Spurious longrange dependence in emerging financial markets
The problem with estimating the long range dependence. Motivation for long memory. Long memory VS structural breaks. Markov switching model. Generating the same statistical properties as the long memory model during processing with structural breaks.
Рубрика  Финансы, деньги и налоги 
Вид  статья 
Язык  английский 
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State University  Higher School of Economics
SPURIOUS LONGRANGE DEPENDENCE IN EMERGING FINANCIAL MARKETS^{ }We gratefully acknowledge financial support through State University  Higher School of Economics scientific fund, grant №08040043. ^{}
S. Bryzgalova, S. Gelman
Contain
 1. Introduction
 2. Motivation for long memory
 3. Long memory VS structural breaks
 4.1 ARFIMAmodel
 4.2 Markov switching (MS) model
 4.3 Tests for predictive power
 Granger  Newbold test
 Diabold  Mariano test
 5. Data
 6. Estimation
 Concluding remarks
 References
1. Introduction
Long memory in financial markets has been a widely discussed issue for the last 40 years. Originally found by Mandelbrot and Wallis (1968), significant autocorrelation in the yields of various financial instruments at low frequency, is one of its manifestations. Further research confirmed that this is not the only paradoxal evidence from the memory of the process  the autocorrelation function itself decays much slower than the one corresponding to standard ARMA  GARCH processes. Long memory presence has been recently extensively used to test the market efficiency and stage of development, for it allows for significant predictability [Rogers, 1997; Eom et al., 2008] and therefore, serves as an indicator of market inefficiency, often applied to emerging markets [Dacorogna et al., 2005; Cajueiro, Tabak, 2006; Eom et al., 2008].
The problem with estimating the long range dependence is that a process with structural breaks generates exactly the same statistical properties as the long memory model, and there is still no test on their differentiation (see Section 3). However, this problem is often neglected in the empirical studies.
We suggest using predictive power of the model to give evidence in favor of the true underlying process. The logic is straightforward: since the long range dependence is supposed to yield superior predictive abilities, we should test them against those of a regimeswitching process. For this purpose we consider emerging markets of BRIC countries. In particular, we estimate ARFIMA and Markov Switching model, build a series of onestep ahead forecasts and them compare them with Granger  Newbold and Diabold  Mariano tests. We show that MS process demonstrates superior predictive powers compared to those of ARFIMA for all the markets. Hence, it is quite likely that traditional long memory estimations are spurious, being due to structural breaks.
The paper is organized as follows. Section 2 discusses the properties of long memory processes and their application in finance, section 3 focuses on the dichotomy between longrange dependence and structural breaks. Section 4 describes the used methodology, section 5 provides the data description. Finally, section 6 presents the results of the estimation, and section 7 concludes, summarizing findings and outlining directions for future research.
financial market markov switching model
2. Motivation for long memory
Significant autocorrelation at low frequencies and a slow decay of the autocorrelation functions are two basic manifestations of the long memory process. Theoretically these unusual patterns could be explained by the heterogeneity of the agents present on the market. In the recent decade there have been proposed a number of models, directly incorporating it into the market structure [Brock, Hommes, 1998; LeBaron, 2000; Abel, 2008]. Finally, there have been developed Heterogeneous Market Hypothesis [Dacorogna et al., 2001] which provides a general framework for these models and postulates major characteristics of the corresponding market structure.
According to HMH, the market is populated by heterogeneous agents of different time horizons and dealing frequency, each having the shortterm memory with a standard exponential decay (in different time scales). It is a known mathematical fact that the superposition of different exponential decays at various scales results in the behaviour close to a hyperbole. Hence, such a market structure could perfectly explain the hyperbolic decay of autocorrelation widely observed in real financial markets.
Long memory processes allow for significant predictability [Rogers, 1997; Eom et al., 2008] and therefore, its presence can be used as an indicator of market inefficiency. Introducing the long memory components directly into the asset pricing theory is rather difficult, but there has emerged a considerable area of research on derivative pricing, namely options, which accounts for possible long memory in stock returns [Jamdee, Los, 2007]. It has been demonstrated that the modified approach results in much better explanation of the market realizations than the standard Black and Scholes formulation.
3. Long memory VS structural breaks
Structural breaks have long been an integral feature of many financial time series. The pioneering work belongs to Chow (1960), who suggested an Ftest for a linear regression model, provided the known date of the break. The limitation of exogeneity was overcome in further studies for endogenous structural breaks: Brown, Durbin and Evans (1975) proposed a procedure based on the cumulative sum of the residuals, known as CUSUMtest. Later studies accounted for possible heteroscedasticity and multiple structural breaks [Andrews, 1993; Andrews, Ploberger, 1994; Bai, Perron, 1998].
Several studies were conducted to check tests validity in the presence of the long memory. In particular, Kramer and Sibbertsen (2000) proved that in the presence of long memory, the null distribution of CUSUM statistics tends to infinity, thus, detecting a spurious structural break. Alas, not only a long memory process can successfully imitate the properties of structural breaks, but the vice versa is also possible. In particular, there have been a number of research papers on the latter effect, including [Ding, Granger, 1996; Andersen, Bollerslev, 1997; Granger, Hyung, 2004; Diebold, Inoue, 2001; Choi, Zivot, 2007].
