How does inclusion of the measurement error in the model, as $$Y_i + \varDelta_i = bX_i + \varepsilon_i$$ affect the standard error of least square estimators $\hat{b}$ of coefficients $b$?
If I obtain a least squares fit from $Y_i = bX_i + \varepsilon_i$ and then later want to incorporate $\varDelta$, is it possible to modify my standard error of $\hat{b}$ to obtain the right value?
(You can assume that $\varDelta \sim \mathcal{N}(0, \sigma^2)$.)
You can move $\Delta_{i} $ to the right-hand side and write $$Y_{i} = bX_{i} + (\varepsilon _{i} - \Delta_{i}) $$.
As long as $\Delta_{i}$ is independent and identically distributed (i.i.d) and uncorrelated with $X_{i}$, the OLS estimate of the $b$ will be BLUE, that is, the estimate of $b$ will be unbiased. So you don't have to modify the standard error of $\hat{b}$.
This is a case of Measurement Error in the Dependent Variable and it is not a problem because of the above explanation. While your estimator isn't biased you will still lose efficiency as you'll have more noise on the left-hand side.
On the other hand, if it was Measurement Error in the Independent Variable, then your estimator will be biased.