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New evidence from the random walk hypothesis for BRICS stock indices: a wavelet unit root test approach
Economic Modelling 43 (2014) 38–41
Contents lists available at ScienceDirect
Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
New evidence from the random walk hypothesis for BRICS stock indices: a wavelet unit root test approach Aviral Kumar Tiwari a, Phouphet Kyophilavong b a b
Research Scholar and Faculty of Applied Economics, Faculty of Management, ICFAI University Tripura, Kamalghat, Sadar, West Tripura Pin-799210, India Faculty of Economics and Business Management, National University of Laos (NUoL), POBOX7322, Vientiane, Laos
a r t i c l e
i n f o
Article history: Accepted 3 July 2014 Available online xxxx JEL classification: G14 C22 Keywords: Stock market Market efficiency Random walk Structural break Wavelet unit root
a b s t r a c t We examine the use of the random walk hypothesis on the BRICS stock indices. Our examination of the stock indices uses a recently developed wavelet-based unit root test by Fan and Gençay (2010) along with a battery of unit root tests. We also examine the sensitivity of the wavelet-based unit root test. Our wavelet-based unit root tests show evidence that rejects the null of the unit root for all of the BRICS countries except for the Russian Federation. Hence, the tests provide support for the predictability of stock market indices in these economies on the basis of historical information. However, there is a need for caution because the results are based on a relatively small sample of only 11 years of monthly observations. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Emerging market economies have recently attracted the attention of researchers from all over the world because of their phenomenal economic growth. The literature verifies that inefficient stock markets are very important to the promotion of higher economic growth. However, at the same time, these stock markets can be detrimental to economic growth as the recent financial crises prove. A random walk describes the movements of stock prices (or any economic/financial series) that cannot be predicted because they can change without limit in the long run. If stock prices follow a random walk process, then any shocks to the stock prices are permanent; and the future returns cannot be predicted on the basis of information on historical prices. However, if stock prices follow a stationary trend, then the price level returns revert to their trend paths over time; and the future returns can be forecasted by using historical prices. Hence, a market is efficient in a weak form if the stock prices reflect all of the available information about the economy, the market, and the specific security; and the
E-mail addresses: [email protected], [email protected] (A.K. Tiwari), [email protected] (P. Kyophilavong).
http://dx.doi.org/10.1016/j.econmod.2014.07.005 0264-9993/© 2014 Elsevier B.V. All rights reserved.
prices adjust immediately to new information. For a long time the confirmation of random walk was considered to be a sufficient condition for market efficiency, but the rejection of the random walk model does not necessarily imply the inefficiency of stock price information. However, the investigation of whether stock prices are following a random walk or a mean reverting process has important implications for investors and the economy. If stock prices are characterized as mean reverting, it implies that the price level will return to its trend path over time that makes the prediction of the future movements in stock prices based on historical information possible. Nonetheless, if stock prices are characterized as following a random walk process, the shocks to stock prices have permanent effects. Therefore, predicting stock prices based on historical information is not possible, which implies that the long run volatility in stock prices will increase over time. A variety of methodological approaches have been used to investigate the random walk properties of stock prices. The most used approach in the recent empirical approaches is to test whether the stock prices contain a unit root. If the evidence supports the nonstationary nature of the data, then the random walk hypothesis is supported; but if the data are stationary, then the random walk hypothesis is rejected. The results of these empirical studies on the stock market are mixed. Some scholars find evidence for the random walk. For
A.K. Tiwari, P. Kyophilavong / Economic Modelling 43 (2014) 38–41
