Abstract— We propose the use of a Neural Network (NN) methodology for evaluating models of time series that exhibit nonlinear mean reversion, such as those stemming from equilibrium relationships that are affected by transaction costs or institutional rigidities. Given the vast array of such models found in the literature, the proposed NN procedure represents a useful graphical tool, providing the researcher with the ability to visualize the data before choosing the most appropriate approach for modeling mean-reversion dynamics with either a Threshold Autoregression (TAR), a Smooth Transition Autoregression (STAR), or any hybrid model. Our case study is involved with understanding the nature of cross-listed stocks (ADRs) and the degree of market integration and efficiency, as captured by the NN methodology. This is done through an analysis of the intradaily price discrepancies of cross-listed French, Mexican and American stocks. The results of the NN methodology are relevant in describing the arbitrage forces that maintain the Law of One Price in these ADR markets, and thus provide a more explicit insight on how these markets are integrated.

I. INTRODUCTION

Actively enforced equilibrium conditions are abundant in

financial markets, such as when the appropriate price of an asset is kept in check by arbitrage forces [1]. In such instances, the time series that results from the difference between the actual and appropriate asset prices will be mean reverting—reflecting the fact that all variables eventually revert back to equilibrium (see [2] for an application to money demand, or [3] for an application to stock markets, and [4] to futures and [5] foreign exchange markets). The degree, shape and speed of the mean reversion process can be used to understand the process by which the market actually enforces equilibrium—whether it is because the market is integrated or somehow conscious of the appropriate asset price or because active arbitrageurs act upon disequilibrium prices [6].

Finding an appropriate model to describe a nonlinear mean reversion process remains an open problem. In this work we propose the use of Neural Networks (NN) as an

intuitive and visually descriptive way of recognizing the appropriate way of modeling equilibrium relationships maintained through arbitrage and other economic enforcement. Naturally, in the real world, we would not expect to see these economic forces act immediately. There must be a ‘no-arbitrage band’ within which arbitrage opportunities are eliminated. An equilibrium that is kept by homogeneous international watchdogs waiting for a commercial opportunity to be profitable may be best described by a Threshold Autoregressive (TAR) process, with distinct thresholds at which economic forces are turned off and on. On the other hand, deviations from the Law of One Price may elicit reactions from many different levels of economic interplay, delayed responses, and heterogeneous arbitrageurs. Such processes would be better described by a Smooth Transition Autoregressive (STAR) model, in which increasingly larger deviations possess increasingly lower half-lives. The NN methodology can help the researcher decipher what model or combinations of models the real-world data presents, as opposed to numerous instances in the literature where the linear model of mean reversion is tested against a particular family of nonlinear models.

For our case study, we analyze the mean reversion properties of the price discrepancy between a stock and its cross-listed equivalent, which is commonly known as an American Depositary Receipt (ADR). An ADR is for all practical purposes the same stock—just traded in a different market and in a different currency. There are at least two reasons why these cross-listed stocks should be priced identically. First, once translated into a common currency they represent identical streams of payments, so integrated markets with a common information set should price them identically. Secondly, even in disintegrated markets, a significant price discrepancy should be arbitraged away—if the markets are at least efficient. The degree to which cross-listed stocks are traded in integrated markets has received significant attention in the past (see [7] - [8]).

The ultimate relevance of analyzing mispricing in the ADR market is to understand the essence of stock price parity enforcement, and the corresponding nature of these international securities. The price discrepancy between the two versions of a cross-listed stock is a representative example of a wide range of disequilibrium time series that are of relevance in the literature [9]. The process of mean reversion to be expected under more realistic circumstances is nonlinear, with higher degrees of mean reversion for large deviations from equilibrium and lower or nonexistent for small deviations [10]. As it relates to our case study, market

The Use of Neural Networks for Modeling Nonlinear Mean Reversion: Measuring Efficiency and Integration in ADR Markets

E. Dante Suarez, Farzan Aminian, and Mehran Aminian, Members, IEEE

E. Dante Suarez is an associate professor in the department of business administration at Trinity University, San Antonio, Texas 78212-7200, USA. Phone: 210-999-7860; Fax: 210-999-8037; Email: esuarez@trinity.edu. F. Aminian is a full professor in the engineering science department at Trinity University, San Antonio, Texas 78212-7200, USA. Phone: 210-999-7562; Fax: 210-999-8037; Email: faminian@trinity.edu. M. Aminian is a full professor in the engineering department at St. Mary’s University, San Antonio, Texas 78228, USA. Phone: 210- 431-2047; Fax: 210-436-3154; Email: maminian@stmarytx.edu.

efficiency occurs when all economically relevant information is quickly reflected on the price of a security, whereas market integration implies that two markets trading securities offering identical streams of payments should price the securities identically at all times [11].

