0
$\begingroup$

I would like to understand how one gets the same coefficient estimate from 2 different model specifications.

Consider single difference estimation model:

$y_{\{Time=1\}}=\alpha+\beta_3 \textbf{Treatment}+\beta_4 y_{\{Time=0\}}+\epsilon, $

where $time:\{0,1\}$ or simply before/after and $\textbf{Treatment}:\{0,1\}$ or simply control/treatment groups.

Now consider double difference estimation model:

$y=\alpha+\beta_1 \textbf{Treatment}+\beta_2 \textbf{Time}+ \beta_3 (\textbf{Treatment}*\textbf{Time}) +\epsilon. $


The source, which I am questioning, claims that one can estimate $ \beta_3$ coefficient using either of above-mentioned models. However when I do simple rearrangement of terms and writing the model while changing group or time I find the following:

Well-known double difference estimator using DID model is the following (suppressing expected values):

$\beta_3=\Delta y_{\{Time=1\}}-\Delta y_{\{Time=0\}}$,

where $\Delta$ is the difference in treatment and control groups.

When I use the single difference model, I get the following for $\beta_3$:

$\beta_3=\Delta y_{\{Time=1\}}-\beta_4 \Delta y_{\{Time=0\}}$,

which shows that unless we put contstraint that $\beta_4=1$, I can not estimate the treatment effect using single difference estimator.


Question

Do I calculate wrongly or miss something? Could someone confirm that both models can result in the same estimate of $\beta_3$ ?

$\endgroup$
0
$\begingroup$

You are correct that the ANCOVA estimator and the DID do not estimate the same parameter. ANCOVA estimates $$(\bar Y^T_{POST}−\bar Y^C_{POST}) − \hat \theta \cdot (\bar Y^T_{PRE} - \bar Y^C_{PRE}),$$ where $\hat \theta$ is the coefficient on the lagged outcome, while DID is $$(\bar Y^T_{POST}−\bar Y^T_{PRE}) − (\bar Y^C_{POST} - \bar Y^C_{PRE})$$

These formulas are given in McKenzie (2012).

You can verify this with yourself with a regression:

. use http://fmwww.bc.edu/repec/bocode/c/CardKrueger1994.dta, clear
(Dataset from Card&Krueger (1994))

. /* fix sample */
. drop if id == 407 // duplicate restaurant
(4 observations deleted)

. xtset id t
       panel variable:  id (strongly balanced)
        time variable:  t, 0 to 1
                delta:  1 unit

. drop if missing(fte)
(19 observations deleted)

. bysort id: keep if _N==2
(19 observations deleted)

. reg fte i.treated##i.t, cluster(id) // DID

Linear regression                               Number of obs     =        778
                                                F(3, 388)         =       1.88
                                                Prob > F          =     0.1318
                                                R-squared         =     0.0091
                                                Root MSE          =     9.0696

                                   (Std. Err. adjusted for 389 clusters in id)
------------------------------------------------------------------------------
             |               Robust
         fte |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     treated |
         NJ  |  -3.104066   1.448499    -2.14   0.033    -5.951955   -.2561769
         1.t |  -2.523333   1.250619    -2.02   0.044    -4.982171   -.0644953
             |
   treated#t |
       NJ#1  |   2.972378   1.334611     2.23   0.027     .3484041    5.596352
             |
       _cons |   20.17333   1.360045    14.83   0.000     17.49935    22.84731
------------------------------------------------------------------------------

. reg fte i.treated L.fte if t==1, cluster(id) // ANCOVA

Linear regression                               Number of obs     =        389
                                                F(2, 388)         =      50.02
                                                Prob > F          =     0.0000
                                                R-squared         =     0.2817
                                                Root MSE          =     7.3454

                                   (Std. Err. adjusted for 389 clusters in id)
------------------------------------------------------------------------------
             |               Robust
         fte |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     treated |
         NJ  |   1.374712   .9578786     1.44   0.152    -.5085701    3.257994
             |
         fte |
         L1. |    .485299   .0485207    10.00   0.000     .3899025    .5806954
             |
       _cons |   7.859902   1.224966     6.42   0.000       5.4515     10.2683
------------------------------------------------------------------------------

. table t treated , c(mean fte) // means

------------------------------
Feb. 1992 |  New Jersey = 1;  
= 0; Nov. |  Pennsylvania = 0 
1992 = 1  |       PA        NJ
----------+-------------------
        0 | 20.17333  17.06927
        1 |    17.65  17.51831
------------------------------

. di (17.518 - 17.069 ) - ( 17.650-20.173 )
2.972

. di (17.518 - 17.650) - .485299*(17.069 -20.173 )
1.3743681
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.