instance, Narayan and Smyth (2006) for 15 European stock markets; Ozdemir (2008) for the Istanbul Stock Exchange National 100 (ISEN 100) index; Marashdeh and Shrestha (2008) for the Emirates Securities Market; Munir and Mansur (2009) for the Malaysian Stock market; Awad and Daraghma (2009) for the Palestinian securities market; Oskooe et al. (2010) for the Iran stock market; and Hasanov and Omagy (2007) for Bulgarian, Czech, Hungarian, and Slovakian stock prices. On the other hand, some scholars do not find evidence for the random walk. For instance, Lima and Tabak (2004) for the Chinese and Singapore stock markets; Tabak (2003) for the Brazilian stock market; Sunde and Zivanomoyo (2008) for the Zimbabwe Stock Exchange; and Uddin and Khoda (2009) for the Dhaka stock exchange. Moreover, a different approach might lead to a different result. For instance, Chancharat and Valadkhani (2007) detect the random walk hypothesis for 16 countries, using Zivot and Andrews (1992) and Lumsdaine Lumsdaine and Papell (1997) unit root models. The ZA test results provide evidence in favor of the random walk hypothesis in 14 countries. However, the LP test results imply that the hypothesis is rejected in five countries. However, in this study, we investigate this issue by using recently developed unit root tests based on a wavelet framework. Hence, our major contribution lies in investigating the random walk hypothesis for the BRICS emerging economies by applying these new wavelet unit root tests. However, we also apply unit root tests that incorporate structural breaks in the data. Our findings are evidence that rejects the null hypothesis of the unit root in the BRICS economies, except for the Russian Federation (but for the Russian Federation the evidence is not strong). The rest of the paper is organized as follows. Section 2 describes the wavelet unit root tests. The econometric results are discussed in Section 3. Section 4 concludes. 2. Methodology
39
where ut is a weakly stationary zero mean error with a strictly ∞
positive long-run variance defined by ω2 ≡ γ0 þ 2∑ γ j ; where γj = j¼1
E(utut − j). They develop the test for the nonzero mean and linear trend cases only. Now assume that the process {yt} is : s
yt ¼ μ þ αt þ yt
ð2Þ
where yst is generated by model (1). If H0: ρ = 1, then {yst } is a unit root process. But, if H0: |ρ| b 1, then {yst } is a zero mean stationary process. If α = 0, then we consider the demeaned series fyt −yg where y ¼ T
−1
T
∑ yt is the sample mean of {yt}. If α ≠ 0, then we work with the t¼1
detrended series
n o t e y where e yt ¼ ∑ Δy j −Δy ; e y is the sample yt −e j¼1
mean of fe yt g in whichΔyt = yt − yt − 1, and Δy is the sample mean of LM Ld {Δyt}. The authors also develop two test statistics, ^S and ^S ; for the T;1
T;1
demeaned and the detrended series, respectively, based on the unit scale DWT wavelet. The two test statistics are defined as follows3:
^SLM ¼ T;1
T=2 X M 2 V t;1 t¼1 T X
ð3Þ 2
ðyt −yÞ
t¼1
and
^SLd ¼ − T;1
T=2 2 X d V t;1 t¼1
ð4Þ
T 2 X e yt −e y t¼1
In the literature, a number of unit root tests have been developed by making several diverse assumptions. Some studies incorporate different numbers of structural breaks, less nonlinearity, and less volatility. However, the earlier tests in the literature are based on a time domain analysis. In this study, we use a newer test developed in the framework of a wavelet analysis. 1 Through the use of the wavelets, we can decompose a stochastic process into wavelet components, each of which is associated with a particular frequency band. Further, the wavelet power spectrum measures the contribution of the variance at a particular frequency band in comparison to the overall variance of the process. Fan and Gençay (2010) develop the wavelet-based unit root tests by decomposing the variance of the underlying process into the variance in its low and high frequency components via the DWT. Specifically, they construct unit root test statistics from the ratio of the energy (variance) from the unit scale to the total energy (variance) of the time series. The following describes the wavelet-based unit root tests of Fan and Gençay (2010).2 They define {y}Tt = 1 as a univariate time series generated by yt ¼ ρyt−1 þ ut ;
ð1Þ
1 There is another test developed by Choi and Phillips (1993), which is based on an alternative spectral approach to time series analysis, called the Fourier spectral analysis. This analysis has advantages of over the tests based on a time domain approach. However, this test makes use of frequency domain estimators of the autoregressive coefficient. Fan and Gençay (2010) show that this test developed by Choi and Phillips (1993) is a special case of their wavelet based tests using the Haar wavelet filter and unit scale MODWT. 2 I am thankful to Y. Fan and R. Gençay for making available the R codes that we use in the study.
d where {VM t,1} and {Vt,1} denote, respectively, the scaling coefficients of the demeaned and the detrended series. These two test statistics are used to test H0: ρ = 1 against H1: |ρ| b 1 in Eq. (1). Under the alternative hypothesis, {yt} is a zero mean stationary process with the longrun variance ω2/(1 − ρ)2 where ω2 is estimated by taking the OLS residuals from a regression of yt on a linear trend and yt − 1. Next they apply a nonparametric kernel estimator with the Bartlett kernel to the residuals.
3. Data and results We use the monthly average stock indices of the stock markets of Brazil (Bovespa Index of São Paulo Stock, Mercantile & Futures Exchange), Russia (MICEX index of MICEX Stock exchange), India (SENSEX index of Bombay Stock exchange), China (SSE composite Index of Shanghai Stock Exchange), and South Africa (Johannesburg Stock Exchange (JSE)) for the period of January 2000 to December 2010. The data is collected from the Financial Indicators database of OECD Stat. Table 1 shows the results of the wavelet-based unit root tests of the test statistics for Eqs. (3) and (4).
