some conditions, we establish the range of parameter values for which the martingale solutions are an important feature. Using another approach that defines stability according to rationalizability of strategies, Evans and Guesnerie (2003) show convergence to the minimum state variables solution with homogeneous agents, but they also show instability in the case of heterogeneous agents. The literature on convergence to rational expectations along the lines of least squares learning (Grandmont, 1998 and Marcet and Sargent, 1989, for example) is also concerned with multiple rational expectations equilibria. Woodford (1990) and Howitt and McAfee (1992) establish the possibility of learning sunspot equilibria given accidental correlations between sunspots and fundamentals.5 While these papers focus on agents learning model parameters over time, our agents know the parameters and are choosing among forecasts constructed from the multiple solutions to the model. den Haan and Spear (1998) explain conditional volatility in real interest rate fluctuations. They construct an optimizing model where agents hold savings in the form of bonds. Agents are heterogeneous, as in the present paper, and receive idiosyncratic shocks. Volatility clustering arises due to borrowing constraints that vary across the business cycle. The organization of the paper is as follows. Section 2 develops the asset pricing model with heterogeneous agents. Sections 3 and 4 describe the different forecasting strategies and their squared forecast errors. Section 5 shows how agents’ choices of strategies evolve over time. Section 6 establishes how convergence of expectations depends on assumptions about agents’ beliefs and willingness to consider new information. Section 7 describes the simulation methodology used in the remainder of the paper. Section 8 provides examples and compares the simulations to some stylized facts. Section 9 presents econometric analysis of the simulations using a GARCH(1,1) model. Section 10 concludes.

While ARCH/GARCH models have proved to be extremely successful empirical econometric techniques, explaining the underlying causes of conditional volatility in financial markets has been a difficult challenge. This paper presents a formal model explaining how such effects arise endogenously when forward-looking agents choose among forecasting strategies. The leading candidate for the source of conditional volatility has long been news of some kind (Engle, 2001 and Engle, 2004). Our results are driven by the arrival of new information, but they do not require any assumption that information arrives at nonuniform rates. We show instead that the process of experimenting with and rejecting sources of information can be a key factor explaining conditional volatility. In fact, our parameter specifying how aggressively agents search for the optimal forecasting strategy is a primary factor in accounting for conditional volatility. Our results do not require any radical departure from rationality. We consider only forward-looking mean–variance optimizing agents who differ only because they choose among three forecasting strategies that are all consistent with the notion of rational expectations. The fundamentalists use the rational expectations solution dominant in that literature. Mysticism follows the same principles, but mistakenly experiments with the idea that the martingale innovations are fundamental. Both fundamentalism and mysticism are, of course, only fully rational if they are adopted by all agents. The reflectivists focus on this point and adopt a forecast taking full account of the behavior of the other two groups of agents. The characteristics of the realized asset price differ dramatically across parameter values. If the martingale variance is small or agents are not very aggressive in pursuing the optimal forecasting strategy, agents tend to agree numerically on the fundamentalist forecast and there is little evidence of volatility clustering. For larger martingale variances and/or more aggressive agents, ARCH/GARCH effects appear in significant fractions of the sample runs. Econometric tests of the simulated data in the latter cases detect ARCH and GARCH effects similar to those found in financial market data. We test for ARCH using Engle's (1982) test and then estimate the GARCH(1,1) model often used in practice. For a range of martingale volatility and a range of agent aggressiveness, the sample statistics indicate that the simulated data is well represented by a GARCH(1,1) model. For the combination of a large martingale innovation variance and very aggressive agents, the evidence points in the direction of ARCH, but in a form more complicated than a GARCH(1,1) model. As we propose in the Introduction to this paper, these results confirm that empirical ARCH/GARCH effects can be an artifact of viewing data generated by heterogeneous expectations from the perspective of a model that assumes a single expectation. One implication of these results is that a test for ARCH can be viewed as a specification test for the assumption that agents agree on a single expectation. ARCH will be observed if the levels of martingale volatility and agent aggressiveness are high enough to make divergent expectations a common feature of the data.From this perspective, the widespread econometric evidence in favor of ARCH/GARCH for variables such as inflation, interest rates, exchange rates, and returns on financial assets presents a challenge to the assumption that agents in models explaining these variables agree on a single expectation.