Bayesian estimation and likelihood-based comparison of agent-based volatility models

In Bayesian modeling observed data,¹\(\varvec{x}= (x_1, \ldots , x_N)\) are related to unobserved parameters/latent variables \(\varvec{\theta }= (\theta _1, \ldots , \theta _K)\) in terms of a joint probability distribution with density \(p(\varvec{x}, \varvec{\theta })\). This density is usually factorized as \(p(\varvec{x}, \varvec{\theta }) = p(\varvec{x}| \varvec{\theta }) p(\varvec{\theta })\), i.e., into the parameter likelihood and prior density. Inference then rests on Bayes rule to obtain the density of the posterior distributionwhere the normalization is given by \(p(\varvec{x}) = \int p(\varvec{x}| \varvec{\theta }) p(\varvec{\theta }) \hbox {d}\varvec{\theta }\). The posterior summarizes the information obtained about the unobserved parameters \(\varvec{\theta }\) and combines the a priori assessment of the modeler, \(p(\varvec{\theta })\), with the information obtained from the data \(p(\varvec{x}| \varvec{\theta })\). Conceptually, Bayesian estimation boils down to a rather mechanical application of Bayes rule, once the full model \(p(\varvec{x}, \varvec{\theta })\) is specified. Below, we will explain how this applies to different agent-based models and discuss, in particular, the role of prior choices in Bayesian modeling.

In practice, the normalization constant \(p(\varvec{x})\) of the posterior density is often intractable, involving an integral over the parameter space. Accordingly, many approximation methods have been proposed which either aim to approximate it with a tractable density or allow to draw posterior samples from its unnormalized density. Hamiltonian Monte-Carlo (HMC) sampling is an example of the latter approach. As a Markov chain Monte Carlo method, it produces a sequence of possibly correlated samples. A comprehensive and readable introduction to HMC and its properties can be found in Betancourt (2017). Here, a rather short overview of the method should suffice.

Consider a target density \(p^*(\varvec{\theta })\), e.g., the posterior distribution \(p(\varvec{\theta }| \varvec{x})\) from a Bayesian model. MCMC aims to construct a transition density \(T(\varvec{\theta }' | \varvec{\theta })\) which leaves the target density invariant, i.e.,Such a transition density can then be utilized to draw a sequence of samples \(\varvec{\theta }_1, \varvec{\theta }_2, \ldots \) with \(p(\varvec{\theta }_2, \ldots | \varvec{\theta }_1) = \prod _{i = 1}^{\infty } T(\varvec{\theta }_{i+1} | \varvec{\theta }_{i})\). The Metropolis–Hastings algorithm uses two steps in order to compute a suitable transition starting at \(\varvec{\theta }_i = \varvec{\theta }\): This so-defined transition density not only leaves the target density invariant, but, under suitable conditions, also ensures that the chain converges to its unique invariant density starting from any initial condition \(\varvec{\theta }_1\) (Bishop 2011).

While, in theory, the Metropolis–Hastings algorithm can produce samples from the desired target density, especially in high dimensions it can suffer from slow convergence. The underlying reason is that most of the probability mass is confined to a small subspace, the so-called typical set. If the proposal density is not well matched to the target density, many steps of the Markov chain are either rejected, e.g., when leaving the typical set, or slowly and randomly move along the typical set. HMC utilizes gradient information in order to generate long sweeps of the proposed states which are nevertheless accepted. To this end, HMC constructs a gradient flow which follows the typical set (Betancourt 2017).

Formally, HMC samples from an augmented state space \((\varvec{\theta }, \varvec{m})\) with densityThe desired marginal density \(p(\varvec{\theta })\) can then be obtained by dropping the \(\varvec{m}\) component of each joint sample. In analogy with physical systems, \(\varvec{m}\) is considered as the momentum of a particle at position \(\varvec{\theta }\). Intuitively, \(\varvec{m}\) controls the speed at which the position is moved along the typical set. The gradient flow following the typical set is obtained by integrating the Hamiltonian \(H(\varvec{\theta }, \varvec{m}) = - \log p(\varvec{\theta }) - \log p(\varvec{m}| \varvec{\theta })\) asHamiltonian dynamics have several well-known properties. In particular, they are time-reversible and conserve volume and total probability \(p(\varvec{\theta }, \varvec{m})\). Thus, no further corrections are necessary when using the state \((\varvec{\theta }', \varvec{m}')\) obtained by integrating the Hamiltonian for some time as a proposal. Due to time-symmetry and conservation of probability, the acceptance probability in Eq. (1) reduces to 1, i.e., the new state is always accepted.

