<p>In Hansen (2019) p.221, I don't understand "Specifically, the fact that <span class="math-container">$(y_i, x_i)$</span> are mutually independent and identically distributed implies that any function of <span class="math-container">$(y_i, x_i)$</span> is iid, including <span class="math-container">$x_i$</span> <span class="math-container">$x_i'$</span> and <span class="math-container">$x_i y_i"$</span>. Is there any theorem related to this? Thank you in advance!</p>

CC BY-SA 4.0
2019-07-27T09:58:21.970Z
You can find this in the chapter on multivariate disitributions in Hogg et al's book on mathematical statistics
Grada Gukovic
CC BY-SA 4.0
2019-07-27T10:49:37.040Z
What does it mean for $f(x_i,y_i)$ to be iid? The distribution of $f(x_i,y_i)$ is independent of what and identical to what?
Angela Pretorius
CC BY-SA 4.0
2019-07-27T12:00:59.953Z
@Angela Richardson. It means that the $z_i \equiv f(x_i,y_i)$ are iid.
Bertrand
CC BY-SA 4.0
2019-07-27T12:09:01.997Z
Identical distributions follows from the "Transformations of Random Variables Theorem".
Bertrand
CC BY-SA 4.0
2019-07-27T12:21:21.870Z
and independence is also a consequence ot the "Transformations of Random Variables Theorem"
Bertrand
CC BY-SA 4.0
2019-07-29T16:51:12.463Z
@AngelaRichardson Independence is a property of random variables not of distributions that random variables follow.
Alecos Papadopoulos
<econometrics>
Any function of jointly iid pair is iid (WLLN)?
Lee
# Any function of jointly iid pair is iid (WLLN)?

(Asked by: username_0 on 2019-07-27)

In Hansen (2019) p.221, I don't understand "Specifically, the fact that $(y\_i, x\_i)$ are mutually independent and identically distributed implies that any function of $(y\_i, x\_i)$ is iid, including $x\_i$ $x\_i'$ and $x\_i y\_i"$. Is there any theorem related to this? Thank you in advance!

**Tags:** econometrics

> **Comments:**
> **username_1**: You can find this in the chapter on multivariate disitributions in Hogg et al's book on mathematical statistics  
> **username_2**: What does it mean for $f(x_i,y_i)$ to be iid? The distribution of $f(x_i,y_i)$ is independent of what and identical to what?  
> **username_3**: @username_2 Richardson. It means that the $z_i \equiv f(x_i,y_i)$ are iid.  
> **username_3**: Identical distributions follows from the "Transformations of Random Variables Theorem".  
> **username_3**: and independence is also a consequence ot the "Transformations of Random Variables Theorem"  
> **username_4**: @AngelaRichardson Independence is a property of random variables not of distributions that random variables follow.  

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