3
It is important to note that when the test statistics are based on the Haar wavelet filter, T=2
∑ ðy2t −y2t−1 Þ2 =2 LM the test statistics reduce to ^ST;1 ¼ 1−t¼1 T . ∑ ðyt −yÞ2 t¼1
40
A.K. Tiwari, P. Kyophilavong / Economic Modelling 43 (2014) 38–41
Table 1 Wavelet-based unit root tests. Stock indices
Brazil Russian Federation India China South Africa
^SLM T;1
^SLd T;1
Test statistics
Test statistics
Lag = 10
Lag = 20
Lag = 30
Lag = 10
Lag = 20
Lag = 30
−52.33029⁎ −36.74062⁎⁎ −49.15789⁎ −155.1594⁎ −97.52691⁎
−49.20166⁎ −33.81057⁎⁎ −44.88037⁎ −153.2756⁎ −96.38076⁎
−45.81438⁎ −33.73754⁎⁎ −41.64277⁎ −144.4671⁎ −96.11615⁎
−229.5457⁎ −168.0725⁎ −174.3253⁎ −208.2685⁎ −413.5443⁎
−215.822⁎ −154.6688⁎ −159.1562⁎ −159.1562⁎ −408.6843⁎
−200.9637⁎ −154.3347⁎ −147.6749⁎ −147.6749⁎ −407.5622⁎
⁎⁎⁎ Denotes significance at the 10% level. ⁎⁎ Denotes significance at the 5% level. ⁎ Denotes significance at the 1% level.
Table 2 Clemente et al. (1998)and Lumsdaine and Papell (1997) unit root tests. Stock indices
t-statistic
TB1
TB2
t-statistic
TB1
TB2
Innovative outliers (IO) −2.772 −3.615 −2.958 −2.642 −3.539
2003m7 2003m4 2003m5 2006m9 2004m6
2005m6 2005m 2005m4 2008m1 2005m4
Additive outlier (AO) −1.133 −3.855 0.104 −2.924 −2.875
2004m1 2001m8 2004m2 2007m1 2005m2
2006m5 2005m10 2006m5 2002m2 2006m6
2004:04 2005:06 2002:02 2006:02 2002:05
2005:12 2008:06 2008:05 2008:06 2008:06
−7.6513⁎⁎⁎ −5.7302 −5.8202 −4.4642 −5.9838
2005:07 2005:10 2003:05 2006:06 2003:01
2008:12 2008:07 2008:05 2009:03 2008:06
Clemente et al. (1998) unit root test Brazil Russian Federation India China South Africa
Lumsdaine and Papell (1997) unit root test Model AA Brazil Russian Federation India China South Africa
−9.8483⁎⁎⁎ −5.7907 −4.6624 −7.5429⁎⁎⁎ −6.3364⁎⁎
Model CC
Note: (1) Critical value for Clemente-Montañés-Reyes unit-root test with double mean shifts, AO and IO model, is – 5.490 at 5% level of significance. (2) In Lumsdaine and Papell (1997) unit root test for Model AA, the 1%, 5% and 10% critical values are – 6.94, – 6.24 and – 5.96 respectively and for Model CC, the 1%, 5% and 10% critical values are – 7.34, – 6.82 and –6.49 respectively. ⁎⁎⁎ Denotes significance at the 1% level. ⁎⁎ Denotes significance at the 5% level. ⁎ Denotes significance at the 10% level.
To test the robustness of our results, we use three different lags: 10, 20 (the optimal choice), and 30. Table 1 clearly shows that the test statistic based on Eq. (3) rejects the null hypothesis for all countries except the Russian Federation, but the test statistic based on Eq. (4) provides sufficient evidence to reject the null hypothesis for all countries despite the choice of the lag length. In the final step, in order to compare our results with the other unit root tests based on the time domain analysis, we use a battery of tests. The battery of unit root tests includes two types of tests: one that does not consider structural breaks in the data such as ADF, PP, ADF-GLS, NP4; and the other that allows for endogenously determined structural breaks in the data such as Lumsdaine and Papell (1997) and Clemente et al. (1998). Clemente et al. (1998) test is based on the framework of the innovative outlier and additive outliers. Lumsdaine and Papell's (1997) test is an ADF-type test that Zivot and Andrews (1992) extend. Both tests incorporate up to two structural breaks in the data series. We present the results of the unit root tests that do not allow structural breaks in Appendix 1, and the results of those unit root tests that allow for structural breaks are presented in Table 2. All of the test statistics that do not allow for structural breaks that are considered in this study do not provide any evidence that rejects the null hypothesis. Table 2 shows that the Clemente et al. (1998) unit root tests (i.e., test statistics based on IO and AO models) do not provide any evidence to reject the null hypothesis; but the application of Lumsdaine and Papell (1997) (results reported in AA and CC models) rejects the null 4 The ADF, PP, DF-GLS, and NP, respectively, denote the test statistics developed by Dickey and Fuller (1981), Phillips and Perron (1988), Elliot et al. (1996), and Ng and Perron (2001).
hypothesis for Brazil, China, and South Africa when we use the AA model. However, when the CC model is used, we find evidence against the null hypothesis only for Brazil.
