st: Difference in Difference vs. Fixed Effects

[Joerg Lang <joerglang0@gmail.com>]

Dear Stalist users,

currently writing my Master thesis and working with
Stata 12, I have the following problem.

I have a dataset on two time periods (2010 and 2012) and two groups
(treatment and control). There is no treatment in the baseline and the
treatment group uptakes the treatment between 2010 and 2012. The uptake is
non-random.
Now, I want to estimate the impact in a difference in difference design.
At first, I estimate the following model:
 y b0+b1Time+b2Treatment+b3Time*Treatment+u

using the -reg command:

-reg y time treatment time*treatment, cluster (h1)

while y is the outcome variable that is between 0 and 1 and h1 is the
household identifier. I use the cluster option to account for the problem
of serial correlation. In a second estimation I also include some other
covariates.

I always thought that this setting and a setting with fixed effects
yield exactly the same result as long as one has only two points in time
(in my case 2010 and 2012).

However, estimating the same model with:

- xtreg y Time Treatment Time*Treatment, fe vce(cluster h1)

gives slightly different results. The difference increases when including
more covariates, which are the same in both cases. As well, there is no
variation in the households. Thus, the same households and the same
variables are used in both estimations.
Obviously, treatment is omitted in the xtreg case since it does not vary
between time.
However, I think that this should not change anything.
My question is:
 Is my model correctly specified or did I overlook something?
And, if my estimation is correct: Why this difference? Is this "normal"? If
so, what does it tell me then, i.e. what is the reason for it?

Since I have already been stuck with this problem for quite a while, any
help or literature suggestions  would be very much appreciated.  Hope
this question is not too trivial for you.

Best regards,

Joerg
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