Estimator of $Y$ in the simple linear regression model

Question

In a statistics textbook I saw that the linear simple regression model is defined as \begin{equation} Y = \alpha + \beta x + e \end{equation} where $x$ is a value of the independent variable, $Y$ is the response, $\alpha$ and $\beta$ are the parameters, and $e$ is a random error with a mean of 0.

Then this textbook talks about estimators. It says $A$ is an estimator of $\alpha$, and $B$ is an estimator of $\beta$. Also, $A+Bx$ is an estimator of $\alpha+\beta x$. However, according to the textbook, $A+Bx$ is an estimator of $Y$. I don't understand that $A+Bx$ can be an estimator of $Y$, since $Y$ is a random variable but not a parameter. Am I wrong?

To me it makes sense only to say that $A+Bx$ is an estimator of the 'mean' of the random variable $Y$. Then, given that the mean of $e$ is zero, $A+Bx$ is an estimator of the mean of $\alpha + \beta x + e$.

Answer 1

You are right*, we are estimating parameters but not (realizations of) random variables. We would predict those instead.

*Except for the last bit where you say

Then, given that the mean of $e$ is zero, $A+Bx$ is an estimator of $\alpha + \beta x + e$.

Note that $\alpha+\beta x+e$ is a random variable and it equals $Y$. If you oppose to calling $A+Bx$ and estimator of $Y$, you should oppose calling it an estimator of $\alpha+\beta x+e$ for the sake of internal consistency.

Answer 2

When we say $A$ is an estimator of $\alpha$ it means that $\mathbb{E}(A)=\alpha$. Since $A+Bx$ is a random variable(because of $\alpha$ and $\beta$) we can write: $$\mathbb{E}(A+Bx)=\mathbb{E}(A)+\mathbb{E}(Bx)+0=\mathbb{E}(A)+\mathbb{E}(B)x+\mathbb{E}(e)=\mathbb{E}(Y)$$ Hope it helps.