demeaning or differentiation in fixed-effect equation I have panel data (with short time-dimension $T>2$) and I consider a simple model of the form:
$$y_{i,t} = x_{i,t} \beta + c_i + \epsilon_{i,t}$$
where $E(x_{i,s}\epsilon_{i,t})=0$ and $\epsilon_{i,t}$ are iid and of mean 0.
To get rid  of the $c_i$ and to estimate the $\beta$, one can consider in particular:


*

*either first differentiating succesive observations and then performing an OLS regression: $\Delta y_{i,t} = (\Delta x_{i,t}) \beta +\Delta \epsilon_{i,t}$

*or demeaning and then performing an OLS regression: $y_{i,t} - \bar{y_{i,t}} = (x_{i,t} - \bar{x_{i,t}}) \beta +(\epsilon_{i,t}-\bar{\epsilon_{i,t}})$
Please note that the demeaning seems to be prefered (it is the "fixed-effect estimator"), and it can be written (for instance here*) that "The [first difference] estimator is slightly less efficient than the [demeaning], as $\Delta \epsilon$ exhibits serial correlation, even if $\epsilon_{i,t}$’s are uncorrelated." For instance demeaning is used in the Stata command xtreg, fe.
My first question is: why should the demeaning approach should be prefered to the differentiating approach?
As a temptative answer, it is indeed easy to see that $\Delta \epsilon_{i,t}$ is correlated with $\Delta \epsilon_{i,t+1}$ as both terms share $\epsilon_{i,t}$. It comes: $cov(\Delta \epsilon_{i,t+1},\Delta \epsilon_{i,t}) = - \sigma_\epsilon<0$
However, both $(\epsilon_{i,t}-\bar{\epsilon_{i,t}})$ and $(\epsilon_{i,t+1}-\bar{\epsilon_{i,t}})$ also share the terme $\bar{\epsilon_{i,t}}$ (and this term includes also $\frac{\epsilon_{i,t}}{T}$ and $\frac{\epsilon_{i,t+1}}{T}$). It comes: $cov(\epsilon_{i,t}-\bar{\epsilon_{i,t}},\epsilon_{i,t+1}-\bar{\epsilon_{i,t}}) = - \frac{2}{T}(1-\frac{1}{T}) \sigma_\epsilon^2<0$
Hence, my understanding is that there is also some correlation after applying demeaning, but less than when applying first-differencing. This is why it is better.
My second (highly related) questions are: am I making a mistake above? If not, what is the impact of these correlations on the estimates of the $\beta$? Thank you in advance!
*see http://statmath.wu.ac.at/~hauser/LVs/FinEtricsQF/FEtrics_Chp5.pdf in page 25 for demeaning, and in page 28 for first differentiating.
 A: Both the first difference transformation and the within transformation generate serial correlation in the estimated residuals even when it is not there in the underlying data generating process. For $T > 2$, the first difference estimator is less efficient than the within if the error term is really iid. For $T = 2$ the first differences estimator and the within estimator are equivalent.
The first difference estimator requires a slightly weaker assumption than strict exogeneity, namely that the transitory component of the error term is uncorrelated with the conditioning variables in the current period, as well as one period before and one period after (rather than all of them). This will not matter in most economic contexts.
A: Why is it more efficient?
Let $e$ denote the transformed error from either transformation. We have that
$$V(\hat{\beta}|X)=(X'X)^{-1}X'V(e|X)X(X'X)^{-1}$$
So looking at $V(e|X)$, we check diagonal and off-diagonal elements. For diagonal elements, we have $2\sigma^2$ for the first-difference, and for the within transform $\sigma^2+V(\overline{\epsilon}(T))$ where the latter term goes towards zero with at a rate of $T^2$. 
For the off-diagonal elements, you have already checked that the within transform becomes more efficient with $T$ for all elements.
Check for example $T=3$, where the estimators differ. On the diagonal, you have for the first-difference $2\sigma^2$. But for the within estimator, the diagonal comes out as $\sigma^2+\frac{1}{4}(2\sigma^2)=\frac{3}{2}\sigma^2$ which is less than $2\sigma^2$.
You can do the same for other elements and for larger $T$.
In that sense, the estimator is more efficient, it provides smaller standard errors for the same data generating process.
Edit: You can calculate this out (I think the exact formula is not so important in this case), but say you will get something like
$$V(\hat{\beta}|x)=\frac{\sum_{t,h}x_{t}x_h \sigma_{t,h}}{2\sum_t x_t^2}$$ and then obviously that thing is declining in $T$ for the within estimator, meaning it makes use of the additional data we have if $T>2$ whereas the difference estimator is constant and thereby not efficient over $T$
