Decomposition to understand differences in data I wish to estimate two separate regression:
$$
y_{it}=\boldsymbol{x_{it}'\beta}+\delta_{t}+v_{it}
$$
and 
$$
\tilde{y}_{it}=\boldsymbol{\tilde{x}_{it}'\gamma}+\mu_{t}+\epsilon_{it}
$$
where the $\boldsymbol{x}$ variables are a vector of regressors,
and the $\delta$ and $\mu$ represent time effects. To be more specific,
$i$ represents country, and $t$ represents time period. I have data
on exactly the same countries, for the same years (data is by country
over time). Both the $y$ variable and the $\boldsymbol{x}$ variables
are the same, and the time dummies are by definition the same. The
origin of the data however,is different for each regression. However, I obtain wildly different results. Given that the specification is
identical in both regressions, the only explanation for different coefficients
is differences in data. To some extent, this is readily verifiable-
I can check to what extent each variable in one dataset tracks the
other- for some variables in $\boldsymbol{x}$ and $\tilde{\boldsymbol{x}}$, I obtain a correlation
of 1, indicating (even if units are different), that this variable
is not generating the difference in results. Instead of doing correlations
of different variables one by one, is there any sort of decomposition
that could tell me what the contribution of each of the mismeasured
variables is towards the differential coefficients? In other words,
consider the difference in the vector:
$$
\left(\boldsymbol{\hat{\beta}_{OLS}}\right)-\left(\boldsymbol{\hat{\gamma}_{OLS}}\right)
$$
Is there any way I could express this difference in vectors as a function
of the differences in $\boldsymbol{x}$ and $\boldsymbol{\tilde{\boldsymbol{x}}}?$
 A: Suppose that the design matrices $\boldsymbol{x}$ and $\boldsymbol{\tilde{x}}$ have already been centred with respect to their time variable.  Then, using the form of the OLS estimator you have:
$$\begin{equation} \begin{aligned}
\hat{\beta} - \hat{\gamma} 
&= (\boldsymbol{x}^\text{T} \boldsymbol{x})^{-1} \boldsymbol{x}^\text{T} \boldsymbol{y} - (\boldsymbol{\tilde{x}}^\text{T} \boldsymbol{\tilde{x}})^{-1} \boldsymbol{\tilde{x}}^\text{T} \boldsymbol{y} \\[6pt]
&= [ (\boldsymbol{x}^\text{T} \boldsymbol{x})^{-1} \boldsymbol{x}^\text{T} - (\boldsymbol{\tilde{x}}^\text{T} \boldsymbol{\tilde{x}})^{-1} \boldsymbol{\tilde{x}}^\text{T} ] \boldsymbol{y} \\[6pt]
&= \mathbf{H}(\boldsymbol{x}, \boldsymbol{\tilde{x}}) \boldsymbol{y}, \\[6pt]
\end{aligned} \end{equation}$$
which is a linear transformation of $\boldsymbol{y}$ using the $p \times n$ matrix:
$$\mathbf{H}(\boldsymbol{x}, \boldsymbol{\tilde{x}}) \equiv (\boldsymbol{x}^\text{T} \boldsymbol{x})^{-1} \boldsymbol{x}^\text{T} - (\boldsymbol{\tilde{x}}^\text{T} \boldsymbol{\tilde{x}})^{-1} \boldsymbol{\tilde{x}}^\text{T}.$$
This matrix is fixed in the regression, since you are conditioning on the values from both the design matrices. If we denote the elements by $\mathbf{H}(\boldsymbol{x}, \boldsymbol{\tilde{x}}) = \{ H_{k,i}  | 1 \leqslant k \leqslant p, 1 \leqslant i \leqslant n \}$ then we can express the deviation in a single coefficient estimate as the linear combination:
$$\hat{\beta}_k - \hat{\gamma}_k = \sum_{i=1}^n H_{k,i} \cdot y_i.$$ 
So, as you can see, the difference between the $k$th coefficient estimates for the two regressions is determined by the response vector  $\boldsymbol{y}$ and the $k$th row of the matrix $\mathbf{H}$.  You can calculate the matrix $\mathbf{H}$ from your data and inspect its rows to see the areas where this is having the most effect on the difference in the coefficient estimates (i.e., the values with high magnitude).