4. Conclusions In this paper, we examine the random walk hypothesis for the BRICS stock indices by using a recently developed test that uses the wavelet. To test the sensitivity of the results of the wavelet-based test, different lag lengths are chosen. Further, we apply a battery of unit root tests that comprises tests that do not incorporate structural breaks in the data and those that do. Our wavelet-based unit root tests show evidence that rejects the null of the unit root for all of the countries except the Russian Federation. However, the unit root tests that do not incorporate structural breaks show the reverse, that is, those tests accept the null of the unit root hypothesis. This unit root analysis incorporates structural breaks in the data and shows that the null hypothesis is rejected for Brazil, China, and South Africa. Hence, our analysis shows two broad findings. First, the wavelet-based tests are able to give a more accurate picture of the data in comparison with the unit root tests in the framework of the time domain even after they incorporate structural breaks in the data. Second, except for the stock market in the Russian Federation, all of the other analyzed stock markets do not follow the random walk behavior during the period studied. However, caution is advised because the reported results are obtained with 11 years of monthly observations. The property of this small sample of wavelet-based tests is unknown.
A.K. Tiwari, P. Kyophilavong / Economic Modelling 43 (2014) 38–41
41
Appendix 1. Traditional unit root tests Stock indices
Unit root tests Constant
Brazil Brazil Russian Federation Russian Federation India India China China South Africa South Africa
Yes – Yes – Yes – Yes – Yes –
Constant and trend
– Yes – Yes – Yes – Yes – Yes
DF/ADF (k)
−0.246788(0) −2.251172 (0) −1.165343(1) −2.090792(1) −0.505256(3) −2.832087(3) −1.175716(0) −2.029412(2) −0.771957(1) −2.054870(1)
DF-GLS(k)
0.520146(0) −1.508432(0) 0.13582691 −2.124642(1) −0.196414(3) −1.645108(3) −0.850158(0) −1.448586(0) 0.126270(1) −1.833228(1)
PP (k)
−0.375946(4) −2.418376(4) −1.064857(3) −1.859441(4) −0.196305(5) −2.667114(5) −1.670433(7) −2.013328(7) −0.569528(4) −2.018494(5)
NP (MZa) (k)
(MZt) (k)
0.69992(0) −4.25841(0) 0.20031(1) −9.25652(1) −0.38876(3) −6.24093(3) −1.91124(0) −4.09048(0) 0.26233(1) −6.84275(1)
0.55405(0) −1.45677(0) 0.13940(1) −2.13397(1) −0.19836(3) −1.75128(3) −0.83686(0) −1.42997(0) 0.17641(1) −1.84899(1)
Note: (1) *denotes significance at the 1% level. (2) “k” denotes the lag length. (3) Selection of lag length in NP test is based on the Spectral GLS-detrended AR based on the SIC and the selection of the lag length (bandwidth), and in the PP test, it is based on the Newey–West with a Bartlett kernel.
References Awad, I.,Daraghma, Z., 2009. Testing the weak-form efficiency of the Palestinian securities market. Int. Res. J. Finance Econ. 32, 7–17. Chancharat, S., Valadkhani, A., 2007. Testing for random walk hypothesis and structural breaks in international stock prices. Working Paper, No. 07–15. University of Wollongong. Choi, I., Phillips, P.C.B., 1993. Testing for a unit root by frequency domain regression. J. Econ. 59, 263–286. Clemente, J., Montanes, A., Reyes, M., 1998. Testing for a unit root in variables with a double change in the mean. Econ. Lett. 59, 175–182. Dickey, D.A., Fuller, W.A., 1981. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072. Elliot, G., Rothenberg, T.J., Stock, J.H., 1996. Efficient tests for an autoregressive unit root. Econometrica 64, 813–836. Fan, Y., Gençay, R., 2010. Unit root tests with wavelets. Econ. Theory 26, 1305–1331. Hasanov, M., Omagy, T., 2007. Are the transition stock markets efficient? Evidence from non-linear unit root tests. Cent. Bank Rev. 7 (2), 1–12. Lima, E.J.A., Tabak, B.M., 2004. Tests of the random walk hypothesis for equity markets: evidence from China, Hong Kong and Singapore. Appl. Econ. Lett. 11, 255–258. Lumsdaine, R., Papell, D., 1997. Multiple trend break and the unit-root hypothesis. Rev. Econ. Statist. 79, 212–218.
