European Journal of Economics, Finance and Administrative Sciences Issue 54, October 2012, pp. 6-12. ISSN 1450-2275 www.europeanjournalofeconomicsfinanceandadministrativesciences.com
Unit root tests in presence of structural breaks: An application to emerging stock markets Mustapha CHAFFAI, High Business School, Sfax University, Tunisia
[email protected] Imed MEDHIOUB, College of Economics and Administrative Sciences, IMAM University, Riyadh, Saudi Arabia
[email protected]
Abstract The question of emergent stock markets efficiency is widely discussed in the financial literature. Although results indicate convergence to the non efficiency markets, many empirical results lead to the verification of weak efficiency form. Throughout the scope of this paper we tent to answer the question of emergent markets efficiency or non efficiency. To do this we have considered a research based on unit root structural break hypothesis. Applying this methodology to the Egyptian, Moroccan, Jordanian and Tunisian markets, covering the period January 1992- December 2008, contrary to the results of classic tests, we accept the weak form efficiency hypothesis. We conclude then that we cannot verify the random walk analysis to these markets without taking into account the existence of structural breaks. Keywords: random walk, efficiency, unit root tests, structural break, emergent markets. JEL classification : G14, G15, C22.
1- Introduction One of the major implications of the random walk of stock prices concerns the independence of successive yields. This condition tends to reflect the efficient functioning of financial markets. The generating data process of returns is then said to be without memory. In the empirical literature several methods were used to test the returns correlations. Detecting serial correlation on stock returns imply the use of robust econometric approach that resolves this problem of correlation. In this context, parametric and nonparametric tests have been developed. In finance, parametric approaches are the most applied. To offer a preliminary formalization of the stock market behavior and the test is directly related to the considered model. Thus, the autocorrelation measure, as proposed by several authors Fama (1965), Tricot and Posse (1988), Poterba and Summer (1988), Fama and al. (1988)) assumed that returns are normally and identically distributed and linked by a linear dependence. Approaches to conditional heteroskedasticity (ARCH) are proposed by Engle (1982), Bollerslev (1986), Zarowin (1990), Gourieroux and Montfort (1992) and Nelson (1990)) the model is an instantaneous distribution of returns which is normal with mean and variance conditioned on the past of the process. Based on the fact that most empirical studies have emphasized the evaluation of standard unit root tests of Dickey and Fuller (1979). -1-
These tests, often lead to the non-rejection of the null hypothesis of unit root. In order to answer the question of efficiency or inefficiency of the emerging markets we intend to conduct a study based on the rejection of the unit root hypothesis taking into account the possibility of the existence of structural breaks in the process. A natural extension of these tests in empirical financial literature is to suppose the existence of more than one structural break under the alternative model with trend break. In this paper, we focus firstly in presenting the results of the classical models of random walk and how it is possible to conclude on the efficiency of the market. Secondly, we present the results generated in response to the innovations in the random walk model and to expose the assumption of quasi-random walk. The next part will be devoted to the validation of the random walk in emerging markets and the determination of the generating process of returns, our study is conducted in reference to the following emerging markets: Egypt, Jordan, Morocco and Tunisia. Our databases is collected from the International Arab Monetary Fund, concern the monthly stock market index for these markets ranging from January 1992 to December 2008. Unit root tests with two breaks confirm the quasi-efficiency hypothesis of these emerging markets and concluding that, emerging markets have the same properties that of developed markets. The remainder of the paper is organized as follows. After introduction, section 2 is devoted to the exposition of the random walk empirical results. Section 3 outlines and explains the use of unit root test to test random walk hypothesis. Section 4 describes our data set and presents and discusses the empirical results. Finally, section 5 provides the conclusion. 