A new research area emerged with the introduction of Markov Switching processes by Hamilton (1988). At each point of time the process can be in one of the specified states of nature, and not only do we estimate each regime parameters, but also the probabilities to be in corresponding states at each moment of time. This idea was used in other specifications of the regime switching models, such as Diebold and Inoue (2001), or Granger and Hyung (2004). These processes can perfectly explain the autocorrelation patterns usually prescribed to long memory evidence.
It is still an open question, how to differentiate between the long memory and regime switching processes. While we do not suggest a formal procedure, we think that in case of financial time series predictive power of the models can be regarded as a valuable criterion. In particular, the main implication of long memory estimations consists in predicting stock returns and testing Efficient Market Hypothesis. Hence, a true long memory process should exhibit a substantial predictive power, which we can test against that of the regime switching process.
4. Methodology
4.1 ARFIMAmodel
With a fractional integration parameter d, the ARFIMA (p,d,q) model is written as
where L stands for the lag operator, Ц (L) and И (L) are polynomials of order p and q correspondingly.
The simplest specification of this model if the socalled fractional Brownian motion (FBM):
with
It is easy to note through a simple factorization why this process exhibits long memory:
where Г stands for gammafunction.
The autocorrelation function for this process demonstrates hyperbolic decay and looks as follows:
An algorithm for the computation of the autocovariance function of a general ARFIMA process is derived in [Sowell, 1992].
We consider three methods to estimate the parameters of ARFIMA:
· Exact Maximum Likelihood, as in [Ibid],
· Modified Profile Likelihood, as in [An, Bloomfield, 1993], and
· Nonlinear Least Squares.
4.2 Markov switching (MS) model
The MS model has been developed by Hamilton (1988). Consider the following AR (1) model
At every period of time the process belongs to one of the regimes, which are characterized by different coefficients and possibly different variance of the disturbances. The state variable y (t) is assumed to follow an ergodic firstorder Markov process and is characterized by the matrix П of transition probabilities from state i to state j.
Once the coefficients of the model and the transition matrix have been estimated, the probability of being in state j, based on the knowledge of complete series can be calculated for each date (the socalled smoothed probabilities), as described in [Kim, Nelson, 1999]. When only information up to a current date is used, the probability estimation is called the filter probability.
To estimate the model we use (quasi) maximum likelihood method [Hamilton, 1988].
4.3 Tests for predictive power
In order to test the predictive powers of one model against the other we use Granger  Newbold (1976) and Diebold  Mariano (1995) tests for 1step ahead predictions.
Consider 2 models estimated at the same data sample, generating the forecasts Forecast errors are defined as
Granger  Newbold test
Let us form new sequences and
Provided that the forecast errors are unbiased and normally distributed, if both models produce equally good forecasts, then x (t) and z (t) should be uncorrelated, and the statistics looks as follows:
where is a sample correlation of x (t) and z (t), H is the number of forecasts generated by each model.
Diabold  Mariano test
This procedure uses a more general setting to test the predictive power of the models, for it does not require the forecast errors to be unbiased, normally distributed or serially uncorrelated.
The mean loss is defined as
If the loss time series is autocorrelated of order q, then provided that both forecasts are equally adequate,
As far as the functional form of g () is concerned, a common approach is to use the square.
5. Data
We conduct our analysis using the data from national indexes of BRIC countries.
· BOVESPA of Brazil. The Bovespa Index is the main indicator of the Brazilian stock market's average performance. The issuing companies of the stocks that compose the Bovespa Index theoretical portfolio are responsible, in average, for approximately 70% of the sum of all BOVESPA's companies' capitalization.
· MICEX of Russia. The MICEX Index is a capitalizationweighted index of the market of most liquid 19 stocks of Russian issuers admitted to circulation on the MICEX (Moscow Interbank Currency Exchange).
· India BSE (100) National for India. India BSE (100) is calculated using the "Freefloat Market Capitalization" methodology. The equity shares of 100 companies from the "Specified" and the "NonSpecified" list of the five major stock exchanges, viz. Mumbai, Calcutta, Delhi, Ahmedabad and Madras have been selected for the purpose of compiling the BSE National Index.
· Shanghai SE Composite for China. Constituents for SSE Composite Index are all listed stocks (A shares and B shares) at Shanghai Stock Exchange.
We have chosen the time interval of 01.200001.2008, since it was rather stable for the countries of interest (no periods of hyperinflation or financial crisis). Logreturns are then computed and used for the analysis.
6. Estimation
An essential feature of ARFIMAmodel consists in the ability to capture both short  and longrange effects of investors' reaction towards news. In addition, autoregressive function for ARFIMAmodel demonstrates not exponential, but hyperbolic rate of decay, which can account for significant autocorrelation at high lags.
We chose the model on the basis of information criteria and autocorrelation fit. The estimation was done in OxMetrics 4,0, results are summarized in Table 1.
Table 1.
ARFIMA (p,d,q) estimates
Country 
Coefficients 

