## 7.8 Prediction: Linear model (Equation) (2)

$$y_{i} = \underbrace{\color{blue}{\beta_{0}} + \color{orange}{\beta _{1}} \times x_{1i} + \color{orange}{\beta _{2}} \times x_{2i}}_{\text{Modell} = \color{green}{\widehat{y}}_{i} = \text{Predicted values}} + \underbrace{\color{red}{\varepsilon}_{i}}_{\color{red}{Error}} = \color{green}{\widehat{y}}_{i} + \color{red}{\varepsilon}_{i}$$

• Q: Why is the linear model called “linear” model?

• Important: Variable values (e.g., $$y_{i}$$ or $$x_{1,i}$$) vary, parameter values (e.g., $$\boldsymbol{\color{blue}{\beta_{0}}}$$) constant across rows

• Important: $$\color{green}{\widehat{y}_{i}}$$ varies across units

Name $$trust2006$$
$$y_{i}$$
$$\boldsymbol{\color{blue}{\beta_{0}}}$$ $$\boldsymbol{\color{orange}{\beta_{1}}}$$ $$victim2006$$
$$x_{1,i}$$
$$\boldsymbol{\color{orange}{\beta_{2}}}$$ $$education2006$$
$$x_{2,i}$$
$$\boldsymbol{\color{red}{\varepsilon_{i}}}$$ $$\color{green}{\widehat{y}_{i}}$$
Daniel 5 ? ? 1 ? 1 ? ?
Leia 5 ? ? 0 ? 1 ? ?
Harrison 7 ? ? 0 ? 5 ? ?
.. .. .. .. .. .. .. .. ..