2- Literature review: the random walk empirical models Theoretically, the random walk hypothesis is related to the notion of efficient market hypothesis. In result, when the stock market prices are governed by a random walk we conclude that prices cannot be predicted. This idea is not new; it was developed since the early works of Maurice Kendal (1953) and Fama (1965). After that date, many studies were concentrated on this concept and many methods were developed in order to test stock market efficiency. Summer (1986), Fama and French (1987) conclude that the changes in stock prices can be considered as a combination of both a random walk and a stationary process. In addition, various studies show that the random walk is not respected in the case of series at daily and weekly frequencies. However, with monthly frequency, Lo and Mackinlay did not find a rejection of the random walk, while other studies, like those of Keim Stambaugh (1987) and Fama and French (1987) found that only 5% of monthly variations can be expected. Rejecting random walk hypothesis does not imply necessary that we reject the efficiency of market since the assumption of risk neutrality is not respected or that the utility functions of individuals are neither separable nor additive. Several empirical attempts have been developed in the literature in order to test the following hypothesis: Should we reject or accept the random walk hypothesis? In this paper we present some methods and results related to this issue. Firstly, empirical methods used to test random walk process were based on F-tests for autoregression coefficients, the Box Pierce Q-test and ratio variance test. The two first tests are based respectively on the autoregression order and cutoff points (see for example, Adler and Lehmann 1983), while the second is based on the presence of serial correlations by using a robust test under heteroscedastic random walk hypothesis (see for example, Huizinga, 1987 and Lo and MacKinlay, 1988). Secondly, the random walk hypothesis is tested via different forms of unit root tests -2-
such as Augmented Dickey Fuller, Philips-Perron, KPSS tests and tests that take into account structural break (s) like Zivot test (1992)1, for one break and Lumsdaine and Papell test (1995), for multiple breaks. P. Fontaine (1990), in performing random walk test to five developed markets namely the U.S., Germany, France, Japan and Great Britain. By using monthly data for the period ranging from 1971 to1987, collected from the MSCI database, he have found the following result: "Contrary to the results based on weekly and annual data, the random walk hypothesis is verified for the five markets for the case of monthly data"
Similarly, H. Alexander (1992) confirmed the findings of P. Fontaine (1990), and then he presented the advantages of the ARCH process in order to understand financial processes by introducing the concept of quasi-random walk in testing efficiency markets. In fact, the efficiency hypothesis is based on how information is integrated into share prices. For a better simplification of the concept of random walk, Granger and Morgenstern have suggested a less restrictive definition than that of FAMA. They replace "identical distribution" by "expectation" and independence by "null covariances". It is related to the ARCH (Autoregressive Conditional Heteroscedasticity) models and finance is the area where most often appear these models. Based on BDS test and using FIEGARCH model2, Ghandi and al. (2005)3 reject the random walk hypothesis for the Tunisian stock market and they conclude that this result is due to the non linear dependence of returns series. In this paper, we are not interested to the tests based on nonlinearity and we concentrate those based on unit roots. 3- Unit root tests in random walk analysis Many studies were interested to the analysis of the Stock market efficiency by considering the random walk hypothesis. In effect, testing the stock market efficiency mean using unit root test to verify whether the random walk hypothesis is verified or not. A large number of works interested to the developed countries such as Middle East countries like Saudi Arabia, Kuwait United Arab Emirates ; East Asian countries : India, South Korea; South American countries: Argentina, Brazil. Many kinds of unit root tests were used in these works: standard unit root tests (ADF, KPSS, and PP), unit root tests with structural breaks (unique or multiple breaks), unit root tests taking into account heteroscedasticity. Awad and Zahran Daraghma (2009) applied ADF and PP unit root to the Palestinian securities market and they accepted the random walk hypothesis. Chancharat and Valadkhani (2007), taking into account the presence of structural breaks, have applied Zivot and Andrews and Lumsdaine and Papell the unit root test for sixteen countries and they found different results from these two kind of tests. In effect, random walk hypothesis was accepted for fourteen markets by applying the Zivot Andrew test whereas it was accepted for only five markets by the Lumsdaine and Papell test. By applying the same tests, Narayan and Smyth (2004) confirmed the weak form efficiency for the stock prices in South Korea.
1
Zivot, E. and Andrews, K. (1992), “Further Evidence On The Great Crash, The Oil Price Shock, and The Unit Root Hypothesis”, Journal of Business and Economic Statistics, 10, pp. 251–70. 2 Fractionally Integrated Exponential Generalized Autoregressive Conditional Heteroscedasticity model. 3 Saadi S.,Gandhi D. and Dutta S. (2006) Testing for nonlinearity and Modeling volatility in emerging capital markets: The case of Tunisia. International Journal of Theoretical and Applied Finance Vol. 9 Nº 7, p. 1021-1050.
