The multiple linear regression model (5.6) can be viewed as a first-order approximation of the general model

\[ \begin{equation*} Y=g(\boldsymbol{x})+\varepsilon \tag{5.42} \end{equation*} \]

where \(\mathbb{E} \varepsilon=0, \operatorname{Var} \varepsilon=\sigma^{2}\), and \(g(\boldsymbol{x})\) is some known or unknown function of a \(d\)-dimensional vector \(\mathrm{x}\) of explanatory variables. To see this, replace \(g(\boldsymbol{x})\) with its first-order Taylor approximation around some point \(x_{0}\) and write this as \(\beta_{0}+\boldsymbol{x}^{\top} \boldsymbol{\beta}\). Express \(\beta_{0}\) and \(\boldsymbol{\beta}\) in terms of \(g\) and \(x_{0}\).