Proof that a and b in linear regression are random variables

Question

Does anyone know how to prove that the variables $a$ and $b$ that are used in linear regression are random variables? For me the assumption would be that these are dependent on the values of $x$ and $y$ which are simply random variables that can take different values as differing observations are made in multiple experiments and that as a consequence, $a$ and $b$ are also random variables. Is there a deeper way of thinking that proves $a$ and $b$ are random?

Answer 1

We can write the model in matrix form, where $a$ and $b$ are both contained in $\beta$. The true model underlying the OLS is the linear projection model $$ y=a+bx_1+e, E[x_1e]=0, E[e]=0 $$ but now we write it as $$ y=x\beta+e, E[xe]=0 $$ It can be proved that the true best linear predictor $\beta=E[xx']^{-1}E[xy]$, and hence $a$ and $b$ are just numbers and nonrandom.

Assume now we have $n$ realizations(observations) of $x$ and $y$. We array it into a dataset $X,Y$, the OLS estimator is $$ \hat\beta=(XX')^{-1}X'Y. $$ If we are to talk about the probablistic properties of $\hat\beta$ such as unbiasedness, we use the expectation conditioning on $X$. Hence the $x$ part in the true model is now assumed to be nonrandom, however there is still a $e$ that is random. For example, the conditional expectation of $\hat\beta$ is \begin{align} &E[(XX')^{-1}X'Y|X]\\ =&E[(XX')^{-1}X'(X\beta+e)|X]\\ =&(XX')^{-1}X'X\beta+(XX')^{-1}X'E[e|X]\\ =&\beta. \end{align} Note that $\hat\beta$, i.e. $\hat a$ and $\hat b$ are random because of $e$.