Unobserved structural shifts and asymmetries in the random walk model for stock returns in African frontier markets

Figures & data

Table 1. Summary of literature review for the individual-based studies

Author(s)	Country/Countries	Period	Method	Main Result: Efficient/non-efficient
Fowdar et al. (2007)	Mauritius	January 1999—December 2004D	ADF and KPSS tests	Inefficient
Sunde and Zivanomoyo (2008)	Zimbabwe	January 1998—November 2008M	ADF tests	Inefficient
Nwosa and Oseni (2011)	Nigeria	January 1986—December 2010M	ADF and PP tests	Inefficient
Chiwira and Muyambiri (2012)	Botswana	December 2011—January 2012W, M	ADF and PP tests	Inefficient
Kamau (2013)	Kenya	January 2008—December 2012D	ADF and PP tests	Efficient
Ogege and Mojekwu (2013)	Nigeria	January 1985—December 2010M	ADF and PP tests	Inefficient
Nwidobie (2014)	Nigeria	January 2000—December 2012	ADF test	Inefficient
Obayagbona and Igbinosa (2014)	Nigeria	January 2006—December 2011M	ADF and PP tests	Inefficient
Yadirichukwu and Ogochukwu (2014)	Nigeria	January 1984—December 2012M	ADF unit root tests	Inefficient
Balparda et al. (2015)	Kenya	January 2001—December 2009D	ADF, PP, KPSS and ERS tests	Inefficient
Grater and Struweg (2015)	South Africa	October 1998—April 2014M	ADF and PP tests	Efficient
Kitso and Ummersingh (2015)	Mauritius	July 2001—July 2014D	ADF, PP and KPSS tests	Efficient
Sule et al. (2015)	Nigeria	1985–2009A	ADF test	Efficient
Phiri (2015)	South Africa	January 2000—December 2014W	ADF, PP, Walter Enders and Clive Granger (1998) and Frédéric Bec, Mélika B.S. Salem, and Marine Carrasco (2004) tests	Inefficient
Njuguna (2016)	Kenya	January 2001—January 2015D,W	ADF and PP tests	Inefficient
Fusthane and Kapingura (2017)	South Africa	January 2005—December 2016M	ADF and PP tests	Efficient

Note: D ≡ daily; W ≡ weekly; M ≡ monthly.

Table 2. Summary of literature review for the panel-based studies

Author(s)	Country/Countries	Period	Method	Main Result: Efficient/non-efficient
Kawakatsu and Morey (1999)	Including Zimbabwe	January 1976 November 1997M	KPSS and DF-GLS	Efficient
Lagoarde-Segot and Lucey (2008)	Including Egypt, Morocco and Tunisia	January 1998—November 2004D	KPSS test	Egypt and Tunisia efficient and Morocco inefficient
Alam and Uddin (2009)	Including South Africa	January 1988—March 2003M	ADF test	Inefficient
Enowbi et al. (2010)	Including Egypt, Morocco, South Africa, Tunisia	January 2000—March 2009D	ADF, PP and KPSS	Efficient
Lee et al. (2010)	Panel of developed and developing including: Botswana, Egypt, Kenya, Mauritius, South Africa	January 1999—May 2007M	LLC, IPS, Breitung, Fisher ADF, Fischer PP, Hadri tests	Inefficient
Nwosu et al. (2013)	Egypt, Kenya, Nigeria, South Africa	January 1998—December 2008W	ADF and KPSS	Inefficient
Adigwe et al. (2017)	Botswana, Egypt, Kenya, Mauritius, Morocco, Nigeria, South Africa, Tunisia, Zimbabwe	January 2013—December 2015M	ADF tests	Inefficient

Note: D ≡ daily; W ≡ weekly; M ≡ monthly.

Figure 1. Stock returns for Frontier markets in Africa (Daily series).

Figure 2. Stock returns for Frontier markets in Africa (Weekly series).

