Estimation in levels vs differences Very basic question. Suppose I have a linear regression model:
$$
y_{it}=\boldsymbol{x_{it}'\beta}+\epsilon_{it}
$$
 where $\left(y,\boldsymbol{x}\right)$ are i.i.d copies, indexed
by individuals $i$ at time $t.$ Under suitable exogeneity assumptions,
the OLS estimate of the parameter vector $\boldsymbol{\beta}$ is
unbiased and consistent. Now, consider the first difference transform,
where $\triangle y_{it}=y_{it}-y_{it-1}$, and estimate instead:
$$
\triangle y_{it}=\boldsymbol{\triangle x_{it}'\beta}+\triangle\epsilon_{it}
$$
In this case, the estimates are still unbiased and consistent, and
converge to the true parameter vector. My question is, in practice,
I have seen wildly different point estimates for the regression in
levels, and in differences. Why is this is case? Is it because this
implies that the conditions were not satisfied in the first place
in the level equation? Or is it because we are estimating a different
model? 
Many thanks!
 A: There was another question about the variance of the FD (first difference) estimator $\Delta\beta$ that you are interested in, under the condition of no serial correlation and homoskedasticity of errors. This question follows the textbook by Adonis Yatchew (2003): Semiparametric regression for the applied econometrician, pages 2-3, so I will also do that. 
You are interested in why and how $\beta$ and $\Delta \beta$ are different. From Yatchew (2003), p.3:
$$\beta - \Delta\beta \quad \xrightarrow{D} \quad \frac{1}{\sqrt{n}} \mathcal{N}\left(0, \frac{3\sigma_{\varepsilon}^2}{2\sigma_{u}^2}\right)= \mathcal{N}\left(0, \frac{3\sigma_{\varepsilon}^2}{2n\sigma_{u}^2}\right)$$ 
where $\sigma_u^2$ is the variance of the explaining variable ($x$ in your case), $\sigma_{\varepsilon}^2$ is that of the residuals. The accepted answer to the other question mentioned above explains some details of how to derive this. 
As you can see, the variance of the estimator declines in $n$ and $\sigma_u^2$ and increases in $\sigma_{\varepsilon}^2$. For very large $n$, $\Delta\beta$ will converge to $\beta$. For smaller $n$, it matters what the variances are. In particular, FD performs better if you have a large variance of the explaining variable and a small one of the residual.
Remember that this is under the assumption of no autocorrelation of errors, i e. $E(\Delta\varepsilon)=0$. If you have serial correlation, this would mess up the regular correlation approach without differences and you can use the FD approach to get rid of the serial correlation.
