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Bruce Hansen's book "Econometrics" defines a random sample as follows: "The observations $(y_i,x_i, z_i)$ are a random sample if they are mutually independent and identically distributed $(iid)$ across $i =1, ..., n.$"

I have two questions about this definition: $1)$ Can two observations be iid or does the concept of iid apply only to random variables? In other words, each observation must be associated with a random variable. And the different observations correspond to the result of the same experiment. $2)$ Since each observation is made up of three elements $(y_i, x_i, z_i)$, then should each observation be considered as the realization of a random vector? So the concept of iid refers here to the independence and identical distribution of random vectors and not random variables?

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  • $\begingroup$ I would assume that each observation was a triplet, and so the realisation of a random variable (in $\mathbb R^3$ or whatever - if it helps to call this triplet a random vector, then you can do so but it is also a random variable). The sample is $n$ such realisations, mutually i.i.d. $\endgroup$
    – Henry
    Commented Jun 28, 2020 at 9:37

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Both independence and having a distribution are properties that can be possessed only by random variables. In your example $y_i, x_i,z_i$ is a random vector, where for each $i$ they jointly have identical distribution and are independent of other such triplets. What we mean by such definition is that we treat the observations as realizations of the random variables.

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  • $\begingroup$ Does the concept of independence apply between random vectors? $\endgroup$ Commented Jun 28, 2020 at 18:26
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    $\begingroup$ @JuanBromas multivariate random variable is a random variable, so it has all the properties random variables have, and same laws of probability apply to it. $\endgroup$
    – Tim
    Commented Jun 28, 2020 at 19:37

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