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	?	?
..	..	..	..	..	..	..	..	..