Marashdeh, H., Shrestha, M., 2008. Efficiency in emerging markets–evidence from the emirates securities market. Eur. J. Econ. Finance Admin. Sci. 12, 143–150. Munir, Q., Mansur, K., 2009. Is Malaysian stock market efficient? Evidence from threshold unit root tests. Econ. Bull. 29 (2), 1359–1370. Narayan, P.K.,Smyth, R., 2006. Random walk versus multiple trend breaks in stock prices: evidence from 15 European markets. Appl. Financ. Econ. Lett. 2, 1–7. Ng, S., Perron, P., 2001. Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554. Oskooe, S.A.P.,Li, H.,Shamsavari, A., 2010. The random walk hypothesis in emerging stock market. Int. Res. J. Finance Econ. Issues 50, 51–61. Ozdemir, Z.A., 2008. Efficient market hypothesis: evidence from a small open-economy. Appl. Econ. 40, 633–641. Phillips, P., Perron, P., 1988. Testing for a unit root in time series regression. Biometrica 75, 335–346. Sunde, T., Zivanomoyo, J., 2008. The random walk hypothesis for the Zimbabwe stock exchange: January 1998–November 2006. J. Soc. Sci. 4 (3), 216–221. Tabak, B.M., 2003. The random walk hypothesis and the behaviour of foreign capital portfolio flows: the Brazilian stock market case. Appl. Financ. Econ. 13, 369–378. Uddin, G.S., Khoda, N., 2009. An empirical examination of random walk hypothesis for Dhaka Stock Exchange: evidence from pharmaceutical sector of Bangladesh. Int. Res. J. Finance Econ. 33, 87–100. Zivot, E., Andrews, D.W.K., 1992. Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. J. Bus. Econ. Stat. 10, 251–270.
Contents lists available at ScienceDirect
Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
New evidence from the random walk hypothesis for BRICS stock indices: a wavelet unit root test approach Aviral Kumar Tiwari a, Phouphet Kyophilavong b a b
Research Scholar and Faculty of Applied Economics, Faculty of Management, ICFAI University Tripura, Kamalghat, Sadar, West Tripura Pin-799210, India Faculty of Economics and Business Management, National University of Laos (NUoL), POBOX7322, Vientiane, Laos
a r t i c l e
i n f o
Article history: Accepted 3 July 2014 Available online xxxx JEL classification: G14 C22 Keywords: Stock market Market efficiency Random walk Structural break Wavelet unit root
a b s t r a c t We examine the use of the random walk hypothesis on the BRICS stock indices. Our examination of the stock indices uses a recently developed wavelet-based unit root test by Fan and Gençay (2010) along with a battery of unit root tests. We also examine the sensitivity of the wavelet-based unit root test. Our wavelet-based unit root tests show evidence that rejects the null of the unit root for all of the BRICS countries except for the Russian Federation. Hence, the tests provide support for the predictability of stock market indices in these economies on the basis of historical information. However, there is a need for caution because the results are based on a relatively small sample of only 11 years of monthly observations. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Emerging market economies have recently attracted the attention of researchers from all over the world because of their phenomenal economic growth. The literature verifies that inefficient stock markets are very important to the promotion of higher economic growth. However, at the same time, these stock markets can be detrimental to economic growth as the recent financial crises prove. A random walk describes the movements of stock prices (or any economic/financial series) that cannot be predicted because they can change without limit in the long run. If stock prices follow a random walk process, then any shocks to the stock prices are permanent; and the future returns cannot be predicted on the basis of information on historical prices. However, if stock prices follow a stationary trend, then the price level returns revert to their trend paths over time; and the future returns can be forecasted by using historical prices. Hence, a market is efficient in a weak form if the stock prices reflect all of the available information about the economy, the market, and the specific security; and the
E-mail addresses: [email protected], [email protected] (A.K. Tiwari), [email protected] (P. Kyophilavong).
http://dx.doi.org/10.1016/j.econmod.2014.07.005 0264-9993/© 2014 Elsevier B.V. All rights reserved.
