Fractional integration in daily stock market indices at Jordan's Amman stock exchange

Fractional integration in daily stock market indices at Jordan's Amman stock exchange

North American Journal of Economics and Finance 37 (2016) 16–37 Contents lists available at ScienceDirect North American Journal of Economics and Fi...

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North American Journal of Economics and Finance 37 (2016) 16–37

Contents lists available at ScienceDirect

North American Journal of Economics and Finance

Fractional integration in daily stock market indices at Jordan’s Amman stock exchange Mohammad Al-Shboul a,1, Sajid Anwar b,c,d,∗ a Department of Finance and Economics, College of Business Administration, University of Sharjah, Sharjah 27272, United Arab Emirates b School of Business, University of the Sunshine Coast, Maroochydore DC, QLD 4558, Australia c School of Commerce, University of South Australia, Adelaide, SA 8001, Australia d Shanghai Lixin University of Commerce, Songjiang District, Shanghai, China

a r t i c l e

i n f o

Article history: Received 23 July 2015 Received in revised form 14 March 2016 Accepted 16 March 2016 Available online 28 March 2016 JEL classification: C58 D53 G01 G02 G14 Keywords: Fractional integration Local whittle Efficient market hypothesis Random walk Log-periodogram Jordan

a b s t r a c t Using daily data on five sectoral indices from 2006 to 2014, this paper aims to investigate the possibility of fractional integration in sectoral returns (and their volatility measures) at Jordan’s Amman stock exchange (ASE). Empirical analysis, using the logperiodogram (LP) and local whittle (LW) based semi-parametric fractional differencing techniques suggest that all sectoral returns at ASE exhibit short memory. However, in the case of volatility measures, we found evidence of long memory. Following the recent literature that argues that structural breaks in a time series could also explain the presence of long memory, we tested the volatility measures for the presence of structural breaks. We found that long memory in some volatility measures could be attributed to the presence of structural breaks. Furthermore, using impulse response functions (IRF) based on ARFIMA, we found that shocks to sectoral returns at ASE exhibit short run persistence, whereas shocks to volatility measures display long run persistence. © 2016 Elsevier Inc. All rights reserved.

∗ Corresponding author at: School of Business, University of the Sunshine Coast, Maroochydore DC, QLD 4558, Australia. Tel.: +61 7 5430 1222. E-mail addresses: [email protected] (M. Al-Shboul), [email protected] (S. Anwar). 1 Tel.: +971 6 5053514. http://dx.doi.org/10.1016/j.najef.2016.03.005 1062-9408/© 2016 Elsevier Inc. All rights reserved.

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1. Introduction One of the most controversial issues in behavioral finance is the modeling of long memory behavior in stock returns. efficient market hypothesis (EMH), proposed by Fama (1965), suggests that the flow of new information influences all investors simultaneously and in response to this information stock prices fluctuate randomly (i.e., stock prices follow a random walk). Accordingly, stock prices cannot be predicted based on historical information (Fama, 1970; Summers, 1986). Stock investors in efficient markets cannot earn abnormal returns as stocks are traded at their fair values. Different from EMH, at least some stock prices may follow a specific pattern in the long run that is inconsistent with the idea of random walk. In such a case, stock prices may be predicted based on the past information and hence stock investors may generate different levels of excess returns. In other words, if stock prices were predictable in the long run, stock investors may be able to develop profitable investment strategies. One possible pattern that stock prices may exhibit in the long run is the long-range dependence (also known as long memory). The other possibility is that stock prices may exhibit a short memory behavior (see Baillie, 1996; Baillie, Bollerslev, & Mikkelsen, 1996; Clark & Coggin, 2011; Fama & French, 1988). Long memory refers to a situation where the current values of stock returns remain significantly correlated with their values in the distant past. This implies that stock returns are not independent over time and hence the future returns can be predicted based on the past returns, which contradicts the weak form of EMH. Stock returns exhibit long memory behavior if their autocorrelation functions decay at slow hyperbolic rates. This possibility involves a fractional order of integration. In simple words, the number of differences required to render a series stationary, i.e., I(0), take a value between 0 and 1. Baillie (1996) argues that a fractionally differenced process can also be viewed as a halfway house between an I(0) and I(1) process.2 On the contrary, a short memory behavior describes the loworder autocorrelation structure of a series whereby the autocorrelation function decays at a faster rate (Assaf, 2006; Bollerslev & Mikkelsen, 1996). Investigation of long memory behavior is important due to several reasons. For example, market efficiency is directly linked to the existence of long memory in the returns series, testing for the presence of long memory can help to determine whether the financial markets are efficient or not (Assaf, 2015). Testing for long memory can also help stock market participants as well as policymakers in (i) developing appropriate risk assessment strategies and (ii) comparing different types of asset return behaviors. Examination of long memory also allows investors to re-evaluate their stock trading strategies. Long memory in volatility series can also play a significant role in derivative pricing as modeling of derivative pricing is associated with the long run volatility structure (Bollerslev & Mikkelsen, 1996). Furthermore, analysis of long memory behavior can contribute to the process of stock prices prediction as well as the overall decision-making process. The existing literature on long memory behavior can be divided in two groups: (i) studies that only test for long memory using various statistical methods and (ii) studies that argue that the presence of long memory in a time series may also be due to structural breaks in the series (Bekaert, Harvey, & Lumsdaine, 2002; Chatzikonstanti & Venetis, 2015; Yau & Davis, 2012). Most existing studies that belong to group 1 (i.e., studies that examine only the possibility of long memory) focus on stock market returns in developed countries.3 However, only a few studies have examined the possibility of long memory in stock market returns in developing and emerging economies. For example, Limam (2003) found evidence of fractional integration in the stock markets of the Arab region (including Bahrain, Egypt, Jordan, Kuwait, Morocco, Oman, Saudi Arabia and Tunisia). Bellalah, Aloui, and Abaoub (2005) and Charfeddine and Ajmi (2013) detected long memory behavior in the Tunisian stock market. Using data from Egypt, Jordan, Morocco, and Turkey, Assaf (2006) found evidence of long memory in the returns series for Egypt and Morocco. Whereas, based on data from January 1997 to December 2007, Rejichi and Aloui (2012) reported evidence of the presence of long memory in the Middle East and

2 The order of integration of a long memory process usually lies between 0 and 1 but it can also be greater than 1 (see Anoruo & Gil-Alana, 2011; Phillips, 1999, 2007). 3 For example, to name a few, see Lo (1991), Ding et al. (1993), Cheung and Lai (1995), and Barros et al. (2012).

