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What's the meaning of indexes in the definition of populational regression?

$$Y_i=\beta_0+\beta_1X_i+e_i$$

When we have a sample the indexes make sense to me because we have each $x_i$ associate with $y_i$, but what's the meaning in populational regression?. To me the definition should be:

$$Y=\beta_0+\beta_1X+e$$

(without index) and in sample, we would have $$y_i=\hat{\beta_0}+\hat{\beta}_1x_i+\epsilon_i$$

Edit: My problem is basically with the idea of ​​"$i$th observation" of $Y$ being $Y_i$.

$Y_i$ is observable quantity. $y_i$ is observed.

So I think the $Y$ indices are potential observations of Y (not observed), it’s like "copies" of $Y$, so $Y_i$ is still a random variable and your observations (in reality) are $y_i$.

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    $\begingroup$ It is a bit frustrating when you change the question after I post an answer and then change it again after I updated my answer. I am happy to respond to comments on my answer but I cant keep checking to see if youve edited the question again.The question is about the meaning of the subscripts, and I have answered that: it indexes the observations. Anything else to do with sample vs population or observed vs unobserved is beside the point. $\endgroup$ – Robert Long Jun 18 at 19:56
  • $\begingroup$ The question remains the same "meaning of the subscripts". I just edited it to expose why I don't understand the indexes in the populational regression $\endgroup$ – Aliss0nR Jun 18 at 20:10
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    $\begingroup$ Ive answered the question multiple times now. The subscript indexes the observation. You say you have a problem with this but you don't say why. $\endgroup$ – Robert Long Jun 18 at 20:20
  • $\begingroup$ @RobertLong I edited it just to explain the problem I had with "The subscript indexes the observation". If you observe $Y$ you have $y_i$ (fixed value and not a random variable) and the definition uses $Y_i$ (random variable). I know that $i$ is to indicate the ordering of the observation, but it didn’t make sense to me when it came to random variables. My point from the start is that index indicates observation and as said when you observe you have $y_i$ $\endgroup$ – Aliss0nR Jun 18 at 20:43
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    $\begingroup$ I'm sorry but I don't understand your point at all. $y$ is never considered as fixed, but even if it was the $i$ subscript still denotes the $i$th observation $\endgroup$ – Robert Long Jun 18 at 20:57
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Edit: Now that the question has been modified following my original answer, which is kept below, my modified answer is:

What's the meaning of indexes in the definition of populational regression?

The usual meaning is to index the observation number, so that $Y_1$ and $X_1$ refer to the 1st observation of the variables $Y$ and $X$. In general $Y_i$ and $X_i$ index the $i$th observation.

So the models:

$$Y_i=\beta_0+\beta_1X_i+e_i$$ and $$ Y=\beta_0+\beta_1X+e $$

are entirely equivalent. The only difference is that in the former we are dealing with individual observations, while in the latter we are dealing with the variables themselves as vectors.

The model:

$$ y_i=\hat{\beta_0}+\hat{\beta}_1x_i+\epsilon_i $$

would usually be used when discussing predictions from either of the models above, though for consistency I would prefer to keep the same upper or lower case notation. $\hat{\beta_0}$ and $\hat{\beta_1}$ are then the estimated ceofficients obtained after fitting either of the above models.


Original answer:

Different people use notation differently. Commonly, the subscripts refer to a particular observation, so

$$Y_i=\beta_0+\beta_1X_i+e_i$$ and
$$ y_i=\hat{\beta_0}+\hat{\beta_1x}_i+\epsilon_i $$ should be equivalent (except that the hats usually represents an estimated value, rather than being used as the description of the model). So $Y_1$ and $X_1$ refer to the 1st observation of the variables $Y$ and $X$. Similarly for $y_i$ and $x_i$: they represent the $i$th observation.

On the other hand,

$$Y=\beta_0+\beta_1X+e$$

could be used when $Y$ is a vector of responses and $X$ is a vector representing a single explanatory variable. It is up to the person writing it out to be clear about what notation they are using and what it means. When dealing with vectors and matrices in regression it is also common to write:

$$Y=X\beta +e$$

where $X$ is a model matrix which will include all explanatory variables including the intercept, and $\beta$ is a vector of coefficients.

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  • $\begingroup$ I would add to this answer that if we are talking about the true population model it is common to write this model as, $$\mu_y = \beta_0 +\beta_1 x$$ where $\mu_y$ is a mean response and we drop the error terms because we have a "true" model. $\endgroup$ – Ariel Jun 18 at 18:13
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    $\begingroup$ @Ariel nice, I didn't know that (or more likely, had forgotten !). Thanks :) $\endgroup$ – Robert Long Jun 18 at 18:23
  • $\begingroup$ What do you mean by "subscripts refer to a particular observation" in populational regression??? Particular observations are samples of Y, they are $y_i$ no $Y_i$ $\endgroup$ – Aliss0nR Jun 18 at 18:49
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    $\begingroup$ I'm not sure what you mean ? The standard use for subscripts on variables is to index a particular observation, regardless of whether or not you use small-case $y$ or upper-case $Y$ or whether you are referring to a sample or the population. As I mentioned, it's up to the writer to be clear about what notation they use. $\endgroup$ – Robert Long Jun 18 at 18:52
  • $\begingroup$ Does this answer your question ? If so please consider marking it as the accepted answer. If not, please let us know why. Also, if you haven't already, please consider upvoting it. $\endgroup$ – Robert Long Jul 24 at 12:03

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