ECON 3818

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# ECON 3818
## Chapter 5
### Kyle Butts
### 29 November 2021

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## Chapter 5: Regression

---
# Review from Last Class

Recall the ways we discussed relationships between two random variables `$X$` and `$Y$`:

Covariance, `$\sigma_{XY}$` (sample analogue: \$s_{XY}\$ )
      
  - Direction matters, but magnitude is hard to interpret
      
Correlation, `$\rho_{XY}$` (sample analogue: \$r_{XY}\$ )

- Direction and magnitude matter
  
  - Correlation is always value between `$[-1,1]$`

---
# Review from Last Class

Recall: 
`$$r=\frac{Cov(X,Y)}{\sqrt{Var(X)}\cdot \sqrt{Var(Y)}}$$`

- Correlation is a function of covariance, just normalizes the magnitudes so we can interpret.

---
# Clicker Question

Suppose you calculate the sample covariance, `$s_{XY}=1.2$`, and the sample standard deviations `$s_X=2$` and `$s_Y=2.5$`. What is the sample correlation, `$r_{XY}$`?

---
# Relationship between X and Y

We often summarize the relationship between `$X$` and `$Y$` using a straight line:

This is called the line of best fit, or the .hi.coral[regression line].

---
# Regression Line Example

In the previous example, the regression line is given by

`$$\widehat{\text{Average SAT Reading}} = 78.87 + 0.7983 * \text{Average SAT Math}$$`

The `$\widehat{ }$` symbol means that we are *predicting* average SAT reading score.

<br/>

.ex[Example:] If a school has an average SAT math score of 600, we would predict their SAT reading score would be

`$$\widehat{\text{Average SAT Reading}} = 78.87 + 0.7983 * 600 = 557.85.$$`

---
# Regression Line

Let `$X$` and `$Y$` be two random variables. A .hi.coral[regression line] is a straight line that describes how the response variable, `$Y$`, changes as the explanatory variable `$X$` changes.

We often use a regression line to predict the value of `$Y$` for a given value of `$X$`, *when we believe the relationship between `$X$` and `$Y$` is linear*.

Since we are predicting `$Y$` with a line, our regression line will be given by:

`$$\underbrace{\coral{\hat{Y}_i}}_{\text{Prediction for } Y_i} = \underbrace{\hat{a}}_{\text{Intercept}} + \underbrace{\hat{b}}_{\text{Slope}} * X_i$$`

`$\coral{\hat{Y}_i}$` is called the .hi.coral[predicted value]

---
# Residual Errors

However, `$\coral{\hat{Y}}$` will not be equal to `$Y$`. Therefore our .hi.kelly[prediction error] for unit `$i$` is given by

$$ \kelly{\varepsilon_i} = Y_i - \coral{\hat{Y}_i} $$

---
# Being the "best"

**Question:** How do we estimate the intercept `$\hat{a}$` and the slope `$\hat{b}$` line of "best fit"?

**Answer:**

1. What about minimizing of `$\sum_i \kelly{\varepsilon_i}$`, *i.e. lower total prediction error*?

- `$\kelly{\varepsilon_i}$` could be positive or negative if we over/under-estimate `$\coral{\hat{Y}_i}$`. If we add up all the residuals, the positive and negatives will sometimes cancel out (same reasoning as variance). This isn't good because we want to be 'penalized' for all prediction errors. 
  
--

2. Therefore we minimize `$\kelly{\varepsilon_i}^2$` because each `$\kelly{\varepsilon_i}^2$` will be positive.

- In general,*the line of best-fit* means the line that minimizes the .kelly[sum of squared errors] (SSE):

---
# Sum of Squared Errors

$$
	SSE = \sum_{i=1}^n (y_i - \hat{a} - \hat{b} X_i)^2
$$

---
# OLS *vs.* other lines/estimators

Let's consider a random dataset