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North African (MENA) stock market returns. Other related studies include Mensi, Hammoudeh, and Yoon (2014), Chiang and Chen (2015) and Do, Brooks, Treepongkaruna, and Wu (2016). A number of research gaps are observed in the studies that deal with developing and emerging economies. First, studies that aim to examine the degree of fractional integration are mostly multicountry studies that do not tell us much about the actual state of affairs in individual countries. Second, most of these studies are based on a limited number of fractional integration estimation techniques, such as the Geweke and Porter-Hudak (1983) estimator, hereafter GPH, the original ratio scale (R/S) methodology of Hurst (1951), variance scale (V/S), and variance ratio tests. Majority of these studies (such as and Assaf, 2006; Limam, 2003; Norouzzadeh & Jafari, 2005; Rawashdeh & Squalli, 2006; Rejichi & Aloui, 2012) focus only on parametric and non-parametric estimation procedures. Only a few studies have made use of the relatively recent semi-parametric estimation techniques such as the log-periodogram and local whittle. Furthermore, most existing studies involving developing and emerging countries are based on the general aggregate stock indices. In recent years, especially in the aftermath of the 2008–2009 global financial crisis (GFC), the literature on long memory behavior has rapidly grown. A number of studies are directed toward investigating whether the evidence of long memory, as reported by previous studies, is spurious or created by the existence of structural breaks in the data series. These studies, as indicated earlier, belong to Group 2. This group of studies has not yet produced conclusive evidence in support of whether long memory behavior is characterized by volatility in the time series or structural breaks, or both. Some Group 2 studies argue that long memory is characterized by structural breaks (e.g., Bisaglia & Gerolimetto, 2008; Choi, Yu, & Zivot, 2010; Kellard, Jiang, & Wohar, 2015; McMillan & Ruiz, 2009; St˘aric˘a & Granger, 2005). However, other studies provide evidence which suggests that, compared to structural breaks, long memory is better explained by volatility of the series (Arouri, Hammoudeh, Lahiani, & Nguyen, 2012; Garvey & Gallagher, 2012; Jung & Maderitsch, 2014; Uludaq & Lkhamazhapov, 2014). These studies find that long memory is partially characterized by structural breaks (Morana & Beltratti, 2004; Yang & Chen, 2014). Almost all of these studies focus on investigating the relevance of long memory against structural breaks, arising from GFC, using data collected from developed economies. This paper focuses on Jordan’s Amman stock exchange (ASE), which was not only affected by GFC but also by economic and political instability arising from other sources. These sources include the Arab spring of 2010–11, wars in Syria and Iraq that intensified over 2012–14 period and more recently by the decline in oil prices. Despite these critical changes, so far, only a few existing studies have investigated the relevance of long memory against the structural breaks in MENA region. For instance, Assaf (2016) reported that the returns and volatility measures in MENA stock markets display weak evidence of long memory in the post-GFC period compared to the pre-GFC period. Using stock returns from the Tunisian stock market, Charfeddine and Ajmi (2013) argue that the long memory behavior observed was a true behavior and was not spuriously caused by structural breaks. However, the sample period of both studies does not cover the most recent structural breaks caused by the after effects of the Arab spring, increased intensity of the war in Syria and Iraq and significant decline in the price of oil. In order to validate their general findings, these studies test for long memory in volatility measures using different methodologies such as the variance scale (V/S) and the Auto Regressive Fractionally Integrated Moving Average-Fractionally Integrated Generalized Auto Regressive Conditionally Heteroskedastic (ARFIMAFIGARCH) and Markov regime-switching ARCH models, respectively. The focus on Jordan’s ASE is appropriate for several reasons. For example, ASE has become more regionally and internationally integrated. Jordanian government has improved regulation of securities market to meet international standards and also expanded and diversified the national economy. To achieve the highest level of integration, ASE has started using international electronic trading, settlement and clearance systems and also strengthened the capital market supervision. These steps have contributed to increased transparency in operations, which has reduced some obstacles to investments (ASE, 2011, 2013). The ASE has some unique features that are quite different from those of other emerging markets. These unique features are the political and economic stability, the free market oriented economy, freedom of press, private enterprise and property system (Assaf, 2006). In 1978, only 66 firms with a total capitalization value of US$94.4 million were listed at ASE. By 2013, the number of listed firms has risen to 240 with the total market capitalization of more than US$18.2 billion

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(ASE, 2013). In addition, there has been a significant increase in the percentage of foreign ownership of ASE listed company stocks over time. Based on these developments, it will be very interesting to test for the presence of long memory in sectoral returns at ASE. This paper makes some important contributions to the existing literature. First, we examine the possibility of long memory at ASE using sectoral data. Specifically, we consider the possibility of fractional integration in the Banking, Insurance, Services, Industry and General indices returns and volatility measures. Second, unlike the existing literature, we employ two relatively recently developed semi-parametric long memory estimation techniques with different statistical properties: (i) the Log-periodogram (LP) and (ii) the local whittle (LW) estimation techniques. In addition, this paper uses daily data collected from ASE. The sample covers the period of January 2006–October 2014. The MENA region has witnessed a number of critical changes during this period, including the 2008–2009 global financial crisis, the Arab spring, the war in Syria and Iraq, and the recent decline in oil prices. These changes may have resulted in structural breaks in ASE listed stock prices and hence we test for structural breaks in all sectoral return series and their volatility measures. The presence of structural breaks can produce spurious long memory results by shifting the fractional differencing parameter away from zero, which leads to inaccurate time series modeling thereby producing inconclusive evidence (Davidson & Sibbertsen, 2005; Kellard et al., 2015; Qu, 2011). Unlike the previous studies, we make use of the modified version of Inclan and Tiao (1994) iterated cumulative sum of squares (ICSS) to identify the number of structural changes and the break dates in our sample. We then use a procedure developed by Shimotsu (2006) to test whether the structural breaks cause spurious persistence in volatility. Finally, we test for persistence of shocks to ASE using the impulse response function (IRF) and the power density function of ARFIMA model. This allows one to show whether persistence in volatility is affected by the existence of structural breaks. Using all estimation techniques, we find evidence of short memory in all sectoral returns. However, the return on insurance and services indices also display negative persistence (i.e., anti-persistence). We find evidence of long memory in volatility measures of all sectoral returns. We also tested whether the evidence of long memory in volatility measures was spurious or not. Our statistical analysis shows that volatility in sectoral returns was partially affected by the presence of structural breaks and the evidence of long memory was spurious. These results continue to hold when IRF and spectral density tests of ARFIMA were used. Furthermore, shocks to sectoral stock returns exhibit persistence in the short run, whereas the volatility measures exhibit persistence in the long run. The rest of the paper is organized as follows. Section 2 describes the methodologies used in this paper. The data and preliminary results are presented in Section 3. The main empirical results are presented and discussed in Section 4. Section 5 contains some concluding remarks. 2. Methodology As indicated earlier, in this paper we use log-periodogram and local whittle estimation based techniques. These techniques are briefly explained in this section. 2.1. Log-periodogram approach (LP) A time series, xt , exhibits long memory process if its spectral density is unbounded at some frequency in the interval [0,␲]. The degree of long memory is denoted by the fractional differencing parameter d, which is modeled by a long memory process using a spectral density that is unbounded at the 0 frequency. If the unboundedness takes place at a non-zero frequency, the series may continue to exhibit long memory behavior. Therefore, a special form of long memory is specified in Eq. (1) where the pole or singularity in the spectrum may take place at the 0 frequency: d xt = (1 − L)d xt = ut

(1)

where L is the lag operator and ut is normally distributed. If d ≥ 0.5 then the series is nonstationary, its variance is infinite, and it exhibits long memory (Granger & Joyeux, 1980). If d = 0 then the series exhibits short memory, suggesting an ARMA structure with autocorrelations decaying at an exponential rate. If d ∈ (0, 0.5) then the series is still stationary

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but the autocorrelations take a long time to decay compared with the case of an I(0) series. Finally, if d ∈ (−0.5, 0) then the series exhibits an anti-persistence or negative persistence. In order to test for long memory under the LP approach, we utilize two methods: (i) the GPH method and (ii) the modified LP (MLP) method developed by Phillips (2007). The GPH approach aims to estimate the fractional differencing parameter, d, based on the slope of the spectral density function around the angular frequency. The parameter d can be estimated based on the first m periodogram ordinates as follows:

  T −1

2 

1  iωj t  xt e  T Ix (ωj ) = 2Tj  

j = 1, . . ., m

(2)

t=0

where ωj = 2j/T for j = 1, . . ., m; m is an integer truncation parameter to narrow the frequencies in the neighborhood of the origin. The GPH estimator of d is given by 1 dˆ = − 2

 m

j=1

(xj − x¯ ) log Ix (ωj )

m

j=1

(xj − x¯ )



,

 

xj = log 2 sin

 ω  j   2

(3)

Phillips (1999, 2007) argues that GPH estimator might be inefficient and asymptotically biased when testing for fractional integration in the non-stationary region. To eliminate these statistical problems, Phillips (1999, 2007) designed a modification for the LP regression by testing for fractional integration in the non-stationary region without relying on Gaussian assumption as GPH did. MLP approach involves modifying xt to reflect the distribution of d under the null hypothesis H0 : d = 1. The MLP estimator of d can be derived from least squares estimation of Eq. (2) with two modifications. The m ¯ where ˛j = log|1 − eiωj | and ˛ ¯ = m−1 j=1 ˛j and the first modification of xt involves using xj = ˛j − ˛,



