Conditional Expectation Function

metrics EC484_Econometrics_Analysis

One of the most important objects to describe the relationship between $y$ and $x$ is conditional expectation.

The Conditional Expectation Function is written as:

$$\mathbb{E}[y|x]$$

which is read as “the expected value of $y$ given $x$”.

$$\mathbb{E}[y|x]=\int u{f}_{y|x}(u|x)du$$

where $f_{y|x}(u|x)$ is the conditional probability density function of $y$ given $x$.

What is probability density function?

Probability Density Function

Note that $\int \dots du$ means we integrate over all possible values of $u$.

Knowing this, we could then define the Law of Iterated Expectations (LIE)