Table 3. Descriptive statistics of time series

Panel A: Daily series	Mean	Median	Maximum	minimum	Std. dev.	j-b (p-value)	Obs.
South Africa	0.023	0.015	6.833	−7.581	1.22	0.00	3548
Botswana	0.006	0.000	3.925	−4.775	0.34	0.00	3548
Mauritius	−0.005	0.000	8.005	−6.382	0.69	0.00	3548
Kenya	0.014	0.000	7.486	−5.141	0.82	0.00	2309
Nigeria	−0.038	0.000	11.758	−9.475	1.17	0.00	3548
Tunisia	0.031	0.005	4.109	−5.004	0.58	0.00	3548
Egypt	0.0046	0.000	7.314	−17.99	1.72	0.00	3548
Morocco	−0.009	0.000	4.464	−4.667	0.69	0.00	3548
Panel B: Weekly series
South Africa	0.108	0.153	16.04	−9.63	2.61	0.00	887
Botswana	0.029	0.073	5.02	−6.73	1.04	0.00	887
Mauritius	−0.02	−0.008	7.91	−15.72	1.91	0.00	887
Kenya	0.06	0.20	15.31	−10.45	2.40	0.00	461
Nigeria	−0.19	−0.05	15.62	−14.24	3.53	0.00	887
Tunisia	0.1553	0.13	8.4943	−13.63	1.63	0.00	887
Egypt	0.02	0.23	19.32	−21.96	4.53	0.00	887
Morocco	−0.05	−0.01	5.36	−9.80	1.67	0.00	887

Table 4. Conventional unit root test results

	ADF		PP		KPSS
Panel A: Daily series	Intercept	Trend and Intercept	Intercept	Trend and Intercept	Intercept	Trend and Intercept
South Africa	−58.52*** [0]	−58.55*** [0]	−59.11*** [22]	−59.24*** [23]	0.29 [22]	0.07 [23]
Botswana	−7.18*** [29]	−7.38*** [29]	−68.45*** [31]	−68.0716*** [31]	0.68** [31]	0.15** [31]
Mauritius	−9.25*** [29]	−9.47*** [29]	−60.62*** [23]	−60.54*** [22]	0.61** [23]	0.05 [22]
Kenya	−28.70*** [0]	−28.70*** [0]	−27.86*** [22]	−27.84*** [22]	0.19 [6]	0.17** [6]
Nigeria	−8.69*** [29]	−8.75*** [29]	−43.13*** [8]	−43.15*** [9]	0.32 [10]	0.14* [10]
Tunisia	−10.77*** [22]	−10.91*** [22]	−46.59*** [9]	−46.64*** [10]	0.62** [6]	0.09 [5]
Egypt	−9.63*** [27]	−9.71*** [27]	−50.46*** [2]	−50.60*** [3]	0.53** [7]	0.27 [6]
Morocco	−9.78*** [27]	−9.93*** [27]	−46.25*** [3]	−46.36*** [1]	0.74*** [12]	0.20** [11]
Panel B: Weekly series
South Africa	−7.49*** [13]	−7.51*** [13]	−31.55*** [6]	−31.53*** [6]	0.08 [5]	0.07 [5]
Botswana	−5.12*** [14]	−5.27*** [14]	−28.52*** [17]	−28.28*** [16]	0.34 [18]	0.07 [18]
Mauritius	−4.90*** [16]	−4.97*** [16]	−26.66*** [13]	−26.64*** [13]	0.23 [15]	0.13* [14]
Kenya	−4.34*** [11]	−4.37*** [11]	−17.93*** [5]	−17.92*** [5]	0.21 [2]	0.18** [2]
Nigeria	−5.99*** [15]	−6.16*** [15]	−28.79*** [14]	−28.75*** [13]	0.4463** [14]	0.08 [13]
Tunisia	−5.20*** [20]	−5.19*** [20]	−27.61*** [10]	−27.60*** [10]	0.20 [11]	0.19** [11]
Egypt	−4.69*** [19]	−4.68*** [19]	−27.87*** [8]	−27.86*** [8]	0.18 [9]	0.20** [9]
Morocco	−4.97*** [20]	−4.96*** [20]	−26.72*** [8]	−26.70*** [8]	0.25 [9]	0.26*** [9]