HMC then proceeds in two steps. At each transition, a new momentum is sampled according to \(p(\varvec{m}| \varvec{\theta })\) which is commonly taken as a Gaussian distribution independent of the current state \(\varvec{\theta }\), i.e., \(\varvec{m}\sim \mathcal {N}(\varvec{0}, \varvec{I})\). Then, the position is moved from \(\varvec{\theta }\) with initial speed \(\varvec{m}\) by following the gradient flow for some time. The final position \((\varvec{\theta }', \varvec{m}')\) is then accepted as the next sample. By conservation of total probability, it holds that \(p(\varvec{\theta }, \varvec{m}) = p(\varvec{\theta }', \varvec{m}')\) potentially leading to a long sweep across the typical set along a level contour of the probability density. Thus, at each step HMC first jumps to a new probability level and then follows the gradient flow at this level, thereby allowing for an efficient exploration of the typical set. In practice, the differential equation describing the Hamiltonian dynamics needs to be integrated numerically and care has to be taken that numerical errors do not accumulate. Fortunately, symplectic integrators can efficiently integrate Hamiltonian systems as numerical errors cancel and simulated trajectories closely approximate the theoretical dynamics. While time-symmetry and volume preservation are retained by symplectic integrators, the total probability is only approximately conserved along numerical trajectories. Thus, in practice, HMC uses a Metropolis–Hastings step to either accept or reject the final position of a trajectory.

Especially in high-dimensional models, i.e., with many parameters, the use of gradient information to guide exploration is crucial to ensure efficient sampling. Note that HMC is restricted to continuous parameter spaces \(\varvec{\theta }\in \mathbb {R}^K\), but could be combined with other methods when discrete parameters are desired. Often, it is advantageous to marginalize over discrete parameters as strong correlations between them can severely hinder efficient sampling.³ In theory, HMC is insensitive to strong correlations between parameters \(\varvec{\theta }\). In practice, the symplectic integrator uses a finite step size to numerically solve the Hamiltonian dynamics. In the case of posterior densities with high curvature, this can prevent the sampler to reach certain parts of the state space. Furthermore, it makes HMC sensitive to the scale of parameters as the step size would need to be adjusted accordingly.⁴ Fortunately, by reparameterizing the model it is often possible to simplify the geometry of the posterior density. Ongoing research explores the geometric aspects of HMC both from theoretical (Livingstone et al. 2019; Ma et al. 2015) and from practical (Betancourt and Girolami 2015) perspectives. Also, a range of novel convergence diagnostics, e.g., based on the stability of the numerical trajectory, have been developed (Betancourt 2017).

Next, we turn to agent-based models for financial markets and show how these can be expressed as statistical models. All models are then implemented and fitted models with the probabilistic programming language Stan (2017). Appendix A provides a short overview. The Stan code for all models is available in the online supplementary material. Implementing agent-based models in a well-tested Bayesian framework has several advantages. On the one hand, inference algorithms such as HMC supported by Stan are well optimized and tested. On the other hand, model specifications can be easily adapted and explored with small changes to their source code. Yet, Stan restricts the user to models with continuous random variables only. While many models, such as the ones presented below, can be approximated in this fashion in the limit of infinitely many agents, more detailed simulations involving many discrete agents are currently beyond the scope of existing toolboxes for probabilistic modeling. In this sense, our approach can be seen as a proof of concept which nevertheless covers several models of interest.

The market in the VS model is populated by N traders. At each time step t, a trader i either buys (\(S_i(t) = 1\)), sells (\(S_i(t) = -1\)) or stays inactive (\(S(t) = 0\)). The normalized net demand from all traders is then given as \(M_t = \frac{1}{N} \sum _{i=1}^N S_i(t)\), and the price adjusts as \(p_{t+1} = \frac{1 + M_t}{1 - M_t} p_t\). An agent’s decision to buy/sell or staying out depends on the perceived mispricing between the current price \(p_t\) and its running average \(p^*_t = \langle p_t \rangle _{\tau }\) which is considered as a proxy for the fundamental price of the asset. The probability of an agent to trade is then given byand a trading agent buys \(S_i(t) = 1\) or sells \(S_i(t) = -1\) at random with equal probability.

In order to obtain a statistical model of volatility, in particular with a continuous latent state as required for HMC sampling, we have adapted the model as follows:where we have used that \(\log (1 + x) \approx x\) for \(|x| \ll 1\).

