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Granger Causality and Transfer Entropy for Financial Returns E.M Vol. 127 (2015) ACTA PHYSICA POLONICA A No. 3-A Proceedings of the 7th Symposium FENS, Lublin, May 14–17, 2014 Granger Causality and Transfer Entropy for Financial Returns E.M. Syczewskaa;∗ and Z.R. Struzikb;c;d aWarsaw School of Economics, Department of Economic Analyses, Institute of Econometrics, Madalińskiego 6/8, 02-513 Warsaw, Poland bRIKEN Brain Science Institute, 2-1 Hirosawa, Wako-shi 351-0198, Japan cGraduate School of Education, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan dInstitute of Theoretical Physics and Astrophysics, The University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland Granger causality in its linear form has been shown by Barnett, Barrett and Seth [Phys. Rev. Lett. 103, 238701 (2009)] to be equivalent to transfer entropy in case of Gaussian distribution. Generalizations by Hlaváčková- Schindler [Appl. Math. Sci. 5, 3637 (2011)] are applied to distributions typical for biomedical applications. The financial returns, which are of great importance in financial econometrics, typically do not have Gaussian distribution. Generalizations leading to the concept of nonlinear Granger causality (e.g. causality in variance, causality in risk), known and applied in econometric literature, seem to be less known outside this field. In the paper an overview of some of the definitions and applications is given. In particular, we indicate some recent econometric results concerning application of the tests in linear multivariate framework. We emphasize importance of other variants of Granger causality, and need of development of methods reflecting features of financial variables. DOI: 10.12693/APhysPolA.127.A-129 PACS: 89.20.–a, 89.70.Cf, 02.50.–r, 05.45.Tp, 02.70.Rr 1. Introduction important generalizations. Second, the question of sta- Granger causality (GC) and transfer entropy (TE) are tionarity and nonstationarity, and of proper choice of two approaches to mutual causation. The transfer en- distribution has been learned by econometricians the tropy, TE, introduced by Schreiber [1], builds on the hard way. This is especially important for financial time concept of (Shannon) entropy, but aims at detecting series — underestimating or overestimating particular dynamic causation links between a pair of variables. values due to a wrong choice of probability distribu- The Granger causality, GC, aside of TE, tends to be tion for the model can lead to losses or to inefficiency. used by researchers in climatology, physiology, neuro- The proper choice of the distribution is especially crucial physiology, multimode laser dynamics, analysis of causal- for risk assessment and volatility forecasting by financial ity in cardio-respiratory interactions, etc. (see [2], p. 5), institutions. partially due to equivalence of the GC and TE con- Hlaváčková-Schindler et al. [2] in addition to entropy cepts, proved by Barnett et al. in [3], and by Hlaváčková- and mutual information measures, describe multivariate Schindler et al. [2]. Let us emphasize that the equiva- GC in vector autoregression model framework, nonlinear lence was shown only for a linear GC test, and originally GC tests, and nonparametric GC measures based on cor- only under assumption of Gaussianity. Later Hlaváčková- relation integral, but still do not cover the GC-TE equiv- Schindler [4] extended the analysis for some probabil- alence in the general sense. On the other hand, econo- ity distributions typical for biomedical phenomena (log- metricians use both linear and nonlinear GC tests, and normal distribution, Gaussian mixtures, generalized nor- tools based on mutual information, entropy etc. It seems mal distribution, Weinman exponential distribution). that absorption by econometricians of methods aimed at Barnett and Bossomaier [5] show TE and GC equivalence detecting causality, developed in the field of neurophysi- for the vector autoregressive model also under assump- ology etc. is stronger than that of methods developed in tion of Gaussianity. This does not cover other concepts of econometrics (especially of nonlinear Granger causality) Granger causality, and does not necessarily cover all dis- in the other direction. tributions used in applied financial research and practice. 