# Pooled OLS with Common Shock

I have the following dataset:

Date        Firm Return, Common_Shock
01/03/2011, 0.01,        .20
01/03/2011, 0.12,        .20
01/03/2011, 0.005,       .20
...
02/03/2011, 0.015,       .10
02/03/2011, 0.0121,      .10
02/03/2011, 0.009,       .10
...
03/03/2011, 0.088,       .21
03/03/2011, 0.04,        .21
03/03/2011, 0.1,         .21


For each $t=1... T$ dates where $T=30$, I have $i=1...N$ firm return observations where $N=500$. On each date T, there is a common shock that can impact the firm returns. This common shock takes the same value for each firm $i$ on each date $t$. Is a Pool OLS the correct regression if I wish to estimate the overall impact of the common shock on firm returns (controlling for lagged returns)? So,

$R^i_t = \sum_j^{J=3}\beta_jR^i_{t-j}+\omega Shock_t + \epsilon_t$

I have some doubts on the limited variations in the common shock since I only have 30 different observations for these shocks.

Update:

I also have other common shock that occurs on different date

Date        Firm Return, Common_Shock  Dummy_Shock_1
01/03/2011, 0.01,        .20           1
01/03/2011, 0.12,        .20           1
01/03/2011, 0.005,       .20           1
...
02/03/2011, 0.015,       .10           1
02/03/2011, 0.0121,      .10           1
02/03/2011, 0.009,       .10           1
...
03/03/2011, 0.088,       .21           1
03/03/2011, 0.04,        .21           1
03/03/2011, 0.1,         .21           1
...
04/20/2011, 0.088,       .30           0
04/20/2011, 0.04,    1.05          0
04/20/2011, 0.1,     .45           0


Then I could run the following:

$R^i_t = \sum_j^{J=3}\beta_jR^i_{t-j}+\omega Shock_t + \delta*Shock\_Type\_Dummy + \epsilon_t$

Would it be better?

Update 2:

I am trying to understand Donald Andrews 2005 Econometrica paper on this subject. It seems that if the common shock is uncorrelated with my error terms then the estimated coefficients are consistent and the t-test, wald, and F are asymptotically valid. Hence, I can run an OLS...