II. CHOOSING AN APPROPRIATE NONLINEAR MEAN REVERSION MODEL

Our case study of high-frequency price discrepancies between ADRs and their underlying securities represents a fertile experiment to test the general applicability of our proposed procedure. The long-run implication of the Law of One Price is that, in a common currency, the prices of the two versions of a cross-listed security generally do not deviate by much. Absence of arbitrage opportunities with no transactions costs implies that the ADR price (PADRt) is equal to the underlying price (PUNDt) adjusted by the spot exchange rate at any given time . The Law of One Price thus holds when the following equation is true: 1⁄ This equilibrium implies that even though each of the price series is non-stationary, the difference between the prices of the logs of the two versions of a cross-listed stock (expressed in the same currency) is a mean-reverting, I(0), process; or, in other words, that the series is cointegrated [12]. Most of the time in the analysis of cointegrated variables, and in discussions of the stochastic evolution of equivalent asset price differences, it is assumed that the forces giving rise to the long-run equilibrium are time invariant and independent of the magnitude of the disequilibrium [13]. Linearity assumptions, however, will not be appropriate if the equilibrium results from the activities of arbitrageurs [14].

If it is indeed the case that two markets are not integrated, but are kept in check by hungry arbitrageurs with common transaction costs, then one would expect the appearance of a ‘no-arbitrage band,’ inside which arbitrage forces are neutralized by lack of arbitrage profitability, and outside of which mean reversion rapidly occurs ([15] - [16]). Such a process can be modeled with a TAR model with three distinct regimes; one—the ‘no-arbitrage band’—in which small price discrepancies can exist indefinitely, and two outside the no-arbitrage band describing significantly positive or negative price differentials. On the other hand, if there are other equilibrium forces at play, such as a latent demand for the cheaper version of a stock, then one could expect a nonlinear mean reversion process with a stronger tendency to revert to the mean as the size of the discrepancy increases. In this case we would expect a process of nonlinear mean reversion with potentially more than two thresholds, which in the limit goes to a STAR series, where larger mispricings gradually exhibit stronger mean reversion [17].

Nonlinear mean reversion can thus occur in multiple ways, and choosing among a vast range of possible models remains the open question we shed light on through this work. In established literature on the matter, one must first test the null hypothesis that the general series is globally cointegrated, and second, whether the reversion to the mean is a linear process, where the degree of mispricing is eliminated by a constant factor every period, or if there is indeed evidence of nonlinear dynamics of a specific type (see [15], [18] - [22]). Once linearity is rejected against a nonlinear model there exists no formal procedure for choosing the most appropriate model amongst a vast number of possible alternatives. As Kapetanios states “the focus of econometric investigation has shifted away from stable linear and unit root processes towards more general classes of processes that include nonlinear specifications. Nevertheless, the tools to distinguish the nature of empirical series are still not fully developed” [23].

Even within the extended family of TAR models, finding an appropriate one is not a straightforward task [24]. In general, an arbitrary number of thresholds may exist. Even in the simplest two-threshold case one must normally assume that the thresholds are symmetrical with respect to the zero-mean of the series, and that the corresponding dynamics for negative and positive outliers are identical. Within this simplest case, how to select the appropriate threshold values remains an open problem, as there is no formal test to do so ([22] and [25]).

Intuitively, one would hope to find thresholds which are—under some definition of optimality—established by an econometric procedure to best fit the data in question. Unfortunately, the two alternatives in the literature cannot be applied to find sensible thresholds for data such as ours, where the bulk of the observations are in the hypothesized equilibrium zone. Namely, the problem with both maximizing the likelihood function of a TAR equation [10], or with minimizing the sum of squared errors of the two separate equations inside and outside of the band [15], is that these optimizing methods do not account for the fact that there are many more observations inside than outside of the band. Hence when optimization takes place, the observations inside the thresholds take all the ‘weight’ of the results.

