ارزیابی مدل بنگاه مدار شکل گیری تمایلات سرمایه گذار در بازارهای مالی

We use weekly survey data on short-term and medium-term sentiment of German investors to estimate the parameters of a stochastic model of opinion formation governed by social interactions. The bivariate nature of our data set also allows us to explore the interaction between the two hypothesized opinion formation processes, while consideration of the simultaneous weekly changes of the stock index DAX enables us to study the influence of sentiment on returns. Technically, we extend the maximum likelihood framework for parameter estimation in agent-based models introduced by Lux (2009a) by generalizing it to bivariate and tri-variate settings. As it turns out, our results are consistent with strong social interaction in short-run sentiment. While one observes abrupt changes of mood in short-run sentiment, medium-term sentiment is a more slowly moving process in which the influence of social interaction seems to be less pronounced. The tri-variate model entails a significant effect from short-run sentiment on prices in-sample, but its out-of-sample predictive performance does not beat the random walk benchmark.

Opinion dynamics in financial markets have been modeled by Topol (1991), Kirman (1993), Lux, 1995 and Lux, 1998 and Alfarano et al. (2008) among others. These models make use of epidemic processes of information transmission between agents that allow for an endogenous formation of expectations. Markets with such interacting speculators give easily rise to speculative bubbles, crashes and excess volatility, and therefore, provide an avenue towards an explanation of these ubiquitous phenomena. Perhaps even more important, a certain number of these agent-based models has also been shown to exhibit more fundamental statistical properties of financial returns: Models like those proposed by Lux and Marchesi, 1999 and Lux and Marchesi, 2000, Iori (2002) or Pape (2007) generate time series that replicate the well-known stylized facts like fat tails and clustered volatility, even up to close numerical proximity of key empirical statistics of financial data (cf. Lux, 2009b, for an overview of this literature). Our aim in this paper is to estimate the parameters of an agent-based model of opinion formation and its impact on prices. This goal means that we attempt to identify structural parameters of an agent-based model (ABM) from aggregate data. The literature on estimation of ABMs has got started only recently and pertinent contributions are still scarce. Early research in this vein has concentrated on regime-switching models with two regimes accounting for periods of dominating chartist or fundamentalist influence in financial markets. Examples include Vigfusson (1997) and Westerhoff and Reitz (2003). While the former reports weakly favorable results for a chartist and fundamentalist framework, the latter authors find that fundamentalists' reactions might not be sufficiently strong to prevent amplification of distortions in the foreign exchange market. More recent contributions have also started to estimate heterogeneous agent models with performance-based switching rules for the choice of strategies or predictor functions. Boswijk et al. (2007) estimate a dynamic asset pricing model with heterogeneous, boundedly rational agents. In this model, a discrete choice-style selection mechanism based on past profits governs agents' switching between a fundamentalist and a chartist predictor. They estimate this model via nonlinear least squares for yearly data of the U.S. stock market from 1871 until 2003 and find that they can reject a benchmark linear asset-pricing model against the nonlinear two-group framework. Belief coefficients are strongly significant and indicate the prevalence of different strategies among market participants. Somewhat less clear-cut evidence for the explanatory power of a similar ABS model is reported for daily stock index data by Amilon (2008). This author points out that replication of stylized facts like volatility clustering hinges on the specification of the noise term while the nonlinearity introduced via the structural ABM model does not contribute much to the overall fit of the model. Franke (2009), in contrast, obtains a better fit of a simple ABM framework to selected moments that capture important stylized facts. A similar approach has been pursued by Goldbaum and Mizrach (2008) who adopt the discrete-choice approach to mutual fund allocation decisions. On the base of 10 years of data on inflow of capital to actively and passively managed funds, they estimate a discrete-choice model on the base of utility differences of investors from the active or passive varieties. They find that about 80% of the variation of fund flows can be explained by the model. A discrete-choice framework for the choice of expectation formation rules for inflation forecasts has been studied by Branch (2004). Using micro data from the Michigan Survey of Consumer Attitudes and Behavior, he finds significant evidence of heterogeneity among respondents. Allowing for the possibilities