# The cesspool of econometrics

I took Oxford’s advanced undergraduate econometrics course. My experience of the course, and really of the entire field, was the following: the concepts are simple, the real challenge is making sense of notation so obfuscatory that you wonder if it’s done on purpose.

In order to arrive at this view, I went through a long and confusing journey, one I wish upon no friend. This document’s structure takes my journey in reverse order.1 I start with what I eventually pinned down as the clear mathematical facts. Once armed with this toolkit, I do my best to explain why standard notation is confusing, and attempt to guess, from context, what econometricians actually mean.

The examples I give really are of standard practice. I give quotes from a few textbooks and our lecture slides, but I promise that you will find the same thing almost everywhere. And the confusing usages are not just a convenience of notation that is readily acknowledged in conversation. When I asked people about this in person, all I got were long, confusing back-and-forths.

# The facts

## Preliminaries

We start with a set of ordered pairs ${ \langle X_1 , Y_1 \rangle,\langle X_2 , Y_2 \rangle,\langle X_3 , Y_3 \rangle, …, \langle X_n,Y_n \rangle}$.

You can think of $X_i$ and $Y_i$ as

• real numbers (facts about each of the the $n$ individuals in the population)
• or as random variables (probability distributions over facts about $n$ individuals in a sample),

all the maths will apply equally. (I will return to this fact and comment on it).

## The CEF minimises $\sum w_i^2$

We then have $% $$As before$w_i$is known, whereas$e_i$is a function of$\beta_0$and$\beta_1$. Here$e_i$is the distance, for observation$i$, between the LRM and the CEF; while$w_i$is the distance between the CEF and the actual value of$Y_i$. We can then call$u_i = e_i + w_i$the distance between the LRM and the actual value.2 We can also see that$E[u_i \mid x_i] =0$is equivalent to$e_i=0$, i.e. the CEF and the LRM occupy the same coordinates. ### Minimisation problem Suppose we want to solve $\min_{\beta_0, \beta_1} \sum (e_i + w_i)^2 \leftrightarrow \min_{\beta_0, \beta_1} \sum (Y_i - \beta_0 -\beta_1 X_i)^2$$

The solution is %

I prove this in appendix B. (It’s possible to prove an analogous result in general using matrix algebra, see appendix C.)

Suppose we specify that $\beta_0$ and $\beta_1$ are equal to these solution values. Now that $\beta_0$ and $\beta_1$ are known, $e_i$ is known too (by the subtraction $e_i = E[Y_i \mid X_i] - \beta_0 -\beta_1X_i$). As before, $w_i$ is known.