Notes: “***”, “**”, “*” denote the 1%, 5% and 10% critical levels

Table 5. Modified unit root test results

	Elliott et al. (2001) tests		Ng and Perron (1996, 2001) tests
	DF-GLS		MZA		MZT		MSB		MPT
Panel A: Daily series	Intercept	Intercept and trend	intercept	Intercept and trend	intercept	Intercept and trend	intercept	Intercept and trend	intercept	Intercept and trend
South Africa	−2.04** [29]	−3.88*** [29]	−3.59 [29]	−6.22 [29]	−1.31 [29]	−1.76 [29]	0.37 [29]	0.28 [29]	6.83 [29]	14.66 [29]
Botswana	−6.93*** [29]	−6.98*** [29]	−20.5*** [29]	−20.97** [29]	−3.20*** [29]	−3.24** [29]	0.16*** [29]	0.1544** [29]	1.20*** [29]	4.35** [29]
Mauritius	−5.51*** [29]	−7.07*** [29]	−13.66** [29]	−22.78** [29]	−2.61*** [29]	−3.37** [29]	0.19** [29]	0.15** [29]	1.79** [29]	4.03*** [29]
Kenya	−28.7*** [0]	−7.43*** [26]	−85.8*** [0]	−89.5*** [26]	−21.1*** [0]	−6.65*** [26]	0.02*** [0]	0.07*** [26]	0.04*** [0]	1.18*** [26]
Nigeria	−8.68*** [29]	−8.61*** [29]	−71.85 [29]	−68.2*** [29]	−5.99*** [29]	−5.84*** [29]	0.08*** [29]	0.09*** [29]	0.35*** [29]	1.34*** [29]
Tunisia	−3.01*** [29]	−5.64*** [29]	−8.99** [29]	−18.98** [29]	−2.12** [29]	−3.05** [29]	0.24* [29]	0.16** [29]	2.72** [29]	4.97** [29]
Egypt	−4.89*** [29]	−7.04*** [29]	−13.38** [29]	−26.23** [29]	−2.56** [29]	−3.62*** [29]	0.19** [29]	0.14*** [29]	1.93** [29]	3.48*** [29]
Morocco	−5.81*** [29]	−7.99*** [28]	−18.5*** [29]	−43.1*** [28]	−3.03*** [29]	−4.62*** [28]	0.16*** [29]	0.11*** [28]	1.37*** [29]	2.22*** [28]
Panel B: Weekly series
South Africa	−1.58 [13]	−2.54 [20]	−2.42 [13]	−2.11 [20]	−1.09 [13]	−0.92 [20]	0.45 [13]	0.44 [20]	10.10 [13]	37.46 [20]
Botswana	−4.87*** [14]	−4.95*** [14]	−28.4*** [14]	−30.1*** [14]	−3.77*** [14]	−3.88*** [14]	0.13*** [14]	0.13*** [14]	0.86*** [14]	3.03*** [14]
Mauritius	−3.74 [16]	−4.80*** [16]	−9.86** [16]	−16.92* [16]	−2.20** [16]	−2.90* [16]	0.22** [16]	0.17* [16]	2.57** [16]	5.39* [16]
Kenya	−0.99 [11]	−2.17 [11]	−1.85 [11]	−4.07 [11]	−0.93 [11]	−1.38 [11]	0.50 [11]	0.34 [11]	12.74 [11]	21.95 [11]
Nigeria	−2.19** [17]	−4.04*** [20]	−4.93 [17]	−11.62 [20]	−1.56 [17]	−2.39 [20]	0.32 [17]	0.21 [20]	4.98 [17]	7.96 [20]
Tunisia	−1.30 [12]	−2.34 [16]	−2.67 [12]	−12.55 [16]	−1.11 [12]	−2.50 [16]	0.42 [12]	0.19 [16]	9.00 [12]	7.26 [16]
Egypt	−3.99*** [19]	−4.47*** [19]	−2.67 [19]	−4.02 [19]	−1.11 [19]	−1.37 [19]	0.42 [19]	0.34 [19]	8.99 [19]	22.15 [19]
Morocco	−4.02*** [20]	−4.16*** [20]	−8.46** [20]	−11.44 [20]	−1.98* [20]	−2.35 [20]	0.23** [20]	0.21 [20]	3.36* [20]	8.21 [20]

Notes: “***”, “**”, “*” denote the 1%, 5% and 10% critical levels. Optimal lags length as determined by the modified AIC reported in [].