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These tests mainly refer to the Dickey and Fuller (1981) and Phillips (1987) works. Let's start firstly by standards unit root tests with the null hypothesis is ρ = 0 that correspond to the general following regression: ∑ According to these tests, most of the time series are characterized by a unit root. Unit root tests are based on regression procedures of augmented Dickey-Fuller (ADF 1984) or Phillips-Perron (PP 1988). Indeed, if ρ = 0, this means that the series is non stationary and it has a unit root. If we consider respectively α = 0, β = 0 and φi = 0 for all i = 1, k, then, the test is reduced to the simple DF test type without constant and drift (type 1). In contrast, if φi ≠ 0, then we have the ADF test with lagged variables. Lagged variables are introduced in the regression to eliminate the problem of autocorrelation errors. Both theoretical and empirical works have shown that ADF tests, present statistical biases due to the autocorrelation problem and the omission of some factors that are not considered in these tests, such as structural change, breaks, conditional heteroscedasticity, etc. According to Kuviat Kowsky, Philips, Schmidt and Shin (1992), traditional tests of DF, ADF, PP, have all tendency to the non rejection of the unit root null hypothesis. Unit root tests with structural change Most empirical studies have emphasized the overvaluation of the DF standard tests DF, which often result to the non rejection of the unit root null hypothesis. Perron (1989) raised this feature and indicated that nonstationarity in the series is due to the existence of a structural break in the trend function. Perron developed three models in the presence of breaks. In the first model, he has specified a linear trend function with a single structural exogenous break in the level or in the intercept. The second considers an exogenous break in the slope. The third model is a combination of the first and the second model. The idea of Perron was as follows: "Most macroeconomic time series are not characterized by the presence of a unit root. Fluctuations are indeed stationary around a deterministic trend function. The only „shocks‟ which have had persistent effects are the 1929 crash and the 1973 oil price shock", (1989, pp.1361). In fact, based on Perron (1989) we have to estimate one of the three following equations: ∑ With : { { When, we obtain the first model of Perron, we obtain the second model and the third model is obtained when and . However, since that date many works demonstrated the failure and the bias of the standard unit root tests and approved the powerful of tests taking into account the existence of breaks. Moreover, many of them, when using Perron model, they have -4-
criticized the procedure of Perron essentially, the point concerning the exogeneity of the break date. In effect, Lumsdaine and Stock (1992), Zivot and Andrews (1992), Perron and Vogelsang (1992) and others modified the Perron model and proved that break date is better considered when it is endogenous. As series can have more than one break, issues are concentrated in the development of multiple breaks. In this context, several studies have been developed that consider two endogenous breaks in level and/or trend such as the works of Lumsdaine and Papell (1997), Papell and Prodan (2003), Lee and Strazicich (2003), etc. For these tests, we apply the same methodology that is considered in the case of one endogenous break. The only difference that we introduce is the two breaks in the model that will be endogenously determined. In other words, there are two possible dates of breaks that should be separated by a sub-sample in order to identify these breaks. The model is written as follows: ∑ Where: { { { { To test unit root hypothesis under these assumptions, Lee and Strazicich (2003), for example, proposed the minimum Lagrange Multiplier (LM) test proposed that determines endogenously the two structural breaks. This test can be generalized to the case of multiple structural breaks. In this case, we will have the same model with the variables and , for i = 1, 2, m, (where m is the number of breaks). In this context, and for further details, we can consult for example the work of Ohara (1999)4. In practice, we should take care when we present the graphics of the studied time series and we must examine carefully the historical events before applying the unit root tests. If, however, from the graph we remark that there is more than one break, we should use the true unit root test in order to avoid the case of model misspecification. 4- Empirical results In this section we present the empirical results concerning unit root tests with one and two breaks. For both tests, the breakpoints dates are assumed to be endogenous. Our analysis focused on the following emerging markets: Tunisia, Morocco and Jordan observed for the period ranging from January 1992 to December 2007 and, Egypt for the period ranging from January 1998 to December 2007.
4
Ohara, H.I. (1999), “A unit root test with multiple trend breaks: A theory and application to US and Japanese macroeconomic time series”, The Japanese Economic Review, Vol. 50, pp. 266-290.
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Data, used in this research and that concern stock price index, were collected from Arab Monetary Fund (AMF) database. As shows the graphics of these series, we should use unit root tests that take into account breaks. 2000
tunisie
12500
1500
maroc
10000 7500
1000 5000 500
2500 1995
2000
2005
1995 10000
jordanie
7500
2000
2005
2000
2005
egypte
7500 5000
5000
2500
2500 1995
2000
2005
1995