An alternative to the threshold model could be one where the degree of mean reversion gradually increases the further away the mispricing is from equilibrium. Such dynamics could stem from a setting in which heterogeneous arbitrageurs react non-synchronously to mispricings (i.e. more intensively as the disequilibrium increases). Such dynamics could be described by a STAR model, where the change in regimes does not occur abruptly but rather occurs gradually [26]. A review of the literature reveals both TAR as well as STAR representations of the mispricing series of equivalent assets (for an example of a TAR model see [4], as for a STAR application see [27] and [28], as well as numerous examples of TAR and STAR descriptions of Real

Exchange Rates, such as [9] and [29]). Evemodeling a mispricing time series with a Tmodel generally claim that their representatthe alternative, there are no studies where thmodels are explicitly tested against each othmore recently [17], [20] and [27]).

III. DATA USED IN OUR CASE

Our case study analyzes the time seridiscrepancy of French and Mexican ADAmerican companies cross-listed in France.construct samples a quote from both markethe exchange rate, every minute. For this, quote from each minute of the trading ovethese observations as the relevant pricesminute, in a discrete fashion. In choosingmust keep in mind the tradeoff between awith more pairs of stocks to be tested, and whose low trading intensity does not pfrequency tests this work proposes. It is ostress that both versions of the dually-listeheavily traded in order for the tests to be app

In our work, the French-American sampperiod from April 7th, 1997, to March 24the 1½ hours of daily trading overlap bmarkets. On the other hand, the Mexicanrepresents 28 trading days sampled betweeof 2002. These represent the seven Mexicanthe US with the deepest markets (as mnumber of minutes with a new quote edescribes the seventeen stock pairs used in detailed description in [31]).

A graph may provide the most intuitive ddata. Fig. 1 shows the difference between TELMEX ADR traded in New York aunderlying TELMEX securities traded iexpressed in dollar terms, for the 28 tradingour sample. As we can see from this graphusually priced very similarly, with mocontained between ten cents of each other; could conceivably be too small for any opportunity. Significant deviations, howevethe data, with some instances in which difffifty cents appear for a few minutes.

The data set used in this paper concenrather than trade data because quotes incorpspreads a potential arbitrageur faces. In ansetting, we would expect the two prices security to be very close to each other at anythere would therefore exist an overlap betwspreads of both stocks, in which case our price discrepancy variable is set to zero. Pand selling of mispriced cross-listed securitionly occur when there is no overlap betwspreads. As it is now defined, the discrepanc

en though studies TAR or a STAR tion is better than hese two types of her (see [30] and

E STUDY ies of the price

DRs, as well as . The data set we

ets, as well as for we take the last

erlap, and define s for that given g securities, one a fuller data set, including stocks

permit the high-of importance to ed stock must be plicable.

mple refers to the 4th, 1998, during between the two n-American data en May and June n ADRs traded in measured by the entered). Table I

this study (see a

description of our the prices of the

and that of the n Mexico City,

g days included in h, both stocks are ost observations a difference that serious arbitrage er, are present in ferences of about

ntrates on quotes porate the bid-ask n efficient market of a cross listed y given time, and ween the bid-ask definition of the

Profitable buying ies, however, can ween the bid-ask cy variable

Fig. 1. The Price discrepancy (D) betweein New York and the underlying security trading days shown on the x-axis.

TABLE I

THE SEVENTEEN CROSS-LISTED STOTO INVESTIGATE THE PROPERTIES OREVERSION AND THE CORRESPONDI

EFFICIENCY

Name of Stock

Mexican Stocks

América Móvil Elektra FEMSA Teléfonos de México (TELMEX) Televisa Televisión Azteca Vitro

French Stocks

Alcatel Alstom AXA Total Fina Elf

American Stocks

American Express General Electric Gillette General Motors McDonald’s Phillip Morris United Technologies

en TELMEX ADR traded traded in Mexico over 28

OCKS USED IN OUR STUDY OF ITS NONLINEAR MEAN ING DEGREE OF MARKET

Y.

