Linear in parameters and/or linear in data

Summary

• The linear regression model assumes that

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+ε_i, i=1,…,n\]

• This can be generalized to an arbitrary regression model

\[y_i=g(β_1,…,β_k,x_{i,2},…,x_{i,l})+ε_i , i=1,…,n\]

• where \(g\) is an arbitrary function of \(l-1\) explanatory variables and \(k\) unknown parameters .
• If the \(x\) -variables are exogenous then

\[E\left( x_i \right)=g(β_1,…,β_k,x_{i,2},…,x_{i,l}) , i=1,…,n\]

• We say that a regression model is linear in parameters if \(g(β_1,…,β_k,x_{2i},…,x_{li})\) is a linear function of \(β_1,…,β_k\) . Examples:
• \(β_1+β_2x_{i,2}+β_3x_{i,3}\) is linear in parameters.
• \(β_1x_{i,2}^{β_2}\) is not linear in parameters.
• We say that a regression model is linear in data if \(g(β_1,…,β_k,x_{2i},…,x_{li})\) is a linear function of \(x_{2i},…,x_{li}\) . Examples:
• \(β_1+β_2x_{i,2}+β_3x_{i,3}\) is linear in data.
• \(β_1+β_2x_{i,2}+β_3x_{i,2}^2\) is not linear in data.
• We say that a regression model is linear if it is linear in parameters.