The Beginner's Guide to React

# Introduction

Source: https://github.com/timlrx/tailwind-nextjs-starter-blog

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.
^[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 are denoted by `$$`.

If you intend to use the `$` sign instead of math, you can escape it (`\$`), or specify the HTML entity (`&dollar;`) [^2]

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

[^2]: \$10 and &dollar;20.

# Deriving the OLS Estimator

Using matrix notation, let $n$ denote the number of observations and $k$ denote the number of regressors.

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

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

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

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

```latex
\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]
```

$$
\mathbf{X} = \left[\begin{array}
  {ccccc}
  x_{11} & . & . & . & x_{1k} \\
  . & . & . & . & .  \\
  . & . & . & . & .  \\
  . & . & . & . & .  \\
  x_{n1} & . & . & . & x_{nn}
\end{array}\right] =
\left[\begin{array}
  {c}
  \mathbf{x}'_1 \\
  . \\
  . \\
  . \\
  \mathbf{x}'_n
\end{array}\right]
$$

The vector of error terms $\mathbf{U}$ 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.

## Least Squares

**Start**:  
$$y_i = \mathbf{x}'_i \beta + u_i$$

**Assumptions**:

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

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

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

**Solution**:  
Hints: $Q$ is a $1 \times 1$ scalar, by symmetry $\frac{\partial b'Ab}{\partial b} = 2Ab$.

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
\end{aligned}
```

$$
\begin{aligned}
  \min Q           & = \min_{\beta} \mathbf{Y}'\mathbf{Y} - 2\beta'\mathbf{X}'\mathbf{Y} +
  \beta'\mathbf{X}'\mathbf{X}\beta \\
                   & = \min_{\beta} - 2\beta'\mathbf{X}'\mathbf{Y} + \beta'\mathbf{X}'\mathbf{X}\beta \\
  \text{[FOC]}~~~0 & =  - 2\mathbf{X}'\mathbf{Y} + 2\mathbf{X}'\mathbf{X}\hat{\beta}                  \\
  \hat{\beta}      & = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y}                              \\
                   & = (\sum^{n} \mathbf{x}_i \mathbf{x}'_i)^{-1} \sum^{n} \mathbf{x}_i y_i
\end{aligned}
$$

## Inline Highlighting

Sample of inline highlighting `sum = parseInt(num1) + parseInt(num2)`

## Code Blocks

Some Javascript code

```js {1,3-4} showLineNumbers
var num1, num2, sum;
num1 = prompt("Enter first number");
num2 = prompt("Enter second number");
sum = parseInt(num1) + parseInt(num2); // "+" means "add"
alert("Sum = " + sum); // "+" means combine into a string
```

Some Java code

```java showLineNumbers
public class HelloWorld {
	public static void main(String[] args) {
		int count = 0;

		for (int i = 0; i < 10; i++) {
			System.out.println("Count: " + count);
			count++;
		}
	}
}
```

Some Python code 🐍

```python
def fib():
    a, b = 0, 1
    while True:            # First iteration:
        yield a            # yield 0 to start with and then
        a, b = b, a + b    # a will now be 1, and b will also be 1, (0 + 1)

for index, fibonacci_number in zip(range(10), fib()):
     print('{i:3}: {f:3}'.format(i=index, f=fibonacci_number))
```


Animi quisquam fugiat eaque. Hic libero deleniti nulla aspernatur repudiandae ab saepe. Aliquid delectus veritatis incidunt assumenda rerum. Culpa atque debitis assumenda ab voluptatem quas. Laborum accusantium reiciendis quidem harum cupiditate eaque officia repellat voluptas. Incidunt nemo praesentium vel officiis sequi molestias ipsum id deleniti.
Reiciendis sint soluta atque ullam est mollitia. Atque voluptatibus accusamus non incidunt placeat. Veniam ratione ipsum. Reiciendis fugiat non tempora voluptatibus error.
Sit ea vel eaque cum soluta eos. Nostrum aliquid quibusdam voluptas est. Totam veritatis omnis quos quo aut sapiente maiores. Perspiciatis consequuntur suscipit voluptatibus fugit illum quidem est saepe quidem.

Lorem ipsum dolor sit amet consectetur adipisicing elit. Libero laudantium sequi illo, veritatis labore culpa, eligendi, quod consequatur autem ad dolorem explicabo quos alias harum fuga sapiente reiciendis. Incidunt, voluptates.

```ts
// Verwende eine for-Schleife, um über alle Zahlen der Liste zu iterieren.
for (let i = 0; i < numbers.length; i++) {
	// DEINE LÖSUNG HIER
}
```

Separating Events from Effects

How was your day?