# Sample space of error term in linear regression

Let $\left(y_{i},x_{i}\right):\Omega\rightarrow\mathcal{Y}\times\mathcal{X}$ be random vectors with $\mathcal{Y}\subset\mathbb{R}$ and $\mathcal{X}\subset\mathbb{R}^{p}$. Consider a model $\left\langle \Theta,P\right\rangle$, with $P:\Theta\times\Omega\rightarrow\mathbb{R}_{+}$ a density over $\Omega$ parametrized by $\theta_{0}\in\Theta\subset\mathbb{R}^p$. In other words, $z_i:=\left(y_{i},x_{i}\right)$ follows $P\left(z;\theta_{0}\right)$. In particular assume that $y_i=x_i^\intercal \theta_0+\epsilon$, with $\epsilon$ a zero mean random variable.

Is there any way to characterize the sample space of $\epsilon$ (i.e., the set of outcomes, which the error term maps from, as $\Omega$ constitutes for $y_i,x_i%$)? I assume there is, since not all three variables can be independent.

• Could you clarify what you mean by "sample space of $\epsilon$"? People seem to use the term "sample space" in various ways, so it's necessary to be more specific concerning what you're asking here.
– whuber
Jan 25 '18 at 14:45

The sample space of $\epsilon$ is the same as $\Omega$. The reason is simply because $\epsilon_i = f(x_i,y_i)$ hence $\epsilon$ is defined on the same sample space as $(x_i,y_i)$.