# Chapter 9 Simple Regression

🚧 Under Construction 🚧

We assume you are familiar with regression or Ordinary Least Square (OLS), but let’s make a quick recap on what regression is all about.

## 9.1 Univariate regression

is a regression between one dependent variable and one independent variable. Most textbooks use a notation $$Y$$ for dependent variable, and use $$X$$ as the independent variable. A univariate regression assume this form: $Y_i=\alpha+\beta X_i+ e_i$ where $$Y_i$$ is your observed dependent variable and $$X_i$$ is your observed independent variable.

There are many versions of the Gauss-Markov assumption, but really there are two main thing we assume for OLS to be unbiased:

• error term $$e_i$$ have a conditional zero mean, i.e., $$E(e_i|X_i)=0$$
• error term $$e_i$$ is independent and identically distributed

We call $$\alpha$$ as intercept. We generally don’t interpret $$\alpha$$ although it is very important to include $$\alpha$$ to avoid bias. We are interested in $$\beta$$, as it shows the general strength of relationship between our $$X$$ and our $$Y$$ is. That is, we often say:

an increase of $$X$$ by 1 unit, is associated with an increase of $$Y$$ by $$\beta$$ unit, assuming everything else constant.

Mathly, $$\beta=\frac{dy}{dx}$$

### 9.1.1 running your univariate regression

suppose you have

In the previous chapter, we have learned how to plot our data using plot(). You can add to your script a line showing your regression result

## 9.2 Multivariate regression

Multivariate regression is just like your univariate regression, but we have more independent variable than just one. Dependent variable still only one. That is:

$Y_i=\alpha+\beta X_i+\gamma Z_i+ e_i$ The interpretation is similar.