# Difference in Difference with multiple treatments with paired data observations (twin studies)

I have a data set of identical twins, such that the data can be considered as paired observations. The data is composed of a survey that was conducted on two separate years, and all twins answer the survey questions in both years. The variables I have are

Stress1: is the stress score of the individual in Year 1 ranging from 1 to 3 with 1 being least stress.

Stress2: is the stress score of the individual in Year 2 ranging from 1 to 3 with 1 being least stress.

WorkStat1: is the work status of the individual in Year 1 and includes 2 categories (1: entrepreneur, 0: not entrepreneur)

WorkStat2: is the work status of the individual in Year 2 and includes 2 categories (1: entrepreneur, 0: not entrepreneur)

I want to study the effect of a change in WorkStat on the change in Stress.

If I understand correctly, the change in WorkStat in the two years translates into a control group (0: Never_Been_Entrepreneur) and 3 treatment groups (1: Became_Entrepreneur, 2: Always_Entrepreneur, and 3: Left_Entrepreneur).

My first question: Is my understanding of the problem in the context of a Diff-in-Diff model correct?

The question that i'm trying to answer is whether becoming an entrepreneur increases stress. so I'm interested in the first treatment group (1: Became_Entrepreneur). How can I conduct this analysis in Stata given that my data is paired (i'm dealing with twins). For example, I have two individuals with twin_id = 1 and person_id equal 1 and 2 respectively for each individual in the twin pair.

Start with the equation for stress $s$:

$$s_{ijt}=\alpha + \beta e_{ijt} + \gamma_t a_{ij}+\varepsilon_{ijt},$$

where $i=1,2$ indexes the twin in pair $j=1,...,N$, and $t=1,2$ is time. The indicator $e$ denotes being an entrepreneur, and $a$ is the unobserved time-invariant ability term. Let's assume that ability is negatively correlated with stress and positively correlated with starting a business, which will create omitted variable bias in estimating $\beta$. Let's also assume that the effect of ability is not constant over time (so we can't just use fixed effects or first-difference panel methods on each person). Perhaps nerds used to to get picked on more in the old days, but now it's cool and less stressful to be one.

The Twin Difference-in-Differences setup where you first you difference each person over time and then difference that for each pair, would be:

$$(s_{1j2}-s_{1j1})-(s_{2j2}-s_{2j1})=\beta \cdot\left[(e_{1j2}-e_{1j1})-(e_{2j2}-e_{2j1}) \right] + (\gamma_2-\gamma_1) \cdot (a_{1j}-a_{2j})+\left[(\varepsilon_{1j2}-\varepsilon_{1j1})-(\varepsilon_{2j1}-\varepsilon_{2j1}) \right].$$

For monozygotic twins, it might be reasonable to assume that unobserved ability $a_{1j}=a_{2j}$, so that term would drop out, allowing you to estimate $\beta$.

Once you get the data in this format (the details would be off-topic here), it's just a simple linear regression with $N$ observations (with no constant) if you are willing to treat this measure of stress as a continuous variable. I would recommend also using het-robust errors.

In response to your comment, if you only difference each twin's data over time (the first difference estimator), the ability affect does not difference out unless $\gamma_2=\gamma_1$ (the effect of ability is constant over time):

$$s_{ij2}-s_{ij1}=\left(\alpha + \beta e_{ij2} + \gamma_2 a_{ij}+\varepsilon_{ij2} \right)-\left(\alpha + \beta e_{ij1} + \gamma_1 a_{ij}+\varepsilon_{ij2} \right)=\beta \cdot (e_{ij2}-e_{ij1}) + (\gamma_2-\gamma_1) \cdot a_{ij}+(\varepsilon_{ij2}-\varepsilon_{ij1}).$$

• @roland Did this help? – Dimitriy V. Masterov Jul 11 '16 at 16:53
• Sorry for the late reply. I'm not sure I understand the assumption that you can't use a fixed effects model because the unobserved individual effects are time varying? Can you please elaborate on that? – finstats Jul 20 '16 at 11:37
• @roland I updated the answer with a better explanation of why first-differences won't work. You can make a similar argument for FEs if you crank through the algebra since $\gamma_t - (\gamma_2+\gamma_1)/2 \ne 0$. – Dimitriy V. Masterov Jul 20 '16 at 17:48