For all series, we remark the evolution over time of the stock market index implying then the importance accorded to these markets last years. In effect, all series has increased but with different characteristics from one market to another. However, they have knowing some declines over this period, with different dates of downturns that every market had its own properties. These graphs show clearly the existence of structural breaks that can be remedied by the use of the Zivot (1992) and Lee and Strazicich (2003) models. The first model considered the existence of one endogenous break, whereas the latter supposed two breaks. Using these type tests has the advantage to better verify the random walk hypothesis under the null hypothesis of unit root under the presence of structural breaks. The estimation results are presented in the following table. Table1: Unit root test with one break t- statistic Break date Tunisia -1,477 -0,28 Morocco -0 ,107 Jordan 0,556 Egypt *Indicate that the break is significant. Table2: Unit root test with two breaks minimal LM tBreak date 1 statistic January 97* Tunisia -4,525 -3,727 December97 Morocco February 00* Jordan -5,586 February 01 Egypt -3,261 * Indicate that the break is significant
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March 2004* April 2006* November 2005* January 2006*
Break date 2 February 02* June 02 June 04* June 04*
The results confirm the interest of our research for using these type-tests, that we obtain significant breaks for all considered series (except Morocco for the case of two breaks). However, regarding the random walk hypothesis, our first objective in this paper, due to the use of Lee and Strazicich procedure, we confirm the rejection of the hypothesis of random walk for Tunisian, Egyptian and Jordanian markets and then conclude the non-efficiency of these three markets. The acceptance of the null hypothesis for these markets can be explained by the presence of more than a break in the process generating the stock market indices. Although, taking into account one single break can lead to false acceptance of the hypothesis of random walk (test not powerful). 5- Conclusion In this analysis, devoted to the study of the weak form of efficiency, we start by presenting the results released at the developed markets, as well as the results obtained for some emerging markets. In this context, previously empirical works showed that the weak form efficiency does not hold for all emerging markets. The first part of this paper sates that the random walk is checked on more emerging markets when analyses are based on classical tests of random walk. Some obtained results considered the use of standard unit root tests without considering the existence of breaks. In the second part, we take into account the existence of breaks and we perform tests according to the methodology of Zivot and Lee and Strazicich on emerging markets data. These markets that have many similarities are the following: Tunisian, Egyptian, Jordanian and Moroccan markets. We find that random walk hypothesis is rejected for three markets among fourth. These results confirm those of Ghandi and al. (2005) for the Tunisian Stock Market. So, the verification of the random walk for emerging markets cannot be achieved without taking into account the possibility of breaks inclusion when using unit root tests for such markets. In addition, the data generating process of stock returns that varies from one market to another leads to suggest that each market has its own characteristics and there are other factors which influence the data generation process of the returns series. A further analysis of the emerging markets characteristics compared to developed markets can be conducted in order to determine the factors that most influence these returns. For doing this, we should present in advance the characteristics, like liberalization, and the evolution of emerging markets and then identify the behavioral factors that have contributed to the inefficiency of these markets. References Chancharat, S. and Valadkhani, A. (2007), “Structural breaks and testing for the random walk hypothesis in international stock prices”, The Journal of the Korean Economy, Vol. 8, No. 1, pp. 21-38. Chaudhuri, K. and Wu, Y. (2003) “Random walk versus breaking trend in stock prices: Evidence from emerging markets”. Journal of Banking and Finance, 27:4, pp. 575-592. DeBondt, Werner F. M. and Richard Thaler (1995), “Does the Stock Market Overreact?” Journal of Finance, 40, pp. 793-805. Dickey, D.A and Fuller, W. A. (1979), “Distributions of the Estimators for Autoregressive Time Series with a Unit Root”, Journal of American Statistical Association, 74(366), pp.427481. Dickey, D.A and Fuller, W.A. (1981), “Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root”, Econometrica, 49(4), pp.1057-1072. Elliott, Graham, Rothenberg, Thomas J. and Stock, James H. (1996), “Efficient tests for an autoregressive unit root”, Econometrica, 64(4), pp. 813-836. Fama, E. (1991), “Efficient capital markets: II”, The Journal of Finance 46, pp. 1575–1617. -7-
Lee, J. and Strazicich, M.C. (2003), “Minimum LM Unit Root Test with Two Structural Breaks”, Review of Economics and Statistics, 63, pp.1082-1089. Lumsdaine, Robin L. and Papell, David H. (1997), “Multiple trend breaks and the unit-root hypothesis,” Review of Economics and Statistics, 79(2), pp. 212-218. Ohara, H.I. (1999), “A unit root test with multiple trend breaks: A theory and application to US and Japanese macroeconomic time series”, The Japanese Economic Review, Vol. 50, pp. 266-290. Ozdemir, Z. A. (2008), “Efficient market hypothesis: evidence from a small open-economy”, Applied Economics, 40, pp.633–641. Pappel, D.H. and Prodan, R. (2003), “The uncertain unit root in US real GDP: Evidence with restricted and unrestricted structural change”, Journal of Money Credit and Banking, Vol. 36, pp. 423-427. Perron, P. (1989), “The great crash, the oil price shock, and the unit root hypothesis”, Econometrica, 57, pp.1361-1401. Perron, P. and Vogelsang, T. J. (1992), “Nonstationarity and Level Shifts with an Application to Purchasing Power Parity”, Journal of Business and Economic Statistics, 10, pp. 301–320. Saadi, S., Gandhi, D. and Dutta, S. (2006) Testing for nonlinearity and Modeling volatility in emerging capital markets: The case of Tunisia. International Journal of Theoretical and Applied Finance, Vol. 9 Nº 7, pp. 1021-1050. Zhen, Zhu (1998), “The random walk of stock prices: Evidence from a Panel of G7 countries,” Applied Economics Letters, 5(7), pp. 411- 413. Zivot, E. and Andrews, K. (1992), “Further Evidence On The Great Crash, The Oil Price Shock, and The Unit Root Hypothesis”, Journal of Business and Economic Statistics, 10, pp. 251–70.
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