Overall, we arrive at the following model dynamics⁶Note that this is a state-space model with a continuous latent state driving the time-varying volatility \(\sigma _{t+1} = \sqrt{\sigma _{\max }^2 \cdot 4 \mathbb {P}(|S(t)| = 1)}\). Indeed, the famous GARCH(1, 1) (generalized auto-regressive conditional heteroscedastic) model (Bollerslev 1986) is of a similar formThe main difference between the VS (in our formulation) and the GARCH model is that the volatility is a function of past prices in the former and past returns in the latter model. Furthermore, due to being founded in an agent-based model all parameters of the VS model are readily interpretable as the sensitivity \(\mu \) of the agents to mispricing and the weighting \(\tau \) of the running price average. In contrast, parameters in the GARCH model are motivated purely from statistical grounds and cannot easily be related to agent behaviors.

From a Bayesian perspective, Eqs. (2) and (3) correspond to the likelihood \(p(\varvec{x}| \varvec{\theta })\), i.e., the conditional probability of the observed data given the model parameters. To complete the model density \(p(\varvec{x}, \varvec{\theta })\), we need to specify a prior distribution on the parameters. The choice of a prior distribution is often considered as subjective (whereas the likelihood has an aura of objectivism). Arguably, from the perspective of modeling the observed data this distinction is of limited relevance. Instead, note that fixing the prior implicitly fixes a distribution on the data space, i.e., obtained as \(p(\varvec{x}) = \int p(\varvec{x}, \varvec{\theta }) \hbox {d}\varvec{\theta }\) by marginalizing over the parameters. A model can be considered as misspecified when it assigns very low probability to the actual observed data. In contrast, a good model should be able to generate similar data with reasonable probability. This viewpoint is in line with Gelman et al. (2017) who argue that the prior can only be understood in the context of the likelihood. Indeed, prior and likelihood act together in shaping the model and expressing our expectations of plausible data.

Here, we propose the use of (weakly) informative priors which take into account our knowledge about the role played by the parameters when generating data from the likelihood model. As an example, consider the parameter \(\tau \in (0, 1)\) of Eq. (2). While it might be natural to simply assign a uniform prior,⁷\(\tau \) controls the time constant of the running price average, i.e.,as \(\tau ^k \rightarrow 0\) for \(k \rightarrow \infty \). Comparing this to an exponentially weighted average in continuous time, i.e.,with time constant \(\rho ^{-1}\) and exponential weighting kernel \(k(s) = \rho e^{- \rho s}\) of unit weight, i.e., \(\int _0^\infty k(s) \hbox {d}s = 1\), we match \(\tau ^k\) with \(e^{- \rho k}\), thus interpreting \(- \frac{1}{\log \tau }\) as the time constant l of the running average. A \(\hbox {Uniform}(0, 1)\) prior on \(\tau \) then corresponds to an \(\hbox {Inv-Gamma}(1, 1)\) on \(l = - \frac{1}{\log \tau }\) putting considerable probability mass on very short time constants below 1 day. In order to obtain a more reasonable distribution, we parameterize the model directly in terms of the time constant l and give it an \(\hbox {Inv-Gamma}(2, 1000)\) prior with a mean of 1000 and unbounded variance. Similarly, \(\mu \) has been given a \(\hbox {Gamma}(3, 0.03)\) which assigns more than \(95\%\) of its probability mass to the interval [20, 250]. Together, these priors inform the VS model to stay away from the boundary \(\mu \rightarrow 0\) or \(\tau \rightarrow 0\) where it becomes trivial, i.e., \(\mathbb {P}(|S_t| = 1) \equiv 1\) and thus \(\sigma _t \equiv \sigma _{\max }\). Accordingly, it cannot be expected to generate data exhibiting pronounced volatility clustering in this case, and indeed, in the original reference Vikram and Sinha (2011), prices were averaged over \(10^4\) time steps, which are well covered by the chosen prior. Implementing both models is straightforward in Stan, and the full code is provided as supplementary material.

Franke & Westerhoff have developed a series of models and have estimated them by moment matching (Franke and Westerhoff 2012, 2011). Here, we follow their presentation and introduce the DCA–HPM model in their terminology.

