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Journal of Economic Interaction and Coordination

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20.08.2014 | Regular Article

Why a simple herding model may generate the stylized facts of daily returns: explanation and estimation

verfasst von: Reiner Franke, Frank Westerhoff

Erschienen in: Journal of Economic Interaction and Coordination | Ausgabe 1/2016

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Abstract

The paper proposes an elementary agent-based asset pricing model that, invoking the two trader types of fundamentalists and chartists, comprises four features: (i) price determination by excess demand; (ii) a herding mechanism that gives rise to a macroscopic adjustment equation for the market fractions of the two groups; (iii) a rush towards fundamentalism when the price misalignment becomes too large; and (iv) a stronger noise component in the demand per chartist trader than in the demand per fundamentalist trader, which implies a structural stochastic volatility in the returns. Combining analytical and numerical methods, the interaction between these elements is studied in the phase plane of the price and a majority index. In addition, the model is estimated by the method of simulated moments, where the choice of the moments reflects the basic stylized facts of the daily returns of a stock market index. A (parametric) bootstrap procedure serves to set up an econometric test to evaluate the model’s goodness-of-fit, which proves to be highly satisfactory. The bootstrap also makes sure that the estimated structural parameters are well identified.

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For recent surveys of this burgeoning field of research, see Chiarella et al. (2009), Hommes (2006), Hommes and Wagener (2009), LeBaron (2006), Lux (2009a) and Westerhoff (2009), among others.

Other evidence is based on laboratory experiments; see, e.g., Heemeijer et al. (2009) or Hommes et al. (2007).

The choice of MSM does not rule out that other estimation methods may be tried as well. For a brief summary of the comparative advantages of MSM, see Franke (2009, pp. 804f). In our opinion, its main merits are the high transparency in the evaluation of a model’s goodness-of-fit, and the relatively low computational cost.

To be exact, by demand we mean the orders (positive or negative) per trading period, not the desired positions of the agents.

For example, individual and presently active traders with a fundamentalist strategy may adopt different values for their fundamental price, they react with different intensities to their trading signal, or they experiment with more complex trading rules which may also be continuously subjected to further modifications. Similarly so for the chartists, which explains the independence of \(\varepsilon ^f_t\) and \(\varepsilon ^c_t\). In short, the two noise variables can be conceived of as a most convenient short-cut of certain aspects that are more specifically (but to some extent also more arbitrarily) dealt with in models with hundreds or thousands of different agents that one would have to keep track of over time (see Farmer and Joshi 2002; LeBaron 2006).

In actual fact, \(\chi = \sigma _c = 0\) results from an estimation of the USD–DEM exchange rate; see Franke and Westerhoff (2011, Section 7). The situation for \(\phi = 0\) and, possibly, \(\sigma _f = 0\) would be formally analogous. In this case, however, the price dynamics would no longer be anchored on the fundamental value.

To see this, define \(n_t = (n^f_t - n^c_t)/2 = x_t N\), write the identity \(n^f_t +n^c_t = 2N\) as \(n^f_t/2 = N - n^c_t/2\) and add \(n^f_t/2\) on both sides of this equation. This yields \(n^f_t = N + n_t\) and, after division by \(2N\), the first part of Eq. (3). The derivation of the second part is analogous.

As usual in this kind of framework, any other feedbacks when his inventory continues to deviate from some target are ignored, which (in a stochastic model) is clearly an inconsistency. It could be removed by adding the risk aversion concept of the market maker (and also the other agents) studied in Franke and Asada (2009). We forgo this option to avoid blurring the central mechanisms of the model.

Randomized demand functions of heterogeneous traders were also considered in Westerhoff and Dieci (2006) and Westerhoff (2008). The idea as such may be traced back to Westerhoff (2003). However, the implied feature of stochastic volatility and its scope for matching certain stylized facts of (daily) returns was not fully elaborated there. More on the particular effects of SSV can be learned from the investigation in Franke (2010), where this principle of heterogeneous noise was incorporated into two other model types.

This is different from the discrete choice approach, which is a constituent part of the Brock–Hommes (1998) model variety. There, the population shares of the agents—and not their rates of change—are directly a function of the state variables of the model. However, introducing an asynchronous updating of strategies in the latter, it becomes essentially the same as the Weidlich–Haag–Lux approach (see the discussion in Franke 2013).

In contrast to the more elaborate treatment in Lux (1995, 1997), this reasoning, which can also be found in Lux (1998, p. 149), is sufficient for an infinite population. A rigorous mathematical argument that begins with a finite population size and the intrinsic noise it implies is spelled out in Franke (2008a; b).

The precise hypothesis is \(d \pi ^{cf}_t/\pi ^{cf}_t = \alpha \, ds_t\) and \(d \pi ^{fc}_t/\pi ^{fc}_t = -\alpha \, ds_t\) for some constant \(\alpha \), which may be unity without loss of generality (since \(s_t\) may be arbitrarily scaled). Integrating these relationships with an integration constant \(\nu \) yields (7).

As it depends on the other parameters in the model, it is a priori not clear what “sufficiently low” would exactly mean. When for the numerical simulations we had to settle down on a specific positive value of \(\nu \), we checked that indeed the upper-bound of unity for \(\pi ^{cf}_t \!\!\), \(\pi ^{fc}_t\) was never reached.

There are several stories about the ways in which \(x_t\) influences the transition probabilities. If the individual agents base their switching decision on the publicly available knowledge of the current majority index, these observations might also involve some noise. We disregard this option for simplicity.

