# Understanding the error term

I am trying to figure out the meaning of these different "hatted" terms in regression analysis. Here is my basic understanding:

• $Y$ the original of population/sample values
• $\hat{Y}$ regression line
• $\beta$ The coefficient, using population values
• $\hat{\beta}$ The coefficient, using sample values.

But I have a question about the error term, is it:

1. $e$ (or $u$) The error term, using the population values, $\hat{e}$ The error term, using sample values (so the error is caused by lack of data)

OR

1. does $\hat{e}$ simply mean the measurement error (error is caused by measurement)?

Also, is there any way to distinguish $Y$ as the population values and $Y$ as the values of a sample.

• Could you please edit your question using LaTeX commands? It is hard to read as is. For instance, use $\hat{Y}$ and $\hat{\beta}$. – Xi'an Dec 14 '14 at 14:58
• You should definitely distinguish random variables from observed values (i.e. don't call them both $Y$'s as if they were the same thing). The usual means is to use $Y_i$ for the random variable and $y_i$ for the observed value on that random variable, but as long as it's absolutely clear which object you are talking about, other notation may be reasonable. – Glen_b -Reinstate Monica Dec 14 '14 at 23:04

$e$ is the population error (the part of $Y$ not explained by the linear combination of $X$ and $\beta$), while $\hat{e}$ are the residuals (the sample part of $Y$ not explained by $X$ and $\hat{\beta}$). Linear regression assumes $e \overset{\rm iid}{\sim} N(0, \sigma^2)$ with $\sigma^2$ constant, and we should check this using the observed sample $\hat{e}$.
The error is any source of variation of $Y$ not included in the model, either from excluded variables or from measurement error, as long as they comply with the distributional assumptions.
We usually use $Y$ to refer generically to the population, and $y_i$ to refer to the $i^{\rm th}$ sampled observations.