A study of co-movements between U.S. and Latin American stock markets: A cross-bicorrelations perspective

Un estudio de comovimientos entre las bolsas de valores de Estados Unidos de Norteamérica y América Latina: Una perspectiva de la bicorrelación cruzada

Semei Coronado ^a, Omar Rojas ^b, Rafael Romero-Meza ^c & Francisco Venegas-Martínez ^d

^a Department of Quantitative Methods, Universidad de Guadalajara, Zapopan, México, semeic@cucea.udg.mx
^b School of Business and Economics, Universidad Panamericana, Guadalajara, México. orojas@up.edu.mx
^c Facultad de Administración y Negocios, Universidad Autónoma de Chile, Región Metropolitana, Chile, rafael.romero@uautonoma.cl
^d Superior School of Economics, Instituto Politécnico Nacional, Ciudad de México, México. fvenegas1111@yahoo.com.mx

Received: March 20^th, 2015. Received in revised form: November 24^th, 2015. Accepted: December 10^th, 2015

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Abstract
This work applies a test that detects dependence between pairs of variables. The kind of dependence is a non-linear one, and the test is known as cross-bicorrelation, which is associated with Brooks and Hinich [1]. We study dependence periods between U.S. Standard and Poor's 500 (SP500), used as a benchmark, and six Latin American stock market indexes: Mexico (BMV), Brazil (BOVESPA), Chile (IPSA), Colombia (COLCAP), Peru (IGBVL) and Argentina (MERVAL). We have found windows of nonlinear dependence and co-movement between the SP500 and the Latin American stock markets, some of which coincide with periods of crisis, leading to an interpretation of a possible contagion or interdependence.

Keywords: Financial crisis; cross-bicorrelations; nonlinear dependence; co-movement; financial markets.

Resumen
Este trabajo aplica una prueba para detectar dependencia entre pares de variables. Este tipo de dependencia es no linear, y la prueba es conocida como bicorrelación cruzada, la cual es asociada a Brooks y Hinich [1]. Estudiamos periodos de dependencia no-lineal entre el índice Standard and Poor's 500 (SP500) de EUA y seis índices de mercados accionarios latinoamericanos: México (BMV), Brasil (BOVESPA), Chile (IPSA), Colombia (COLCAP), Perú (IGBVL) and Argentina (MERVAL). Hemos encontrado ventanas de dependencia no-lineal y de co-movimiento entre el SP500 y los mercados accionarios latinoamericanos, algunas de las cuales coinciden con períodos de crisis, lo cual da paso a posibles interpretaciones de contagio o interdependencia.

Palabras clave: Crisis financiera; bicorrelaciones cruzadas; dependencia no lineal; co-movimiento; mercados financieros.

1. Introduction

The study of the transmission of shocks from one country to another and the correlations between several countries and co-movements that cannot be explained by strong economic arguments has attracted the attention of researchers in economics and finance, as well as of practitioners. Research on this topic has significant effects, on both asset pricing, allocation and forecasting, as well as on other elements cf. [2-4].

Recently, co-movements between financial markets have been studied, with emphasis on the return on stock market indexes, through econometric or time series models that allow for a better understanding of the behavior of markets, especially through periods of crisis. A lot of this research has studied interdependencies and co-movements from the point of view of contagion, cf. [5-10]. Several approaches are adopted to analyze the co-movements between financial markets, some of which use quantitative tools borrowed mostly from physics and computational sciences, cf. [11-13].

According to [6], if two markets are highly correlated, and the correlation does not increase in one of the markets after a financial crisis, but on the contrary, there is a continuous variation in its co-movement, then both markets are highly interdependent and contagion cannot be considered as the cause of the relationship between the two markets. Thus, for these authors the privileged dimension is a linear dependency. However, there is an important line of research that emphasizes the need for an empirical verification of nonlinear univariate and multivariate dependencies. Several arguments have been put forward in favor of this route of investigation, according to [14]. On the one hand, if after running a regression there is an indication of nonlinear dependence in the error terms of a standard model, the most common being the linear one, it can be argued that the standard model does not represent the data sufficiently well. On the other hand, if the evidence of nonlinear behavior is found in the first moment, the conditional mean, the formulation of a trading scheme built on this finding would be conceivable. This would ensure that benefits greater than a passive trading plan are obtained.

In order to explore these -less known and less obvious- nonlinear relationships between financial markets and how they co-move, in this work we use the Brooks and Hinich [1] nonlinearity test, that uses a measure of the dependence between pairs of variables called the cross-bicorrelations between time series, using bivariate autoregressive vectors in high frequency data. According to [15] these tests can be viewed as natural multivariate extensions of Hinich's portmanteau bicorrelation and whiteness statistics, but in this case the test examines nonlinear characteristics for pairs of variables. The advantage in using the cross-bicorrelation test is that it addresses the specific window frames in which the nonlinear dependence is present and also signals the direction of the nonlinear dependence, which is not provided by the Granger causality test

Both univariate [16-24] and multivariate [25-27] tests have been successfully applied to analyze the nonlinear behavior of different financial and economic time series. However, to the best of our knowledge, this is the first time that such a multivariate nonlinear test is used to uncover how stock markets co-move.