T

x eiωj t . second modification involves calculation of the sample periodogram, Ix (ωj ) = 1/ 2Tj t=1 t √ 2 The limiting distribution of the estimated d is given by (m(dˆ − d)) → N(0,  /24). This distribution is similar to GPH distribution in the stationary region. MLP method is consistent in the case of both d < 1 and d > 1 fractional alternatives and allows for d to be tested in the intervals (0.5, 1) and greater than 1. The time series is considered no longer covariance stationary if d ∈ (0.5, 1). However, it is still mean-reverting with shocks affecting disappear in the long run. Finally if d is more than 1, the series is nonstationary (i.e., it represents explosive behavior) and non-mean-reverting. In this paper, we also use the MLP method. 2.2. Local whittle estimation (LW) It has been argued that the LW approach is more efficient than LP approach to testing for fractional differencing of time series because the LW estimator is akin to the Maximum Likelihood (ML) in the frequency domain (Robinson, 1995b). The LW method was originally developed by Kunsch (1987) and relies on the frequency domain Gaussian likelihood function restricted to vicinity of origin. The LW estimators are consistent for d ∈ (0.5, 1) and asymptotically normally distributed for d ∈ (0.50, 0.75). In the stationary region, LW estimator is consistent with LP and has the same asymptotic normal distribution (Robinson, 1995a,b). The LW estimator is a band of frequencies that degenerate to zero. The long memory parameter estimates of d can be defined by dˆ = argmind (log C(d) − 2d/m where C(d) = (1/m)

n

I(ωj )ωj2d , j=1

(4)

and ωj = 2j/T is the frequency.

As the length of the series degenerates to zero for the bandwidth m, the term (1/m) + (m/T) also degenerate to zero. Ix (ωj ) is the periodogram of the time series xt which can be written as

 2 T    iωj t  Ix (ωj ) = 1/2T  xt e    t=1

(5)

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By taking the finiteness of the four moments and other conditions into account, Robinson (1995a,b) showed an asymptotic relationship as

 1 √ m(dˆ LW − d0 ) → N 0, 4

for T → ∞

(6)

where d0 is the true value of d. Although the estimates of LW method are consistent in the region (0.5, 1) and asymptotically normally distributed in the region (0.5, 0.75), Phillips and Shimotsu (2004) argue that LW estimator may follow a non-normal limit distribution in the region (0.75, 1). Moreover, the LW estimator may converge to unity and become inconsistent when d is greater than 1 (i.e., in the non-stationary region). Therefore, long memory parameter estimation based on LW is not regarded as a good general-purpose estimator. In an attempt to circumvent the inconsistency and unreliability for inference when d may be greater than 0.75, Shimotsu and Phillips (2005) extends the LW model by designing an “exact” form of the LW estimator. This exact form of LW (ELW hereafter) provides the best estimation of the parameter d in stationary and non-stationary regions. The main contribution of ELW is the implementation of a correction for degrees of freedom of xt by minimizing the objective function



1 1 log(Gωj−2d ) + I(1−L)d x (ωj ) m G m

Qm (G, d) =

(7)

j=1

By minimizing the above objective function with respect to G, Shimotsu and Phillips derived the following ELW estimator dˆ = argmind ∈ (1 ,2 ) R(d)

(8)

where 1 and 2 are the lower and upper bounds of d and ˆ R(d) = log G(d) − 2d

1 log ωj , m m

1  2d ˆ G(d) = ωj Ix (ωj ) m

j=1

j=1

m

(9)

Under certain conditions, such as d ∈ [1 , 2 ] with 1 − 2 ≤ 9/2, the asymptotic relationship can be specified by

 1 √ m(dˆ ELW − d0 ) → N 0, 4

for T → ∞

(10)

Shimotsu (2010) extended the ELW approach by developing a new method, known as the “Twostage Exact Local Whittle” method (2ELW hereafter). In addition offering the inherent desirable properties of ELW estimation, 2ELW approach also accommodates for an unknown mean and a polynomial time trend. Specifically, the objective function of ELW is modified by using a tapered LW estimator to follow an N(0, 0.25) asymptotic distribution for d ∈ (−0.5, 2) or d ∈ (−0.5, 0.25) when the data series contain a polynomial trend. In order to estimate the parameter d to include a wider range, Shimotsu (2010) derived an estimator of the unknown mean as follows: (d) ˜ = ω(d)¯x + (1 − ω(d))x1

(11)

where ω(d) is a twice continuously differentiable weighted function that equals to the value of 1 for d ≤ 1/2 and the value of zero for d ≥ 3/4. The 2ELW estimator of d can be derived by applying the ELW estimation technique on the time series x¯ − (d). ˜ This paper attempts to test for the presence of long memory in ASE returns by making use of GPH, MLP, ELW and 2ELW methods. 3. Data The sample used in this study consists of the daily data on ASE indices (banks, insurance, industry, services and the general index) from the beginning of 2006 to the end of 2014. The indices are used

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to calculate the continuously compounded rate of return. Because this paper also considers possible long memory behavior in measures of volatility, we make use of both absolute and squared values of the returns. The reason for considering volatility measures is that persistence in volatility may be attributed to a shift in investors’ risk-return profiles, which tends to have a significant influence on the long-run investment behavior (Assaf, 2006; Ding, Granger, & Engle, 1993). All data are collected from the Amman stock exchange database. 4. Empirical results 4.1. Descriptive statistics and autocorrelations A summary of the descriptive statistics for stock market returns on each of the five sectors are reported in Table 1. The average return on all indices is approximately zero. All return series exhibit significant variation as indicated by the standard deviations. The results of Jarque–Bera (J–B) test reported in Table 1 suggest that all variables are normally distributed. The return series are found to be significantly auto-correlated as indicated by Ljung–Box values for AR(1)–AR(40). In other words, the return series exhibit time-dependence. There is strong evidence of autocorrelation in the returns series. We tested all data series for stationarity using the augmented Dickey–Fuller (ADF) (Dickey & Fuller, 1981), the KPSS (Kwiatkowski, Phillips, Schmidt, & Shin, 1992), and the Ng–Perron (NP) (Ng & Perron, 2001). The results based on a constant and time trend are reported in Table 1. The results of ADF and KPSS tests suggest that the data series are stationary. However, NP test results suggest non-stationarity. Table 1 Summary statistics and diagnostic testing results.

Mean St. Dev. Skewness Kurtosis J–B (1) (2) (20) Q(40) rs (1) rs (2) rs (20) Qs (40) ADF returns KPSS returns NP MZa MZt MSB MPT

Banks

Insurance

Industry

Services

General

0.000 0.013 0.121 5.825 560.3*** 0.170 −0.017 −0.028 114.35*** 0.409 0.288 0.098 2360.5*** −3.299*** 0.066

0.001 0.032 −0.517 559.1 227.6*** 0.130 −0.039 −0.011 293.74*** 0.498 −0.001 −0.001 416.92*** −4.954*** 0.099

0.001 0.014 −0.256 152.24 100.20*** 0.259 0.075 −0.033 206.66*** 0.480 0.386 0.344 7781.75*** −5.893*** 0.093

0.000 0.018 −0.189 4.136 220.89*** −0.117 −0.008 −0.004 50.776*** 0.497 0.000 −0.001 414.53*** −7.11*** 0.051

0.000 0.012 −0.159 6.554 887.1*** 0.170 −0.017 −0.028 119.3*** 0.459 0.243 0.131 2158.32*** −4.488*** 0.086

−5.212 −1.588 0.305*** 17.389

−2.355 −1.062 0.451*** 37.684

−26.441 −3.636 0.138*** 3.447

−14.268 −1.585 0.111 11.627

−9.456 −2.162 0.229*** 9.693

This table reports the summary statistics of sectoral returns. J–B is Jarque–Bera normality test statistic with 2 df with the corresponding p-values; (k) is the sample autocorrelation coefficient at lag k and Q(k) is the Box–Ljung portmanteau statistic based on k-squared autocorrelations; rs (k) are the sample autocorrelation coefficients at lag k for squared returns and Qs (40) is the Box–Ljung portmanteau statistic based on 40-squared autocorrelations. The maximum lags order of ADF test was chosen using Schwert criterion. ADF critical values for H0 : the series is trend stationary are: (1%: −3.480; 5%: −2.831; 10%: −2.545). The critical values used are adapted from Elliott, Rothenberg, and Stock (1996). The maximum lag order of KPSS test was chosen using Schwert criterion and the auto-covariances weighted by Bartlett kernel. KPSS critical values for H0 : the series is trend stationary are: (10%: 0.119; 5%: 0.146; 2.5%: 0.176; 1%: 0.216). For both tests, the maximum lags are 24. The last test used is the NP test which includes an intercept, and the lag order was chosen using the modified AIC (MAIC). The critical values of NP test are: for MZa (1% = −23.8; 5% = −17.3; 10% = −14.2); for MZt (1% = −3.42; 5% = −2.91; 10% = −2.62); for MSB (1% = 0.143; 5% = 0.168; 10% = 0.185); for MPT (1% = 4.03; 5% = 5.48; 10% = 6.67). *** Null hypothesis rejected at the 1% level of significance.