of vector autoregressive, adaptive and naive expectations, Branch finds that the respondents' choice of predictors reacts negatively to mean squared prediction error. ABM models with contagious interpersonal communication have been estimated by Gilli and Winker (2003), Alfarano et al. (2005), Klein et al. (2008), Franke (2008) and Lux (2009a). Gilli and Winker (2003) estimate the “ant” model of pairwise exchange of information of Kirman (1993) for foreign exchange data and find evidence for bi-modality, i.e. changes between dominance of both underlying opinions (chartist and fundamentalist predictors in the application to a foreign exchange market). Alfarano et al. (2005) estimate the parameters of a closely related model with asymmetric switching propensities and find that different markets are governed by different prevailing tendencies towards fundamentalist or chartist behavior. Klein et al. (2008) attempt to estimate the more involved model by Lux and Marchesi (1999) that combines the chartist-fundamentalist dichotomy with social interactions among agents. While they do not report parameter estimates, they provide results on the estimated fraction of chartists as it develops over time. Results appear to be in good harmony with historical perceptions of the financial history over the last 60 years. Closest to our current paper are the recent contributions by Franke (2008) and Lux (2009a) who attempt to estimate the parameters of models of social opinion formation among agents for economic sentiment data. Our goal in this paper is to go one (or two) steps beyond a previous paper (Lux, 2009a) that introduced a method for identification of the parameters of microscopic opinion processes from aggregate data. This paper, however, was confined to estimation of the parameters of a model for a univariate time series, namely the diffusion index form (number of optimistic individuals minus number of pessimistic individuals) of a business climate survey. While the same model and estimation methodology could be applied for financial sentiment data (which often share the format of diffusion indices), a univariate model would only allow us to cover one of the building blocks of the above asset pricing models. As a minimum requirement, however, for an empirical validation of a stochastic behavioral asset pricing model one would like to study the joint dynamics of asset prices and sentiment. We will, therefore, extend our previous model into this direction and provide parameter estimates for a simple version of a simultaneous system. Since our underlying time series cover two sentiment variables, one for the short horizon and one for the medium-term horizon, we can even go one step further and study two interacting opinion processes together with the time development of the asset price. Since this amounts to studying the dynamics of a tri-variate series, we proceed in this paper from the 1D case of Lux (2009a) to the 2D and 3D cases. As in the previous paper, the methodology presented below could be applied to a wide variety of hypothesized opinion dynamics interacting with objective economic variables. In order to demonstrate the practical use of estimated agent-based models, we also perform an out-of-sample forecasting experiment based on our estimated models. Apart from the still relatively sparse literature on estimation of agent-based models our approach could also be linked to a much more voluminous strand of empirical research: Empirical models of survey measures of sentiment or investment (business, consumer) climate. Since we will use a diffusion approximation to our underlying opinion process, we could also interpret our exercise as estimation of a time series model motivated by an agent-based model. Sentiment indices could be considered as risk factors in the asset pricing equation in such a setting. Our research would, then, explore the added explanatory power of nonlinearities introduced through the interaction of agents while previous research has mainly used linear models for modeling sentiment and its explanatory power as a risk factor for stock price movements (cf. Brown and Cliff, 2004 and Schmeling, 2009). The rest of the paper is structured as follows: In Section 2 we introduce our stochastic framework of sentiment dynamics and simultaneous price changes. Section 3 provides details on our estimation methodology, maximum likelihood estimation based on a numerical approximation of the transient density of the underlying stochastic process. Section 4 gives details on the sentiment data we use as well as an overview on previous findings on the interaction between sentiment and returns within a non-behavioral VAR framework. In Section 5 we present results for univariate population dynamics and diffusion processes for each one of our three time series. In 6 and 7 we proceed to various combinations of 2D models and the full-fletched model of three simultaneous stochastic processes. Section 8 summarizes our findings and concludes. The Appendix provides details on the numerical approximation schemes for the dynamics of the transient density.

