I have a data set with 4000 companies and I have calculated a liquidity measure of each of the company in the dataset as
Where, Turnover is the monthly average ratio of daily volume to shares outstanding for a given stock. Liquidity (measured by this ratio) Liquidity is highly persistent. And my analysis shows that in these indiviual companies in my dataset depict strong first order autocorrelation. So, as suggested by literature I should transform this liquidity measure of each indiviual company by AR(2). By following AR(2) process equation
Where, Ct i is a measure of liquidity for stock i at month t, x is the number of lags included in the autoregressive process, and ut i is the residuals in liquidity for stock i at month t.
Part where is need help most. Where,β1 is similar to the market beta of the CAPM except for additional term that is realted to the trading cost in the denominator. The remaining systematic risk components are associated with liquidity. β2 represents liquidity commonality, that is, the co-movement between individual stock liquidity and market liquidity.β3 measures the co-movement between stock returns and market liquidity.β4 captures the co-movement between individual liquidity and market returns.
Calculating liquidity betas for indiviual betas for indiviual stocks based on eqs. 3 to 6 could increase the power of test by providing ample observations but cost of doing so is that betas estimated at indiviual stock level have higher level of noise. To mitigate the measurement error problem calcute betas at portfolio level and then assign these portfolio betas to indiviual stocks to cross sectional regressions at the indiviual stock level (Fama and French,1992).
So, beginning of each year in our sample period ten portfolios are formed according to their level of liquidity(measured by ratio mentioned above).
I would really appreciate if you help me understand this decile portfolio formation part. I'm using R. Link to the paper that presented this model http://pages.stern.nyu.edu/~lpederse/papers/liquidity_risk.pdf