When we are talking about the least squares assumptions, one of the assumpions is that (X,Y) are i.i.d. What bothers me is that if we take an example population and research distributions of age and salaries and lets imagine that distribution of salaries (Y) in the population looks somewhat like chi squared distribution with df=10 and distribution of age (X) resembles exponential distribution. So how can we have an i.i.d. variable when sampling?
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$\begingroup$ Random draws with replacement from the same population are iid. $\endgroup$– AmitCommented May 3, 2017 at 13:39
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$\begingroup$ So is that supposed to mean that when we are talking about iid and saying that (Y, X) are iid, we mean that that all X´s are identical and all Y´s are identical, not that X and Y are identical? $\endgroup$– A. KutilainenCommented May 3, 2017 at 14:00
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1$\begingroup$ There is no point of regression of Y on X if X and Y are iid. $\endgroup$– AmitCommented May 3, 2017 at 14:42
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$\begingroup$ Such questions are much better fitted to Cross Validated. Take a look and you will find quite a few similar ones with good answers. $\endgroup$– Richard HardyCommented May 3, 2017 at 16:11
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$\begingroup$ Thank you for the tip. I assumed it would be suitable place since this is subject in our econometrics course, which is part of economics studies. $\endgroup$– A. KutilainenCommented May 3, 2017 at 16:22
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1 Answer
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The "identically - independently distributed" assumption (or any restriction thereof) on a sample, refers to the relation between random variables in different observations, not between the variables forming the same observation.
So in a multidimensional sample "of size $n$"
$$\{(y_1, x_{11}, x_{21}),(y_2, x_{12}, x_{22}),...,(y_n, x_{1n}, x_{2n})\}$$
the assumption applies between the variables of observation $i$ and all variables in any other observation. So $y_1$ is assumed independent from, say $y_2$, but it is also assumed indepedent from $x_{12}$. Also $x_{11}$ is assumed independent from $x_{12}$ but also from $x_{22}$, etc.
Certainly, $y_i$ is assumed stochastically dependent with $x_{1i}, x_{2i}$, while also $x_{1i}$ and $x_{2i}$ can be stochastically dependent for the same $i$, i.e. inside each observation.
answered May 3, 2017 at 14:49