Memory property in heterogeneously populated markets

This paper focuses on the long memory of prices and returns of an asset traded in a nancial market. We consider a microeconomic model of the market, and we prove theoretical conditions on the parameters of the model that give rise to long memory. In particular, the long memory property is detected in an agents' aggregation framework under some distributional hypotheses on the market's parameters.


Introduction
During last years quantitative studies of nancial time series have shown several interesting statistical properties common to many markets. Among the others, long memory is one of the most analyzed. This concept raised by time series empirical analysis in terms of the persistence of observed autocorrelations. The long memory property is ful lled by a time series when the autocorrelation decays hyperbolically as the time lag increases. Therefore, this statistical feature is strongly related to the long run predictability of the future phenomenon's realizations.
Long memory models were introduced in the physical sciences since at least 1950, when some researches in applied statistics stated the presence of long memory within hydrologic and climatologic data. The earliest studies on this eld are due to Hurst (1951Hurst ( , 1957, Mandelbrot and Wallis (1968), Mandelbrot (1972), and McLeod and Hipel (1978) among others.
In this paper a theoretical microeconomic structural model is constructed and devel-oped. We rely on time series of assets traded in a nancial market and we address the issue of giving mathematical proof of the exact relation between model parameters evidencing the presence of long memory.
The literature on structural models for long-memory is not wide. Some references are Willinger et al. (1998), Box-Steffensmaier and Smith (1996), Byers et al. (1997), Tschernig (1995), Mandelbrot et al. (1997). The keypoint of the quoted references is to assume distributional hypotheses on parameters of models in order to detect the presence of long memory in time series.
We adopt the approach of the structural model of Kirman and Teyssiere (2002) is based on the assumption that the market is populated by interacting agents. The interaction among agents leads to an imitative behavior, that can affect the structure of the asset price dynamics. Several authors focus their research on describing the presence of an imitative behavior in nancial markets (see, for instance, Avery and Zemsky (1998), Chiarella et al. (2003), Bischi et al. (2006)).
The traditional viewpoint on the agent-based models in economics and nance relies on the existence of representative rational agents. Two different behaviors of agents follow from the property of rationality: rstly, a rational agent analyzes the choices of the other actors and tends to maximize utility and pro t or minimize the risk.
Secondly, rationality consists in having rational expectations, i.e. the forecast on the future realizations of the variables are assumed to be identical to the mathematical expectations of the previous values conditioned on the available information set.
Thus, rationality assumption implies agents' knowledge of the market's dynamics and equilibrium, and ability to solve the related equilibrium equations. Simon (1957) argues that it seems to be unrealistic assuming the complete knowledge about the economic environment, because it is too restrictive. Moreover, if the equilibrium model's equations are nonlinear or involve a large number of parameters, it can be hard to nd a solution.
An heterogeneous agent systems is more realistic, since it allows the description of agents' heterogeneous behaviors evidenced in the nancial markets (see Kirman (2006) for a summary of some stylized facts supporting the agents' heterogeneity assumption). Moreover, heterogeneity implies that the perfect knowledge of agent beliefs is unrealistic, and then bounded rationality takes place (see Hommes, 2001). Hommes (1997, 1998) propose an important contribution on this eld.
The authors introduce the learning strategies theory to discuss agents' heterogeneity in economic and nancial models. More precisely, they assume that different types of agents have different beliefs about future variables's realizations and the forecast rules are commonly observable by all the agents. Brock and Hommes (1998) consider an asset in a nancial market populated by two typical investor types: fundamentalists and chartists. An agent is fundamentalist if he/she believes that the price of the aforementioned asset is determined by its fundamental value. In contrast, chartists perform a technical analysis of the market and do not take into account the fundamentals. In this paper, heterogeneity is assumed to be involved within each single agent, that wears simultaneously two hats: the forecast of the assets' prices are driven by technical analysis of the market (chartist approach) but also by the fundamentals' value (fundamentalist point of view).
In our model each agent performs price forecasts following a short term approach, but the collective behavior can exhibit long memory property. In this context, we extend some existing results (see Zaffaroni 2004Zaffaroni , 2007aZaffaroni , 2007b about the arise of the long memory property due to the aggregation of micro units, by enlarging the class of probability densities of agents' parameters. The contribution of cross-correlation parameters among the agents to the long memory of the aggregate is shown. Furthermore, it is also evidenced that the presence of long memory in the asset price time series implies that the log returns have long memory as well.
The rest of this paper is organized as follows: section 2 introduces the model; section 3 provide the proof of long memory property of the prices. Section 4 provides the analysis of the returns, and section 5 is devoted to the conclusions. The Appendix contains some well-known de nitions and results, for an easier reference.

