This section discussed data and methodologies adopted in this study.
Data
The data set is comprised of monthly, daily and weekly returns of KSE and PSX. The KSE data set consists of the period from 02 January 2004 to 10 January 2016. As Pakistan Stock Exchange was established on 11 January 2019, PSX data set is comprised of the period from 11 January 2016 to 30 April 2019. The data about PSX and KSE were collected from the official Web site of PSX. To calculate the market anomalies, closing returns are used.Footnote 3
Modelling framework
This study evaluates the impact of market anomalies with dummy variables in a multiple regression analysis using least squares, ARCH and EGARCH-in-mean models. Breusch–Godfrey serial correlation LM test is used to check the serial correlation in the return series.Footnote 4 Stock returns may have nonsymmetric properties, and due to time-varying variance in the series, result would be in the form of inefficient estimates. This study resolves this problem by constructing ARCH and EGARCH-in-mean models. Wald test has been employed to evaluate the joint significance of all dummies/anomalies coefficients. Furthermore, Box–Jenkins (ARIMA) technique is employed by the study to evaluate the best fit of time series model to the past values of that time series.
The tests employed by the study are connected with parametric and nonparametric groups in order to examine different hypotheses mentioned above. The daily returns are being calculated as follows:
$$R_{t} = \ln \left( {\frac{{l_{t} }}{{l_{{\left( {t - 1} \right)}} }}} \right) \times 100$$
(1)
where It refers to the index price and Rt refers to the stock returns on any day (t).
Day-of-the-week effect
The study constructed the following equation to evaluate the day-of-the-week effect by employing five dummies from Monday to Friday:
$$R_{t} = \omega_{1} D_{1} + \, \omega_{2} D_{2} + \, \omega_{3} D_{3} + \, \omega_{4} D_{4} + \, \omega_{5} D_{5} + \, \phi_{t}$$
(2)
where Rt is the return as discussed in Eq. (1). ω1, ω2, ω3,…, ω5 are dummy coefficients which indicate mean returns of each day of the week. D1, D2, D3,…,D5 are dummy variables for each day of the week, which are either 0 or 1. Φt is the white noise or error term for any day (t).
$${\text{Hypothesis}}\left( {H_{0} } \right):\omega_{1} = \omega_{2} = \omega_{3} = \omega_{4} = \omega_{5} = 0$$
H0 emphasizes that all the days of the week have joint/similar return patterns. If the empirical results reject this hypothesis, it would mean that there exists seasonality in the stock market. The study also tests the significance of daily returns, i.e. ω1, ω2, ω3,…,ω5, to evaluate to what extent they differ from zero. The signs of these coefficients indicate whether their difference is zero, positive or negative.
Weekend effect
The study empirically evaluated these two hypotheses to check the impact of weekend effect:
Trading-time hypothesis This hypothesis emphasizes that Monday mean returns are higher than other days of the week:
$$R_{t} = \psi_{1} + \psi_{2} D_{2t - 1} + \psi_{3} D_{3t - 1} + \psi_{4} D_{4t - 1} + \psi_{5} D_{5t - 1} + \phi_{t}$$
(3)
where Rt is the mean return on any day (t). ψ1 is the expected Monday mean returns; ψ2, ψ3, ψ4 and ψ5 are dummy coefficients and represent the difference between expected Monday mean returns and the returns on other days of the week. D2, D3, D4 and D5 are dummy variables for each day of the week, which are either 0 or 1:
$${\text{Hypothesis}}\;\left( {H_{0} } \right){:}\;\psi_{2} = \psi_{3} = \psi_{4} = \psi_{5} = 0.$$
If this hypothesis is significant, it means that there exists a variation between Monday returns and returns on other days of the week.
Calendar-time hypothesis This hypothesis emphasizes that returns on Monday are three times higher than the returns on other days of the week.
where Rt are mean returns on any day (t). Ϫ1 is the expected one-third mean Monday returns. Ϫ2, Ϫ3, Ϫ4 and Ϫ5 are dummy coefficients and show the difference between mean returns of other days of the week and one-third of Monday returns. D2, D3, D4 and D5 are dummy variables, which are either 0 or 1.
If this hypothesis is significant, it means that there exists a variation between one-third of Monday mean returns and returns on other days of the week.
This study investigated the calendar-time and trading-time hypotheses for KSE and PSX as Pakistan Stock Exchange now has fully electronic trading system with T + 3 settlement period.
Month-of-the-year/January effect/portfolio rebalancing hypothesis
To evaluate the January effect and month-of-the-year effect, the study used the following equation:
$$R_{t} = \delta_{1} + \delta_{i} D_{it} + \phi_{t}$$
(5)
where Rt represents the monthly return on any month (t). δ1 is the expected January mean returns. δit is the dummy coefficient and shows the difference between expected January returns and returns in other months. Dit is the dummy variable for each month-of-the-year, which is either 0 or 1:
$${\text{Hypothesis}}\;\left( {H_{0} } \right):\delta_{2} = \delta_{3} = \delta_{4} = \delta_{5} = , \ldots , = \delta_{12} = 0.$$
If the empirical results reject this hypothesis, it means that there exists a variation between expected January mean returns and returns in other months of the year.
July effect/tax-loss selling hypothesis
The study utilized this test to examine the tax-loss selling hypothesis in KSE and PSX:
$$R_{t} = \delta_{1} + \delta_{i} D_{it} + \phi_{t}$$
(6)
where Rt is the monthly return on any month (t). δ1 is the expected July mean returns. δit is dummies coefficient and shows the difference between expected July returns and returns in other months. Dit is the dummy variable for each month-of-the-year, which is either 0 or 1.
$${\text{Hypothesis}}\;\left( {H_{0} } \right){:}\;\delta_{2} = \delta_{3} = \delta_{4} = \delta_{5} = , \ldots , = \delta_{12} = 0.$$
The rejection of this hypothesis means that there exists a variation between expected July returns and returns in other calendar months.
In Pakistan, financial year is closed on June 30. Therefore, the study evaluated July effect. This study examined tax-loss selling hypothesis and portfolio rebalancing hypothesis in the context of KSE and PSX.