Nonlinear Models

Polynomial Models §

$$Y_i={\beta }_0+{\beta }_1{X}_{1i}+{\beta }_2{X}_{1i}^2+\cdots +{ϵ}_i$$

• This is still bivariate! it’s just $Y$ vs $X_1$

Logarithmic Models §

1. Linear-Log: $Y_i={\beta }_0+{\beta }_1\ln X_i+{ϵ}_i$
2. Log-Linear: $\ln Y_i={\beta }_0+{\beta }_1{X}_i+{ϵ}_i$
3. Log-Log: $\ln Y_i={\beta }_0+{\beta }_1\ln X_i+{ϵ}_i$ Be careful since:

• Units of coefficients are different
• Variables must all be positive
• There isn’t a statistical test to choose which model to use

Example: Wage vs Height §

Regression Table:

1. No Log: “Wage increases by \0.412$per$\text{inch}$ increase in height”
2. Linear-Log: “Wage increases by \frac{29.316}{100}=\0.293$per$1%$ increase in height”
3. Log-Linear: “Wage increases by $0.033\times 100=3.3\%$ per $\text{inch}$ increase in height”
4. Log-Log: “Wage increases by $2.362\%$ per $1\%$ increase in height”