普通最小二乘法

普通最小二乘法就係建基於誤差值之上嘅。用呢種做法估計 ${\displaystyle \beta }$ ，首先要計個殘差平方和（residual sum of squares；RSS）出嚟，RSS 係指將所有誤差值嘅平方加埋得出嘅數：

${\displaystyle RSS=\sum _{i=1}^{n}e_{i}^{2}\,}$ 

${\displaystyle \beta _{0}}$  同 ${\displaystyle \beta _{1}}$  嘅數值可以用以下嘅算式計出嚟：

${\displaystyle {\widehat {\beta }}_{1}={\frac {\sum (x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum (x_{i}-{\bar {x}})^{2}}}}$ 

${\displaystyle {\widehat {\beta }}_{0}={\bar {y}}-{\widehat {\beta }}_{1}{\bar {x}}}$ 

當中 ${\displaystyle {\bar {x}}}$  係 ${\displaystyle x}$  嘅平均值，而 ${\displaystyle {\bar {y}}}$  係 ${\displaystyle y}$  嘅平均值。

假設個總體嘅誤差值有一個固定嘅變異數，呢個變異數可以用以下呢條式估計：

${\displaystyle {\hat {\sigma }}_{\varepsilon }^{2}={\frac {RSS}{n-2}}.\,}$ 

呢個數係所謂嘅均方誤差（mean square error），個分母係樣本大細減個模型要估計嘅參數嘅量。呢個迴歸模型當中有兩個未知嘅參數（${\displaystyle \beta _{0}}$  同 ${\displaystyle \beta _{1}}$ ）^[1]。

而啲參數估計嘅標準誤差（standard error）係：

${\displaystyle {\hat {\sigma }}_{\beta _{1}}={\hat {\sigma }}_{\varepsilon }{\sqrt {\frac {1}{\sum (x_{i}-{\bar {x}})^{2}}}}}$ 

${\displaystyle {\hat {\sigma }}_{\beta _{0}}={\hat {\sigma }}_{\varepsilon }{\sqrt {{\frac {1}{n}}+{\frac {{\bar {x}}^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}}}={\hat {\sigma }}_{\beta _{1}}{\sqrt {\frac {\sum x_{i}^{2}}{n}}}}$ 

有咗上面呢個模型，個研究者手上有會有 ${\displaystyle \beta _{0}}$  同 ${\displaystyle \beta _{1}}$  嘅估計值，將來可以用呢條式嚟預測 ${\displaystyle Y}$  嘅數值。