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The mixed results underline the need to go beyond the usual I(1)/I(0) framework and hence we test for fractional integration. Fig. 1 shows the autocorrelation functions of the returns, absolute returns and squared returns for each of the five sectoral indices. Most autocorrelations of the sectoral returns are small in size and lie within the respective 95% confidence intervals. All sectoral returns appear to show evidence of short memory, because the autocorrelation functions decay at a fast rate over a few lags. Furthermore, the autocorrelation functions of the returns for Banks, Industry and the General index exhibit a meanreverting process with positive persistence, while the returns for ASE’s Insurance and Services indices show an anti-persistence behavior. These results are consistent with the estimated autocorrelations reported in Table 1. Unlike the sectoral returns autocorrelations, the autocorrelation functions of the volatility measures (i.e., the squared and absolute returns) of three out of five sectoral returns are high and remain

Fig. 1. Autocorrelations of sectoral indices returns and their volatility measures.

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positive and statistically significant over several lags. This suggests the presence of long memory in volatility measures as the autocorrelation functions decay at a very slow hyperbolic rate over several lags. However, in the case of Insurance and Services sectors, the autocorrelations of the volatility measures are not high and decay at a fast rate. These results are consistent with those reported in Table 1. In overall terms, there is evidence of long memory in volatility measures. However, the long memory behavior may have been spuriously generated by the existence of structural breaks in the time series (Arouri et al., 2012; Diebold & Inoue, 2001). 4.2. Log-periodogram (LP) In order to test for the presence of long memory in sectoral returns, we first use the GPH approach, which is based on semi-parametric LP methodology. The results of GPH estimation are reported in Table 2. This technique requires selection of a number of harmonic ordinates to include in the spectral regression. Since the results of GPH are sensitive to the number of ordinates and the power values, we use a range of values between 0.50 and 0.80. Under GPH, if a very small number of ordinates are selected, the value of d does not account for autocorrelation. However, if a very large number of ordinates are selected then the value of d can potentially capture certain forms of autocorrelation (Assaf, 2006; Chueng & Lai, 1995). The empirical results reported in Table 2 show that the null hypothesis of d = 0 is not rejected for all selected powers and for all return series, suggesting no evidence of long memory. This implies that autocorrelations of the stock returns decay exponentially at a fast rate, meaning that stock returns revert to their respective mean at a very fast rate (i.e., the return series exhibit mean-reversion). As the power increases, the estimated values of d decrease and eventually take negative values. Baring the Insurance sector, the negative values are statistically insignificant. This suggests that the ASE sectoral returns would exhibit anti-persistent behavior, meaning that the return values change sign and converge to their respective equilibrium means (Assaf, 2006). These results suggest that efficient market hypothesis holds in the case of ASE.4 The empirical results concerning the volatility measures of the sectoral returns are reported in Table 2. The results show that absolute and squared returns (i.e., the volatility measures) are significantly different from zero for all powers and for all indices. This suggests the presence of long memory in volatility measures of most sectoral returns as the null hypothesis of d = 0 is rejected. The exceptions are the absolute returns in the case of Insurance sector and the squared returns in the case of Services sector where the estimated values of d are not found to be statistically different from zero (which suggests the presence of short memory behavior). The presence of long memory in volatility measures as reported in Table 2 implies that shocks are persistent in the volatility processes, which is inconsistent with random walk behavior (Coleman & Sirichand, 2012). The tendency of short memory behavior in absolute returns on insurance sector index is consistent with relatively low level of volatility. A report commissioned by the Jordan Insurance Federation (2011) suggests that the insurance sector, which has been steadily growing over time, consists of a large number of family controlled firms. The Jordanian policy makers have attempted to enhance the level of protection offered to the insurance industry from the associated risks, which has contributed to general stability of the sector. Other factors that can account for the low level of insurance sector volatility include Jordan’s low disposable income, religious issues, the cultural attitudes toward insurance and the lack of public awareness of the availability different insurance products. The squared returns on the services sector index show evidence of short memory with antipersistent behavior. The services sector is known to be the main driver of Jordan’s economy over the past 10 years (The World Bank, 2013). However, this sector has underperformed in the last few years. However, a few sub-sectors of this sector have registered significant growth (Owen & Dorsey, 2012). The deterioration of the services sector would have affected the behavior of investors resulting in substitution of long-term strategies in favor of short-term strategies, thereby creating short memory anti-persistent behavior. The Jordanian economy is a service-oriented economy where the

4 The weak form of the efficient market hypothesis suggests that asset prices reflect all available historical information and hence asset prices fluctuate like a white noise thereby reflecting the unpredictable behavior of asset returns.

Powers

Banks

Insurance

Industry

Services

General

Returns 0.5 0.55 0.6 0.65 0.7 0.80

0.120 (1.01) [1.02] 0.062 (0.66) [0.66] 0.054 (0.64) [0.72] −0.013 (−0.21) [−0.21] −0.059 (−1.17) [−1.17] −0.024 (−0.65) [−0.68]

0.165c (1.73) [1.40] 0.083 (0.98) [0.89] 0.041 (0.55) [0.54] 0.023 (0.35) [0.37] −0.091c, * (−1.78) [−1.8] −0.264a, * (−8.20) [−7.59]

0.343b, * (2.61) [2.92] 0.125 (1.21) [1.33] −0.059 (−0.76) [−0.78] −0.056 (−0.92) [−0.91] −0.007 (−0.16) [−0.14] 0.088a, * (2.79) [2.55]

0.099 (0.83) [0.84] 0.086 (0.99) [0.92] 0.076 (1.09) [0.10] 0.045 (0.76) [0.73] −0.009 (−0.19) [−0.18] −0.041 (−1.31) [−1.19]

0.219* (1.30) [1.86] 0.100 (0.81) [1.07] −0.013 (−0.14) [−0.16] −0.042 (−0.57) [−0.67] −0.027 (−0.47) [−0.53] 0.018 (0.43) [0.521]

Absolute returns 0.5 0.55 0.6 0.65 0.7 0.80

0.518a, * 0.321a, * 0.383a, * 0.369a, * 0.335a, * 0.304a, *

(5.24) [4.41] (3.74) [3.43] (5.87) [5.03] (6.56) [5.96] (7.13) [6.61] (9.35) [8.77]

0.046 (0.76) [0.39] 0.024 (0.58) [0.25] 0.041 (1.33) [0.53] 0.036 (1.56) [0.59] 0.032c (1.86) [0.63] 0.076a, * (6.29) [2.19]

0.658a, * (5.41) [5.59] 0.545a, * (5.68) [5.82] 0.463a, * (6.12) [6.08] 0.412a, * (6.70) [6.67] 0.352a, * (7.08) [6.95] 0.276a, * (7.86) [7.96]

Squared returns 0.5 0.55 0.6 0.65 0.7 0.80

0.560a, * 0.269a, * 0.379a, * 0.343a, * 0.324a, * 0.296a, *

(4.24) [4.76] (2.46) [2.87] (4.57) [4.99] (5.01) [5.55] (5.80) [6.40] (8.33) [8.54]

0.005c (1.71) [0.04] 0.004b (1.99) [0.04] 0.006a (3.99) [0.08] 0.008a (7.45) [0.13] 0.015a (13.49) [0.30] 0.063a (21.22) [1.84]

0.774a, * 0.666a, * 0.546a, * 0.448a, * 0.379a, * 0.276a, *

(8.04) [6.58] (7.87) [7.11] (7.45) [7.18] (7.51) [7.25] (7.82) [7.48] (7.73) [7.96]