Ticker Symbol

Average Price in Sample

(in USD) AMX $ 14.27 EKT $ 10.00 FMX $ 35.99 TMX $ 31.45

TV $ 28.41 TVZ $ 4.75 VTO $ 2.43 ALA $ 25.81 AXA $ 37.41 TOT $ 51.71 AXP $ 79.92 GE $ 93.87 G $ 69.24

GM $ 49.33 MCD $ 43.05 MO $ 43.05 UTX $ 79.68

precisely measures this difference. In this fashion, we incorporate the stocks' and foreign exchange bid-ask spreads into the discrepancy variable by redefining it as follows:

⁄⁄ ⁄ ⁄0 … … … … … … … … … . … 2

with its relative version defined as

0.5 0.5 .⁄ 3 In these equations, BNY and ANY are the domestic bid and ask prices, BPa and APa are foreign bid and ask prices, SB is the spot bid price and SA is the ask exchange rate at time t.

IV. A NEURAL NETWORK DESCRIPTION OF A DISEQUILIBRIUM TIME SERIES

Neural networks (NN) represent a nonlinear regression model for mapping an input space to an output space (see [32], [33] and [34]). In contrast with methodologies in which the researcher chooses a model that may work under certain assumptions, NNs are fitted without the researcher’s preconceived notion of what the appropriate descriptive model should be. This is an important point to stress in our methodology—that instead of assigning an arbitrary model of nonlinear mean reversion, it captures the time series dynamics present in the data, in a process that could be compared to data mining. NNs have emerged as an important tool to study complex problems in finance (see [24] and [34] - [37] for interesting examples). One problem of interest is data modeling and forecasting in which a data set of size N denoted by , , , , … … , , is available, but the underlying mapping function from the inputs to the outputs 1, is unknown. A NN finds an approximate model , to by adjusting its free parameters , to learn the desired input-output relationship described by the data set. In this notation, and represent the set of weights and biases in the network, respectively. To achieve this goal, a neural network minimizes the mean square error or performance function given by 1 , . 4

The values of , , … … … , that minimize represent the optimal parameter values associated with the neural network model. It can be shown that an important characteristic of a neural network trained by minimizing the sum-of-square error function is that its output approximates the average of the target data available for a given input [33].

The work presented here to model arbitrage opportunities is based on a two-layer NN. This network, which uses tansig and linear transfer functions for the hidden and output layer

neurons respectively, is discussed in detail in the references described at the beginning of this section. Furthermore, we train the neural networks using a Bayesian regularization technique [32] which effectively eliminates the possibility of over-fitting the data. This technique penalizes complex models with many weights and biases and chooses the number of such parameters for best generalization. In addition, by minimizing (4), the regularization technique does not allow the values of these adjustable parameters to become arbitrarily large, which would be an indication of over-fitting.

We begin our analysis by reiterating what has been conclusively established in the literature: that the two versions of a cross-listed stock behave as one in the long-run ([38] - [39]), and that short-lived and seldom profitable arbitrage opportunities appear in high-frequency data [40]. We employ NN to model nonlinear mean reversion processes—such as the one reflected by the difference in prices between two cross-listed stocks, which in equilibrium remains close to zero. The proposed procedure is designed to pick up a wide variety of nonlinear specifications, and to graphically bring to light the particular nonlinear description that best fits the reversion to the mean of cointegrated data. To estimate the slope or reversion to the mean associated with a given value of , which represents the relative price differential a stock experiences in two different markets at time , the degree of mean reversion is defined as in the following equation:

1 1 . 5

This equation calculates the consecutive slopes associated with price discrepancies up to time and averages the results. When 1, reversion to the mean for a given is calculated based on just one future value while bigger values of take into account a larger set of future values. As a result, small values of focus on very short behavior of price discrepancies while larger values of allow a bigger window of time to observe its variation. The standard deviation of decreases exponentially as increases. In this work, we set 8 so that the NN can accurately estimate from a small sample of with similar values. A small standard deviation causes the

values corresponding to similar values to cluster together making their estimations easier for the NN. For a given cross-listed stock, we can calculate where 1 for a given using equation (5), where is the final trading time for the two markets. Since equation (5) requires m consecutive price discrepancies, the last trading time for which we can calculate mean reversion is . The data set with inputs

and target values is then used as training data for our neural network. The NN used in our study has a two-layer feed forward architecture with four neurons in the hidden layer having tansig transfer functions and one linear

output neuron. Once the NN is trained to minimize the mean square error described in equation (4), it learns to map a given value to the average of the corresponding target values available in the training set. The NN is then simulated to generate for the range of observed in the input data and the results are plotted.

