Can I recover the level of a dummy in the constant? In the following country-level panel data equation
$$ Y_{it} = c + \lambda_t + X_{it}\beta + e_{it} $$
I use time dummies to capture the year-fixed effects, $\lambda_t$. Obviously, one dummy must be left out, to avoid multicollinearity. Say I leave out $\lambda_1$. This means that when I estimate the model, the estimated constant will include
$$ c + \lambda_1 $$
From my regressions, I can produce a plot of $\lambda_t$, showing the year fixed effects. However, the level of this series includes the constant $c$. Is there a way I can identify "$c$", in order to produce the true series of year effects, and not just a relative one?
$c$ is supposed to be the (expected?) average value of $Y_{it}$ when $X_{it}$ is zero? Is this helpful in any way for my goal?
 A: There is no such thing as the "true series of year effects". They depend on the units in which everything is measured, which in turn will affect $c$. 
What is key to realise here is that year effects represent deviations with respect to the base category. Thus, what is invariant are those deviations. In other words, if you are estimating $T$ year effects, there is always only $T-1$ pieces of independent observations. 
To see this more clearly, you could estimate an alternative model where instead of selecting $t=1$ as the base, you select $t=T$ as the base (i.e. you omit that dummy). Then you will get an estimate of $\lambda_t$ for $t=1,...,T-1$. Yet, again, you have $T-1$ independent results. Naturally, the selection of the base should not matter, so whichever you assume should provide you with the same deviations.
Another way to see this is to estimate the model with all the year effects, plus a constant. This is possible if you estimate the model with the following constraint (originally suggested in this paper): 
$$ \sum_{t=1}^T\lambda_t=0 $$
There you have all the year effects, plus a constant. Yet, there are only $T-1$ degrees of freedom. 