0.393a, * 0.324a, * 0.275a, * 0.237a, * 0.199a, * 0.180a, *

(3.66) [3.34] (4.07) [3.46] (4.63) [3.61] (4.97) [3.84] (5.44) [3.93] (7.15) [5.20]

−0.039 (−1.23) [−0.33] −0.031 (−1.44) [−0.33] −0.021 (−1.39) [−0.27] −0.010 (−0.89) [−0.15] 0.002 (0.27) [0.04] 0.056a (10.86) [1.63]

0.649a, * (7.63) [5.52] 0.448a, * (6.03) [4.79] 0.504a, * (7.79) [6.62] 0.418a, * (7.27) [6.76] 0.372a, * (7.89) [7.34] 0.290a, * (8.45) [8.37] 0.560a, * (5.89) [4.76] 0.352a, * (4.51) [3.76] 0.430a, * (7.28) [5.65] 0.329a, * (6.35) [5.33] 0.304a, * (7.18) [6.00] 0.251a, * (8.02) [7.22]

This table reports that results of d estimates using the GPH test for stock returns and their volatility measures (absolute and squared returns). Statistical significance of the estimated d values is tested by t and z tests. The null hypothesis is H0 : d = 0. The tabulated values of t test are reported in the usual brackets whereas the tabulated values of z-statistic are reported in the squared brackets. The number of ordinates used when estimating the values of d are as follows: (power: ordinate) (0.50: 41); (0.55: 60); (0.60: 86); (0.65: 125); (0.7:181); (0.8: 378), respectively. “a, b and c”, respectively, denote significance of the t-statistic at the 1%, 5% and 10% level. *

Statistical significance of the z-statistic at less than 5% level.

M. Al-Shboul, S. Anwar / North American Journal of Economics and Finance 37 (2016) 16–37

Table 2 The spectral regression function of fractional integration (GPH).

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services sector accounted for about 67% of the GDP in 2013 (ASE, 2013). A significant proportion of the long term investment in Jordanian economy is directed toward education and financial services sectors. The long-term investment in these subsectors can contribute to economic stabilization as well as a decrease in volatility of the services sector resulting in short memory behavior. In summary, although evidence of short memory is reported in sectoral returns, strong evidence of long memory is detected in volatility measures. The estimated values of d show evidence of fractionally integrated processes, indicating that autocorrelation functions decay hyperbolically to zero. Given the mixed nature of these results, further analysis is carried out using the modified log-periodogram (MLP) approach. Specifically, we utilize the MLP approach developed by Phillips (2007). This method requires the selection of a number of powers to be included in the spectral regression. Like GPH estimation, we used a range of powers between 0.50 and 0.80 and utilized both t and z tests. The estimated results, reported in Table 3, indicate that the estimated values of the returns lie in the range 0.00–0.50. Hypothesis testing suggests that the estimated values are significantly different from 1 but not significantly different from zero. The return series of all sectors exhibit evidence of mean reversion and their autocorrelations decay at a fast rate. As the power increases, the values of d decline at a slower rate and become negative. This suggests a mean-reverting process with an anti-persistence component. Based on these results, it can be argued that fluctuations in stock prices in the future would have a short-term impact on sectoral returns. The estimated results for all sectoral returns are qualitatively similar to those reported in Table 2, where GPH method was used. It should be noted in Tables 2 and 3 that, as the power increases, the estimated z-values are larger than the corresponding t-values for GPH and MLP estimates of d. The best justification for this phenomenon is that the test results are distorted by the sample size and the corresponding critical values are obtained from two different distributions. The empirical results presented in Table 3 also show evidence of long memory behavior in volatility measures. Except for the insurance industry absolute returns and the services sector squared returns, the null hypothesis (i.e., H0 : d = 0) is consistently rejected, at all powers, for all sectoral indices. In general, we observe that, as the power increases, the estimated values of d (for both volatility measures) continue to show long memory behavior even though the estimated value of d changes from nonstationary region (0.5, 1) to covariance stationary region (0, 0.5). In other words, stock prices quickly adjust as they began reverting to their respective means at a fast hyperbolic rate. Finally, the null hypothesis of d = 1 can be rejected. For both volatility measures, for all powers, the estimated value of z-statistic shows no evidence of explosive behavior. In other words, no case of variance stationary exists. The estimated values of d are consistently less than 1 for all return series, which suggests mean-reversion behavior. In the case of insurance industry absolute returns and services sector squared returns, the estimated values of d are less than zero, which indicates mean reversion with anti-persistence. However, the null hypothesis of d = 0 is not rejected for all powers whereas the null hypothesis of d = 1 is rejected for all powers. This suggests that no case of variance stationary exists within our sample. The absolute returns in the insurance industry continue to exhibit short memory behavior. These findings are also consistent with existing studies such as Rawashdeh and Squalli (2006) and Assaf (2006). Since almost all ASE sectoral returns exhibit mean-reverting behavior, it is likely that shocks to stock prices at ASE were not large enough to have a significant long term impact. In overall terms, the results of MLP estimation are similar to those reported for the GPH estimators in Table 2. 4.3. Local whittle (LW) The results of the LW estimation are reported in Table 4. These results are based on testing for fractional differencing in the stationary and non-stationary regions using m = n0.5 , n0.55 , n0.60 , n0.65 , n0.70 and n0.80 number of frequencies. We use different values of the truncation parameter to account for all possible estimations of d in the regions of (0.5, 0.75) and (0.75 and 1) as required by LW approach. It has been argued that estimation results are sensitive to the choice of m (Charfeddine, 2014; Hassler & Olivares, 2013). Until now, the existing literature has not identified the optimal value of the truncation parameter. Accordingly, we use different values of m to estimate the true values of d. We assume that the true value of m lies in range m = n0.5 –n0.60 .

Power

Banks

Insurance

Industry

Services

General

Returns 0.5 0.55 0.60 0.65 0.70 0.80

0.262b,* (2.25) [−7.28] 0.154b,* (1.68) [−10.14] 0.122* (1.50) [−12.63] 0.038* (0.62) [−16.71] −0.022* (−0.44) [−21.4] −0.003* (−0.08) [−30.41]

0.467a,* (4.13) [−5.25] 0.280a,* (2.98) [−8.62] 0.200* (1.50) [−11.5] 0.149* (1.12) [−14.8] 0.005* (0.09) [−20.82] −0.21a,* (−6.10) [−36.59]

0.375a,* (2.83) [−6.17] 0.146* (1.40) [−10.23] −0.045* (−0.57) [−15.02] −0.046* (−0.75) [−18.17] 0.000* (0.01) [−20.93] 0.092a,* (2.90) [−27.53]

0.308a,* (2.77) [−6.82] 0.234a,* (2.82) [−9.18] 0.192* (1.80) [−11.62] 0.132* (1.21) [−15.07] 0.054* (1.14) [−19.79] −0.007* (−0.21) [−30.52]

0.406a,* (3.12) [−5.86] 0.242b,* (2.36) [−9.08] 0.100* (1.28) [−12.93] 0.044* (0.67) [−16.61] 0.039* (0.74) [−20.12] 0.065* (1.67) [−28.31]

Absolute returns 0.5 0.55 0.60 0.65 0.70 0.80

0.312a,* (2.43) [−6.79] 0.146* (1.48) [−10.23] 0.241a,* (3.21) [−10.92] 0.240a,* (4.03) [−13.20] 0.235a,* (4.76) [−16.02] 0.243a,* (7.33) [−22.94]

−0.011* (−0.17) [−9.98] −0.019* (−0.42) [−12.2] 0.008* (0.25) [−14.26] 0.013* (0.51) [−17.14] 0.015* (0.79) [−20.62] 0.066a,* (5.30) [−28.3]

0.688a,* 0.579a,* 0.482a,* 0.420a,* 0.357a,* 0.275a,*

(6.07) [−3.08] (5.73) [−5.04] (6.12) [−7.44] (6.67) [−10.08] (6.94) [−13.46] (7.80) [−21.95]

0.238b,* (2.14) [−7.51] 0.202a,* (2.52) [−9.56] 0.182a,* (3.05) [−11.77] 0.165a,* (3.48) [−14.51] 0.145a,* (3.99) [−17.89] 0.150a,* (5.98) [−25.76]

