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Fix a few minor typos

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Alexander Zeilmann
2021-12-20 21:02:50 +01:00
parent e05191f1c3
commit c0c50b4e2d
9 changed files with 89 additions and 89 deletions
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104
data/blog/deriving-ols-estimator.mdx
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@@ -9,12 +9,12 @@ summary: 'How to derive the OLS Estimator with matrix notation and a tour of mat
# 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 in any pages.
KaTeX and its associated font is included in `_document.js` so feel free to use it on any page.
^[For the full list of supported TeX functions, check out the [KaTeX documentation](https://katex.org/docs/supported.html)]
Inline math symbols can be included by enclosing the term between the `$` symbol.
Math code blocks is denoted by `$$`.
Math code blocks are denoted by `$$`.
The dollar signal displays without issue since only text without space and between two `$` signs are considered as math symbols.[^2]
@@ -30,23 +30,23 @@ The vector of outcome variables $\mathbf{Y}$ is a $n \times 1$ matrix,
```tex
\mathbf{Y} = \left[\begin{array}
{c}
y_1 \\
. \\
. \\
. \\
y_n
{c}
y_1 \\
. \\
. \\
. \\
y_n
\end{array}\right]
```
$$
\mathbf{Y} = \left[\begin{array}
{c}
y_1 \\
. \\
. \\
. \\
y_n
{c}
y_1 \\
. \\
. \\
. \\
y_n
\end{array}\right]
$$
@@ -54,45 +54,45 @@ The matrix of regressors $\mathbf{X}$ is a $n \times k$ matrix (or each row is a
```latex
\mathbf{X} = \left[\begin{array}
{ccccc}
x_{11} & . & . & . & x_{1k} \\
. & . & . & . & . \\
. & . & . & . & . \\
. & . & . & . & . \\
x_{n1} & . & . & . & x_{nn}
{ccccc}
x_{11} & . & . & . & x_{1k} \\
. & . & . & . & . \\
. & . & . & . & . \\
. & . & . & . & . \\
x_{n1} & . & . & . & x_{nn}
\end{array}\right] =
\left[\begin{array}
{c}
\mathbf{x}'_1 \\
. \\
. \\
. \\
\mathbf{x}'_n
{c}
\mathbf{x}'_1 \\
. \\
. \\
. \\
\mathbf{x}'_n
\end{array}\right]
```
$$
\mathbf{X} = \left[\begin{array}
{ccccc}
x_{11} & . & . & . & x_{1k} \\
. & . & . & . & . \\
. & . & . & . & . \\
. & . & . & . & . \\
x_{n1} & . & . & . & x_{nn}
{ccccc}
x_{11} & . & . & . & x_{1k} \\
. & . & . & . & . \\
. & . & . & . & . \\
. & . & . & . & . \\
x_{n1} & . & . & . & x_{nn}
\end{array}\right] =
\left[\begin{array}
{c}
\mathbf{x}'_1 \\
. \\
. \\
. \\
\mathbf{x}'_n
{c}
\mathbf{x}'_1 \\
. \\
. \\
. \\
\mathbf{x}'_n
\end{array}\right]
$$
The vector of error terms $\mathbf{U}$ is also a $n \times 1$ 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.
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
@@ -107,7 +107,7 @@ $$y_i = \mathbf{x}'_i \beta + u_i$$
4. $Var(\mathbf{U}|\mathbf{X}) = \sigma^2 I_n$ (Homoskedascity)
**Aim**:
Find $\beta$ that minimises sum of squared errors:
Find $\beta$ that minimises the sum of squared errors:
$$
Q = \sum_{i=1}^{n}{u_i^2} = \sum_{i=1}^{n}{(y_i - \mathbf{x}'_i\beta)^2} = (Y-X\beta)'(Y-X\beta)
@@ -120,22 +120,22 @@ Take matrix derivative w.r.t $\beta$:
```tex
\begin{aligned}
\min Q & = \min_{\beta} \mathbf{Y}'\mathbf{Y} - 2\beta'\mathbf{X}'\mathbf{Y} +
\beta'\mathbf{X}'\mathbf{X}\beta \\
& = \min_{\beta} - 2\beta'\mathbf{X}'\mathbf{Y} + \beta'\mathbf{X}'\mathbf{X}\beta \\
\text{[FOC]}~~~0 & = - 2\mathbf{X}'\mathbf{Y} + 2\mathbf{X}'\mathbf{X}\hat{\beta} \\
\hat{\beta} & = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y} \\
& = (\sum^{n} \mathbf{x}_i \mathbf{x}'_i)^{-1} \sum^{n} \mathbf{x}_i y_i
\min Q & = \min_{\beta} \mathbf{Y}'\mathbf{Y} - 2\beta'\mathbf{X}'\mathbf{Y} +
\beta'\mathbf{X}'\mathbf{X}\beta \\
& = \min_{\beta} - 2\beta'\mathbf{X}'\mathbf{Y} + \beta'\mathbf{X}'\mathbf{X}\beta \\
\text{[FOC]}~~~0 & = - 2\mathbf{X}'\mathbf{Y} + 2\mathbf{X}'\mathbf{X}\hat{\beta} \\
\hat{\beta} & = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y} \\
& = (\sum^{n} \mathbf{x}_i \mathbf{x}'_i)^{-1} \sum^{n} \mathbf{x}_i y_i
\end{aligned}
```
$$
\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
\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}
$$