prices adjust immediately to new information. For a long time the confirmation of random walk was considered to be a sufficient condition for market efficiency, but the rejection of the random walk model does not necessarily imply the inefficiency of stock price information. However, the investigation of whether stock prices are following a random walk or a mean reverting process has important implications for investors and the economy. If stock prices are characterized as mean reverting, it implies that the price level will return to its trend path over time that makes the prediction of the future movements in stock prices based on historical information possible. Nonetheless, if stock prices are characterized as following a random walk process, the shocks to stock prices have permanent effects. Therefore, predicting stock prices based on historical information is not possible, which implies that the long run volatility in stock prices will increase over time. A variety of methodological approaches have been used to investigate the random walk properties of stock prices. The most used approach in the recent empirical approaches is to test whether the stock prices contain a unit root. If the evidence supports the nonstationary nature of the data, then the random walk hypothesis is supported; but if the data are stationary, then the random walk hypothesis is rejected. The results of these empirical studies on the stock market are mixed. Some scholars find evidence for the random walk. For
A.K. Tiwari, P. Kyophilavong / Economic Modelling 43 (2014) 38–41
instance, Narayan and Smyth (2006) for 15 European stock markets; Ozdemir (2008) for the Istanbul Stock Exchange National 100 (ISEN 100) index; Marashdeh and Shrestha (2008) for the Emirates Securities Market; Munir and Mansur (2009) for the Malaysian Stock market; Awad and Daraghma (2009) for the Palestinian securities market; Oskooe et al. (2010) for the Iran stock market; and Hasanov and Omagy (2007) for Bulgarian, Czech, Hungarian, and Slovakian stock prices. On the other hand, some scholars do not find evidence for the random walk. For instance, Lima and Tabak (2004) for the Chinese and Singapore stock markets; Tabak (2003) for the Brazilian stock market; Sunde and Zivanomoyo (2008) for the Zimbabwe Stock Exchange; and Uddin and Khoda (2009) for the Dhaka stock exchange. Moreover, a different approach might lead to a different result. For instance, Chancharat and Valadkhani (2007) detect the random walk hypothesis for 16 countries, using Zivot and Andrews (1992) and Lumsdaine Lumsdaine and Papell (1997) unit root models. The ZA test results provide evidence in favor of the random walk hypothesis in 14 countries. However, the LP test results imply that the hypothesis is rejected in five countries. However, in this study, we investigate this issue by using recently developed unit root tests based on a wavelet framework. Hence, our major contribution lies in investigating the random walk hypothesis for the BRICS emerging economies by applying these new wavelet unit root tests. However, we also apply unit root tests that incorporate structural breaks in the data. Our findings are evidence that rejects the null hypothesis of the unit root in the BRICS economies, except for the Russian Federation (but for the Russian Federation the evidence is not strong). The rest of the paper is organized as follows. Section 2 describes the wavelet unit root tests. The econometric results are discussed in Section 3. Section 4 concludes. 2. Methodology
39
where ut is a weakly stationary zero mean error with a strictly ∞
positive long-run variance defined by ω2 ≡ γ0 þ 2∑ γ j ; where γj = j¼1
E(utut − j). They develop the test for the nonzero mean and linear trend cases only. Now assume that the process {yt} is : s
yt ¼ μ þ αt þ yt
ð2Þ
where yst is generated by model (1). If H0: ρ = 1, then {yst } is a unit root process. But, if H0: |ρ| b 1, then {yst } is a zero mean stationary process. If α = 0, then we consider the demeaned series fyt −yg where y ¼ T
−1
T
∑ yt is the sample mean of {yt}. If α ≠ 0, then we work with the t¼1
detrended series
n o t e y where e yt ¼ ∑ Δy j −Δy ; e y is the sample yt −e j¼1
mean of fe yt g in whichΔyt = yt − yt − 1, and Δy is the sample mean of LM Ld {Δyt}. The authors also develop two test statistics, ^S and ^S ; for the T;1
T;1
demeaned and the detrended series, respectively, based on the unit scale DWT wavelet. The two test statistics are defined as follows3:
^SLM ¼ T;1
T=2 X M 2 V t;1 t¼1 T X
ð3Þ 2
ðyt −yÞ
t¼1
and
^SLd ¼ − T;1
T=2 2 X d V t;1 t¼1
ð4Þ
T 2 X e yt −e y t¼1
In the literature, a number of unit root tests have been developed by making several diverse assumptions. Some studies incorporate different numbers of structural breaks, less nonlinearity, and less volatility. However, the earlier tests in the literature are based on a time domain analysis. In this study, we use a newer test developed in the framework of a wavelet analysis. 1 Through the use of the wavelets, we can decompose a stochastic process into wavelet components, each of which is associated with a particular frequency band. Further, the wavelet power spectrum measures the contribution of the variance at a particular frequency band in comparison to the overall variance of the process. Fan and Gençay (2010) develop the wavelet-based unit root tests by decomposing the variance of the underlying process into the variance in its low and high frequency components via the DWT. Specifically, they construct unit root test statistics from the ratio of the energy (variance) from the unit scale to the total energy (variance) of the time series. The following describes the wavelet-based unit root tests of Fan and Gençay (2010).2 They define {y}Tt = 1 as a univariate time series generated by yt ¼ ρyt−1 þ ut ;
ð1Þ
1 There is another test developed by Choi and Phillips (1993), which is based on an alternative spectral approach to time series analysis, called the Fourier spectral analysis. This analysis has advantages of over the tests based on a time domain approach. However, this test makes use of frequency domain estimators of the autoregressive coefficient. Fan and Gençay (2010) show that this test developed by Choi and Phillips (1993) is a special case of their wavelet based tests using the Haar wavelet filter and unit scale MODWT. 2 I am thankful to Y. Fan and R. Gençay for making available the R codes that we use in the study.
