Topic 4 Properties of Estimators

So far, we have been talking about population quantities such as \(f_{Y|X}\) (conditional pdf/pmf), \(\mathbb{E}[Y]\) (expected value of \(Y\)), or \(\mathbb{E}[Y|X]\) (expected value of \(Y\) given \(X\)).

In practice, most often we do not know what these population quantities are equal to (with the exception of some trivial cases like flipping a coin or rolling a die).

A fundamental challenge is that it is uncommon that we observe the entire population.

Instead, we will take the approach that we have access to a sample of data from the original population. We’ll use the sample to try to estimate whatever population quantities we are interested in as well as develop the tools to conduct inference, paying particular interest to questions like: how precisely can we estimate particular population quantities of interest?

The topics considered in this section fall broadly under the topic of statistics (a reasonable definition of statistics is that it is the set of tools to learn about population quantities using data). Some of this material may be familiar from courses that you have taken before, but this section provides a fairly advanced discussion of these issues with a particular eye towards (i) inference issues that are important econometrics and (ii) prediction problems. Many of the tools that we cover in this section will be used throughout the rest of the course.