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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Modeling financial time series at different time scales is still an open challenge. The choice of a suitable indicator quantifying the distance between the model and the data is therefore of fundamental importance for selecting models. In this paper, we propose a multiscale model selection method based on the Jensen–Shannon distance in order to select the model that is able to better reproduce the distribution of price changes at different time scales. Specifically, we consider the problem of modeling the ultra high frequency dynamics of an asset with a large tick-to-price ratio. We study the price process at different time scales and compute the Jensen–Shannon distance between the original dataset and different models, showing that the coupling between spread and returns is important to model return distribution at different time scales of observation, ranging from the scale of single transactions to the daily time scale.

The complexity of market behavior has fascinated physicists and mathematicians for many years [

A specific challenge is the modeling of how the return distribution changes at different time scales [

In this work, we propose to perform multiscale model selection for financial time series by using the Jensen–Shannon distance [

The paper is organized as follows. In Section 2, we illustrate the definitions of the Jensen–Shannon divergence and distance, and we characterize the unavoidable bias, due to the finiteness of the data sample. In Section 3, we illustrate the statistical models of mid-price and spreads dynamics developed in [

Distance or divergence measures are of key importance in a number of theoretical and applied statistical inference and data processing problems, such as estimation, detection, compression and model selection [

Let _{KL}_{KL}_{L}_{L}_{1}, _{2} ≥ 0, _{1} + _{2} = 1 are the weights of the probability distributions, _{L}_{JS}_{1} = _{2} = 1/2. Endres _{L}_{JS}

In this paper, we are interested in using the Jensen–Shannon distance as a method for selecting among a set of models the one that best describes a given dataset. We are concerned with the case when our data is represented by a discrete time series of length

To be more specific, consider the random variable, _{1}, ⋯, _{k}_{1}, ⋯, _{k}_{1}, ⋯, _{k}_{i}_{i}_{1}, ⋯, _{k}_{1}/_{k}/N_{m}_{m}_{m= 1} = _{m}_{m}

In order to select the best model that describes the data at all aggregation scales, we compute the Jensen–Shannon distances for various values of _{JS}_{m},_{m}_{m}_{JS}_{m}_{m}_{JS}_{1,}_{m}_{2,}_{m}_{1} = _{2} =

It is important to stress that even if we knew the _{m}_{JS}_{m}_{m}_{m}_{JS}_{JS}_{JS}_{m}_{m}_{JS}_{1,}_{m}_{2,}_{m}

The concept of a systematic bias of the numerical values of Jensen–Shannon divergence, _{JS}^{2}) involves the unknown probabilities _{1}, ⋯, _{k}

Grosse _{JS}_{1}, _{2}) between two i.i.d. sequences of length _{JS}

In this section, we present a toy example of the use of Jensen–Shannon distance for model selection. The purpose of the section is mostly didactical and serves to show the multiscale procedure and the issues related to the finiteness of the sample that will be present also in the real financial case described in the next section.

Let us consider a process, which at scale _{m= 1} is described by _{B}_{B}_{m}_{m,1}, ⋯, _{m,k}_{B}_{m}_{m}_{m,1}/_{m}_{m,k}_{m}_{m,i}

The probability distribution of empirical frequencies is given by the multinomial distribution:

In principle, one can compute exactly the moments of the distances, _{JS}_{m}_{m}

The computational problem with these expectations are the values of ^{5}.

To handle this problem, we compute these expectations by means of Monte Carlo simulations, and we replace ensemble averages with sample averages, _{r}

We first consider the problem of the finite sample bias in the computation of the Jensen–Shannon divergence and distance. Specifically, we compute
^{2}, 10^{3}, 10^{4}, 10^{5}, 10^{6}, as reported in

As expected, the bias decreases with ^{e}^{6} gives ^{−7}, ^{−7}.

In the case of the Jensen–Shannon distance, we do not have any analytic result and limit ourselves to a power-law fit of the initial part of the curve. For the case ^{6}, the fit gives ^{−3},

In order to illustrate how to perform model selection with the Jensen–Shannon distance, we consider the case of an (artificial) sample generated from the binomial model with _{B}_{B}

Specifically, suppose that
_{1}, _{2}, ⋯, we generate synthetic samples of length

In this section, we use the above multiscale procedure, based on the Jensen–Shannon distance, in order to select the best statistical model in the particular case of models describing the high frequency price dynamics of a large tick asset. The models used here were introduced by Curato and Lillo [

In financial markets, there are two important prices at each time _{ASK}_{BID}_{ASK}_{BID}_{mean}_{ASK}_{BID}_{ASK}_{BID}

We compare three different models for price dynamics in transaction time proposed in [_{B}

It is well known that the spread process, _{2,2} (ℝ):
_{4,4} (ℝ):
_{11}, _{21}, _{1}, _{4}, that can be estimated from the data.

_{B}_{B}_{B}_{B}_{1}, _{4}.

For the next section, it is useful to quantify the number of possible states of the variables,

Our problem is now how to select the model that reproduces the data better. We focus our selection problem on the ability of the models to reproduce the price-change process at different time scales. The selection problem does not involve the spread process,

We study this selection problem by means of the concepts developed in Section 2.1. First, we compute the Jensen–Shannon distance between two realizations of the real process. To this end, we divide the sample into two non-overlapping samples, each of length _{JS}_{r}_{r}_{B}_{B}_{B}

Our analysis has been performed by using the Jensen–Shannon distance. However, other distances between probability distributions exist, such as the Kolmogorov–Smirnov distance [

One important issue for the study of price dynamics is the selection and validation of a statistical model against the empirical data. Usually, financial time-series models that work well at a fixed time scale do not work comparably well at different time scales. The Jensen–Shannon distance analysis that we have performed enables us to perform an accurate test of goodness of our statistical models and to select among a pool of competing models. We have performed the same model selection procedure with different statistical distances. We find that their power to discriminate between different competing models is not larger than that of Jensen–Shannon distance. Moreover for the Jensen–Shannon distance, we have a good control of the finite sample properties.

Our analysis demonstrates that, for large tick assets, the coupling between mid-price dynamics and spread dynamics is important to account for the mid-price dynamics from the time scale of a single transaction to the time scale of one trading day.

We believe that the described method, based on the Jensen–Shannon distance, could be used also in contexts different from the financial one investigated in this work. This method could be useful each time we want to perform a multiscale test for a model against the empirical data samples.

The authors acknowledge partial support by the grant, SNS11LILLB “Price formation, agents heterogeneity, and market efficiency”.

The authors declare no conflict of interest.

Expectations and standard deviations of the Jensen–Shannon distance between two samples of the binomial model with the same parameter _{B,1} = _{B,2} = 0.5 (red squares) and with different parameters (green diamonds and blue triangles). The black circles are an estimation of the Jensen–Shannon distance between a sample and the true model.

Dynamics of the mid-price, _{mean}

Mean and standard deviation of _{JS}_{B}_{JS}