In choosing an appropriate number of lags to analyze our data, we should avoid lags which are too long as we lose the same number of observations as the chosen lag. Since our observations cannot go into the next day, this is a problem that becomes exacerbated in the relatively short daily sample periods for French and American cross-listed stocks. To provide a common basis for comparison of all stock pairs in our case studies, we choose a lag-length of 8 minutes to calculate the corresponding degree of nonlinear mean reversion. Our ultimate goal is not to find a most appropriate description of this data but rather to find a parsimonious model in which our approach can be tested.

V. RESULTS Our proposed NN approach allows us to provide a

representation of the market forces that maintain the Law of One Price in the ADR markets, with a graphical representation that sheds light on the most appropriate model for describing mean-reversion dynamics and the corresponding institutional forces that could potentially give rise to such statistics. The three figures of this section depict the compiled analysis for each three sets of stock-pairs in our sample. To avoid possible random idiosyncrasies of each one of the stock-pairs in the sample, we analyze the price discrepancies of all Mexican stocks at once, and do the same for French and American stocks. To appropriately bundle stocks of the same nationality, we proceed to use our relative price discrepancy measure , and therefore avoid overlapping series of significantly different sizes.

Fig. 2 presents the NN analysis for the seven Mexican stock-pairs in our sample. In this figure, the X-axis now measures the percentage price discrepancy as a function of the stock price. From this graph, it is apparent that very little—if any—mean reversion occurs within ± 1.5% of the price of the stock. In other words, there exists a no-arbitrage band inside which arbitrage is hindered by transaction costs (such as the cancellation/subscription of the ADR), and therefore small price discrepancies can live indefinitely long. Outside this band, however, active arbitrage induces a rapid reversion to equilibrium, with larger price discrepancies inducing an increasingly stronger degree of mean reversion.

In Fig. 2 (and similarly all other graphs analyzed in this section), the price discrepancy of the aggregated Mexican stocks is shown as differences between the price in New York and the price in Mexico City, and expressed on the X-axis. The Y-axis measures the expected change from such a price discrepancy in the next eight minutes. Naturally, all graphs display a line that goes from the north-

west to the south-east, as a positive price discrepancy is to be followed by a negative change in its value.

The French stock-pair conglomerate depicted in Fig. 3 provides us with another interesting story. Here, the band of inaction does not seem to be quite completely inactive, as a small degree of mean reversion is present even for the smallest price discrepancies. The degree of mean reversion, however, is increasing with the size of the price discrepancy, suggesting that this particular data set may be best described by a STAR model. This fact notwithstanding, there may be a significant change in the dynamics for instances in which the price in New York is more than ten percent higher than in Paris (in the left of the graph), for it is with these largest shown disequilibria that arbitrage is most profitable, and therefore instances that the market must quickly correct. The different behavior shown on the other side of the graph, where Paris prices are higher and the degree of mean reversion is actually lower for the highest discrepancies, may represent a random error caused by the small number of observations, but further analysis is warranted.

Fig. 4 presents the NN mean reversion analysis for the American stock pairs. This graph clearly portrays the benefits of our proposed approach since it allows us to visually interpret the way in which American and French Depositary Receipt markets are integrated. For price discrepancies between -6% and 2% of the price of the stock, there is little if any observed mean reversion, suggesting that there is a corresponding no-arbitrage band in which equilibrium forces are hindered. Outside this band, reversion to the mean is an increasing function of the size of the

Fig. 2. A neural network model of mean reversion as a function of relative price discrepancy for all Mexican stocks.

Fig. 3. A neural network model of mean reversion as a function of relative price discrepancy for all French stocks.

discrepancy, thereby suggesting that there exist heterogeneous arbitrageurs or a latent world demand for the underpriced version of a cross-listed stock. Such a mean-reverting process would be best described by a TAR model, and the NN methodology we propose would help the researcher find the most appropriate thresholds.

