#StackBounty: #estimation #sampling #sample #consistency Which is the relation between population/probability space/sampling?

Bounty: 50

I am trying to understand the relation between population/probability space/sampling. My arguments are divided in 3 sub-questions which trace my attempt to link in a logical way the three concepts. I am using an Economic example, but I regard my question as generic.


Consider a target population of individuals; for each individual in this population we believe that $$income=beta*education+u$$

where $income, education, u$ are real numbers. $u$ collects all the variables affecting income in addition to education.

We are interested in learning about $beta$.


Sub-question 1: Imagine to extract at random an individual $m$ from the population and observe her income, education, and additional features (denoted by the random variables $Y_m, X_m, U_m$). Intuitively, I understand why we can say that the income, education, additional features of the extracted individual are random variables: we don’t know a priori which individual will be picked from the urn representing the whole population and, hence, we attach a probability to each potential outcome. More formally, to define a random variable we need a probability space $(Omega, mathcal{F}, Pr)$. Is $Omega$ set equal to the population?


Sub-question 2: If the answer to question 1 is YES, then, if we knew the entire population (i.e., if we knew $(Omega, mathcal{F}, Pr)$) and suppose $E(X_m^2)neq 0$ and $E(X_mU_m)=0$, we could easily compute the exact value of $beta$ as
$$
beta=E(Y_mX_m)/E(X_m^2)
$$
Correct?


Sub-question 3: The problem is that we don’t know the entire population (i.e., we don’t know $(Omega, mathcal{F}, Pr)$) and so we try to approximate in some good way $E(Y_mX_m)$ and $E(X_m^2)$ by appropriately taking a subset of the whole population (sampling). For example, a way to appropriately take a subset of the whole population is the following: for $m=1,…,M$:

  • We draw at random an individual from the urn containing the entire population, we label him/her with the index $m$, and we register his/her income and education level (denoted by the random variables $Y_m, X_m)$. The additional features affecting income remain unobserved (denoted by the random variables $U_m$).

  • We put back in the urn individual $m$.

The sampling scheme just described implies that
$$
(i) hspace{1cm}{Y_m, X_m, U_m}_{m=1}^M text{ are i.i.d. across $m$}
$$

We then define
$$
hat{beta}=frac{frac{1}{M}sum_{m=1}^MY_mX_m}{frac{1}{M}sum_{m=1}^M X_m^2}
$$
By $(i)$, $frac{1}{M}sum_{m=1}^MY_mX_mrightarrow_p E(Y_mX_m)$ and $frac{1}{M}sum_{m=1}^M X_m^2rightarrow_p E(X^2_m)$.

Hence,
$$
hat{beta}rightarrow_p beta
$$
Correct?


Sub-question 4: Suppose also that $U_m$ is a continuous random variable. Does this imply stating that the population is very large or a continuum or infinite?


Get this bounty!!!

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