Exact Maximum Likelihood 
Modified Profile Likelihood 
Nonlinear Least Squares 

Coefficient St. error 
Coefficient St. error 
Coefficient St. error 

d^{**} 0,042528 0,01776 
d^{**} 0,06291 0,01785 
d^{*}  0,0394184 0,02881 

AR5^{*}  0,042037 0,02206 
AR5^{*}  0,041956 0,02207 
AR5^{*}  0,0411425 0,02248 

AR6 0,003650 0,02195 
AR6 0,003775 0,02196 
AR6 0,0063301 0,02219 

AR7^{*}  0,036474 0,02194 
AR7^{*}  0,036313 0,02196 
AR7^{*}  0,0296034 0,02211 

AR8  0,019318 0,02194 
AR8  0,019134 0,02196 
AR8  0,014776 0,02201 

AR9^{*}  0,031507 0,02191 
AR9^{*}  0,031309 0,02193 
AR9^{*}  0,0262247 0,02194 

AR10^{**} 0,059842 0,02196 
AR10^{**} 0,06006 0,02197 
AR10^{**} 0,0620966,02201 

Brazil 
MA1^{*} 0,0393780 0,03630 

20002008 
AIC  5, 20955043 
AIC  5, 20955043 
AIC  5, 20955043 

Coefficient St. error 
Coefficient St. error 
Coefficient St. error 

d^{**}  0,164521 0,08484 
d^{*}  0,137394 0,07922 
d^{**}  0,167749 0,08379 

AR1^{**} 0,178968 0,08876 
AR1^{*} 0,152034 0,08294 
AR1^{*} 0,184778 0,08780 

AR2^{*} 0,054598 0,03608 
AR2^{*} 0,0456001 0,03594 
AR2^{*} 0,0519264 0,03550 

AR3  0,0079348 0,02835 
AR3  0,0134459 0,02841 
AR3  0,0055267 0,02823 

AR4^{** }0,0629145 0,02894 
AR4^{** }0,0571543 0,02865 
AR4^{**} 0,0601306 0,02866 

Russia 
AR9^{**} 0,0515149 0,02401 
AR9^{** }0,0486693 0,02374 
AR9^{**} 0,0520668 0,02385 

20002008 
AIC  4,73139209 
AIC  4,71767168 
AIC  4,752945 

Coefficient St. error 
Coefficient St. error 
Coefficient St. error 

d^{*}  0,0232635 0,02193 
d  0,0165652 0,03583 
d  0,0077324 0,04165 

AR1^{**} 0,119743 0,04388 
AR1^{**} 0,113240 0,04354 
AR1 0,0008142 0,04352 

AR4^{**} 0,062277 0,02350 
AR4^{**} 0,0612208,02346 
AR4^{**} 0,0562914 0,02340 

AR9^{**}  0,0465497 0,02225 
AR9^{** }0,0463685 0,02226 
AR9^{**} 0,0447982 0,02211 

India 
AR10^{*} 0,0360177 0,02198 
AR10^{*} 0,0362668 0,02200 
AR10^{* }0,0422150 0,02438 

20002008 
AIC  5,3351933 
AIC  5,3201407 
AIC  5,34626565 

Country 
Coefficients 

Exact Maximum Likelihood 
Modified Profile Likelihood 
Nonlinear Least Squares 

Coefficient St. error 
Coefficient St. error 
Coefficient St. error 

d^{*} 0,118024 0,06802 
d^{* }0,118046 0,06805 
d^{*} 0,112709 0,06684 

AR1^{***} 0,583834 0,1365 
AR1^{***} 0,583824 0,1365 
AR1^{***} 0,556809 0,1399 

AR4^{**} 0,057311 0,02594 
AR4^{**} 0,057308 0,02594 
AR4^{**} 0,060619 0,02579 

AR5^{***}  0,083117 0,02278 
AR5^{***}  0,083118 0,02279 
AR5^{***}  0,080423 0,02232 

AR10^{*} 0,031185 0,02237 
AR10^{*} 0,031183 0,02238 
AR10^{*} 0,0290795 0,02198 

China 
MA1  0,0690887 0,1074 
MA1  0,06909 0,1075 
MA1  0,0665491 0,1133 

20002008 
AIC  5,60483872 
AIC  5,59682986 
AIC  5,62015626 

Emerging markets are characterized by long memory (significant value of d at least at 10%), which is regarded by the previous researchers as a sign of market inefficiency. Using similar estimates, they described the current stage of development in countries like BRIC as relatively low and the degree of predictability as rather substantial.
Table 2 summarizes MSestimation results for the same data.
Table 2.
Markov switching estimates
Country 
Model Specification 
Transition matrix 

