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Global and Local Stationary Modelling in Finance: Theory and Empirical Global and local stationary modelling in finance : theory and empirical evidence Dominique Guegan To cite this version: Dominique Guegan. Global and local stationary modelling in finance : theory and empirical evidence. 2007. halshs-00187875 HAL Id: halshs-00187875 https://halshs.archives-ouvertes.fr/halshs-00187875 Submitted on 15 Nov 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Documents de Travail du Centre d’Economie de la Sorbonne Global and Local stationary modelling in finance : Theory and empirical evidence Dominique GUEGAN 2007.53 Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647 Paris Cedex 13 http://ces.univ-paris1.fr/cesdp/CES-docs.htm ISSN : 1955-611X Global and Local stationary modelling in finance: Theory and empirical evidence Guégan D. ∗ April 8, 2007 Abstract In this paper we deal with the problem of non - stationarity encoun- tered in a lot of data sets coming from existence of multiple season- nalities, jumps, volatility, distorsion, aggregation, etc. We study the problem caused by these non stationarities on the estimation of the sample autocorrelation function and give several examples of mod- els for which spurious behaviors is created by this fact. It concerns Markov switching processes, Stopbreak models and SETAR processes. Then, new strategies are suggested to study locally these data sets. We propose first a test based on the k-th cumulants and mainly the construction of a meta-distribution based on copulas for the data set which will permit to take into account all the non-stationarities. This approach suggests that we can be able to do risk management for portfolio containing non stationary assets and also to obtain the dis- tribution function of some specific models. Keywords: Non-Stationarity - Distribution function - Copula - Long-memory - Switching - SETAR - StopBreak Models - Cumulants - Estimation. theory. JEL classification: C32, C51, G12 ∗E.N.S. Cachan, Département d’Economie et Gestion, Centre d’Economique de la Sor- bonne (CES), Antenne Cachan, UMR 7184, 61 Avenue du President Wilson, 94235, Cachan Cedex, France, e-mail : [email protected] 1 1 Introduction Adequate framework to study the behavior of stock returns, and more gen- erally asset prices, is important for several reasons. First, the nature of asset prices behavior is necessary to get robust forecasts. Second the knowledge of the probabilistic properties of these asset prices is fundamental to the for- mulation of the concept of risks: indeed the measurement of risks depends heavily on properties of the empirical distribution such as stationarity, long tailedness, finiteness of the second and higher order moments. Third, various tests for the empirical validity of financial models and the application of these models rely on the robustness of statistical tools which can be deficient in specific context. Fourth, several important pricing models for stock options and other similar financial instruments usually require explicit estimates of stock return variances. The usefulness of these models depends largely on the adequacy and the stationarity of almost the second order moments. Indeed, to model real data sets using classical stochastic processes imposes that the data sets verify almost the second order stationarity condition. This sta- tionarity condition concerns the unconditional moments of the process. It is in that context that most of the models developed from the sixties’ have been studied; We refer to the ARMA processes (Brockwell and Davis, 1988), the ARCH, GARCH and EGARCH models (Engle, 1982, Bollerslev, 1986, Nelson, 1990), the SETAR process (Lim and Tong, 1980 and Tong, 1990), the bilinear model (Granger and Andersen, 1978, Guégan, 1994), the EX- PAR model (Haggan and Ozaki, 1980), the long memory process (Granger and Joyeux, 1980, Hosking, 1981, Gray, Zang and Woodward, 1989, Beran, 1994, Giraitis and Leipus, 1995, Guégan, 2000), the Markov switching process (Hamilton, 1988). For all these models, we get an invertible causal solution under specific conditions on the parameters, then the forecast points and the forecast intervals are available. Thus, the stationarity assumption is the basis for a general asymptotic theory for identification, estimation and forecasting. It guarantees that the increase of the sample size leads to more and more information of the same kind which is basic for an asymptotic theory to make sense. Moreover, non-stationarity modelling has also a long tradition in economet- rics. This one is based on the conditional moments of the data generating process. It appears mainly in the heteroscedastic and volatility models, like the GARCH and related models, and stochastic volatility processes (Ghysels, Harvey and Renault (1997)). This non stationarity