# Expected value of simple linear regression with matrices

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?

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.