Seminal works analyzing Latin American stock markets as [24] and [22] use the univariate test, as is the case of the bicorrelation. In our case, we implement a bivariate test, the cross-bicorrelation that allows us to study co-movement between pairs of variables. Thus, the cross-bicorrelation is a multivariate extension of the bicorrelation, and the cross-bicorrelation can capture most types of dependence between pairs of series of the third-order statistics.

In this paper, we use the non-linearity test proposed by [1] in order to uncover the cross-covariances and cross-correlations between U.S. Standard and Poor's 500 (SP500), used as a benchmark, and six Latin American stock market indexes: Mexico (BMV), Brazil (BOVESPA), Chile (IPSA), Colombia (COLCAP), Peru (IGBVL) and Argentina (MERVAL). We found windows of nonlinear dependence between the SP500 and the Latin American stock markets, some of which coincide with periods of crisis, leading to an interpretation of possible contagion or interdependence.

The organization of the document is the following: Section 2 presents the information collected. Section 3 describes the methodology used. Section 4 reports the empirical results. Finally, the main conclusions are presented in Section 5.

2. The data

For this study we consider daily returns of seven stock market indexes, namely the U.S. Standard and Poor's 500 (SP500) is taken as a baseline market for comparison against six Latin American stock market indexes: Mexico (BMV), Brazil (BOVESPA), Chile (IPSA), Colombia (COLCAP), Peru (IGBVL) and Argentina (MERVAL). Daily closing prices from January 2^nd, 2003 to January 8^th, 2015, for a total of 3025 observations of each index were obtained from Bloomberg. The data was sampled for this period of time in order to capture the effects that the U.S. might have had on the Latin American equity markets during the sub-prime financial crunch and to have a broad view of other possible cross-bicorrelation phenomena. Prices were converted into a continuous rate of returns, taking natural log differences between consecutive daily closing prices of equity markets. Table 1 presents summary statistics for these returns. The statistics are consistent, as expected, with some of the characteristic particularities of financial variables [28,29]. In particular, the kurtosis indicates that return distributions are leptokurtic. Furthermore, the Jarque-Bera statistic (JB) confirms returns not normally distributed. Although the results of the KPSS test for seasonality are not listed, it does not reject the null hypothesis of seasonality or the results of the ADF and PP tests, under the null hypothesis of a unit root. Both tests with a 5% significance level are available upon request.

3. Methodology

Brooks and Hinich [1] claim that the cross-bicorrelation test would allow a researcher identify any existence of nonlinear dependence between two pairs of variables. The size of the sample series is N, with two stationary variables and . As we are working with the first percentual logged differences and small sub-samples of the total series, to assume stationarity is more than reasonable. Each series is separated into equal length non-overlapping moving time windows or frames, where t is an integer and k represents the k-th window and, both series are jointly covariance stationary, which have been standardized. The test's null hypotheses states that the two variables and have no dependence and in fact are pure white noise. The alternative hypothesis states that the series have cross-covariances, defined as or any of the cross-bicovariances, defined as , different from zero.

Under the null hypothesis, and are zero for every except when. According to the test, there is dependence between a pair of variables if or for at least one value of r or a pair of values of and, respectively. Next, we present the statistics that give the sample cross-correlation and the sample cross-bicorrelation, respectively

for , and

where .

We can interpret the cross-bicorrelations as the degree of relation of the value of one variable with the value of the cross-correlation of the two variables. The second-order test does not include current elements, and is executed on the errors terms of an fit to clean out the univariate autocorrelation arrangement. Thus current correlations will not be reason for rejecting the null hypothesis. To perform the third-order test, we apply the test on the errors terms of a model having a current term in one of the equations (the order p of the and models is chosen to optimize the Schwartz (BIC) criterion). The pre-whitening step is grounded on the elimination of any presence of linear correlation or cross-correlation. Therefore any outstanding dependence between the variables should be classified as nonlinear. Let where (for our case of study we use , and thus we have 121 non-overlapped windows of length 25 days). The corresponding test statistics for non-zero cross-correlations and cross-bicorrelations are

And

for , respectively. In these statistics is the number of times that the correlations are verified and is the number times that the cross-bicorrelations are probed. Following [15], we state that and are asymptotically with and degrees of freedom, respectively, as

4. Empirical results

In Table 2, we report the results for the cross-bicorrelation test. All tests are run taking SP500 as the benchmark for comparison, since the effects of the U.S. on Latin America are the ones we wanted to test. We present the number and percentage of significant (at the 5% level) cross-bicorrelation windows, correlation for all windows and the correlation for the largest window. As can be seen, the countries with the most significant cross-bicorrelation windows are Brazil (BOVESPA), Argentina (MERVAL) and Peru (IGBVL), with 28.1%, 25.6% and 24.0% of significant windows, respectively. On the other hand, Colombia (COLCAP), Chile (IPSA) and Mexico (BMV) are the countries with less significant windows (14.0%, 14.9% and 16.5%, respectively). As for the correlations for all windows, most countries present a correlation between 0.70 and 0.79, with the exception of Argentina (0.55). Furthermore, the country with an episode of largest correlation (0.50) was Argentina, whereas the one with the lowest correlation for a single episode was Peru (-0.01). These results shed light on the degree of dependence and co-movement between economies.

