Data and Sample-

The study is based on the data of 49 firms listed on the Karachi Stock Market (KSE), which is the largest stock exchange in terms of size and volume in the country for the period January 2000 to December 2012.
These 49 companies were selected out of the total number of 779 firms listed on the KSE.
The firms were chosen on the basis that they had a continuous listing on exchange for the entire period of analysis, or had the highest capitalization or turnover in their sectors in the period under study. Consequently, the sample has a slight bias towards large companies.
From 2000 to 2012, the daily data on closing prices of the stocks and the KSE100 index were obtained from the Karachi Stock Exchange Data Portal, which is the primary source of all stock data in the country.
Monthly stock returns for each stock are calculated using this data. Monthly market returns were calculated from the broadly-based, capitalization weighted KSE 100 index as the KSE 100 Index was taken as the market portfolio.
The three months treasury-bill rate was used as risk-free rate. The data on six-month treasury-bill rates was obtained from the Market Treasury Bills Auction Results from the State Bank of Pakistan data Portal.

Methodology-
Betas are estimated using the single period, discrete time Market Model (MM) developed by Markowitz (1959) Shape (1963).The MM is expressed as,
Rit = αit + βiRmt + εit, -----------------(1)
Where Rit is the expected return on an asset “,i”, Rmt is expected return on market portfolio, proxied by the stock market index and βi is the “beta” or the measure of risk or market sensitivity parameter. Rit is measured as the continuously compounded return on the i'th security during time t, while Rmt is the corresponding return on the market. In equation 1, α and β are unknown firm specific parameters assumed to be constant over time, while ɛit is a random error which is distributed identically and independently, i.e. IN.(0,δi²)
The beta and Rmt in equation 1 are defined as, βi =cov (Ri,Rm) / var(Rm) and Rmt =Rmt-Rf, Where Rf is the risk free rate.
The beta is be obtained by the estimation of equation (1) by OLS, in which a beta estimate is derived by regressing 60 monthly relatives of stock “i” return upon the corresponding values for the market portfolio. In this study, the beta is modeled through the OLS rolling regression method proposed by FAMA and MACBETH (1973). It supposes that betas are constant over short time intervals, usually 5 years. Each month a regression of the market model is carried out using the last 60 observations. For each beta, only 1 observation on 60 is new and therefore this overlapping problem lead to autocorrelation in the beta time-series. The auto correlation problem is avoided by taking log of Rit and Rmt.

In this study betas are estimated for single securities and for equally weighted portfolios containing two and ten securities respectively. The portfolios are formed by ranking betas of securities, from the lowest to highest values and then grouping 2 or 10 securities with the smallest estimates of beta to form a portfolio. The first portfolio consequently would of all weighted portfolios have the smallest estimated portfolio beta, since the portfolio beta is an average of the estimates of the individual securities.
The logic of using portfolio instead of individual stocks is that, according to the diversification principle, the portfolio return is fully explained by its beta and the excess market return. The term εit is only a disturbance term and not the specific return of the portfolio i which has been eliminated in the diversification process when portfolios are formed. The use of portfolios also improves the quality of the beta’s estimation.
The OLS betas were used to estimate Dimson betas. In an effort to decide upon the priors necessary for applying the Dimson Aggregated Coefficients Technique (1979) for estimating unbiased systematic risk measures, a simple aggregation of significant t-stats on lagged and leading market terms was carried out.
This criteria led to the decision that at two lagged and one leading term would is necessary or is the optimal choice for adjusting for thin trading bias.(see table)