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In addition to modelling of time-varying efficiency, AMH further requires determining the market condition that favours efficiency and inefficiency. Thus, this study investigates whether return predictability or market efficiency relation varies under different market conditions as postulated by Lo (2004). Therefore, it was hypothesised that market efficiency and investors’ decision on stock investment is influenced by general stock market conditions. To evaluate how the market conditions affect return predictability in

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the selected African stock markets as propounded by AMH, the monthly27 _{measures of}

return predictabilities are regressed on dummies of market conditions as explained below.

4.3.2.1 Measures of Return Predictability

Test statistics of linear and nonlinear dependence tests or the associated p-values are natural measures of return predictabilities (Kim et al., 2011 and Urquhart, 2016). Following Urquhart and McGroarty (2016), p-value of VR and BDS tests are used as proxies for the linear and nonlinear return predictability. This measure is similar to the absolute value of VR and portmanteau tests t-statistics used by Kim et al., (2011) and Zhou and Lee (2013); however, the P-values are easier to understand and interpret. High or large P-values indicate low predictability and vice versa. The p-values of joint VR test and BDS test, generated by implementing the tests in two-year rolling window, rolled forward by one-month, are adopted as monthly measures of linear and non-linear predictability. When the window is rolled forward by one month, the first window covers first trading day of January 1998 to last trading day of December 1999 while the second window covers February 1998 to January 2000 and the last window starts from March 2016 to February 2018.

4.3.2.2 Measures of Market Condition

AMH links fluctuation in efficiency to changes in market conditions, although it did not itemise the exact makeup of market conditions or its expected relation with return predictability. From the literature, where the stock market price or return behaviour or trend is considered, the market conditions may be defined as bullish or bearish. The terms bull and bear conditions are the primary ways of describing market situation in the investing world. These conditions are adopted because they described the path of the market which is a major force influencing investment portfolio. Fabozzi and Francis

27 _{Step size is 1-month (windows roll forward by 1-month) (Kim et al. 2011 and Urquhart & McGroarty,} 2014, 2016), unlike annual measures (with 1-year step size). Different sizes serve a robustness purpose.

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(1977) identified various definitions of bull and bear market conditions. To identify these market conditions, the first definition separates returns data into up and down months when returns are positive and negative, respectively (Fabozzi & Francis 1977; Urquhart & McGroarty, 2016). This categorisation accordingly does not take trend into consideration; hence, the definitions of the bull, bear and normal market conditions by Klein and Rosenfeld (1987) are also considered. A window is deemed bullish or bearish when its mean return is greater or less than 50 percent of the market standard deviation obtained over entire windows. Any window that does not fall into the bull or bear category is categorised as normal month. Note that for a month to qualify as bullish, there must be two or more consecutive substantial movements (Klein & Rosenfeld, 1987). Since the monthly measures of return predictabilities are calculated on two-year window basis, the steps in determining the market conditions are also on window basis and are as stated as follows:

i. Calculate μ (mean return) for each of the windows as monthly average return; ii. Define as Up market when a window’s μ is positive and Down market when window’s μ is negative; iii. Calculate δ (standard deviation) of the entire (windows’) monthly average returns in (i); iv. Define as Bull market when μ in step (i) is > 0.5 of the δ in step (iii) for 2 or more consecutive windows; v. Define as Bear market when μ in step (i) is < 0.5 of δ in step (iii) for 2 or more consecutive windows; and vi. Define as Normal market any month (window) that does not fall into Bull or Bear market (Urquhart & McGroarty, 2016).

Further, Kim et al. (2011) identify subprime mortgage global financial28 _{crisis, which}

covered 2008 to 2009 as one of the fundamental conditions influencing return predictability. Financial crisis tends to impart on the behaviour and psychology of market

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operators and affect the movement in stock returns (Kim & Shamsuddin, 2008; Lim, Brooks & Kim, 2008). The incidence of a market crash or financial crisis is one more probable cause of market inefficiency. The reason is that market participants are usually swamped by panic during that chaotic financial atmosphere and this would adversely influence their ability to price assets efficiently (Lim & Brooks, 2011). Hence, this condition, which produced a uniform of 19 months of financial crisis for each of the five markets (2007:12–2009:6) is also incorporated in this study. The crisis periods are guided by Kim et al. (2011).

The study also implements Anderson, Bollerslev, Diebold and Labys’s (2003) realised volatility as a surrogate for market risk and a control variable (Kim et al., 2011). Realised volatility is obtained in this study as the square root of squares of the two- year’s window returns. This is done by squaring daily returns over a window, adding them up and obtaining the square root of the sum (Urquhart & McGroarty, 2016) and repeating the same for all windows. The value is regressed against predictability without necessarily categorising the value into high or low. Brailsford and Faff, (1996, p. 419) and Brooks (2014, p. 424) note that “the conclusion arising from this growing body of research is that forecasting volatility is a notoriously difficult task”. Therefore, this study employs the realised volatility. Realised volatility has become popular in recent times because it is less noisy than, for example, the daily squared or absolute returns and it is an unbiased and highly efficient estimator of return volatility (Andersen, Bollerslev, Diebold & Labys, 2001; Barndorff-Nielsen & Shephard, 2001, 2002).

4.3.2.3 Dummy Regression Model for Predictability and Market Condition Relation

Moreover, after the generation of return predictabilities and dummies of market conditions as dependent and independent variables respectively, the regression models are estimated. For comparative29 _{purpose, the dummy regression models for return}

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predictability and different definitions of market conditions (up and down & bull, bear and normal) are specified respectively and the best model is selected using information criteria. Such that:

𝑅𝑃_𝑡= 𝛽₁𝑈𝑃 + 𝛽₂𝐷𝑊 + 𝛽₃𝐹𝐶 + 𝛽₄𝑉𝑂𝐿 + 𝜀_𝑡 (26) 𝑅𝑃𝑡= 𝛽1𝐵𝑈 + 𝛽2𝐵𝐸 + 𝛽3𝑁𝑂 + 𝛽4𝐹𝐶 + 𝛽5𝑉𝑂𝐿 + 𝜀𝑡 (27)

𝐻₀: 𝛽_𝑖 = 0 … … 𝐻₁: 𝛽_𝑖 ≠ 0

𝑅𝑃_𝑡 is time 𝑡 return predictability (P-values of VR and BDS tests). 𝑈𝑃₁ is the dummy, which is equal to 1 if 𝑡 is UP and 0 if nut; 𝐷𝑊 is the dummy, which takes the value of 1 if 𝑡 is Down and 0 if not; 𝐵𝑈 is the dummy, which is equal to 1 if 𝑡 is Bull and 0 if not and so on. 𝐹𝐶₃ is the dummy for global financial crisis which takes the value of 1 when 𝑡 is any month between 2007:12 to 2009:6. 𝛽𝑖(𝑖 = 1, … ,5) are the coefficient estimates of market

conditions and 𝜀_𝑡 is stochastic error term. AR term (lagged dependent variable) is included as a regressor to ensure the residuals mimic white noise. Significant negative (positive) 𝛽_𝑖 is used to determine the market condition that is associated with high (low) predictability or inefficiency.