# Definition indepedence and identically distributed (iid)

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?

• 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. Jun 28, 2020 at 9:37

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.