Fig. 4. A NN model of mean reversion as a function of relative price discrepancy for all American stocks.

VI. CONCLUSION A general consensus as to how to evaluate whether a time

series reflects nonlinear mean reversion has been reached in the literature. Once this fact is established, however, the researcher is faced with an unresolved question: How to choose from a potentially infinite number of alternative nonlinear models. Instead of answers or admissions of a lack thereof, what we find in the literature are tests which compare a linear representation of the data (which has been commonly already proven to be inappropriate) to a nonlinear model that has been proposed as appropriate [41].

In the process of analyzing a given phenomenon involving nonlinear mean reversion, the researcher may have either a strong conviction for the appropriate model to fit to the dataset or an interest in establishing that the data can be modeled relatively well by a particular family of

nonlinear descriptions. In such circumstances, the literature provides well-defined methodologies for estimation, but in the more common situation where the researcher has no a priori knowledge of the appropriate model, research can be stranded in a position where no established methodology has a clear path of analysis. Theoretical formulations generally attempt to distil time series that are generated by clear and specific mathematical processes, but real-world data may never be so unsoiled; there is no reason to expect that the mean-reverting dynamics of a financial time series may in fact be the hybrid result of a combination of models.

TABLE II

THE BENEFITS OF NONLINEAR MODELING OF THE DATA, AS CAPTURED BY THE PROPOSED NN METHODOLOGY.

Cross-listed stock

Correlation between neural-network and actual data outputs

Correlation between linear

model and actual data outputs

American stock 0.35 0.2 French stock 0.58 0.51 Mexican stock 0.55 0.36 ALA 0.54 0.48 AXA 0.49 0.42 AMX 0.72 0.68 AXP 0.43 0.39 EKT 0.62 0.57 FMX 0.83 0.79 G 0.41 0.35 GE 0.47 0.31 GM 0.34 0.07 MCD 0.46 0.37 MO 0.59 0.49 TMX 0.69 0.45 TOT 0.55 0.50 TV 0.62 0.53 TVZ 0.62 0.56 VTO 0.50 0.35 UTX 0.54 0.46 Ultimately, an appropriate and general methodology for

choosing the best fitting nonlinear model from a large set of alternatives may be established [42]. Until then, our proposed NN methodology can be a welcomed tool for those working with financial data that does not perfectly fit any one theoretical framework. In the process of analyzing the mean reversion properties of the price discrepancies of the ADRs of our case study, one may be tempted to fit a common model to all stock pairs, but as we can see from the evidence presented, more than one model may be applied even to one single price discrepancy series. The NN methodology developed in this work provides the researcher with a first glance at the data he or she is dealing with, so that further analysis can be guided. Table II reflects the statistical gains of expressing our time series with the open nonlinear representation of the neural network as opposed to

a linear representation. This technique is expected to be particularly relevant for modeling financial time series in situations where deviations of prices from equilibrium values depend on discrete transaction costs and where market regulators follow intervention rules based on threshold values of control variables ([43] - [45]).

As we can see from Table II, most of these time series are highly nonlinear, with improvements in the correlation between the modeled output and the data of almost 500%, as in the case of the General Motors (GM). In terms of the goodness of fit, the NN representations of the time series will be more or less successful depending on the degree of nonlinearity present in the data (for details on the calculation of the correlation coefficients, refer to [32]). Furthermore, the NN analysis now provides a significantly clearer picture of the degree to which these markets are disintegrated, as well as about the procedures that maintain the Law of One Price. In all cases, the increase in the goodness of fit is significant, and in some the improvement is quite large.

The dynamics of the nonlinear mean reversion observed in our dataset confirm previous results that these markets are not fully integrated but that they are relatively efficient. In the tradition of Kleidon and Werner’s [8] study of market integration, we have proposed this alternative methodology for not only classifying markets in relatively arbitrary categories of integration and efficiency, but rather provide a quantification of the degree to which they are efficient and integrated. The NN analysis has therefore provided us with a better understanding of the corresponding degree of market integration and market efficiency displayed.

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