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# Re: st: Stata implementation of difference-in-differences with binary outcomes

 From Nils Braakmann To statalist@hsphsun2.harvard.edu Subject Re: st: Stata implementation of difference-in-differences with binary outcomes Date Sun, 18 Apr 2010 20:51:59 +0200

```Just to add one point: Using  a linear probability model is relatively
innocuous in a DiD-setting as the model is saturated (and consequently
non-parametric) in its main part. To elaborate on that: The main issue
with the linear probability model ist that it is linear and unbounded
while the data generating process is non-linear and the outcome is
bounded between "0" and "1". The linearity in the LPM may leed to
predictions that are outside of the [0,1]-range. This basically only
bites when looking at continuous covariates, where the linearity (or
any other functional form) assumption matters. With dummies (and their
interactions) you are always looking at mean differences, hence no
over- or underprediction.

Now, with DiD you are essentially comparing four means, which are all
bounded between 0 and 1. To make this more concrete, say, you're
interested in modeling employment shares. Employment in the treatment
group before treatment (1st period) is "a" (some number between 0 and
1), after treatment it is "b". Similarly, in the control group, the
1st period outcome is "c" and the 2nd period outcome is "d". Note that
all outcomes are bounded somewhere between "0" and "1" The
corresponding DiD-estimate is t = (b-a) - (d-c), which is also always
bounded between "0" and "1". Now write this as regression using D as
the treatment group indicator and T as the second period indicator
(both dummies):
y = c + (a-c)*D + (d-c)*T + [(b-a) - (d-c)]*(D*T) + error = g_0 +
g_1*D + g_2*T + g_3*(T*D) + error

The expected values for the both groups and the two periods are:

E[y|D=0,T=0] = g_0 = c
E[y|D=0,T=1] = g_0 + g_2 = c + (d-c) = d
E[y|D=1,T=0] = g_0 + g_1 = c + (a-c) = a
E[y|D=1,T=1] = g_0 + g_1 + g_2 + g_3 = a + (a-c) + (d-c) + [(b-a) - (d-c)] = b

These are all always bounded between "0" and "1". The case becomes
more complicated if you add continuous covariates, though, as these
may lead to overpredictions. From my experience, however, this usually
does not matter much in practice.

Best,
Nils

On Fri, Apr 16, 2010 at 5:14 PM, Steve Samuels <sjsamuels@gmail.com> wrote:
> I agree with Maarten.
>
> "(possibly with the -vce(robust)- "
> I'd say "necessarily, " or just use the "robust" option, in order to
> assure correct standard errors and tests.
>
> In my experience, a the estimated DiDs and their CIs do not have
> boundary problems: the possible range is -2 to +2, with the average
> usually close to the middle.
>
> Steve
>
> On Fri, Apr 16, 2010 at 10:57 AM, Maarten buis <maartenbuis@yahoo.co.uk> wrote:
>> --- On Fri, 16/4/10, C Engelbrecht wrote:
>>> But what if the outcome variable is binary? How should I
>>> model the difference of two latent variables, as is the
>>> case in Probit / Logit? The usual DID is based on
>>> differencing Y across these groups, but what should we
>>> do now that we only have a latent Y*?
>>
>> Difference in difference is all about getting at a causal
>> effect, which is usually difined as a difference in
>> averages. This also exists and is meaningful when the
>> dependent variable is binary, that is the risk difference.
>> You can calculate it using a linear probability model,
>> which is just a fancy name of using -regress- on a binary
>> variable (possibly with the -vce(robust)- option.
>>
>> There is often some uneasyness in specifying "the effect"
>> as linear in the probability metric, as that can
>> eventually lead to predictions outside the range [0, 1].
>> However, if you define the effect interms of odds ratios
>> or probit coefficients, you won't get the causal effects
>> either, see for example: Mood 2010, Allison 1999, or
>> Neuhaus and Jewell 1993.
>>
>> So my guess would be that the linear probability model
>> is in this case the lesser of two evils.
>>
>> Hope this helps,
>> Maarten
>>
>> Allison, Paul D. 1999. "Comparing Logit and Probit
>> Coefficients Across Groups." Sociological Methods &
>> Research 28:186–208.
>>
>> Mood, Carina. 2010. "Logistic regression: Why we cannot
>> do what we think we can do, and what we can do about
>> it." European Sociological Review 26:67–82.
>>
>> Neuhaus, John M. and Nicholas P. Jewell. 1993. "A
>> Geometric Approach to Assess Bias Due to Omited
>> Covariates in Generalized Linear Models." Biometrika
>> 80:807–815.
>>
>> --------------------------
>> Maarten L. Buis
>> Institut fuer Soziologie
>> Universitaet Tuebingen
>> Wilhelmstrasse 36
>> 72074 Tuebingen
>> Germany
>>
>> http://www.maartenbuis.nl
>> --------------------------
>>
>>
>>
>>
>>
>>
>> *
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>> *   http://www.ats.ucla.edu/stat/stata/
>>
>
>
>
> --
> Steven Samuels
> sjsamuels@gmail.com
> 18 Cantine's Island
> Saugerties NY 12477
> USA
> Voice: 845-246-0774
> Fax:    206-202-4783
>
> *
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>

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```