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I am currently trying to run a regression working out the change in the mental well-being of ethnic individuals in a set of countries after a migration policy was imposed. I have two waves of data, one before the policy and one after the policy. Essentially the control group are non-ethnic individuals in the countries while the treatment group are the ethnic individuals in the country. The model is:

〖Mental Wellbeing〗_cit=δ_0+δ_1 Country+δ_2 〖Ethnic〗_ic+δ_3 〖PostPolicy〗_t+δ_4 (Country  〖PostPolicy〗_t )+ δ_5 (Country  〖Ethnic〗_ic ) δ_6 (〖Ethnic〗_ic  〖PostPolicy〗_t  Country)+ω_ict+ε_ict

where $Country$ is the name of the country analysed, $Ethnic$ is a dummy variable for whether the individual is an ethnic = 1 or non-ethnic = 0, $PostPolicy$ is for the individuals interviewed in the second period, after the policy had been placed (1 if in second wave and 0 otherwise).

Given this the coefficient of interest is $\delta_6$ as it shows the difference in the well-being of ethnic and non-ethnic individuals after the policy had been imposed. It captures the difference-in-difference-in-differences.

I wondered whether my model makes sense and whether the regression is correct.

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  • $\begingroup$ How many countries do you have? Also, is the immigration policy uniform? In other words, is the “post policy” period the same across countries? $\endgroup$
    – Thomas Bilach
    Commented Apr 6, 2022 at 23:07
  • $\begingroup$ there are 9 countries and the policy is uniform $\endgroup$
    – Lily Tee
    Commented Apr 7, 2022 at 4:31
  • $\begingroup$ You state that the country variable is a country identifier, or simply the "names" of the countries you're analyzing. But this should actually be a binary treatment variable indexing those countries where the immigration policy was introduced. The second treatment group is a subgroup within your countries that is believed to be more/less sensitive to the immigration reform. Does this align with your study? Also, what is $\omega_{ict}$? I assume it's a vector of individual level control variables, but I can't be sure. $\endgroup$
    – Thomas Bilach
    Commented Apr 8, 2022 at 22:14
  • $\begingroup$ Wict is a vector of individual controls. I was wondering if my analysis is also correct so that the treatment group are the ethnics in the UK and then the ethnics in the other 8 countries. Whilst the control group are the non-ethnic individuals in the UK and in the other 8 countries. I'm specifically looking at the UK as the policy was implemented in the UK. I want to see the effect of the policy in the UK but also in the 8 other countries. $\endgroup$
    – Lily Tee
    Commented Apr 10, 2022 at 10:35
  • $\begingroup$ So this is only a UK policy? $\endgroup$
    – Thomas Bilach
    Commented Apr 10, 2022 at 14:32

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I wondered whether my model makes sense and whether the regression is correct.

Seems okay, but you're missing one of the double interactions.

Assuming the policy is uniformly adopted, then the model requires the standard triple interaction term, and all double interactions. The ethnic dummy must also be multiplied with the post-policy indicator. For brevity, I re-specified your model below using abbreviated variable names:

$$ MW_{ict} = \alpha + \delta_ 1E_i + \delta_ 2C_c + \delta_3 P_t + \delta_4 (E_i \times P_t) + \delta_5 (C_c \times P_t) + \delta_6 (C_c \times E_i) + \delta_7 (E_i \times C_c \times P_t) + \epsilon_{ict}. $$

If executed properly, software should return eight coefficients, including the regression constant.

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  • $\begingroup$ Oh yes, that makes sense. Thank you $\endgroup$
    – Lily Tee
    Commented Apr 10, 2022 at 10:36
  • $\begingroup$ I was also wondering what the assumptions are for a 2-time period DiD analysis? I know you can't test the parallel trends assumption but what would we just assume the interviews of the individuals are done randomly? Is this all that we can assume? Or is there more to this? I am slightly confused based on what I have read $\endgroup$
    – Lily Tee
    Commented Apr 21, 2022 at 13:45
  • $\begingroup$ Also I was wondering if i need to include an individual and time fixed effects within the model $\endgroup$
    – Lily Tee
    Commented May 4, 2022 at 10:08
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    $\begingroup$ yes, otherwise it's not a DID estimator. For more on the assumption required for triple DID, or DDD, see academic.oup.com/ectj/article/25/3/531/6545797?login=false or the blog series causalinf.substack.com/p/triple-differences-part-1 $\endgroup$
    – Marti
    Commented Aug 27 at 4:21

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