Table 6. KSS unit root tests without FFF

	Raw	De-meaned	De-trended
Panel A: Daily series
South Africa	−5.78*** [7]	−5.93*** [7]	−5.93*** [7]
Botswana	−7.86*** [10]	−7.84 *** [10]	−7.84*** [10]
Mauritius	−6.41*** [10]	−26.54*** [10]	−26.64*** [10]
Kenya	−0.95 [11]	−0.95 [11]	−0.94 [11]
Nigeria	−17.74*** [9]	−17.76*** [9]	−17.73*** [9]
Tunisia	−9.38*** [13]	−9.49*** [13]	−9.49*** [13]
Egypt	−3.86*** [5]	−3.91*** [5]	−3.90** [5]
Morocco	−4.61*** [12]	−4.54*** [12]	−4.55*** [12]
Panel B: Weekly series
South Africa	−6.061*** [6]	−6.06*** [6]	−6.39*** [6]
Botswana	−2.58** [6]	−2.58** [6]	−2.56 [6]
Mauritius	−7.59*** [6]	−7.59*** [6]	−7.54*** [6]
Kenya	−4.81*** [6]	−4.81*** [6]	−4.82*** [6]
Nigeria	−6.71*** [5]	−6.71*** [5]	−6.50*** [5]
Tunisia	−3.66*** [6]	−3.66*** [6]	−3.65** [6]
Egypt	−6.07*** [8]	−6.07*** [8]	−6.18*** [8]

Notes: “***”, “**”, “*” denote the 1%, 5% and 10% critical levels. Critical values are derived from Kapetanois et al. (2003) as follows. For the raw series −2.82(1%), −2.22 (5%), −1.92(10%), for the de-meaned series −3.48(1%), −2.93(5%), −2.66(10%), for the de-trended series −3.93(1%), −3.40(5%), −3.13(10%). Optimal lags length as determined by the Schwarz criterion reported in [].

Table 7. KSS test with FFF

	raw series		de-meaned series		de-trended series
Panel A: Daily series	t-stat	k	t-stat	k	t-stat	k
South Africa	−2.94*** [16]	5	−3.10** [16]	5	−3.71* [16]	5
Botswana	−7.86*** [10]	5	−7.84 *** [10]	5	−7.84*** [10]	5
Mauritius	−26.40*** [10]	2	−26.53*** [10]	2	−26.63*** [10]	2
Kenya	−0.85 [22]	2	−0.44 [22]	2	−0.48 [22]	2
Nigeria	−17.07*** [13]	3	−17.07*** [13]	3	−16.93*** [13]	3
Tunisia	−9.38*** [13]	1	−9.49*** [13]	1	−9.46*** [13]	1
Egypt	−3.43*** [6]	1	−3.48*** [6]	1	−3.47** [6]	1
Morocco	−4.62*** [12]	1	−4.35*** [12]	1	−4.54*** [12]	1
Panel B: Weekly series
South Africa	−6.06*** [6]	4	−6.38*** [6]	4	−6.39*** [6]	4
Botswana	−2.99*** [4]	5	−2.97*** [4]	5	−2.97 [4]	5
Mauritius	−7.60*** [6]	5	−15.92*** [6]	5	−7.55*** [6]	5
Kenya	−4.80*** [6]	4	−4.80*** [6]	4	−4.79*** [6]	4
Nigeria	−6.70*** [5]	5	−6.64*** [5]	5	−6.50*** [5]	5
Tunisia	−3.66*** [6]	5	−3.65*** [6]	5	−3.65** [6]	5
Egypt	−8.14*** [4]	1	−8.14*** [4]	1	−8.24*** [4]	1
Morocco	−9.09*** [5]	5	−9.02*** [5]	5	−9.02*** [5]	5

Notes: “***”, “**”, “*” denote the 1%, 5% and 10% critical levels. Critical values are derived from Kapetanois et al. (2003) as follows. For the raw series −2.82(1%), −2.22 (5%), −1.92(10%), for the de-meaned series −3.48(1%), −2.93(5%), −2.66(10%), for the de-trended series −3.93(1%), −3.40(5%), −3.13(10%). Optimal lags length as determined by the Schwarz criterion reported in [].

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