2. Typical features of financial variables The concentration on the linear form of the GC (and the Gaussianity assumption) in biomedical appli- Financial variables have specific features, which led to cations seems to be somewhat misleading. First of all, development of particular modeling tools from the field of the Granger causality concept as described in papers financial econometrics. Let Pt denote price of a financial by C.W.J. Granger is richer than that (covers linear instrument at time t. Most financial variables of interest and nonlinear causality, causality in spectral domain, (stock indices, exchange rates, stock and options etc.) causality based on information concepts), and has several are nonstationary. Their returns, defined as difference of natural logarithms of prices, rt = ln(Pt) − ln(Pt−1), are stationary in mean, but typically show changes of volatility in time (termed volatility clustering). Changing ∗corresponding author; e-mail: [email protected] volatility is modeled e.g. with ARCH or GARCH-models, the first equation of which describes (and forecasts) mean (A-129) A-130 E.M. Syczewska, Z.R. Struzik of the variable, and the second equation describes de- The GED probability density function is (see Tsay [15], pendence between changing volatilities with use of condi- p. 122): tional variance of the first equation. The ARCH (“autore- ν exp(−0:5jx/λjν ) f (x) = ; (2) gressive with conditional heteroskedasticity”) model was GED λ2(1+1/ν)Γ(1/ν) introduced by Robert F. Engle [6] in 1982; GARCH (gen- where Γ(·) is the gamma function, and λ = eralized ARCH) — by Timothy Bollerslev [7] in 1986, and [2(−2/ν)Γ(1/ν)=Γ(3/ν)]1=2. This distribution has heavy many more detailed GARCH-type specifications have tails when ν < 2 and reduces to a Gaussian distribution been introduced by various authors to cover specific fea- when ν = 2. tures of financial instruments and particular markets. To illustrate discrepancies, especially in the tails, be- The GARCH family models originally estimated by the tween Gaussian and the empirical distribution of the maximum likelihood method with assumption of Gaus- daily returns of the USDPLN exchange rate and the sianity of error terms, prove to give more accurate results WIG20 returns, see Figure for daily quotations from if non-Gaussian, often skewed, probability distributions stooq.pl (Jan. 4, 2000–Oct. 31, 2014)‡. The results of the are applied, especially for daily and higher frequency data Granger causality test (based on the VAR model) for the (see Alexander [8]). two variables [16] show that during crisis, Oct. 01, 2007– Returns of financial instruments typically have skewed Feb. 27, 2009, the null hypothesis of no causality from and leptokurtic distributions. According to Mandelbrot, USDPLN to WIG20 is weakly rejected, the same for who first noticed volatility clustering in financial returns causality from WIG20 towards USDPLN. The results time series [9], returns should be modelled with use of † change somewhat after the crisis. Neither the returns nor Pareto-Lévy processes . In finance and in econometrics the error terms of the VAR models are Gaussian. returns are most often described with use of the follow- ing distributions: Gaussian mixtures, t-Student, or more complicated: general error distribution (GED in short) or USDPLN generalizations of the last two (see Osińska [12], pp. 170– ● ● ● ●●●●● ●●●● ●●●●●● 0.04 ●●●●●●● 172). The GED (General Error Distribution) has been ●●●●●●●●● ●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● introduced in [13] by Subbotin, and later generalized by ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● 0.00 ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●● Theodossiou [14] to a Skewed GED. The reason is that ●●●●●●●●● ●●●● ●●●●●●●●● ●●●●● some stable distributions are not suitable for statistical ● ● Sample Quantiles testing of an econometric model, and that more complex ● −0.06 distributions are intended to better reflect actual features −2 0 2 of financial variables. The skewed generalized t distribu- tion can be expressed as (see Osińska [12]): Theoretical Quantiles k fSGDT(x) = C 1 + WIG20 ν − 2 ν+1 k) k ● ● x + µ ●●● −k ●●●●●● ●●●●●●●●●● × [θ(1 + sgn(x − µ))λ] ; (1) ●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●● σ ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● 0.00 ●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●● where B(·; ·) — the beta function, k; ν; λ, µ, σ — param- ●●●●●●●●●●● ●●●●●●●●●● ●●●●●● ●●●●●●● ●●●●●● eters of the distribution (k > 0; ν > 2; −1 < λ < 1; µ — ●●●● ●●●● z Sample Quantiles ● ● the mean; σ — the standard deviation); sgn(z) = jzj ; −0.10 the coefficients are defined as −2 0 2 1 ν −1:5 3 ν − 20:5 C = 0:5B ; B ; S(λ)σ−1; Theoretical Quantiles k k k k Fig. 1. Comparison of the quantiles of daily returns k 1=k 1 ν 3 ν − 2 θ = B( ; )0:5B( ; )0:5S(λ)−1; with theoretical Gaussian distribution. ν − 2 k k k k " In order to be aplicable in financial econometrics, ques- 2 ν − 12 S(λ) = 1 + 3λ2 − 4λ2B ; tion of TE and GC equivalence should include general- k k ization to the types of distributions typically applied in #0:5 financial econometric research. 3 ν − 2−1 1 ν −1 × B ; B ; : In this paper, examples of causation inference in the k k k k case of realistic data models
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