d where {VM t,1} and {Vt,1} denote, respectively, the scaling coefficients of the demeaned and the detrended series. These two test statistics are used to test H0: ρ = 1 against H1: |ρ| b 1 in Eq. (1). Under the alternative hypothesis, {yt} is a zero mean stationary process with the longrun variance ω2/(1 − ρ)2 where ω2 is estimated by taking the OLS residuals from a regression of yt on a linear trend and yt − 1. Next they apply a nonparametric kernel estimator with the Bartlett kernel to the residuals.
3. Data and results We use the monthly average stock indices of the stock markets of Brazil (Bovespa Index of São Paulo Stock, Mercantile & Futures Exchange), Russia (MICEX index of MICEX Stock exchange), India (SENSEX index of Bombay Stock exchange), China (SSE composite Index of Shanghai Stock Exchange), and South Africa (Johannesburg Stock Exchange (JSE)) for the period of January 2000 to December 2010. The data is collected from the Financial Indicators database of OECD Stat. Table 1 shows the results of the wavelet-based unit root tests of the test statistics for Eqs. (3) and (4).
3
It is important to note that when the test statistics are based on the Haar wavelet filter, T=2
∑ ðy2t −y2t−1 Þ2 =2 LM the test statistics reduce to ^ST;1 ¼ 1−t¼1 T . ∑ ðyt −yÞ2 t¼1
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A.K. Tiwari, P. Kyophilavong / Economic Modelling 43 (2014) 38–41
Table 1 Wavelet-based unit root tests. Stock indices
Brazil Russian Federation India China South Africa
^SLM T;1
^SLd T;1
Test statistics
Test statistics
Lag = 10
Lag = 20
Lag = 30
Lag = 10
Lag = 20
Lag = 30
−52.33029⁎ −36.74062⁎⁎ −49.15789⁎ −155.1594⁎ −97.52691⁎
−49.20166⁎ −33.81057⁎⁎ −44.88037⁎ −153.2756⁎ −96.38076⁎
−45.81438⁎ −33.73754⁎⁎ −41.64277⁎ −144.4671⁎ −96.11615⁎
−229.5457⁎ −168.0725⁎ −174.3253⁎ −208.2685⁎ −413.5443⁎
−215.822⁎ −154.6688⁎ −159.1562⁎ −159.1562⁎ −408.6843⁎
−200.9637⁎ −154.3347⁎ −147.6749⁎ −147.6749⁎ −407.5622⁎
⁎⁎⁎ Denotes significance at the 10% level. ⁎⁎ Denotes significance at the 5% level. ⁎ Denotes significance at the 1% level.
Table 2 Clemente et al. (1998)and Lumsdaine and Papell (1997) unit root tests. Stock indices
t-statistic
TB1
TB2
t-statistic
TB1
TB2
Innovative outliers (IO) −2.772 −3.615 −2.958 −2.642 −3.539
2003m7 2003m4 2003m5 2006m9 2004m6
2005m6 2005m 2005m4 2008m1 2005m4
Additive outlier (AO) −1.133 −3.855 0.104 −2.924 −2.875
2004m1 2001m8 2004m2 2007m1 2005m2
2006m5 2005m10 2006m5 2002m2 2006m6
2004:04 2005:06 2002:02 2006:02 2002:05
2005:12 2008:06 2008:05 2008:06 2008:06
−7.6513⁎⁎⁎ −5.7302 −5.8202 −4.4642 −5.9838
2005:07 2005:10 2003:05 2006:06 2003:01
2008:12 2008:07 2008:05 2009:03 2008:06
Clemente et al. (1998) unit root test Brazil Russian Federation India China South Africa
Lumsdaine and Papell (1997) unit root test Model AA Brazil Russian Federation India China South Africa
−9.8483⁎⁎⁎ −5.7907 −4.6624 −7.5429⁎⁎⁎ −6.3364⁎⁎
Model CC
Note: (1) Critical value for Clemente-Montañés-Reyes unit-root test with double mean shifts, AO and IO model, is – 5.490 at 5% level of significance. (2) In Lumsdaine and Papell (1997) unit root test for Model AA, the 1%, 5% and 10% critical values are – 6.94, – 6.24 and – 5.96 respectively and for Model CC, the 1%, 5% and 10% critical values are – 7.34, – 6.82 and –6.49 respectively. ⁎⁎⁎ Denotes significance at the 1% level. ⁎⁎ Denotes significance at the 5% level. ⁎ Denotes significance at the 10% level.
