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Since there is the possibility that there are some factors result in whether a data point belong to treatment group, I think we should add a dummy variable indicating whether a data point belong to treatment group when using difference-in-differences approach.

However, this was not done in research generally, e.g. this paper: Chhaochharia, Vidhi, and Yaniv Grinstein. "CEO compensation and board structure." The Journal of Finance 64.1 (2009): 231-261.

Are there reason for not adding the dummy variable? Would colinearity be an issue? (I don't think it is an issue in the above example.)

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    $\begingroup$ Not quite sure what your question is all about. Could you please add more information? I have an access to the aforementioned article and in methodology chapter, part C they do use $\mathrm{dummy}(\mathrm{board}) \cdot \mathrm{dummy}(\mathrm{time})$, isn't it exactly what you're asking? Also, it could be beneficial if you found a similar issue in article that is available for anyone to read. $\endgroup$
    – bajun65537
    Apr 14 '20 at 10:11
  • $\begingroup$ I think they might should have also included just dummy(board) (without any interaction). $\endgroup$
    – Aqqqq
    Apr 14 '20 at 11:59
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    $\begingroup$ @Aqqqq but if they estimate a FE model (see my answer) they don't need this dummy board. I don't know their paper. $\endgroup$
    – emeryville
    Apr 14 '20 at 12:08
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It depends on the estimated model. Let me use the example provided in Mostly Harmless Econmetrics.

The experiment

Suppose we are interested in the effect of the minimum wage on employment. Card and Krueger (1994) use a dramatic change in the New Jersey state minimum wage. On April 1, 1992, New Jersey raised the state minimum from \$4.25 to \$5.05. Card and Krueger collected data on employment at fast food restaurants in New Jersey in February 1992 and again in November 1992. Fast food restaurants are big minimum-wage employers. They also collected data from the same type of restaurants in eastern Pennsylvania, just across the Delaware river. The minimum wage in Pennsylvania stayed at \$4.25 throughout this period.

Using their data, they computed diferences-in-diferences estimates of the effects of the New Jersey minimum wage increase. That is, they compared the change in employment in New Jersey to the change in employment in Pennsylvania around the time New Jersey raised its minimum.

The Fixed Effects Model

Differences-in-differences (DD) is a version of fixed-effects estimation using aggregate data. To see this, let

  • $y_{1ist} =$ fast food employment at restaurant i and period t if there is a high state minimum wage.

  • $y_{0ist} =$ fast food employment at restaurant i and period t if there is a low state minimum wage.

Let $D_{st}$ be a dummy for high-minimum-wage states, where states are index by $s$ and observed in period $t$. In the absence of a minimum wage change, employment is determined by the sum of a time-invariant state effect ($\gamma_s$) and a year effect ($\lambda_t$) that is common across states. Assuming that $E[y_{1ist} - y_{0ist}|s,t)$ is a constant, denoted $\delta$, we have:

$$Y_{ist}= \gamma_s + \lambda_t + \delta D_{st} + \epsilon_{ist} ~~~~~~ (1) $$ (I corrected $\lambda_s$ here.)

The Regression Model

We can use regression to estimate equations like (1). Let $NJ_s$ be a dummy for restaurants in New Jersey and $d_t$ be a time-dummy that switches on for observations obtained in November (i.e., after the minimum wage change). Then

$$Y_{ist}= \alpha + \gamma NJ_s + \lambda d_t + \gamma(NJ_s \times d_t) + \epsilon_{ist} ~~~~~~ (2) $$

is the same as (1) where $NJ_s \times d_t = D_{st}$.

Your Question

Regarding your question, a key difference is that equation (2) includes the dummy variable indicating whether a data point belongs to treatment group ($NJ_s$ the dummy for restaurants in New Jersey). This dummy is not present in model (1) but the two models are equivalent.

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  • $\begingroup$ Thank you for your answer. However, in the example by you, I think the dummy variable indicating whether a data point belongs to treatment group is included in the equation (1), it is $\gamma_s$. $\endgroup$
    – Aqqqq
    Apr 18 '20 at 12:05
  • $\begingroup$ The dummy is $NJ_s$ and $\gamma$ is the parameter associated with the dummy. $\endgroup$
    – emeryville
    Apr 18 '20 at 12:36
  • $\begingroup$ But the characteristic specific to the control entity and treatment entity is still accounted for in equation (1), because of $ \lambda_t$ in $Y_{ist}= \gamma_s + \lambda_t + \delta D_{st} + \epsilon_{ist} $, unlike in the instance where it is not done (e.g. in the paper I mentioned). $\endgroup$
    – Aqqqq
    Apr 18 '20 at 20:16
  • $\begingroup$ I don't know the paper you mentionned but in equation (1), $\lambda_s$ is a fixed effect, it's the time-invariant state effect (I corrected a small typo). $\endgroup$
    – emeryville
    Apr 18 '20 at 22:51
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    $\begingroup$ I get your point. In this particular example with two states the $\gamma_s$ fixed effect (sorry for the typo) is equivalent to a dummy variable. But if you think about a more general case with more than 2 states, then $\gamma_s$ will be a categorical variable with more than 2 levels, and the $NJ_s$ could still be a dummy equal to 1 for restaurants in New Jersey, and 0 otherwise. Does it make sense? $\endgroup$
    – emeryville
    Apr 19 '20 at 16:46
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There are several reasons why that’s not done.

For starters, DiD is most often estimated using panel FE regression, in order to have fixed effects that can control for time invariant unobservables and that can’t be reconciled with an another time invariant dummy. Also, in cases where you have only 1 treatment and 1 control fixed effects would in essence be equivalent to a dummy indicating treatment status and in other cases where you have multiple treated and controls fixed effects are superior because they don’t impose restriction on all treated and non treated time invariant unobservables being same.

Next what matters in DiD most is the common trend assumption. In fact DiD is used precisely in cases where treatment assignments is not random. The whole advantage of DiD is that conditional on the treated and control following common trend you don’t need to care about how the treatment was assigned (unlike RCT, or RD/fuzzy RD where you would try to make treatment assignment random or care about probability of being assigned treatment)

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