<p>The "identically - independently distributed" assumption (or any restriction thereof) on a sample, refers to the relation <em>between random variables in different observations</em>, not between the variables forming the same observation.  </p>

<p>So in a multidimensional sample "of size $n$"</p>

<p>$$\{(y_1, x_{11}, x_{21}),(y_2, x_{12}, x_{22}),...,(y_n, x_{1n}, x_{2n})\}$$</p>

<p>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.</p>

<p>Certainly, $y_i$ is assumed stochastically <strong>dependent</strong> with $x_{1i}, x_{2i}$, while also $x_{1i}$ and $x_{2i}$ can be stochastically dependent <em>for the same $i$</em>, i.e. inside each observation.</p>

CC BY-SA 3.0
2017-05-03T14:49:11.160Z
2017-05-03T15:16:53.350Z
2017-05-03T15:16:53.350Z
Alecos Papadopoulos
<p>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?</p>

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2017-05-03T13:39:58.523Z
Random draws with replacement from the same population are iid.
Amit
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2017-05-03T14:00:44.097Z
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?
A. Kutilainen
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2017-05-03T14:42:54.363Z
There is no point of regression of Y on X if X and Y are iid.
Amit
CC BY-SA 3.0
2017-05-03T16:11:55.453Z
Such questions are much better fitted to [Cross Validated](http://stats.stackexchange.com). Take a look and you will find quite a few similar ones with good answers.
Richard Hardy
CC BY-SA 3.0
2017-05-03T16:22:54.727Z
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.
A. Kutilainen
<econometrics><statistics><regression>
Independent identically distributed
A. Kutilainen
# Independent identically distributed

(Asked by: username_0 on 2017-05-03)

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?

**Tags:** econometrics, statistics, regression

> **Comments:**
> **username_1**: Random draws with replacement from the same population are iid.  
> **username_0**: 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?  
> **username_1**: There is no point of regression of Y on X if X and Y are iid.  
> **username_2**: Such questions are much better fitted to [Cross Validated](http://stats.stackexchange.com). Take a look and you will find quite a few similar ones with good answers.  
> **username_0**: 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.  

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## 1 answers

### Answer by username_3 on 2017-05-03. Score: 3 | Accepted answer

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.


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