Sorry if this is too basic.

In an OLS regression given by

$y=ax+b$

$b$ intercept, $a$ the slope.

Then $a,b$ are not numbers but random variables.

I find this confusing since I start with data points $(x_1,y_1),(x_2,y_2),..(x_n,y_n) ; x_i \neq x_j $ when $i \neq j $. Then we find the line of best fit, which would give us actual numbers $a,b$, so that these seem to be constants, not (Random) variables.

Is the reason these are called variables that I am selecting just one of many possible values $y_j$ for a given variable $x_j$?

  • 1
    Estimators of slope and intercept are random variables.because they're functions of the responses, which are random variables. – Glen_b Sep 23 at 23:57
  • @Glen_b: So the point is that for each data set $(x_i,y_i)$ for the same variables X,Y ( of same size) I would get different values for the slope, the intercept? – gary Sep 24 at 0:02
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    New samples would indeed lead to different estimates (because - even assuming fixed x's - you'd have different realizations of each of the n corresponding y's) – Glen_b Sep 24 at 1:09
  • Thanks, Glen_b , should I delete the question or do you want to answer it. Or should I? – gary Sep 24 at 1:10
  • I wasn't sure whether that's what you were seeking (which is why I commented, figuring you'd clarify the question if you needed something else), I am happy to post it as an answer (or you can if you prefer). – Glen_b Sep 24 at 1:12
up vote 5 down vote accepted

Estimators of slope and intercept are random variables because they're functions of the responses, which are random variables.

New samples would lead to different estimates (because - even assuming fixed $x$'s - you'd have different realizations of each of the n corresponding sets of $Y$'s)

If we set the situation up to make the variables and their realizations a little more distinct, the situation may become clearer; taking the $x$'s as fixed (for simplicity of exposition), you have $$Y_i = ax_i + b + \epsilon_i$$ where $\epsilon_i$ is the error term. You draw a sample with that set of $x$'s and you observe a corresponding set of $y$'s, corresponding to a particular realization of the $\epsilon_i$. Let's call that set of observed $y$ values $\mathbf{y}^{(1)}$. We repeat our sampling procedure at the same set of x-values and obtain a new set of responses, $\mathbf{y}^{(2)}$ and we keep going, up to $\mathbf{y}^{(k)}$ say. Each realization will have its own slope and intercept, so realization $j$ has $a^{(j)}$ and intercept $b^{(j)}$, which will be a function of $\mathbf{y}^{(j)}$.

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