In Fig. 1 and Fig. 2, we plot the (-values of the significant cross-bicorrelation windows between SP500 and Mexico, Brazil and Colombia (Fig. 1), and Chile, Peru and Argentina (Fig. 2). These windows correspond to the ones reported in Table 2 and 3. It is clear how there are two main periods of nonlinear dependence between U.S. and Latin American stock markets: 2008-2009 and 2011. Brazil and Argentina showed cross-bicorrelations with the U.S. in 2003.

In Fig. 3 we plot the normalized prices for the SP500, BOVESPA and COLCAP indexes. The SP500 is the benchmark and it is compared with BOVESPA that presents the most significant cross-bicorrelation windows (28.1%) and COLCAP that presents the least significant percentage of windows (14.0%). We also plot the returns and the (-values of the significant cross-bicorrelation windows. As can be seen in the prices and returns plots, it is clear how for the year 2008 and the year 2011, the SP500 falls in prices and has a higher volatility before the other indexes.

5. Conclusions

In this document we have successfully applied the Brooks and Hinich [1] cross-bicorrelation test to uncover the cross-covariances and cross-correlations between U.S. Standard and Poor's 500 (SP500), used as a benchmark, and six Latin American stock markets indexes: Mexico (BMV), Brazil (BOVESPA), Chile (IPSA), Colombia (COLCAP), Peru (IGBVL) and Argentina (MERVAL). We found windows of nonlinear dependence between SP500 and the Latin American stock markets, some of which coincide with periods of crisis, giving way to an interpretation of possible contagion or interdependence. Using a different but related methodology, [23] found that there are several periods where there were international financial crises that present strong univariate nonlinearity for several Latin American financial markets.

We have identified some moments of cross-bicorrelations that might be interesting to explore from a deeper economic point of view and it is left as a work in progress. Furthermore, following [23], it would be interesting to run a test including overlapped windows in a rolling scheme. Thus, like [23] it would be possible to identify the start, the end, the intensity and persistence of the cross-bicorrelation instead of just the bicorrelation.

Acknowledgements

The authors would like to acknowledge Jorge Ahumada García (MSc in Economics student, ITAM) for his help providing some of the data used in this paper and Itzel Cano (Universidad Panamericana) for her help formatting some of the tables. All errors remain the sole responsibility of the authors.

The authors are grateful for the support of FONDECYT (Project 1111034).

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S. Coronado, obtained a PhD. degree in Business and Economics from the University of Guadalajara, in Guadalajara, Mexico. He is a research professor in the Department of Quantitative Methods at the University of Guadalajara. He is currently a member of the Mexican National System of Researchers (Level I, CONACYT) where his research areas of interest are time series, emerging market finance and applied statistics. ORCID: 0000-0002-7945-7155

O. Rojas, received his PhD in Mathematics from La Trobe University, Melbourne, Australia. He is associate professor and research Director at the School of Business and Economics at Universidad Panamericana, Guadalajara, Mexico. His research areas of interest are: nonlinear time series and multivariate statistical methods applied to business. He is a member of Mexican National System of Researchers (Level C, CONACYT). ORCID: 0000-0002-0681-3833

R. Romero-Meza, received a Dr. of Business Administration (DBA) from Boston University. He is research professor at Facultad de Administración y Negocios at Universidad Autónoma de Chile, and he is also Director of Global Council of PKF International Finance. Dr. Romero-Meza has over twenty academic articles in applied economics, applied economics letters, macroeconomic dynamics, economic modelling, the Journal of Economic Asymmetries among others. He participates as referee in several journals: Energy Economics, Economics Letters, INNOVAR, Emerging Markets. His research and teaching interests are in emerging markets, efficient markets, nonlinear time series, and corporative finance. ORCID: 0000-0001-5108-2681

F. Venegas-Martinez, was a PhD. researcher in Finance at Oxford University. He received his PhD in Mathematics and a second PhD in economics from Washington State University. He is a professor in Instituto Politécnico Nacional, Mexico. Dr. Vengas-Martínez is a member of the Mexican National System of Researchers (Level III, CONACYT). He participates in more than 20 editorial boards and scientific research journals, in Mexico and internationally. Dr Venegas-Martínez has over a hundred academic articles in The Brazilian Journal of Probability, Journal of the Inter-American Statistics and Econometrics, Journal of Economic Dynamics and Control, International Journal of Theoretical and Applied Finance, Journal of Economic Modeling, Journal of Development Economics, among others. His research areas of interest are stochastic process, econometrics, time series, and economic development. ORCID: 0000-0002-1528-5593