Expected value notation

Question

I come across the notions $\mathrm{E}[wage]$ and $\mathrm{E}[wage_i]$. I would like to clarify if they have different meanings.

1. Take the first expression. $\mathrm{E}[wage]$ could represent the average of all elements in $wage$ in the population. Here, it is my guess that $wage$ is the vector of all theoretical $N$ observations. Otherwise I do not know what the first expression is trying to say.

2. We consider $y_i$ as a random variable. It has a distribution. The mean of this distribution is $\mathrm{E}[wage_i]$. Hence, $\mathrm{E}[wage_i]$ means that we take the average of all theoretical observations $wage_i$ can take.

Could I then conclude that $\mathrm{E}[wage]$ and $\mathrm{E}[wage_i]$ refer to the same thing?

Answer 1

Very often it is assumed that the stochastic variables under consideration are identically distributed. So $wage_i$ follows a distribution $F_i(.)$ but since this $F_i(.)= F(.)$ for all $i$ the expectation for $wage_1$ and $wage_2$ will be the same and so for all $i\not = j$. This very often leads to the adoption of the notation where the index is simply left out.

Note though that in some contexts authors choose to leave out index simply because it is too cumbersome to write. This is clearly the case in a setting of wage regressions where

$$\mathbb E[wage_i \lvert \mathbf x_i ] = \mathbf x_i^\top \boldsymbol \beta$$

is very often simply written as

$$\mathbb E[wage \lvert \mathbf x ] = \mathbf x^\top \boldsymbol \beta$$

without this implying that wages are identically distributed. So here it is simply short hand notation for the perhaps more precise notation that includes the index.