Squared returns 0.5 0.55 0.60 0.65 0.70 0.80

0.369a,* (2.51) [−6.22] 0.117b,* (2.04) [−10.57] 0.262a,* (3.04) [−10.61] 0.241a,* (3.55) [−13.18] 0.253a,* (4.45) [−15.64] 0.248a,* (6.99) [−22.76]

−0.017a,* (−3.4) [−10.0] −0.012a,* (−3.51) [−12.1] −0.007a,* (−2.55) [−14.5] −0.001* (−0.40) [−17.38] 0.008a,* (4.96) [−20.75] 0.060a,* (18.87) [−28.49]

0.820a,* 0.673a,* 0.538a,* 0.434a,* 0.364a,* 0.267a,*

(6.90) [−1.78] (6.87) [−3.91] (6.67) [−6.64] (6.81) [−9.82] (7.19) [−13.30] (7.37) [−22.22]

−0.019* (−0.64) [−10.05] −0.016* (−0.8) [−12.17] −0.010* (−0.71) [−14.52] −0.001* (−0.14) [−17.39] 0.008* (1.11) [−20.75] 0.060a,* (12.03) [−28.5]

0.585a,* 0.378a,* 0.430a,* 0.343a,* 0.307a,* 0.245a,*

(4.85) [−4.10] (4.04) [−7.45] (5.73) [−8.19] (5.56) [−11.4] (6.21) [−14.5] (7.05) [−22.9]

0.440a,* (4.35) [−5.53] 0.273a, * (3.46) [−8.71] 0.372a,* (6.16) [−9.03] 0.282a,* (5.4) [−12.47] 0.267a,* (6.29) [−15.4] 0.227a,* (7.29) [−23.4]

This table reports that results of d estimates using the modified log periodogram (MLP) regression fractional differencing test, as proposed by Phillips (2007), for stock returns and their volatility measures (absolute and squared returns). The statistics used to test for the significance of d values are t and z statistics. The null hypothesis of t is H0 : d = 0 and the null for z is H0 : d = 1. The tabulated values of t test are in the (brackets) and the tabulated value of z is in the [squared brackets]. The number of ordinates used for estimating the values of d are as follows: (power: ordinate) (0.50: 41); (0.55: 60); (0.60: 86); (0.65: 125); (0.7: 181); (0.8: 378), respectively. In the case of t statistic, “a and b” Statistical significance of the 1%, 5% and 10%, respectively. In the case of z statistic, “*” signifies significance at less than 5% level. Absolute and squared returns are measures of sectoral return volatility.

M. Al-Shboul, S. Anwar / North American Journal of Economics and Finance 37 (2016) 16–37

Table 3 The modified log-periodogram regression fractional differencing (MLP).

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Table 4 Local whittle (LW) based estimates of fractional differencing parameter d for sectoral return series and volatility series. n

0.4

0.5

0.55

0.6

0.65

0.7

0.8

Returns LW Banks Insurance Industry Services General

1672 1672 1672 1672 1672

0.287 0.559 0.922 0.309 0.522

0.641 0.623 0.764 0.522 0.734

0.593 0.577 0.639 0.621 0.646

0.633 0.639 0.596 0.664 0.633

0.642 0.690 0.638 0.658 0.645

0.656 0.710 0.732 0.714 0.699

0.802 0.729 0.895 0.814 0.834

2ELW Banks Insurance Industry Services General

1672 1672 1672 1672 1672

0.289 0.576 0.942 0.302 0.512

0.628 0.656 0.758 0.580 0.775

0.646 0.642 0.642 0.655 0.659

0.658 0.643 0.604 0.668 0.646

0.687 0.665 0.644 0.675 0.670

0.694 0.668 0.733 0.699 0.718

0.775 0.675 0.931 0.775 0.838

Absolute returns LW 1672 Banks 1672 Insurance 1672 Industry Services 1672 1672 General

0.878 0.708 1.165 0.748 1.068

0.794 0.814 0.997 0.767 0.872

0.953 0.870 1.026 0.885 1.001

0.959 0.909 1.016 0.901 0.975

0.963 0.936 1.008 0.921 0.971

0.967 0.948 1.004 0.953 0.974

0.965 0.969 0.985 0.972 0.970

2ELW Banks Insurance Industry Services General

1672 1672 1672 1672 1672

0.617 0.666 1.302 0.603 1.081

1.075 0.687 1.225 0.953 1.183

0.907 0.692 1.137 0.922 1.010

1.029 0.706 1.122 0.937 1.097

1.046 0.720 1.115 0.924 1.052

1.067 0.763 1.122 0.946 1.080

1.148 0.906 1.133 1.028 1.125

Squared returns LW Banks Insurance Industry Services General

1672 1672 1672 1672 1672

0.827 0.428 1.295 0.427 0.911

0.723 0.553 1.056 0.531 0.831

0.797 0.599 1.069 0.599 0.939

0.913 0.653 1.045 0.656 0.971

0.933 0.704 1.025 0.710 0.934

0.943 0.753 1.019 0.775 0.954

0.944 0.883 0.991 0.904 0.959

2ELW Banks Insurance Industry Services General

1672 1672 1672 1672 1672

0.548 0.450 1.311 0.415 0.920

0.993 0.538 1.245 0.581 1.060

0.631 0.582 1.169 0.606 0.909

0.971 0.624 1.125 0.642 1.037

0.997 0.669 1.104 0.680 0.971

1.042 0.722 1.111 0.719 1.004

1.142 0.883 1.122 0.885 1.070

This table reports the estimated values of d based on the LW methods for stock return series. The number of frequencies used for d is calculated based on powers: 0.5, 0.55, 0.60, 0.65, √ 0.70 and 0.80, and the sample period n = 1672. The confidence interval can be computed by adding and subtracting 1.96 × 1/ 4m from the point estimates.

The results presented in Table 4 show that, for the LW methods, the estimated values of d for all frequencies are (i) located within the region of 0.5–0.75 and (ii) statistically significant for all sectoral return series. Based on the 95% asymptotic confidence interval, these results suggest evidence of short memory.5 While not directly comparable, these results are inconsistent with the empirical results of Shimotsu (2010) who provided evidence of long memory behavior for a number of the US macroeconomic indicators (GNP, real per capita GNP, and employment). However, the results presented in this paper are in line with the results of Hassler and Wolters (1995) who reported that inflation was are I(d) with d ∈ (0, 1).

5

√ Following Shimotsu (2010), the confidence interval is constructed by adding or subtracting 1.96 × 1/ 4m to the estimates.