In the FW model, the market is populated with two types of agents, namely fundamental and chartist traders. The fraction of fundamental traders at time step t is denoted by \(n_t^f \in [0, 1]\). The corresponding fraction of chartist traders is then given by \(n_t^c = 1 - n_t^f\). The log price, denoted by \(p_t\), adjusts to the average demand from fundamental \(d^f\) and chartist \(d^c\) traders asThe demand is composed of a deterministic and stochastic component. It is assumed that fundamental traders react to mispricing, i.e., the difference between \(p_t\) and the (known) fundamental price \(p^*\), whereas chartist traders react to past price movement, i.e., \(p_t - p_{t-1}\). According to Franke and Westerhoff (2012), the demand dynamics is modeled aswith parameters \(\phi , \xi > 0\) specifying the sensitivity to price differences for the fundamental and chartist traders. Note that these demands are unobserved as only their weighted sum effects the price. While such a dynamics could be modeled by means of a stochastic latent state, in the present case it is possible to marginalize out the demand. As the sum of two normally distributed random variables is again normal, the combined demand gives rise to a stochastic model for the log return \(r_t = p_t - p_{t-1}\)The volatility \(\sigma _t = \mu \sqrt{(n_{t-1}^f)^2 \sigma _f^2 + (n_{t-1}^c)^2 \sigma _c^2}\) now depends on the fraction of chartist vs fundamental traders and changes over time. Franke & Westerhoff (2012) call this structured stochastic volatility, in analogy with structural models in economics, as the parameters of the agent-based model are grounded in behavioral terms and therefore economically meaningful.

The model is then completed by an update equation for the fraction of traders in each group. Here, we consider the DCA–HPM specification of Franke and Westerhoff (2012) which is given byThe parameter \(a_t\) denotes the relative attractiveness of the fundamental over the chartist strategy. It includes a general predisposition \(\alpha _0\) and herding \(\alpha _n > 0\) as well as mispricing \(\alpha _p > 0\) effects. We chose this specification for two reasons: Overall, the model dynamics is fully specified by Eqs. (5), (6) and (7). The parameters of the model are given by \(\varvec{\theta }^{FW} = (\mu , \phi , \sigma _f, \xi , \sigma _c, \beta , \alpha _0, \alpha _n, \alpha _p, p^*)\). Note that \(\beta \) and \(\mu \) are redundant as they simply control the scale of \(\alpha _0, \alpha _n, \alpha _p\) and \(\xi , \phi , \sigma _f, \sigma _c\), respectively. Thus, throughout we fix them at \(\beta = 1\) and \(\mu = 0.01\) as in the simulation exercise of Franke and Westerhoff (2012).

When estimating the model on real stock returns below, we do not know the fundamental price. When simulating from the model or matching moments as in Franke and Westerhoff (2012), the fundamental price can be considered as fixed. Yet, when fitting the model to actual return data more flexibility in order to estimate reasonable values for the unobserved fundamental price is needed. Here, we consider two specifications of the model: First, as in the VS model presented above, the fundamental price is derived as an average of past prices. Note that in order to be faithful and comparable to the VS model, we compute a running average of past prices and not log prices. Secondly, following Lux (2018), we assume that the log fundamental price is time varying as a Brownian motionThis not only introduces another parameter \(\sigma _*\) but also turns the model into a stochastic volatility model, i.e., the volatility \(\sigma _t\) now includes a stochastic component. To see this, note that \(\sigma _t\) depends on \(a_{t-2}\) via \(n_{t-1}^f\) and the attractiveness in turn includes the stochastic fundamental log price \(p^*_{t-2}\). Thus, the fundamental price plays a similar role as the log volatility \(h_t\) in a classical discrete time stochastic volatility (SV) model (Kim et al. 1998)that we include as a benchmark alongside the GARCH(1, 1) model. Again, in contrast to the purely phenomenological dynamics in Eq. (8) the parameters of the FW model are interpretable in terms of behavioral trades of agents. Furthermore, the model specification combines aspects of local and stochastic volatility in that its volatility \(\sigma _t\) depends on the random fundamental price as well as past prices via Eqs. (6) and (7).

Nevertheless, implementing the model in Stan is readily possible. As before, the full code of the FW model—for both specifications—is provided as a supplementary material. Interestingly, along the same lines a variant of the VS model with a random walk specification for the fundamental price can be defined. For comparison, we have also implemented this model using both specifications. Note that the time-varying log fundamental prices \(p^*_t\) do not appear as a T-dimensional vector (where T denotes the number of observed time steps) in the parameter block. Instead, we have used a non-centered parameterization, i.e.,  is computed from as a transformed parameter. Formally, we can express this as follows:instead ofThis is a standard example of a reparameterization which does not change the model, but helps when HMC sampling as the innovation parameters \(\epsilon _t^*\) all have unit scale and are a priori independent, no matter which variance \(\sigma _*^2\) is currently sampled.

