How does this remove autocorrelation? The first method to test for the day of the week effect is OLS regression. This method has been used by many empirical researchers testing for a day of the week effect.


I am struggling to understand how the sum of the lagged values of the return equation being added removes autocorrelation from the error term.
 A: Often not including lagged values of dependent variable or independent variables will induce autocorrelation structure in residuals when these values should have been included.
A: As @Analyst pointed out the inclusion of lagged dependent variables excludes one source of regression error autocorrelation. The autocorrelation can still be present if the lags of dependent variables are included. Here is the mathematical illustration. Suppose the true model is the following
$$Y_t=\alpha+\beta_0X_t+\beta_1X_{t-1}+u_t,$$
where $Eu_t|(u_{t-1},...,X_t,X_{t-1})=0$, meaning that $u_t$ is not autocorrelated and it does not correlated with the regressors.  Suppose you are estimating the model
$$Y_t=\alpha+\beta_0X_t+v_t$$
then
$$EX_tv_t=EX_tu_t+\beta_1EX_tX_{t-1}$$
Now if $EX_tX_{t-1}\neq 0$ then you have the ommited variables problem and the autocorellation is the least of your worries, since the OLS estimates in this case are inconsistent. Now if $X_t$ is not autocorrelated then $EX_tv_t=0$ and OLS estimates are consistent and asymptotically normal (if $Eu_t^2<\infty$ and $EX_t^2<\infty$). 
But
\begin{align*}
Ev_tv_{t-1}&=E(u_t+\beta_1X_{t-1})(u_{t-1}+\beta_1X_{t-2})\\
&=Eu_tu_{t-1}+\beta_1Eu_{t-1}X_{t-1}+\beta_1Eu_tX_{t-2}+\beta_1EX_{t-1}X_{t-2}\\
&=\beta_1Eu_tX_{t-2}
\end{align*}
and this might be non zero giving the autocorrelation problem. 
So to sum up the claim in the citation is not entirely correct. If lags are omitted this can lead to omitted variable bias and that is the first reason to include them.