Brazil 

Russia 

India 

China 

Although not all the coefficients in the estimated model are significant, the overall tendency is that the algorithm differentiates between positive and negative dynamics on the market, the latter characterized by a substantial increase in volatility.
Having estimated the models, we divide our sample in 2 parts, taking 7 years for the estimation period and 1 year (250 observations) for the forecast period. Note, that it is rather short compared to the dominating approach in the literature (about 1/3 of the sample). However, robust long memory estimation requires a very large data set; hence, we had to allow only for 250 observations. Each period we estimate models and make 1step ahead forecast For ARFIMAmodel we use EMLestimates.^{}, then update the information and repeat the procedure. This yields 250 static forecasts for both models.
Finally, we carry out the tests on the predictive power of our models.
Table 3.
Granger  Newbold test
Brazil 
Russia 
India 
China 

GNstatistics 
3,24451 
0,288673 
1,5006 
1,6777 

Hypothesis 
H1 (1%) 
H0 
H0 
H1 (10%) 

For Brazil and China we tend to reject the null hypothesis of equal predictive power. However, for Russia and India the statistics lies within the confidence interval. However, Granger  Newbold test is rather restrictive, for it assumes unbiased normally distributed forecast errors with no serial autocorrelation, while Diabold  Mariano test is free from these drawbacks. To conduct this test, we assumed that the first model is ARFIMA, and the second  MS, so that a strictly positive value of the statistics means worse predictive power of the former.
Table 4.
Diebold  Mariano test for different orders of autocorrelation included
q 
Brazil 
Russia 
India 
China 

q = 1 
8,804563 
7,734737 
5,85846522 
6,911763 

q = 2 
7,389438 
7,258142 
4,91231773 
7,109308 

q = 3 
6,740247 
5,706779 
4,45101673 
6,427741 

q = 4 
6,49862 
5,425833 
4,07456239 
4,057996 

q = 5 
6, 20797 
5,149708 
4,01762569 
5,482122 

q = 6 
6,055783 
4,884736 
4,02561498 
5,66679 

q 
Brazil 
Russia 
India 
China 

q = 7 
5,90206 
4,814111 
4,0016626 
5,845283 

q = 8 
5,029722 
4,6045 
4,005025 
5,936899 

q = 9 
5,798005 
4,470787 
3,99979397 
5,719408 

q = 10 
5,969835 
4,416624 
3,98522941 
5,491889 

Regardless of the number of autocorrelation coefficients included MS models demonstrate superior predictive power to that of the ARFIMAmodel for any country. This means that long memory models do not actually have significant predictive power as it was postulated in the literature, since even the simplest regimeswitching model outperforms them. Next, the quality of long memory estimation is not likely to lead to strong conclusions, such as market inefficiency. Hence, we are more inclined to think that it is actually regimeswitching nature of the process, not a longrange dependence that drives the market dynamics.
Concluding remarks
In this paper we empirically investigate validity of the long memory specification for BRIC countries. It has been noted in the literature that processes with structural breaks are often confused with those exhibiting long memory. We suggest assessing the validity of the model by its predictive power. We estimate ARFIMA and Markov Switching models, and build series of onestep ahead forecasts. Predictive powers of the models are compared using Granger  Newbold and Diebold  Mariano tests. We demonstrate that even the simplest specification of MSmodel shows superior predictive power to that of ARFIMA for all the countries. This evidence raises doubts concerning the use of the long memory presence as an indicator of good time series prediction and a sign of market inefficiency, while the regimeswitching nature of the process is more likely to be present in the data. However, one should carefully interpret the conclusions of this paper, since fractional integration is only one of the means to capture longrange dependence. Alas, most of other techniques do not allow for predictability of the series, and ARFIMAmodel stays one the most popular means of analysis.
Despite poor predictive power, long range dependence has a clear advantage of representing the movements in a heterogeneous agents market, which is quite a realistic assumption. On the other hand, it is not clear what causes sudden changes in the regime of the model. We think that the solution might lie in the combination of agents' heterogeneity and changes in that structure. For instance, a substantial negative shock could undermine the trust of longterm investors, and they temporarily abstain from transactions, so that the market is driven mostly by shortterm investors. It seems that dynamic structure of market participants could perfectly reconcile heterogeneous agents' framework and regimeswitching behavior of the index. We believe that further research will be able to capture these effects.
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