Symmetrically to point (c) in the proposition, a sufficiently positive predisposition parameter \(\alpha _o\) would establish a unique equilibrium value of \(x = x^{fd}\) where fundamentalism takes over. As has just been stated, this situation will be of no concern to us.

A mathematical proof is omitted.

Reckoning 250 days per year. Specifically, the empirical sample period is January 1980 to March 2007 (just before the financial crisis began to unfold).

Detailed descriptions of the statistical properties of asset prices can be found in Cont (2001), Lux and Ausloos (2002), or Lux (2009b).

Generally, one might also include a negative skewness of stock returns. Stylized small-scale asset pricing models, such as the present one, do not, however, provide for any asymmetry in this respect.

That is, at lag \(\tau \) the mean of the three autocorrelation coefficients for \(\tau - 1\), \(\tau \), \(\tau + 1\) is computed, except for \(\tau = 1\), where it is the average of the first and second coefficient. It may also be noted that volatility clustering, which describes the tendency of large changes in the asset price to be followed by large changes, and small changes to be followed by small changes, is closely related to these long-term dependencies between the returns.

To reduce the thus arising bias, even the identity matrix could be a superior weighting matrix; see Altonji and Segal (1996).

We checked that the weighting matrix resulting from our bootstrap procedure is indeed positive definite.

For the normally distributed \(\varepsilon _t\) with variance \(\sigma ^2_t\) in (4), (5), this means, more precisely, that for each simulation run at time \(t\) the same random number \(\tilde{\varepsilon }_t\) is drawn from the standard normal distribution \(N(0,1)\) and \(\varepsilon _t\) is set as \(\sigma _t \, \tilde{\varepsilon }_t\).

We use the Nelder-Mead simplex search algorithm (see Press 1986, pp. 289–293) and restart it upon convergence several times until no further noteworthy improvement in the minimization occurs.

Admittedly, the value \(\nu = 0.57\) in Franke and Westerhoff (2011) is psychologically not very convincing.

This value can be slightly reduced to \(\widehat{J} = 6.98\) by treating \(\nu \) as a free parameter, too. We then get a higher value \(\nu = 0.067\) which, however, is something that we had sought to avoid. Besides, given the random seed \(\tilde{a}\), a marginal improvement, \(J = 7.16\), can also be obtained by a lower value of the flexibility parameter, \(\nu = 0.033\).

In Franke and Westerhoff (2011, 2012b), a nonparametric bootstrap was employed. There we also discussed statistical measures that could characterize the matching of the single moments.

To perfectly imitate the original estimation, one would also have to take into account that different return series \(r^c_t\) (in obvious notation) give rise to different weighting matrices in the loss function. Unfortunately, this would mean carrying out an extra bootstrap for each of the 1,000 artificial samples. We refrain from this additional computational effort and employ the original weighting matrix \(W\) from (12), (13) for all of the re-estimations.

Concerning symbol \(p\), there should be no confusion with the log prices \(p_t\), which by now will have disappeared from the scene.

The density functions in this and the next diagram are estimated using the Epanechnikov kernel; see Davidson and MacKinnon (2004, pp. 678–683) for the computational details.

In fact, among the 1,000 estimations there is only one case where \(p^a\) is slightly below 5 %.

Since presently a set of 1,000 estimations on an average personal computer takes between 27 and 31 hours, an increase in \(S\) would require a parallel computing device.

See Lee and Ingram (1991, p. 202).

The estimates \(\{ \widehat{\theta }^a \}\) in (15) only take the sample variability in the simulations into account but not the variability arising from different realizations of the data generation process.

On the basis of a number of explorations, we are confident that the intervals continue to be bounded and so the conclusion remains valid if \(\nu \) is also treated as a free parameter.

Even though the model may be misspecified, a pseudo-true parameter vector \(\theta ^o\) is a well-defined concept. If \(m^o\) is the expected moment vector of the true model of the stock market, \(\theta ^o\) satisfies \(J[m(\theta ^o),m^o] \le J[m(\theta ),m^o]\) for all admissible \(\theta \), where \(m(\theta ) = lim_{S \rightarrow \infty } E[m^a(\theta ;S)]\) (assuming ergodicity, the expected values converge to the same limit for all random number sequences). This definition corresponds to that in Hnatkovska et al. (2011, p. 6), where the expected moments of the model can be analytically computed.

A conclusion that actually was strongly insinuated to us by some severe readers of a previous version of the paper.

Incidentally, large parts of this paper were written before we even started with FW (2011, 2012b). Unfortunately, after its completion it took a rather thorny path through the journal landscape.

In fact, we are now aware that we did not fully exploit the information that could be obtained from a nonparametric bootstrap, where the critical point is a recentring of the loss function [cf. the discussion in Franke (2012), Sect. 2.2]. The parametric bootstrap in the present paper does not suffer from this kind of problem.

The present authors do not exempt themselves from this.

For a specific system, this question is answered by an explicit (elaborate) mathematical analysis in Lux (1997, Sects. 4.1 and 4.2).

Incidentally, the argument remains the same if \(\alpha _x \le 0\), although we would then have the opposite of herding.

For which Andrews (2004) proposes the concept of a block–block bootstrap.

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Titel: Why a simple herding model may generate the stylized facts of daily returns: explanation and estimation
verfasst von: Reiner Franke
Frank Westerhoff
Publikationsdatum: 20.08.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: Journal of Economic Interaction and Coordination / Ausgabe 1/2016
Print ISSN: 1860-711X
Elektronische ISSN: 1860-7128
DOI: https://doi.org/10.1007/s11403-014-0140-6

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