16.11 Parameter Interpretation

Given a change in $x$, how much do we expect $y$ to change by?

Case 1: Level-Level Model

$$Y_i = \beta_0 + \beta_1X_i+\epsilon_i$$

• Dependent Variable: $Y$
• Independent Variable: $X$
• Interpretation of $\beta_1$: If we change $X$ by one unit, we’d expect $Y$ to change by $\beta_1$

Case 2: Log-Level Model

$$\log (Y_i) = \beta_0 + \beta_1X_i+\epsilon_i$$

• Dependent Variable: $\log Y$
• Independent Variable: $X$
• Interpretation of $\beta_1$: If we change $X$ by one unit, we’d expect $Y$ to change by $100 \beta_1$ percent.

Case 3: Level-Log Model

$$Y_i = \beta_0 + \beta_1 \log(X_i)+\epsilon_i$$

• Dependent Variable: $Y$
• Independent Variable: $\log X$
• Interpretation of $\beta_1$: If we increase $X$ by one percent, we’d expect $Y$ to increase by $\beta_1 /100$ units.

Case 4: Level-Log Model

$$\log (Y_i) = \beta_0 + \beta_1 \log(X_i)+\epsilon_i$$

• Dependent Variable: $\log Y$
• Independent Variable: $\log X$
• Interpretation of $\beta_1$: If we change $X$ by one percent, we’d expect $Y$ to change by $\beta_1$ percent.

Adapted from here