appears also in a different way with structural changes models like the Markov switching model (Hamil- 2 ton, 1988), the Stopbreak model (Diebold and Inoue, 2001, Breidt and Hsu, 2002, Granger and Hyung, 2004) and the SETAR model (Tong, 1990), for instance. It can also be observed for linear models with time varying coeffi- cients (Nicholls and Quinn, 1982, Tsay, 1987). Thus, using stationary unconditional moments suggest a global stationarity for the model, but using non-stationary unconditional or conditional mo- ments suggest that this global stationarity fails and that we only observe a local stationary behavior. The growing evidence of instability in the stochastic behavior of stocks, of exchange rates, of some economic data sets like growth rates for instance, characterized by volatility, explosions or jumps inside the variance or on the levels of the prices imposes to discuss the assumption of global stationar- ity and its consequence in modelling, particularly in forecasting and risk management strategy. There exist a voluminous literature concerning the modelling of stock returns, and most of the published research work relies on the stationary assumption that is embedded in almost conventional sta- tistical methods: it is assumed that the distributional shapes are invariant and the variance finite. Thus we can address several questions with respect to these remarks. 1. What kinds of non-stationarity affect the major financial and economic data sets? How can we detect them? 2. What is the impact of evidence of non stationarity on the statistics computed from the global non stationary data sets? 3. How can we analyze data sets in the non stationary global framework? Does the asymptotic theory work in non-stationary framework? 4. Can we use a local modelling to avoid a global non-stationarity? How to detect it? Based on the fact that we cannot expect market return variability to be a constant value over time, the conventional concept of risk cannot be defined on a stationary basis. Thus the purpose of studying the non constancy of market return variability will permit to gather better insights into the complex mechanism underlying the evolution of these returns in order to better forecast and to define a robust risk management strategy. These questions begin to be discussed in some papers in the economic lit- erature. For some of these questions, the answers are known, for others, 3 very few works exist, we refer for instance to Lobato and Savin, 1998, Engle and Smith, 1999, Granger and Terasvirta, 1999, Diebold and Inoue, 2001, Granger and Hyung, 2004, Mikosch and Starica, 2004. In this paper we discuss all these problems and we propose new strategies to detect non sta- tionarity based on the cumulants spectral density function and on copulas, and modellings to solve them in order to propose a new approach for risk measures. Several interesting topics in empirical finance awaiting future re- search are also discussed. In the next Section, we recall the notion of global stationarity and short and long memory behaviors. In Section three, we introduce the different types of non stationarities observed in real data sets and we examine what are the specific behaviors observed in presence of non stationarity on real data sets. In Section four, we study the asymptotic behavior of the sample ACF in presence of non stationarity. In Section 5, we introduce the notion of local stationarity. Section 6 is devoted to new results concerning the detection of local stationarity and the building of a kind of meta-distribution. This last approach permits to compute the empirical distribution function for a whole non-stationary sample using a local approach and also to give new possi- bilities to build the distribution function for specific models characterizing by certain non stationarities. Finally Section 7 concludes. In the different annexes, we found proof of the proposition 4.1 and some simulations that illustrate problems discussed all along the paper. 2 Some notations and recalls In order to find a model which characterizes a real data set, we use a theoret- ical framework assuming that the observations of any data set are dependent random variables obtained at different moments of time under stationarity and corresponding to the realization of a process (Yt)t. We first recall the notion of stationary stochastic process. A stochastic process is a sequence of random variables (Yt)t defined on a probability space (Ω, A,P ). Then ∀t fixed, Yt is a function Yt(.) on the space Ω, and ∀ω ∈ Ω fix, Y.(ω) is a function on Z. The functions (Y.(ω))ω∈Ω defined on Z are realizations of the process (Yt)t. A second order stochastic process 2 (Yt)t is such that, ∀t, EYt < ∞. For a second order stochastic process, the mean µt = EYt exists, ∀t, as the variance and the covariance.
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