Given the long lasting interest in sentiment data in financial economics, it might come as a surprise that there is hardly any empirical work estimating and testing behavioral models for such data. Of course, under an efficient market perspective, such data would represent a relatively unimportant noise component. However, evidence exists for a certain impact of sentiment on prices. While for U.S. data, Granger causality in the short-run seems to run from returns to sentiment (Brown and Cliff, 2004), our German data indicate a causal relationship in the opposite direction (Lux, 2010). Even for the U.S., however, some predictive power of sentiment has been found for longer horizons (Brown and Cliff, 2005). The significant influence of sentiment on returns in the German data motivated us to adopt a behavioral, agent-based framework for the dynamics of short- and medium-run sentiment. In order to estimate the parameters of such models, we could take stock of an approach proposed in Lux (2009a) using a numerical maximum likelihood procedure. In terms of methodology, the contribution of this paper is the extension of this estimation technique to higher dimensions. Monte Carlo simulations in Lux (2009a) showed that this method performed well even in relatively small samples like the present one. Choosing appropriate methods from the large range of finite difference approximations for partial differential equations allowed an extension of the univariate approach to 2D and 3D. Materially, we found evidence for strong social interaction in short-term sentiment and only moderate social influences in medium-run sentiment. With moderate interaction, estimation of the agent-based model appears somewhat cumbersome because of almost collinear behavior of some parameters. As we have seen, in this case, a more parsimonious Ornstein–Uhlenbeck process provides practically the same fit to the data. One could, therefore, use a simple macroscopic equation instead of the full microscopic Markov process. We believe that results like this one are important in that they provide an indication of the necessary degree of complexity of behavioral models in different scenarios (an issue very much neglected in economics where theoretical models are mostly either based on the assumption of a representative agent or an infinite population). This also means that there is not necessarily a conflict between agent-based modeling and modelling of aggregate data via structural equations. On the contrary, combining an analytical approach for the derivation of appropriate approximations to the agent-based stochastic Markov process with an approach for empirical estimation of the later, we were able to show that under certain conditions two very different functional specifications can yield practically undistinguishable results. However, this congruence crucially depends on the parameter values and might break down (as it does here for S-Sent) if other parameters lead to another type of system behavior. Overall, most of our results were in nice coincidence with the previous VAR results of Lux (2010). The social dynamics estimated for S-Sent and M-Sent, therefore, seems a potential candidate for a data generating process of our sample. While our forecasting exercise turned out disappointing results for price forecasts, this might be due to potential non-stationarity of the returns process during our out-of-sample period that coincides with the last three years of financial turmoil. Due to the computational demands of the present approach, all forecasts were conducted on the base of the fixed parameter estimates from the in-sample period. Note that more successful forecasts from VAR models were based on iterative updating of both paramaters and the selection for the appropriate VAR subset model, and did indeed find quite some variation of the influence of S-Sent vs M-Sent on returns throughout pretty much the same out-of-sample period. We might hope that moving towards iterative updating of parameters for the present approach might also help to capture better the subtleties of joint causation from various sentiment indicators on returns. Current attempts at developing a fully parallel algorithm for the present approach will allow us to revisit the forecasting capacity of the opinion models with an iterative approach in the near future. We believe that the present paper and its predecessor (Lux, 2009a) could provide an avenue to empirical estimation of a broad range of agent-based models. While we noted that we reached the limits of current computational power at our 3D applications, we also note that we have only used a small range of numerical schemes so far. Methods using adaptive adjustment of meshes or refined methods for parallelization of tasks might allow us to dramatically reduce computation time in future applications. Further research in this direction should be of high priority.