To test the robustness of our results, we use three different lags: 10, 20 (the optimal choice), and 30. Table 1 clearly shows that the test statistic based on Eq. (3) rejects the null hypothesis for all countries except the Russian Federation, but the test statistic based on Eq. (4) provides sufficient evidence to reject the null hypothesis for all countries despite the choice of the lag length. In the final step, in order to compare our results with the other unit root tests based on the time domain analysis, we use a battery of tests. The battery of unit root tests includes two types of tests: one that does not consider structural breaks in the data such as ADF, PP, ADF-GLS, NP4; and the other that allows for endogenously determined structural breaks in the data such as Lumsdaine and Papell (1997) and Clemente et al. (1998). Clemente et al. (1998) test is based on the framework of the innovative outlier and additive outliers. Lumsdaine and Papell's (1997) test is an ADF-type test that Zivot and Andrews (1992) extend. Both tests incorporate up to two structural breaks in the data series. We present the results of the unit root tests that do not allow structural breaks in Appendix 1, and the results of those unit root tests that allow for structural breaks are presented in Table 2. All of the test statistics that do not allow for structural breaks that are considered in this study do not provide any evidence that rejects the null hypothesis. Table 2 shows that the Clemente et al. (1998) unit root tests (i.e., test statistics based on IO and AO models) do not provide any evidence to reject the null hypothesis; but the application of Lumsdaine and Papell (1997) (results reported in AA and CC models) rejects the null 4 The ADF, PP, DF-GLS, and NP, respectively, denote the test statistics developed by Dickey and Fuller (1981), Phillips and Perron (1988), Elliot et al. (1996), and Ng and Perron (2001).
hypothesis for Brazil, China, and South Africa when we use the AA model. However, when the CC model is used, we find evidence against the null hypothesis only for Brazil.
4. Conclusions In this paper, we examine the random walk hypothesis for the BRICS stock indices by using a recently developed test that uses the wavelet. To test the sensitivity of the results of the wavelet-based test, different lag lengths are chosen. Further, we apply a battery of unit root tests that comprises tests that do not incorporate structural breaks in the data and those that do. Our wavelet-based unit root tests show evidence that rejects the null of the unit root for all of the countries except the Russian Federation. However, the unit root tests that do not incorporate structural breaks show the reverse, that is, those tests accept the null of the unit root hypothesis. This unit root analysis incorporates structural breaks in the data and shows that the null hypothesis is rejected for Brazil, China, and South Africa. Hence, our analysis shows two broad findings. First, the wavelet-based tests are able to give a more accurate picture of the data in comparison with the unit root tests in the framework of the time domain even after they incorporate structural breaks in the data. Second, except for the stock market in the Russian Federation, all of the other analyzed stock markets do not follow the random walk behavior during the period studied. However, caution is advised because the reported results are obtained with 11 years of monthly observations. The property of this small sample of wavelet-based tests is unknown.
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Appendix 1. Traditional unit root tests Stock indices
Unit root tests Constant
Brazil Brazil Russian Federation Russian Federation India India China China South Africa South Africa
Yes – Yes – Yes – Yes – Yes –
Constant and trend
– Yes – Yes – Yes – Yes – Yes
DF/ADF (k)
−0.246788(0) −2.251172 (0) −1.165343(1) −2.090792(1) −0.505256(3) −2.832087(3) −1.175716(0) −2.029412(2) −0.771957(1) −2.054870(1)
DF-GLS(k)
0.520146(0) −1.508432(0) 0.13582691 −2.124642(1) −0.196414(3) −1.645108(3) −0.850158(0) −1.448586(0) 0.126270(1) −1.833228(1)
PP (k)
−0.375946(4) −2.418376(4) −1.064857(3) −1.859441(4) −0.196305(5) −2.667114(5) −1.670433(7) −2.013328(7) −0.569528(4) −2.018494(5)
NP (MZa) (k)
(MZt) (k)
0.69992(0) −4.25841(0) 0.20031(1) −9.25652(1) −0.38876(3) −6.24093(3) −1.91124(0) −4.09048(0) 0.26233(1) −6.84275(1)
0.55405(0) −1.45677(0) 0.13940(1) −2.13397(1) −0.19836(3) −1.75128(3) −0.83686(0) −1.42997(0) 0.17641(1) −1.84899(1)
Note: (1) *denotes significance at the 1% level. (2) “k” denotes the lag length. (3) Selection of lag length in NP test is based on the Spectral GLS-detrended AR based on the SIC and the selection of the lag length (bandwidth), and in the PP test, it is based on the Newey–West with a Bartlett kernel.
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