Again, we complete the model with weakly informative priors for all parameters. As few insights are available about the proper choice of the attractiveness parameters \(\alpha _0, \alpha _n\) and \(\alpha _p\), we assign weakly informative priors, e.g., , which restrict the scale of the parameter yet, being heavy-tailed, allow substantially larger values.⁸ In case of the standard deviation parameters \(\sigma _f, \sigma _c\) and \(\sigma _*\), we impose stronger priors and resort to the observed data to set the proper scale. While not being purely Bayesian, this choice restricts the model to reasonable scales accounting for the fact that volatility could be measured in arbitrary units, e.g., percent per year. Figure 1 illustrates prior predictive checks, i.e., sample data generated according to the model with parameters drawn from the prior, for the FW model using a moving average specification for the fundamental price.⁹ While the scale of the data appears well matched, the model only occasionally produces volatility clustering as pronounced as in the real data. Nevertheless, we found these priors effective in simulation studies as well as when estimating the model on stock data. Furthermore, Appendix C contains additional robustness checks confirming the above rationality behind our choice of priors.

Alfarano, Lux & Wagner model a financial market populated by N chartist traders as well as additional fundamental traders. The excess demand from the population of fundamental traders is given bywith total trading volume \(T_f\) and log fundamental price \(p_f\). As in the FW model, prices are exclusively considered in logarithmic terms, i.e., to simplify notation the log price is denoted by p.

Chartists traders are either in an optimistic or pessimistic state. Optimists are assumed to buy a certain amount \(T_c\) at each time step, while pessimists sell amount \(T_c\). Denoting the number of optimistic traders as n, the market sentiment \(x = 2 \frac{n}{N} - 1 \in [-1, 1]\) is defined and the excess demand from chartist traders given asThen, assuming a Walrasian pricing mechanism log market prices are adjusted asNote that the FW model assumed a very similar log price adjustment, albeit formulated in discrete terms in Eq. (4).

Lux (2018) now assumes instantaneous price adjustment, i.e., \(\beta \rightarrow \infty \), and the market is cleared by matching fundamental and chartist demand. In this case, the market price is found to beand the corresponding returns can be expressed aswhere the last line follows when assuming a Brownian motion for the log fundamental price, i.e., \(p_{f,t} = p_{f,t-1} + \sigma _f \epsilon _{f,t}\) with \(\epsilon _{f,t} \sim \mathcal {N}(0, 1)\). From a statistical perspective, market returns are distributed asIn contrast to the FW model, this model does not exhibit stochastic volatility. Furthermore, we do not need to model the fundamental price. Indeed, assuming instantaneous price adjustment market prices cannot deviate persistently from the fundamental price and it becomes essentially an observed quantity (compare Eq. 14).

Now, the market sentiment changes according to a herding process originally proposed by Kirman. Here, each agent randomly switches from pessimistic to optimistic or vice versa with transition rates \(\pi ^+ = a + b n\) and \(\pi ^- = a + b (N - n)\), respectively. Parameter a expresses a general tendency to switch state, while b models the herding with the switching probability increasing in the size of the contrarian population. As detailed in Alfarano et al. (2008), these transition rates lead to a sentiment dynamics with nonvanishing fluctuations even in the limit of an infinite population. The sentiment dynamics is then governed by the following Langevin equation:In our implementation, the continuous time equation (Eq. (16)) has been Euler-discretized with \(\Delta t = 1\) day, i.e., \(x_t = - 2 a x_{t-1} + \sqrt{2 b (1 - x_t^2)} \; \epsilon _t\), where \(\epsilon _t \sim \mathcal {N}(0, 1)\), and then coupled to the observed returns via Eq. (15). As in Lux (2018), we have fixed \(\frac{N T_c}{T_f}\) as one and assume weakly informative truncated standard normal priors on the remaining parameters a, b and \(\sigma _f\). As illustrated in Fig. 1, the model produces return time series similar to the FW model, albeit at a somewhat larger scale. It appears that a prior favoring smaller values for \(\alpha \) and \(\beta \) would be beneficial. The prior chosen here provides a compromise between such an informative prior and the uniform one suggested in Lux (2018) which with high probability exhibits almost independent returns without clustering (not shown). Again, the full Stan code for the ALW model is provided as a supplementary material.

