MCO. Mínimos Cuadrados Ordinarios

Modelo de Regresión Lineal Simple

$$Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i$$

$$\displaystyle \min\sum_{i=1}^n\hat{\varepsilon}_i^2 = \min\sum_{i=1}^n(Y_i-\hat\beta_0-\hat\beta_1 X_i)^2$$

Condiciones de primer orden:

$$\left\{ \begin{align} \displaystyle \dfrac{\partial\sum\hat\varepsilon_i^2}{\partial\hat\beta_0}= & -2\sum(Y_i-\hat\beta_0-\hat\beta_1 X_i)=0 \\ \displaystyle \dfrac{\partial\sum\hat\varepsilon_i^2}{\partial\hat\beta_1} = & -2\sum X_i(Y_i-\hat\beta_0-\hat\beta_1 X_i)=0 \end{align} \right.$$

Despejamos $\hat\beta_0$ de la primera condición: $$\sum Y_i-n\hat\beta_0-\hat\beta_1\sum X_i = 0$$

$$\begin{align} \hat\beta_0 = & \dfrac{1}{n}\left(\sum Y_i-\hat\beta_1\sum X_i\right) \\ = & \bar Y- \hat\beta_1\bar X \end{align}$$

De la segunda condición, dividimos por $n$ : $$\begin{align} \dfrac{1}{n}\sum X_i Y_i = & \dfrac{1}{n}\hat\beta_0\sum X_i + \dfrac{1}{n}\hat\beta_1\sum X_i^2\\ =& (\bar Y- \hat\beta_1\bar X)\bar X + \dfrac{1}{n}\hat\beta_1\sum X_i^2 \\ =& \bar X \bar Y - \beta_1 \bar X^2 + \hat\beta_1\dfrac{1}{n}\sum X_i^2\\ =& \bar X \bar Y + \hat\beta_1\left(\dfrac{1}{n}\sum X_i^2-\bar X^2\right) \end{align}$$

y despejamos $\hat\beta_1$

$$\begin{align} \hat\beta_1 = \dfrac{\dfrac{1}{n}\sum X_i Y_i-\bar X \bar Y}{\dfrac{1}{n}\sum X_i^2-\bar X^2} = \dfrac{cov(X,Y)}{var(X)} \end{align}$$