12.11 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