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I want to test the impact of X on Y. The dependent variable Y is being employed. Now, I want to see if the impact of X is different for those employed in agriculture (A) and non-agriculture (N) sectors. What is the right model to do this?

Do I need to include some interactions?

Or should I define separate dependent variables for A and N? How do I define those variables? Which one of the following: 1- Create a dummy variable for group A (N) which takes value 0 for group N (A) and also not-working group 2- Create a dummy variable for group A (N) which takes value 0 for group N (A) and drop the observation for not-working group 3- Create a dummy variable for group A (N) which takes value 0 for not-working group and drop the observation for group N (A)

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For individual $i$, let $Y_i$ and $X_i$ be the relevant variables and let $D_i$ be the dummy for agricultural sector: 1 if in agriculture and 0 if not.

Then you can run the following regression: $$ Y_i = \alpha_0 + \alpha_1 D_i + \beta_0 X_i + \beta_1 X_i\times D_i + \varepsilon_i. $$ Assume that $\mathbb{E}[\varepsilon_i|D_i, X_i] = 0$.

Then if a person is not employed in agriculture, we have: $$ \mathbb{E}[Y_i|X_i = x, D_i = 0] = \alpha_0 + \beta_0 x. $$ On the other hand for a person employed in agriculture, you obtain: $$ \mathbb{E}[Y_i |X_i = x, D_i = 1] = (\alpha_0 + \alpha_1) + (\beta_0 + \beta_1) x. $$

So you only need to create one dummy. The value of $\alpha_1$ give the difference in agriculture and non-agriculture employment when $x = 0$. The value of $\beta_1$ gives the difference in the marginal effect of $X$ on $Y$ between agriculture and non-agriculture.

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