Consider
$Y = \theta X + \Delta$
where $\theta$ is the unknown matrix to be found, $Y, X$ are the data matrices of finite length $T$
$Y = [y_0 \ y_1 \ \dots \ y_T], \ X = [x_0 \ x_1 \ \dots \ x_T]$
and $\Delta$ is an unknown deterministic perturbation matrix, which has the same structure of $Y,X$ (can be Hankel or Toeplitz)
$\Delta = [\delta_0 \ \delta_1 \ \dots \ \delta_T]$.
When the perturbation is not present, the solution $\theta$ is found by Least Squares
$\theta = Y X^\dagger$
where $X^\dagger = X^\top (XX^\top)^{-1}$. When $\Delta$ is measurable, so it is known,
$\theta = (Y -\Delta)X^\dagger$.
The question is: how to correctly estimate $\theta$ when we don't have any information about $\Delta$ (I can only assume a structure and some bounds on the norm)?
Maybe total least squares (LS) can be useful in this way? Are there methods of Robust LS that can guarantee a correct estimate having finite samples of data?
Notice that by construction $\Delta = Y -\theta X$, so if the basic elements of $\Delta$ are in $\mathbb{R}$ then the OLS estimator or MLE exists in $\mathbb{R}$. If you want a consistent OLS or MLE estimator, since the OLS or MLE is $\hat\theta=\theta_0+(X'X)^{-1}X'\Delta$, you must assume that plim $n^{-1}(X'\Delta)=0$ and that $\lim_{n\to \infty}n^{-1}X'X=Q$ where $Q$ is not singular.