The model
The basic features of the market model, that we are going to set up, are the existence of two groups of agents, with heterogeneity inside each group.
Let us consider a market with N agents that can make an investment either in a risk free or in a risky asset. Furthermore, the risky asset has a stochastic interest rate ρ t ∼ N(ρ , σ 2 t ) and the risk free bond has a constant interest rate r. We suppose that ρ > r for the model to be consistent.
Let P i,t be the estimate of the price of the risky asset done by the agent i at time t .
The change of the price at time t + 1 forecasted by the i-th agent, conditioned to his information at time t , I t , is given by ∆ P i,t +1 |I i,t .
Let us assume that the market is not ef cient, i.e. we can write the following relationship: where E is the expected value operator, as usual.
In this model, we suppose that the behavior of the investors is due to an analysis of the market data (by a typical chartist approach) and to the exploration of the behavior of market's fundamentals (by a fundamentalist approach). Moreover, the forecasts are in uenced by an error term, common to all the agents: ated to the fundamentalist point of view and u t is a stochastic term representing an error in forecasts.
As a rst step we assume that all the agents have the same weight in the market and that the price P t of the asset in the market at time t is given by the mean of the asset price of each agent at the same time. So we can write The chartists catch information from the time series of market prices. The forecast of the change of prices performed by the agent i is assumed to be given by the following linear combination: (4)  The fundamental variablesP i,t can be described by the following random walk: The fundamental prices observed by the agent i at time t , P i,t , are assumed to be biased by a stochastic error: . . , N, are parameters drawn by sampling from the cartesian product (1 − ξ , 1 + ξ ) N , ξ > 0, equipped with the relative product probability measure. The de nition ofᾱ i,t takes into account the fact that the error in estimating depends on the adjustment performed by each agent of the market price.
More precisely, the observation of the fundamental prices is affected by the subjective opinion of the agents on the in uence on the fundamental of the market price.
If β i > 1, then agent i guesses that market price is responsible of an overestimate of the fundamental prices. Otherwise, the converse consideration applies.
Moreover, the forecasts of the fundamentalist agents is based on the fundamental prices and his/her forecast on market prices at the previous data. So we can write Let us de ne d i,t to be the demand of the risky asset of the agent i at the date t . Thus the wealth invested in the risky asset is given by P t +1 d i,t and, taking into account the stochastic interest rate ρ t +1 , we have that the wealth grows as (1 The remaining part of the wealth, (W i,t − P t d i,t ) is invested in risk free bonds and thus gives (W i,t − P t d i,t )(1 + r) (Cerqueti and Rotundo, 2003). The wealth of the agent i at time t + 1 is given by W i,t +1 , and it can be written as The expression of W i,t +1 can be rewritten as Each agent i at time t optimizes the mean-variance utility function where E and V are the usual mean and variance operators and thus: Each agent i maximizes his expected utility with respect to his demand d i,t , conditioned to his information at the date t . For each agent i the rst order condition By the rst order conditions we obtain Let X i,t be the supply function at time t for the agent i. Then Let us denote By (2), (4), (7) and (9) we get: where L is the backward time operator.
Condition (3) and equation (11) allow to write the market price as 3 Long term memory of prices This section shows the long term memory property of market price time series.
Equation (12) evidences the contribution of each agent to the market price formation.
Each agent is fully characterized by her/his parameters, and it is not allowed to change them. Parameters are independent with respect to the time and they are not random variables, but they are xed at the start up of the model in the overall framework of independent drawings.
The heterogeneity of the agents is obtained by sampling α i , i = 1, . . . , N from the cartesian product R N with the relative product probability measure. No hypotheses are assumed on such a probability up to this point.
In order to proceed and to examine the long term memory property of the aggregate time series, the following assumption is needed: This Assumption thus introduces a correlation in the way in which actual prices P t play a role in the fundamentalists' and chartists' forecasts, and meets the chartists' viewpoint that market prices re ect the fundamental values. Moreover, a relationship between the parameters of the model describing the preferences and the strategies of the investors, α i and ν, and the interest rates of the risky asset and risk free bond (combined in the parameter c) is evidenced.
By a pure mathematical point of view, since ρ > r (and, consequently, c < −1), the variation range of α i is, in formula (13), respected.
We assume that Assumption (A) holds hereafter.
Equation (14) xes the role of the parameters of the model in the composition of the price.
The theoretical analysis of the long term memory of the time series (14) is carried on through two steps: • long memory is detected for each component of P t ; • the terms are aggregated. Fixed i = 1, . . . , N, let γ i,t be a stationary stochastic process such that
Proof. First of all, we need to show that 1 δ i, j is the usual Kronecker symbol, e.g. δ i, j = 1 for i = j; δ i, j = 0 for i = j.