In order to check our model implementation, we simulated the FW model with parameters as given in table 1 of Franke and Westerhoff (2012), i.e., \(\mu = 0.01, \beta = 1, \phi = 0.12, \xi = 1.50, \alpha _0 = -0.327, \alpha _n = 1.79, \alpha _p = 18.43, \sigma _f = 0.758, \sigma _c = 2.087\) and \(p^*\) being a moving average of past prices with a length scale of \(l = 300\). Then, we re-estimated the model parameters on the simulated price series of \(T = 2000\) time steps shown in Fig. 2.

The resulting samples from the posterior distribution are shown in Fig. 3. Overall, we have run four chains starting from independent random initial conditions and drawn 400 samples from each, after discarding an initial transient of another 400 samples as warm-up. Compared to other studies, the number of samples appears very low, but the high quality of the samples is clearly visible in the trace plots. The model appears to have converged after just about 50 samples, and all chains produce almost uncorrelated samples from the same distribution. This is also confirmed by standard convergence diagnostics such as Gelman & Rubin’s \(\hat{R}\), which compares the variance between and within chains, or the number of effective samples, which is based on the sample autocorrelation (not shown). If desired, more samples can easily be drawn as the shown estimation runs in a few minutes on a standard laptop.

Figure 4 shows the resulting posterior distributions together with the true parameters that generated the data. We found that about at least 1000 observed prices are necessary for the posterior to reliably cover the true parameters. Interestingly, preliminary runs on considerably longer time series of 5000 observations suggest that posterior uncertainty reduces only slightly. This suggests that the model is rather flexible with different parameter settings giving rise to similar return dynamics. Accordingly, the information that observed returns provide about the underlying parameters is limited. Appendix B provides further discussion of this point in the context of the GARCH model.

Indeed, for the even more flexible models where the fundamental price is assumed to follow an (unobserved) random walk, some parameters could not be recovered at all from simulated data. Figure 5 shows the actual and inferred latent sentiment dynamics on data simulated from the ALW model. Estimates are shown as the posterior mean together with the 95% credibility bands around it. Here, only one chain has successfully recovered the actual sentiment time series, whereas another chain exhibits short excursions away from the actual sentiment. As the probability of both chains is vastly different, with the correct one being substantially higher, this appears to be a problem of the sampling algorithm which is getting stuck in a local mode of the posterior. It might well be that other algorithms, such as the sequential Monte Carlo methods employed by Lux (2018), are less susceptible to this problem. On the other hand, HMC is highly effective in sampling the global parameters of the model which is a major bottleneck for sequential Monte Carlo methods (Livingstone et al. 2019; Monnahan et al. 2017).

Furthermore, the FW model with the random walk specification for the fundamental price shows an even deeper non-identifiability. The top panel of Fig. 6 compares the estimated fraction of fundamental traders for two different chains of samples to the actual simulated time series. While one chain stays close to the actual values, the other estimate appears to be a mirror image. Interestingly, in this case, the likelihood of both chains is almost identical. Indeed, the lower panel of Fig.  6 shows that the estimated volatilities are almost identical as well and correspondingly both estimates assign very similar probability to the observed returns. This suggests a multimodal posterior where each chain samples from a well-defined, yet different, mode of the distribution. Furthermore, in at least one of the modes we find that \(\sigma _c < \sigma _f\)!

Accordingly, the model offers two very different explanations for the observed volatility dynamics. In the first scenario, the number of chartists is usually low and rises sharply in volatile market phases (as intended by the model). In contrast, in the second scenario the number of chartists is usually high and drops in volatile market phases. Volatility is then driven by the high demand uncertainty of fundamental traders. Interestingly, both scenarios lead to very similar estimates and predictions for the volatility \(\sigma _t\). The model accomplishes this by assuming a very different trajectory for the unobserved Brownian motion of the fundamental price leading in turn to very different mispricings and demands of the fundamental traders. Thus, the seemingly innocuous assumption that the fundamental price follows a Brownian motion apparently introduces a symmetry into the model. This not only makes the unobserved latent states of the model unidentifiable, but also reveals that our understanding of agent-based models and their generated time series dynamics is far from complete. In particular, we cannot use the model in order to unambiguously uncover the fundamental price as in Majewski et al. (2018) where the Kalman filter computes a uni-modal posterior approximation. Furthermore, the herding parameters are hard to interpret as they act differently in different modes. Further work is needed in order to characterize and ideally remove this unidentifiability.

