Deriving the OLS Estimator

Deriving the OLS Estimator

Introduction

Parsing and display of math equations is included in this blog template. Parsing of math is enabled by remark-math and rehype-katex. KaTeX and its associated font is included in _document.js so feel free to use it on any page. 1

Inline math symbols can be included by enclosing the term between the $ symbol.

Math code blocks are denoted by $$.

If you intend to use the $ sign instead of math, you can escape it (\$), or specify the HTML entity ($) 2

Inline or manually enumerated footnotes are also supported. Click on the links above to see them in action.

Deriving the OLS Estimator

Using matrix notation, let nn denote the number of observations and kk denote the number of regressors.

The vector of outcome variables Y\mathbf{Y} is a n×1n \times 1 matrix,

\mathbf{Y} = \left[\begin{array}
  {c}
  y_1 \\
  . \\
  . \\
  . \\
  y_n
\end{array}\right]
Y=[y1...yn]\mathbf{Y} = \left[\begin{array} {c} y_1 \\ . \\ . \\ . \\ y_n \end{array}\right]

The matrix of regressors X\mathbf{X} is a n×kn \times k matrix (or each row is a k×1k \times 1 vector),

\mathbf{X} = \left[\begin{array}
  {ccccc}
  x_{11} & . & . & . & x_{1k} \\
  . & . & . & . & .  \\
  . & . & . & . & .  \\
  . & . & . & . & .  \\
  x_{n1} & . & . & . & x_{nn}
\end{array}\right] =
\left[\begin{array}
  {c}
  \mathbf{x}'_1 \\
  . \\
  . \\
  . \\
  \mathbf{x}'_n
\end{array}\right]
X=[x11...x1k...............xn1...xnn]=[x1...xn]\mathbf{X} = \left[\begin{array} {ccccc} x_{11} & . & . & . & x_{1k} \\ . & . & . & . & . \\ . & . & . & . & . \\ . & . & . & . & . \\ x_{n1} & . & . & . & x_{nn} \end{array}\right] = \left[\begin{array} {c} \mathbf{x}'_1 \\ . \\ . \\ . \\ \mathbf{x}'_n \end{array}\right]

The vector of error terms U\mathbf{U} is also a n×1n \times 1 matrix.

At times it might be easier to use vector notation. For consistency, I will use the bold small x to denote a vector and capital letters to denote a matrix. Single observations are denoted by the subscript.

Least Squares

Start:
yi=xiβ+uiy_i = \mathbf{x}'_i \beta + u_i

Assumptions:

  1. Linearity (given above)
  2. E(UX)=0E(\mathbf{U}|\mathbf{X}) = 0 (conditional independence)
  3. rank(X\mathbf{X}) = kk (no multi-collinearity i.e. full rank)
  4. Var(UX)=σ2InVar(\mathbf{U}|\mathbf{X}) = \sigma^2 I_n (Homoskedascity)

Aim:
Find β\beta that minimises the sum of squared errors:

Q=i=1nui2=i=1n(yixiβ)2=(YXβ)(YXβ)Q = \sum_{i=1}^{n}{u_i^2} = \sum_{i=1}^{n}{(y_i - \mathbf{x}'_i\beta)^2} = (Y-X\beta)'(Y-X\beta)

Solution:
Hints: QQ is a 1×11 \times 1 scalar, by symmetry bAbb=2Ab\frac{\partial b'Ab}{\partial b} = 2Ab.

Take matrix derivative w.r.t β\beta:

\begin{aligned}
  \min Q           & = \min_{\beta} \mathbf{Y}'\mathbf{Y} - 2\beta'\mathbf{X}'\mathbf{Y} +
  \beta'\mathbf{X}'\mathbf{X}\beta \\
                   & = \min_{\beta} - 2\beta'\mathbf{X}'\mathbf{Y} + \beta'\mathbf{X}'\mathbf{X}\beta \\
  \text{[FOC]}~~~0 & =  - 2\mathbf{X}'\mathbf{Y} + 2\mathbf{X}'\mathbf{X}\hat{\beta}                  \\
  \hat{\beta}      & = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y}                              \\
                   & = (\sum^{n} \mathbf{x}_i \mathbf{x}'_i)^{-1} \sum^{n} \mathbf{x}_i y_i
\end{aligned}
minQ=minβYY2βXY+βXXβ=minβ2βXY+βXXβ[FOC]   0=2XY+2XXβ^β^=(XX)1XY=(nxixi)1nxiyi\begin{aligned} \min Q & = \min_{\beta} \mathbf{Y}'\mathbf{Y} - 2\beta'\mathbf{X}'\mathbf{Y} + \beta'\mathbf{X}'\mathbf{X}\beta \\ & = \min_{\beta} - 2\beta'\mathbf{X}'\mathbf{Y} + \beta'\mathbf{X}'\mathbf{X}\beta \\ \text{[FOC]}~~~0 & = - 2\mathbf{X}'\mathbf{Y} + 2\mathbf{X}'\mathbf{X}\hat{\beta} \\ \hat{\beta} & = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y} \\ & = (\sum^{n} \mathbf{x}_i \mathbf{x}'_i)^{-1} \sum^{n} \mathbf{x}_i y_i \end{aligned}

Footnotes

  1. For the full list of supported TeX functions, check out the KaTeX documentation

  2. $10 and $20.

Share:

Related Posts

Release of Tailwind Nextjs Starter Blog v2.0

Release of Tailwind Nextjs Starter Blog v2.0

Release of Tailwind Nextjs Starter Blog template v2.0, refactored with Nextjs App directory and React Server Components setup.Discover the new featur...

read more
New features in v1

New features in v1

An overview of the new features released in v1 - code block copy, multiple authors, frontmatter layout and more

read more
Introducing Tailwind Nextjs Starter Blog

Introducing Tailwind Nextjs Starter Blog

Looking for a performant, out of the box template, with all the best in web technology to support your blogging needs? Checkout the Tailwind Nextjs S...

read more
Images in Next.js

Images in Next.js

In this article we introduce adding images in the tailwind starter blog and the benefits and limitations of the next/image component.

read more
Markdown Guide

Markdown Guide

Markdown cheatsheet for all your blogging needs - headers, lists, images, tables and more! An illustrated guide based on GitHub Flavored Markdown.

read more
O Canada

O Canada

The scenic lands of Canada featuring maple leaves, snow-capped mountains, turquoise lakes and Toronto. Take in the sights in this photo gallery exhib...

read more
Sample .md file

Sample .md file

Example of a markdown file with code blocks and syntax highlighting

read more