M. Al-Shboul, S. Anwar / North American Journal of Economics and Finance 37 (2016) 16–37

29

The estimated values of the long memory parameter, d, for volatility measures are reported in Table 4. We first discuss the results of LW estimation. In the case of absolute returns, the estimated values are significant and close to 1 (in the range of 0.75–1). Interestingly, the null hypothesis of trend stationarity (i.e., H0 : d = 0) is rejected in the case of all absolute sectoral returns. Expect for the insurance industry, there is evidence of long memory behavior. In other words, volatility in almost all sectoral returns decays at a very slow rate. In the case of the insurance sector volatility, there is evidence of mean reverting behavior. Table 4 also contains the results of 2ELW estimation. The null hypothesis of trend stationarity (i.e., H0 : d = 0) is rejected in the case of all squared returns. There is evidence of long memory behavior in all sectoral returns except for the squared returns in insurance and services sectors. The insurance and services sector squared returns exhibit short memory behavior. The estimated results based on the three methods are fairly close to each other. These results confirm the results reported in Tables 2 and 3. Furthermore, these findings are generally consistent with Shimotsu (2010) and Crato and Rothman (1994). In summary, the results of LW estimation confirm the empirical results based on LP methods. It is worth mentioning that the estimated values of d, reported in Tables 2–4, vary considerably across estimation techniques and the estimated values are greater than 0.5 for some sectoral returns. In fact, these estimated values of d are inconsistent with the usual findings. This inconsistency can perhaps be attributed to the bias inherent in GPH and MLP estimators and/or misspecification of the long memory process that arises when structural breaks are not taken into account (Charfeddine, 2014; Granger & Hyung, 2004). The choice of the bandwidth and truncation parameter can also affect the results of long memory estimation (especially in the case of the semi-parametric LW group of methods). In this paper, we find that the estimated values of d, using LW and 2ELW methods, are somewhat different from the estimated values obtained from GPH and MLP methods. For example, when we set m = n0.05 , the estimated values of d for sectoral returns using GPH and MLP methods, respectively, range from 0.099 to 0.467 and −0.051 to 0.192. In general, the estimated values of d obtained from GPH and MLP methods and lower than those obtained from LW methods. The long memory estimation results presented here using semi-parametric methods are sensitive to the choice of bandwidth and truncation parameters. Additionally, there is some evidence of model misspecification. In summary, although different values of, d, are reported in this paper, irrespective of the bandwidth and truncation parameter values, the empirical results based on all semi-parametric methods confirm the presence of high persistence in volatility in sectoral returns, when volatility is measured by squared returns. The presence of high persistence in volatility can also be attributed to extreme sensitivity of long memory models to the presence of structural breaks.6 Our results are consistent with the general findings of Charfeddine and Guégan (2012) and Charfeddine (2014). 4.4. Testing for long memory and structural breaks In this section, we report the results of testing for the relevance of long memory against structural breaks. The existing literature suggests that neglecting the existence of structural breaks can lead to evidence of spurious long memory results (Baillie & Morana, 2009; Perron & Qu, 2007). However, before testing for the relevance of long memory against structural breaks in sectoral returns at ASE, we first test whether structural breaks exist in the squared returns. In order to perform the test, we make use of the adjusted version of the iterated cumulative sum of squares (ICSS) algorithm test proposed by Inclan and Tiao (1994). The ICSS test has been extensively used in the existing literature to identify the structural breaks volatility measures (Arouri et al., 2012; Charfeddine, 2014; Hammoudeh & Li, 2008). The number of breaks and the estimated break dates are reported in the top panel of Table 5. There is evidence of at least one common structural break in their squared returns (i.e., the unconditional variances). We detect five breaks in the squared return for the Banks, two breaks for Insurance, four for Services, five breaks for Industry, and four breaks in the General index. Eleven of these identified breaks are associated with the 2008–09 GFC. This suggests that the ASE is not immune from global shocks. We

6 Charfeddine (2014) pointed out that GPH-based methods are less sensitive to the presence of structural breaks compared to LW methods.

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Table 5 Long memory against structural breaks. No. of breaks

Break dates

Panel A: Testing for structural breaks using ICSS algorithm results for structural breaks 5 2007:08:31; 2008:08:08; 2009:04:30; 2010:05:11; 2013:10:15 Banks 2011:11:18; 2011:11:22 2 Insurance 4 2007:09:07; 2008:08:07; 2011:08:25; 2011:08:29 Services 6 2007:10:31; 2008:06:12; 2009:05:04; 2011:01:17; 2012:08:08; 2013:10:17 Industry General 5 2007:08:29; 2008:08:07; 2009:06:17; 2011:08:25; 2013:09:30 dˆ

d¯ b=2

W b=4

b=2

Panel B: Estimating the relevance of Long memory against structural breaks 0.553 0.626 0.568 0.641 Banks Insurance 0.421 0.422 0.636 2.224 Services 0.410 0.685 0.861 13.285* Industry 1.012 0.828 0.960 9.478* General 0.615 0.503 0.935 0.388

Zt

u

−1.467 −0.956 −1.389 −2.403 −1.933

0.318* 0.304* 0.205* 0.164* 0.296*

b=4 2.897 5.691 10.526* 11.380* 3.290

This table reports the results of statistical tests of long memory hypothesis against structural change in the squared returns. b denotes the number of subsamples. W, Zt and u are the estimated values of the Wald, Phillips–Perron, and KPSS tests, respectively. The critical values for the Wald test are: 2 (1) is 3.84 and 2 (3) = 7.82, respectively, at 0.05 level. Zt critical 5% = −2.92 and 1% = −3.46. The maximum lag order of KPSS test was chosen using Schwert criterion and the Auto-covariances weighted by Bartlett kernel. KPSS critical values for H0 : the series is trend stationary are: (10%: 0.119; 5%: 0.146; 2.5%: 0.176; 1%: 0.216). *

Rejection of the null hypothesis of constancy of the long memory parameter d at the 5% level.

also find that seven of these structural breaks are associated with the Arab spring of 2010–11. Finally, four of the identified breaks occurred during 2012–14 period, which covers the period of wars in Syria and Iraq. The presence of structural breaks in the squared returns of sectoral indices suggests that the evidence of long memory preened in Tables 2–4 might be spurious. In order to investigate this issue, we now test for the relevance of long memory against structural breaks in measures of volatility in ASE sectoral indices returns. In order to test for the relevance of long memory against structural breaks, we use a procedure that was developed by Shimotsu (2006). In order to perform this procedure, the sample period was divided into b = 2 and b = 4 subsamples. The results are reported in the bottom panel of Table 5. These results show that the use of a number of subsamples increases the power of the test but it does not increase the number of significant cases. The results of the Wald (W) show that the null hypothesis of the constancy of estimated values of d is rejected across subsamples for the volatility measures of industry and services indices. This suggests that the evidence of long memory in volatility measures of two sectoral returns presented earlier was spurious. However, the squared returns for banks, insurance and the general index do not show evidence of spurious long memory behavior. Based on these results, it can be argued that presence of structural breaks only partially affects our long memory results. The results of KPSS test (i.e., u ) suggest that the presence of long memory in volatility measures of all sectoral returns are spurious as the hypothesis of stationarity cannot by rejected. However, the Philips–Perron test (zt ) does not reject the null hypothesis of I(d). In summary, based on Shimotsu’s procedure, there is evidence of spurious long memory in ASE sectoral returns volatility measures. In order to further investigate the tendency of long memory in sectoral returns on ASE, we also used spectral density analysis. The spectral density process describes the relative importance of random components of the series at different frequencies. Spectral densities are calculated based on both Auto Regressive Moving Average (ARMA) and ARFIMA models. Fig. 2 shows the spectral densities of (i) the ARMA parameter estimates and (ii) the long-run ARFIMA parameter estimates and (iii) the shortrun ARFIMA parameter estimates. Fig. 2 consists of 15 panels. Each panel contains two graphs using ARIMA and ARFIMA models. The top graph shows spectral densities implied by ARMA and ARFIMA long memory models, whereas the bottom graphs show the plot of spectral densities implied by ARMA and ARFIMA short memory models. Spectral densities are plotted against different frequencies when d > 0.

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Fig. 2. Spectral density (based on ARFIMA and ARMA models).

Fig. 2 shows that spectral densities of the return series for Banks, Industry and the General index, based on ARMA model, are at their highest level when frequency is 0. These densities taper off (but not to zero) as frequency increases, suggesting a positive asymptote. The long-run ARFIMA models, which aim to capture the long memory behavior when frequency is zero, present a different picture. The spectral densities of both ARMA and the long-run ARFIMA models capture the high-frequency components thereby displaying evidence of long memory behavior in the estimates. However, there is also some evidence of short memory. The spectral densities implied by the ARMA and the shortrun ARFIMA parameter estimates illustrate the ability of the short-run ARFIMA parameters to capture

