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I have a panel regression for countries. There are two groups of countries, rich ($k=1$) and poor ($k=0$). The equation is:

$$ Y_{ikt} = c_k + \lambda_{kt} + X_{it}\beta{k} + e_{ikt} $$

$\lambda_{kt}$ is a year fixed effect which depends on the type of country. I can estimate this using the full sample, as follows:

$$ Y_{ikt} = c + D_k \alpha + \lambda_{t} + \lambda_t D_k + X_{it}\beta + X_{it}D_k\gamma + e_{ikt} $$

where $D_k$ is a dummy for rich and poor. Like in this other question, the year fixed effects are estimated with year dummies. Assume we use $t=1$ as base year. As in that question, the estimated constant will contain

$$ c + \lambda_1$$

From this regression I can create a graph of the year fixed effects, one line for each type of economy (rich or poor). Notice these lines can go in the same plot, because I know the difference in their relative levels, given by the coefficient of the dummy $\lambda_1$ itneracted with $D_k$.

Now, instead of estimating the full sample, we can simply estimate the equation separately for each group. Just as in the equation linked above, the constant on each of them will be

$$ c_k + \lambda_1 $$

I can produce a graph for the year fixed effects, but I cannot put them in the same plot because I do not know the relationship between $c_k$ in the two equations.

Now, in theory these methods are equivalent. Then, there must be a way the second method will tell me the relation between the two constants, so I can put the two lines in the same graph. But I see no way to do this. It's like the second method has lost information, but if so, why are they equivalent (if they are)?

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No, they are not equivalent. The joint method has only one variance of the error term to estimate. The separated method has several variances to estimate, without assuming anything about them. Thus, the degrees of freedom are different. The only way to identify a common trend is to impose the extra restriction of identical variance. This way, the constant term is:

$$ c_k = c + \alpha D_k + \lambda_1 $$

Thus, in the baseline regression ($D_k=0$), the constant is

$$c + \lambda_1$$

In the alternative regression, this is

$$ c + \alpha + \lambda_1$$

Thus, you can then deduce $\alpha$, and subtract from the latter in order to produce a common base. Notice that without imposing identical variance to the error you cannot consistently do the above, because nothing warrants that the mean of the error term is the same. Only with that extra asumption the two methods become identical.

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