Estimation of Sentiment Effects in Financial Markets: A Simulated Method of Moments Approach

Abstract

We take the model of Alfarano et al. (J Econ Dyn Control 32:101–136, 2008) as a prototype agent-based model that allows reproducing the main stylized facts of financial returns. The model does so by combining fundamental news driven by Brownian motion with a minimalistic mechanism for generating boundedly rational sentiment dynamics. Since we can approximate the herding component among an ensemble of agents in the aggregate by a Langevin equation, we can either simulate the model in full at the micro level, or via an approximate aggregate law of motion. In the simplest version of our model, only three parameters need to be estimated. We explore the performance of a simulated method of moments (SMM) approach for the estimation of this model. As it turns out, sensible parameter estimates can only be obtained if one first provides a rough “mapping” of the objective function via an extensive grid search. Due to the high correlations of the estimated parameters, uninformed choices will often lead to a convergence to any one of a large number of local minima. We also find that the efficiency of SMM is relatively insensitive to the size of the simulated sample over a relatively large range of sample sizes and the SMM estimates converge to their GMM counterparts only for large sample sizes. We believe that this feature is due to the limited range of moments available in univariate asset pricing models, and that the sensitivity of the present model to the specification of the SMM estimator could carry over to many related agent-based models of financial markets as well as to similar diffusion processes in mathematical finance.

Appendices

Appendix 1: Pseudo Code

Below is the pseudo code for the SMM Monte Carlo simulations.

Algorithm: SMM Monte Carlo simulations
1: for i = 1 to 400, do
2:       Set random seed 1: seed1(i).
3:       Generate pseudo-empirical time series with sample size $T_{emp}$.
4:       Set random seed 2: seed2(i).
5:       Set weighting matrix W1 equal to identity matrix.
6:       Grid search to find 10 best parameter sets $\theta _{k}, k=1, \ldots , 10$.
7:       for k = 1 to 10, do
8:             Based on $\theta _{k}$, simulate time series with sample size $T_{sim}$ and calculate weighting matrix W2                  using Newey West estimator.
9:             Minimize the objective function to obtain the optimized parameter $\widehat{\theta }_{k}$.
10:       end for
11:       Output the best optimized parameter set among the 10 optimized $\widehat{\theta }_{k}$.
12: end for

Appendix 2: Tests for Stationarity and Ergodicity

We follow the runs test procedure of Grazzini (2012) to test the stationarity and ergodicity of the sentiment dynamics and raw returns from the agent based model as well as the generated moments. For the stationarity test, we generate a time series with sample size 2,000,000 and adopt the runs test procedures with different test window lengths 1, 10, 50, 100, 500, 1000, 5000 and 10,000. We repeat the test for 100 runs. As reported in Table 8, under the bimodal setting, the probabilities of rejecting stationarity at the level of 5% are much higher than 0.05 for the second and fourth moment and the autocovariances of squared returns for small test windows. Increasing the test window, probabilities of rejection of stationarity decrease towards their nominal size. It can be concluded that sentiment, raw returns as well as all the moment conditions are stationary but for the second and fourth moment and autocovariances of squared returns, the long-range dependence of volatility makes smaller samples appear non-stationary. Similar results can be found under the unimodal setting.

Table 8 Stationarity tests with sample size of 2 million for different sizes of test windows

Table 9 Ergodicity tests for different sample sizes

For the ergodicity test, we generate time series with different sample sizes 5000, 10,000, 20,000, 100,000 and 200,000. We divide these time series into 100 sub-samples to create the first test sample. Matching the sample size of these sub-samples, we also generate 100 time series (with different random seeds) to create the second test sample. The two test samples are used to test the ergodicity property of the sentiment index $x_{t}$, raw returns and the 15 moment conditions. We again repeat the test 100 times. Table 9 reports the result of the ergodicity tests. Regardless of the parameter values, the number of rejections is smaller than the nominal size of the test for raw returns and all the moment conditions. For sentiment, we find sizable numbers of rejections for smaller sample size reflecting the strong autocorrelation of this variable.