Simple Linear Regression

Econometrics Statistics 回归分析（regression analysis）是通过其他变量来预测某一变量的值

Basic Model

$$y={\beta }_0+{\beta }_{1}x+ϵ$$

其中 $ϵ$ 是一个误差变量，也是新引用出来的 $\hat{y}=b_0+b_{1}x$

在这里 $\hat{y}$ 是一个预测值或者说拟合值。利用到第四章中所讲过的最小二乘法（least squares methods）就是利用一群点（我们样本里的数据）来尝试拟合出一条直线，越精确越好。

也就是说，系数 $b_0$ 和 $b_1$ 必须使离差平方和

${\sum }_{i=1}^n(y_1-\widehat{y_i})$

接下来是一些最小二乘直线系数： $b_1=\frac{s_{xy}}{s_x^2}$ This is coefficient

$b_0=\hat{y}-b_1\hat{x}$

Inside,

$s_{xy}=\frac{{\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{n-1}$ This is Covariance

$s_{x^2}=\frac{{\sum }_{i=1}^n(x_i-\overline{x}{)}^2}{n-1}$ This is Variance

We seldom calculate those stuff by hand. It’s too hard.

---

In the course ECON3003 Econometric II taught by professor Chen Jia, we first dive into a new notation, which is Simple Linear Regression in Matrix Notation, combined with some background knowledge from Linear Algebra MOC.