I am currently taking a regression course and ended up with someting that I cannot figure out to solve. So if anyone who can help me further I would appreachiate that.

Suppose we have $y_i=\beta_0+\beta_1x_i+\varepsilon_i, \;\;\; i=1,...,n$ where we have $\varepsilon\sim N(0,\sigma^2_\varepsilon)$. Suppose further that $x$ is measured with error, such that $v_i=x_i+\delta_i,\;\;\; i=1,...,n$. Where $v_i$ are the observed (i.e. measured) values of the predictor, $x_i$ are the unobserved values, and $\delta_i\sim N(0,\sigma_\delta^2)$ are i.i.d. error terms, $i=1,...,n$.

  1. We shall show that the regression of $y$ onto $v$ yields estimates $(\tilde{\beta}_0,\tilde{\beta}_1)$ whose expected value can be written as $$E\binom{\tilde{\beta}_0}{\tilde{\beta}_1}=A\binom{\beta_0}{\beta_1}$$ for some matrix $A$.

So what should I do? Should I find that matrix or?


1 Answer 1


Let $X$ be the design matrix for the explanatory variable and $V$ be the design matrix for the measured values of $X,$ so that one column of $V$ consists of $1$s and the other is the vector $(v_i).$ The "regression of $y$ onto $v$" thereby is given by the estimate

$$\tilde\beta = (V^\prime V)^{-}V^\prime y = (V^\prime V)^{-} V^\prime(X\beta + \varepsilon).$$

Find the expectation of this estimate in two steps, first with respect to $\varepsilon$ (conditional on $V$) and then with respect to $V.$ Because (since $\varepsilon$ is independent of $V$) $E[\varepsilon\mid V] = 0$ and conditional expectation is linear,

$$E[\tilde\beta] = E_V E_{\varepsilon\mid V}\left[(V^\prime V)^{-} V^\prime(X\beta + \varepsilon)\mid V\right] = E_V\left[(V^\prime V)^{-} V^\prime(X\beta\right)].$$

Again because expectation is linear,

$$ E_V\left[(V^\prime V)^{-} V^\prime(X\beta\right)] = E_V\left[(V^\prime V)^{-} V^\prime\right] X\beta.$$

Assuming this expectation is well-defined, we may compare it to the question by taking

$$A = E_V\left[(V^\prime V)^{-} V^\prime\right]\, X,$$

the expectation of the regression parameter estimate is

$$E[\tilde \beta] = A\beta.$$

When $V$ has a Normal distribution as given in the question and there are at least three observations and $X^\prime X$ is invertible, the $V$ expectation does exist and is finite. I suspect that the demonstration of these special facts was not intended to be part of this exercise.


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