Let us examine
The terms of the series are positive, and so it is possible to exchange the order of the sums: (19) In the limit as N → +∞ and setting x := λ i , y := λ j , (19) becomes: (20) where F is the joint distribution over x and y.
Taking the mean w.r.t. the time and by using the hypothesis (16), we get By using the distributional hypothesis on λ i , for each i, we get Now, the rate of decay of the autocorrelation function related to A 1 is given by (23).
By using the results in Rangarajan and Ding (2000) on such rate of decay and the Hurst's exponent of the time series, we obtain the thesis. Theorem 2. Let us assume that u t is a stationary stochastic process, with

The common component
Moreover, let us assume that there exists a, b ∈ (0, +∞) such that the parameters λ i are drawn by a B(a, b) distribution.
Proof. The proof is similar to the one of Theorem 1.

The component associated to the perception of the fundamentals' value
t is a term typically linked to the perception of the fundamentals' value by the agents.
The stability of the gaussian distribution implies that In particular, Γ t is a stationary stochastic process.
By (26) and (27), we can write The long memory property is formalized in the following result.
Proof. The proof is similar to the one provided for Theorem 1. In order to treat this case, we need to point out that P t is a stationary process, since it can be viewed recursively as a sum of stationary processes. Therefore, the following result holds: Theorem 4. Suppose that λ i are parameters drawn by a B(a, b) distribution, for each i = 1, . . . , N, and a, b > 0. Then, as N → +∞, the long term memory property for A 4 t holds, with Hurst's exponent H 4 , with the following distinguishing: • b > 1 implies H 4 = 1/2: • b ∈ (0, 1) and the following equation holds: imply H 4 = (1 − b)/2. In this case it results H 4 < 1/2, and the process is mean reverting.
Proof. The proof is similar to the one provided for Theorem 1.

Aggregation of the components
In this part of the work we want just summarize the results obtained for the disaggregate components of the market's forecasts done by the investors.
Theorem 5. Suppose that λ i are sampled by a B(a, b) distribution, for each i, with b ∈ R. Then, for N → +∞, we have that P t has long memory with Hurst's exponent H given by Proof. It is well-known that, if X is a fractionally integrated process or order d ∈ [−1/2, 1/2], then X exhibits the long term memory property, with Hurst's exponent H = d + 1/2. Therefore, using Proposition 1, by Theorems 1, 2, 3 and 4, we obtain the thesis. Remark 1. Theorem 5 provides the long term memory measure of P t . The range of the Hurst's exponent includes as particular case H = 1/2, that correspond to brownian motion. Thus the model can describe periods in which the ef cient market hypothesis is ful lled as well as periods that exhibit antipersistent behavior. Moreover, the long term memory property can not be due to the occurrence of shocks in the market. This nding is in agreement with the impulsive nature of market shocks, not able to drive long-run equilibria in the aggregates.

Analysis of returns
This section aims at mapping the long memory exponent of price time series generated by the model into long memory of log-returns. In order to achieve this goal, we analyze the effect of log-transformation of a long-memory process. Dittman and Granger (2002) provide theoretical results on the long memory degree of nonlinear transformation of I(d) processes only if the transformation can be written a nite sum of Hermite polynomials. Therefore they cannot be used for examining logreturns, which the logarithms is involved in.
The same authors provide further results through numerical analysis. Let {X t } t be I(d), Y t = g(X t ) with g(·) a transcendental transformation. Numerical estimates of the degree of long memory of Y t , d , suggest the following behaviour: Remark 2. From the usual results on differencing, we remark that if log(P t ) is I(d) then the log-returns time series r t = log(P t ) − log(P t −1 ) is d = d − 1.
We can state the following

Conclusions and further developments
In this paper a theoretical microeconomic model for time series of assets traded in a nancial market is constructed. The market is assumed to be populated by heterogeneous agents. We provide mathematical results concerning the presence of long memory in prices and log-returns.
Our work extends Zaffaroni (2004Zaffaroni ( , 2007aZaffaroni ( , 2007b, discussing the long term memory property in an agents' aggregation framework by enlarging the class of probability densities of agents' parameters.
Moreover, we study the shift of the memory property from the asset price time series to the log-returns. In particular, it is also evidenced that the presence of long memory in the asset price time series implies that the log returns have long memory as well.
The model allows also for the correlation between the agents and its approach can be useful for modeling also other kind of interaction between the agents.