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low-frequency components in the fractionally differenced series. In summary, ARFIMA models are more appropriate due to their ability to capture both short and the long memory behaviors. The graphs presented in Fig. 2 show that, as compared to ARMA model, in the case of most industry sectors, ARFIMA model is able to capture both low and high-frequency components. However, in the case of insurance and services sector returns, the top and the bottom graphs (in the relevant panels of Fig. 2) indicate that spectral densities of both ARFIMA and ARMA models have a negative coefficient, suggesting the presence of anti-persistence. In the top graphs, ARMA and ARFIMA long-memory spectral densities are at their lowest level at frequency 0. However, these densities monotonically increase as the number of frequencies increase. Although, the ARFIMA long-memory spectral densities are pulled upwards to capture the high-frequency components, the estimated values of spectral densities of ARMA and ARFIMA long-memory models remain finite. This suggests the presence of long-memory behavior. In other words, the high-frequency random components are the most important components of the process. However, the results of spectral densities for ARFIMA longmemory and ARMA presented in the bottom graphs show the ability of the ARFIMA short-memory model in capturing the short and long-run effects in Insurance and Services sectors. In summary, in general, both ARFIMA and ARMA models yield similar spectral density curves. In other words, both models perform equally well in capturing the fractional-difference parameter for both long and short-term effects. These results are also consistent with the results reported in Tables 2–4. Fig. 2 shows the spectral densities of volatility measures (i.e., the squared returns and absolute returns) of all sectoral indices returns. These densities describe the relative importance of the random components at different frequencies for given ARMA and ARFIMA models. Generally speaking, the top and the bottom graphs in each panel of Fig. 2 show similar pattern for the squared and absolute returns for all indices. However, the spectral densities are very different. These graphs show that spectral densities of the volatility measures for Banks, Industry and the General index, based on the ARMA model, are at their highest level at frequency 0. However, as the frequency increases, these densities decline at a fast rate, which indicates short memory behavior. The spectral densities of the long-run ARFIMA models capture the high-frequency components thereby displaying evidence of long memory behavior in the estimates. The spectral densities of ARFIMA models are at their highest level when frequency level is zero. However, as the frequency increases, the spectral densities gradually decease, suggesting a positive asymptote. However, there is also some evidence of short memory. In contrast to the long-run ARFIMA parameter estimates, the spectral densities implied by the short-run ARFIMA parameter estimates capture the low-frequency components in fractionally differenced series. In summary, ARFIMA models are more appropriate due to their ability to capture both short and the long memory behaviors. Fig. 2 shows that, as compared to ARMA model, in the case of most industry sectors, ARFIMA model is able to capture both low and high-frequency components. The spectral densities of insurance and services sector volatility measures tell a different story. In the top and bottom panels of Fig. 2, spectral densities of ARFIMA show a downward trend whereas ARMA based densities show an upward trend. In the case of ARFIMA models, the densities are lowest at frequency 0 but these densities monotonically increase with the number of frequencies. This suggests that high-frequency random components are the most important part of the random components. However, in the case of ARMA, only the low frequency random components are the most important random components, meaning that ARMA only captures the short run effects. In the case of the volatility measures, it is clear that the short memory ARFIMA model is able to capture both the short and long run effects. These results are consistent with our earlier results based on GPH, MLP and LW techniques. 4.5. Persistence In order to test for persistence in shocks to the sectoral returns and their volatility measures, following Hassler and Kokoszka (2010), we use impulse response functions (IRF). In the first stage, we selected an appropriate ARFIMA model. This selection requires the correct specification of ARMA structure to obtain the final ARFIMA specification. After estimating all possible models for p = 0–3 and

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q = 0–3, we selected ARFIMA (1, d, 1) model because the associated value of the Akaike Information Criterion (AIC) was the lowest. In the second stage, after estimating the appropriate AFRIMA (1, d, 1) model, we make use of the IRF. The IRF refers to the response of a series to a unit shock in error term of the ARFIMA model. In other words, the impact of the value of d on the shock duration is considered. Persistence refers to the time that is necessary for a shock to the series to completely evaporate. Persistence can be measured by the degree of changes in the current returns series that cause permanent changes in the future returns series (see Arouri et al., 2012; Elder & Villupuram, 2012; Hull & McGroarty, 2014). If stock market participants are able to forecast the nature of volatility in stock returns then, by employing appropriate strategies, they can earn higher returns (Andrie & Hasler, 2015). Determining whether the shocks are permanent or temporary is an important issue. If shocks are permanent and cause a dramatic reduction in investment returns, then strategies that could reserve this situation are highly desirable. However, in case of temporary shocks, long term strategies are not needed because the market would most likely return to its original position in the near future. If d is less than one,

Fig. 3. Impulse response functions (IRF) for all return series and their volatility measures.

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shocks will be temporary, whereas if it is equal to or greater than one, shocks are considered to be permanent and relevant series tends to exhibit an explosive behavior. ARFIMA based IRFs that measure persistence in all sectoral returns at ASE are presented in Fig. 3. The IRF graphs for sector returns, squared returns and absolute returns for each of the five sectors indicate that the impact of the shock on all sectoral returns decreases at a fast rate in the first five months and almost disappears after 7 to 8 months. In the case of the volatility measures, except for the insurance and services sectors, the impact of the shock appears to decay at a relatively slower rate. In the case of the insurance and services sectors, the impact of the shock appears to decay at a relatively fast rate. In summary, in all cases, the impulse response sharply decreases over time.

5. Concluding remarks Using daily data on sectoral stock indices returns and their volatility measures (i.e., absolute and squared returns), this paper aims to provide an in-depth analysis of the possibility of fractional integration in Jordan’s Amman stock market (ASE). The empirical analysis presented in this paper is based on two semi-parametric estimation techniques. The first technique relies on spectral regressions such as the log-periodgram regression proposed by Geweke and Porter-Hudak (1983) as well as the modified log-periodgram regression (MLP) proposed by Phillips (2007). The second technique relies on the frequency domain Gaussian likelihood function restricted to the vicinity of origin also known as the local whittle (LW) estimation technique. In addition to using the exact LW (ELW), we also make use of the two-stage exact local whittle (2EWL) estimation techniques proposed, respectively, by Shimotsu and Phillips (2005) and Shimotsu (2010). The empirical analysis presented in this paper suggests that sectoral returns at ASE exhibit a tendency of fractional integration. This result holds for all estimation techniques applied on both the sectoral returns as well as their volatility measures. Specifically, the sectoral returns show evidence of short memory while volatility measures exhibit long memory behavior. Long memory tendencies suggest that autocorrelation of stock prices die at a slower hyperbolic rate thus violating the random walk hypothesis. We also tested for the presence of structural breaks in our sample and found evidence of structural breaks in volatility measures of the sectoral returns. These breaks are associated with (i) 2008–09 GFC, (ii) the Arab spring of 2010–11, (iii) wars in Iraq and Syria and (iv) a sharp decline in the price of oil in recent years. The presence of structural breaks in volatility measures implies that the evidence of long memory in volatility measures may be spurious as the sectoral returns volatility was affected by the presence of structural breaks. We also tested for the relevance of structural breaks in volatility measures against long memory in all sectors and found mixed results. This allows us to conclude that long memory behavior found in volatility measures of sectoral indices returns is partially caused by the structural breaks. Based on the analysis presented in this paper, it can be argued that ASE stock exchange exhibits characteristics that are similar to relatively mature markets. As the sectoral returns exhibit short memory behavior, which is a special case of mean reversion process, ASE can be described as a fairly stable stock market where investing environment is good. The empirical results presented in this paper are likely to offer investors, intending to price financial risk and derivatives, with a very clear understanding of the behavior of ASE stock prices. Furthermore, derivative investors who rely on forecasted stock prices may also benefit from the results of this paper. Although investment in derivative securities at ASE is relatively small and limited, the Jordanian stock market offers attractive financial risk management opportunities. It may be possible to identify the types of shocks deriving stock prices at ASE. One can also attempt to identify whether the stock returns at ASE deviate far from their initial equilibrium values and whether this deviation increases in the short or the long run. The analysis presented in this paper suggests that persistence in exogenous shocks at ASE may have a direct impact on market efficiency and hence pose a serious challenge to the supporters of random walk theory. As the results show that stock returns at ASE are inconsistent with random walk theory, it may be useful to investigate other aspects of fractional integration behavior at ASE. It may be useful to also investigate fractional integration tendency using more recent recently developed financial modeling techniques, such as, the ELW fractionally cointegrated systems developed by Shimotsu (2012) and Okimoto et al.

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(2014) and the Markov regime-switching technique. It may also be useful to test for market efficiency and stability using ASE data.

Acknowledgements The authors are extremely grateful to two anonymous reviewers for very detailed comments and suggestions. These comments have led to a significant improvement to the overall quality of this paper. However, the authors are solely responsible for all remaining errors and omissions. Part of this work was completed when the first author was on sabbatical leave in 2014–15 funded by Al-Hussien Bin Talal University (AHU), Jordan. The generosity of AHU is greatly appreciated.

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