Why is correction of EoV not necessary when predicting? I'm reading or trying to understand the generals of error-in-variable models when I stumbled over the statement, that a correction of a bias, caused by an error in the predictor, is not necessary when the model is used for prediction.
To be honest, it is relative well described but nevertheless I don't get it.
A first question is: There is stated that it is not necessary when prediction is not based on regression coefficients. How else is prediction working?
Nevertheless, can someone please explain it (even more) simple?
Thanks a lot in advance!
 A: A short answer to your question would be that prediction here is related to predicting the response $y$. Hence it is true that the coefficients of the true relationship you are looking for and no estimated correctly, but as the error in the variable is mostly a problem of inference you do not get problems with predictions. Or put it more formally. The real relationship you are interested in is:
$$y=\beta_1 + \beta_2x+u$$
but you cannot measure $x$ but only $z=x+e$, with $e$ and $u$ being i.i.d. normal distributed and independent from each other. Hence you will estimate, based on the equation
$$y=\gamma_1 + \gamma_2z+\epsilon$$
As shown in the wikipedia entry you are refering to, we will have that the parameter vector $\boldsymbol{\gamma}$ is different from $\boldsymbol{\beta}$. So you cannot identify the $\beta_1$ you are looking for.
But if your interest is only in predicting the $y$: $\hat{y}= E[y|x]$ you are not in trouble as the model with the gammas as coefficients fulfills the assumptions needed, especially that $z$ is independent of $\epsilon$ and $E[\epsilon]=0$
Obviously your prediction using the proxy $z$ will have a higher variability as $var(\epsilon)>var(u)$ but this does not mean that the predictions are wrong in expectation. An heuristic and intuitive argument for that is, that the combination of $z$ and the estimated $\boldsymbol{\gamma}$ leads to very similar outcomes as the $x$ and teh estimated $\boldsymbol{\beta}$ although $\boldsymbol{\gamma}$ and $\boldsymbol{\beta}$ differ.
A small and simple $\texttt{R}$ skript shows this easily:
#real variable
x<-rnorm(1000,0,3)

#proxy
z<-x+rnorm(1000,0,0.5)

plot(x~z)

#true relationship
y=1+3*x+rnorm(1000,0,1)

# we can estimate only with the proxy
proxy<-lm(y~z)
real<-lm(y~x)

par(mfrow=c(1,2))
plot(y~proxy$fitted.values)
plot(y~real$fitted.values)

As a result of the code we will get two pretty similar pictures of fitted values, where only the variation among the conditional mean varies.