Thus, currently identification of some of the models discussed here is plaqued by two major issues: on the one hand, of algorithmic nature with the sampler getting stuck in poor local optima of the posterior density; on the other hand, multimodality of the model likelihood leading to several posterior modes with vastly different interpretation for the model parameters. Here, we leave such investigations for future studies and instead focus on model predictions. In particular, a detailed comparison of the efficiency, e.g., in terms of effective sample size per wall clock time, and reliability of different algorithms for volatility models with complicated posterior geometries is beyond the scope of this paper. Instead, we focus on the models itself and continue with a principled model comparison in the next section.

To circumvent these identification issues, we now focus on predictions where identifiability is of no concern. In order to also circumvent the problem of the sampler getting stuck in a local minimum, we run several Markov chains for each model and only use the chain with the highest in-sample likelihood for predicting. Then, we compare models based on their probability assigned to held-out data. In the context of time series, models are commonly compared based on rolling look-ahead predictions, i.e., using the predictive distribution for future returns \(r_{T + \tau }\) or volatilities \(\sigma _{T + \tau }\) based on the T previous time points. Time is then rolled forward, and the model is evaluated again. In general, this approach requires refitting the model for each prediction.

Here, for computational reasons, we instead resort to leave-one-out (LOO) predictions, i.e., predicting current returns in the context of past and future returns, excluding the current one. With the method of Pareto smoothed importance sampling (PSIS), the corresponding predictive likelihoodscan be estimated from posterior samples (Vehtari et al. 2017). Note that in contrast to the full posterior, the LOO likelihood is conditioned on all but the ith data point. In order to estimate the LOO likelihood from posterior samples, the ith data points need to be effectively removed before evaluating the prediction. In general, this is far from trivial and we refer the reader to Vehtari et al. (2017) for details about how PSIS estimates LOO likelihood. Here, we just note that the method smoothens the estimate by fitting a generalized Pareto distribution to the tail of observed log likelihoods. Thereby, a more stable estimate as well as a useful diagnostics is obtained. In particular, large tail exponents in the Pareto distribution suggest that estimates could have unbounded variance and should not be trusted. Indeed, this happens especially for influential data points and no reliable estimate can be obtained in this case.

At least for such data points, results need to be validated by direct estimation of the predictive likelihood, i.e., refitting the model on some of the data points and predicting the held-out ones. To this end, we also implemented a direct variant of K-fold cross-validation where the data are partitioned into K parts. LOO then corresponds to N-fold cross-validation which would require fitting the model N times on all but one data point and predict the left-out one. Here for computational reasons, we have chosen to hold out every 20th data point instead. The model is then fitted 20 times, on all but the remaining data points and asked to predict the unobserved ones. (Figure 7 shows the resulting schema for \(K = 3\).) The choice \(K = 20\), i.e., to hold out every 20th data point, provides a good compromise between computational performance (low K) and almost independent predictions (high K). Using information theory, it has been shown that volatility is only weakly dependent across a wide range of stochastic volatility models (Pfante and Bertschinger 2019). Thus, we can reasonably expect that volatilities are nearly independent after a couple of weeks. Numerical results in the next section confirm this assumption with estimates based on LOO and 20-fold cross-validation being statistically indistinguishable.

Accordingly, we have implemented all models in a way that allows for unobserved or missing returns. To this end, we pass in all returns \(r_1, \ldots , r_T\), observed and unobserved, together with a binary vector indicating at each time whether the return is considered missing or not. From a Bayesian perspective, missing data points are simply unobserved latent variables that can be inferred together with other latent variables/parameters. Thus, for each missing return a corresponding parameter \(\epsilon _{\text {miss}}\) is introduced and sampled alongside the other model parameters. To enhance sampling efficiency, the missing returns are represented in terms of innovations which are then transformed to actual returns via \(r_t = \mu _t + \sigma _t \epsilon _{\text {miss}}\) akin to the non-centered parameterization explained in Eq. (9). As the model density is defined on \(r_t\) and not the underlying parameters \(\epsilon _{\text {miss}}\), we need to account for the resulting change of measure by multiplying the density with the absolute Jacobian \(\frac{\partial r_t}{\partial \epsilon _{\text {miss}}}\) of the transformation.

Finally, we have fitted all models on price data from the S&P 500 stock market index. As a benchmark, a standard GARCH(1, 1) and SV model have been included for comparison. The corresponding fits from January 2009 to December 2014 and January 2000 to December 2010 are shown in Figs.  8, 9, 10, 11 and 12. The estimated model volatility is overlaid on the actual market returns.

