confidence intervals in linear regression

Question

I am trying to understand the confidence interval for linear regression parameters. At this link Derive Variance of regression coefficient in simple linear regression an answer is provided. However, I did not understand in the derivation fully. Do we consider the regression parameters as random variables?

That is what are the random variables in the following regression: $$ y=\beta_1 x+\beta_0 + \epsilon $$ From what I understand is that just the $\epsilon$ is random variable with mean 0 and variance $\sigma^2$. Considering $\beta_1, x, \beta_0$ as constants $y$ also becomes random variable with mean $\beta_1 x+\beta_0$ and variance $\sigma^2$, from which confidence interval can be calculated.

However,for regression coefficients confidence intervals are also calculated. Can you please clarify the random variables in regression please?

Thanks.

Answer 1

I am from a different domain, and use somewhat different language, but maybe this will help.

Imagine doing an experiment. $x$ is a set of given values, an "independent variable". Not random. For each of these values you measure a dependent variable, $y$. Presumably, $y$ depends on $x$ in a deterministic (non-random) way, but your measurements are also affected by some noise, $\epsilon$. Therefore, $y$ is a random variable.

When you calculate coefficients of linear regression, $\hat{\beta}_0$ and $\hat{\beta}_1$ (which are estimates of "true", non-random $\beta_0$ and $\beta_1$), their values depend on $y$, and therefore they are also random. If you repeat your experiment once again, for the same $x$ values, you will get (slightly, or not slightly) different $y$, and then will calculate different $\hat{\beta}_0$ and $\hat{\beta}_1$.

Finally, note that $\hat{\beta}_0$ and $\hat{\beta}_1$ are expressed in terms of $y$ linearly. Therefore, variances of $\hat{\beta}_0$ and $\hat{\beta}_1$ are proportional to $var(y) = var(\epsilon)$.

Answer 2

Error is the only random variable. The X's are assumed to fixed but if you assume linearity you can generalize to other values of X. Of course extrapolation is very risky and rarely justified.