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Suppose I have a regression of the form: $$ Y_{it}=\beta X_{it}+\beta_{2}X_{it}\times D_{i}+\alpha_{i}+\epsilon_{it} $$ In the above, we have information by country-year on $Y_{it}$ and two regressors: $X_{it}$ and $D_{i}$. $X_{it}$ is time varying, but $D_{i}$ is fixed for each country over time. $\alpha_{i}$ measures the country fixed effect.

My question is: can $\beta_{2}$ be identified? As a fixed effects procedure uses only within-country variation, and as $D_{i}$ does not vary over time, how is $\beta_{2}$ separately identified from $\beta?$ I ran the regression and did find that it was identified. Am I missing something?

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  • $\begingroup$ They don't have the same values. For countries where D=1, they are the same, but for countries where D=0, they are not. $\endgroup$
    – ChinG
    Commented Mar 3, 2019 at 19:40
  • $\begingroup$ Hi: It does seems identifiable to me intuitively. because the effect of $D_i$ is additive to the effect of X_it. The coefficient on $X_{it}$ is either $\beta$ or $\beta + \beta_{2}$ depending the value of $D_{i}$ so yes, you can get estimates and the $\beta_2$ will representing the change in the slope of the regression line due to a country having a $D_i$ equal to 1. But let's not trust me. Let's wait for another answer.. $\endgroup$
    – mark leeds
    Commented Mar 3, 2019 at 20:03
  • $\begingroup$ I was thinking about this more and perhaps a clearer way to write the model is to use Dit except that Dit=Di for a given i and all t.. To me, this makes the identifiability of the model more straightforward.. – $\endgroup$
    – mark leeds
    Commented Mar 4, 2019 at 18:50
  • $\begingroup$ Hi @markleeds . Yes it will be identified if cross sectional variation. However, if the fixed effects is used, then only within country varition is used, and D does not change within country. So for countries where D=1, the two variables are exactly collinear, and for countries where D=0, the interaction term=0. $\endgroup$
    – ChinG
    Commented Mar 5, 2019 at 17:07
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    $\begingroup$ Did more research and it seems to be that the use of the term "fixed effects" is referring to a systems model type of fit, so I mis-understood. So I take back everything I said and my apologies for noise. I checked and, if you define it that way and use systemfit in R, it will choke. So, yes, I think it's not identifiable. $\endgroup$
    – mark leeds
    Commented Mar 5, 2019 at 23:25

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