This example demonstrates the difference between a theoretical observation and a realized observation. A theoretical observation is a random variable with a probability distribution, while its realization is a number. Some statistical concepts have different word concept from its realization. For example, an estimator is a random variable, while an estimate is it realization. But this is an exception. Most statistical concepts - and 'observation' is one of them - have only one word to denote both. In the previous chapter, observations were realized (given the data), but now they denote theoretical (random) data.
This an excerpt from an econometrics book. I would like to focus on the estimator part. I need two clarifications.
I am not happy now because the answer of @whuber here What is the relation between estimator and estimate? seems to contradict with this excerpt which says an estimator is a random variable. According to this excerpt, as I interpret it, the OLS estimator is a random variable because it depends on X and y, which are random, and hence it has a sampling distribution. Therefore it has a variance (that we estimate). OLS estimator is a statistic. This thread also claims that the estimator is a random variable Why is an estimator considered a random variable?
Given what is above, is it true that when talking about the variance, we should always talk about the variance of the estimator and not the variance of the estimate? An estimate is not a random variable. But if you search google, you will find countless hits that talk about the "variance of the OLS estimate".
It may help this thread if we keep the discussion within the bounds of econometrics and not get into very complex mathematical perspectives.