How can we address the error associated with estimators in linear regression? I am dealing with a noisy dataset that has a certain amount of error for every estimator. Say, xi's are plus/minus a constant.
I am wondering if there is a way to handle these errors or at least see their effect on the resulting regressed line. 
 A: If your $x$ variables (predictors, features, IVs) have errors, the ordinary least squares estimators will be biased. 
[This assumes that the x's the y's are based on are the 'true' without-error x's, $y=\beta_0+\beta_1x+\epsilon$ but we only observe $x^o=x+\eta$]

Above is a plot of some 'true' $x$ (blue) and observed $x$ with error (orange) with the fitted least squares lines for each (the population line is dashed in red but visually coincident with the blue line in the lower/left half of the plot. In this example the error in the $x$ is about the same size as the error in the $y$, and this increase in the variation of $x$ "spreads" the points out, flattening the ordinary least squares estimate a little (it's small because the variance of the observation error in $x$ is small relative to the variance of $x$).
We see some indication of the 'attenuation' effect (bias toward $0$) that is seen with simple-regression if one uses least squares on such data.
In multiple regression the bias is not necessarily toward 0 but that is the more usual situation.
For additional information, see https://en.wikipedia.org/wiki/Errors-in-variables_models and questions with the tag errors-in-variables. You might also get some value from searches related to model II regression.
