[[toc]]

Econometrics. A field where the concepts are simple, but 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 statement, 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.

In my view, it’s a pretty scathing indictment of the field that I spent about ten times longer engaging in this interpretative guesswork than I spent understanding the underling concepts.

# 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.