Volatility estimates are shown as the posterior mean together with the \(95\%\) credibility bands around it. The posterior of the volatility \(\sigma _t\) at time step t is based on data points from returns \(r_t\) observed over all T data points, i.e., \(p(\sigma _t | r_1, \ldots , r_T)\) for \(t = 1, \ldots , T\). In the terminology of time series models, this is known as the smoothing distribution. Comparing the volatility estimates and predictions of the different models, a few remarks are in order:Having compared the models graphically, we proceed with a more principled assessment of model fit. In particular, Table 1 contains the predictive log likelihoods estimated using LOO and 20-fold cross-validation, respectively. As explained above, each model has been fitted using 8 randomly initialized chains and predictions are based on the best chain only. As expected, for the GARCH and SV model the estimates of predictive likelihood based on LOO and 20-fold CV are statistically indistinguishable. Similarly, the estimates for the VS and FW model using a random walk specification for the fundamental price are very similar.

For the other models, the situation is slightly different for several reasons:As the LOO estimates are unreliable in some case, we focus on the cross-validation results in the following. Tables 2 and 3 provide a more detailed pairwise comparisons between all models. Here, for each model combination the difference in total predictive log likelihood and the standard deviation of this difference is given. Note that mean differences are consistent with Table 1, but the standard deviations are computed specifically for each pairwise comparison. To this end, model predictions are compared pointwise for each data point predicted by both models to obtain mean and standard deviation of likelihood differences directly. For visual convenience, each column shows the advantage of the column model over the row model. The only models which are never significantly worse than any other model, i.e., with a likelihood difference of two standard deviations below zero, across both considered data periods are FW_walk and VS_walk. All other models are significantly outperformed by some model over some time period. In particular, the GARCH model is outperformed by every other model except for the ALW model on data including the crisis year of 2008. Thus, the best agent-based models beat this standard benchmark model and are on par with a simple SV model. This is encouraging as such models are routinely fitted to volatility data and shows that agent-based models provide viable alternatives with several advantages. First, in contrast to purely phenomenological SV models they have interpretable parameters related to trader behavior, allowing to uncover market sentiment or trader strategies over time. Secondly, they not only model the conditional volatility dynamics, but also provide non-trivial predictions for conditional returns. (Compare the return distributions of Eqs. (3) and (5) for instance.) We leave investigations to which extent conditional predictions of such models are able to identify trends and rallies in stock prices to future work.

To get an even better understanding of the relative performance between several models, Figs. 13 and 14 provide a running comparison between the best agent-based model, FW_walk and the SV as well as the ALW model, both within sample and on predicted data points. The top panel of Fig. 13 shows the cumulative advantage cum_adv\(^{\hbox {FW}/\hbox {SV}}_t\) of the FW_walk model over the SV model within sample log likelihood, i.e.,The lower panel shows the corresponding cumulative log likelihood difference on predicted data points. The estimates of predictive log likelihood are obtained by combining data from all 20 folds, i.e., predictive likelihoods of time points 1, 21, ...are obtained from fold 1 which held out the corresponding returns, predictive likelihoods of time points 2, 22, ...are obtained from fold 2 and so on. As before, the mean difference is shown alongside with the 95% credibility band around it. The last value of the lower panel at time T corresponds to the total predictive log likelihood difference and is recorded in Table 3. Interestingly, the SV model is better in-sample with a negative cumulative advantage of the FW model of about \(- 20 \pm 5\). Out-of-sample, i.e., on the predicted data points, the situation is reversed with the FW model now being better for a total cumulative advantage of \(17 \pm 5\). Thus, the FW model is less flexible in fitting each data point in sample, yet providing better predictions. By Occam’s razor, such a model should be preferred being both simpler—in statistical terms of effective parameters—yet more reasonable in terms of predicting the actual dynamics.

Figure 14 shows the corresponding comparison between the FW and the ALW model. Here, the strong overfitting of the ALW model is apparent with a huge advantage in-sample (upper panel), but much weaker predictions (lower panel). Interestingly, most of the predictive performance is lost during the year 2008 of the financial crisis. This suggests that the ALW model is unable to accurately reflect the unusual return dynamics unfolding during the financial crisis. Indeed, on the data sample from 2009 to 2014 excluding the critical year its predictive performance is on par with the other models as noted in the last column of Table 2. We take this as strong evidence that the assumption of instantaneous price adjustment is problematic and especially during the crisis persistent mispricing has to be assumed.