Multiple Regression Analysis: Further Issues

Multiple Regression Analysis: Further Issues

Functional Form

• Logarithmic Functional Forms

  • Provides convenient percentage/elasticity interpretation.

  • Slope coefficients of logged variables are invariant to rescalings.

  • Taking logs can eliminate or mitigate problems with outliers.

  • Helps to secure normality and homoskedasticity.

  • Caution on when to log:

    • Variables measured in years or percentage points should not be logged.

    • Logs not applicable if variables are zero or negative.

    • Reversing the log operation poses challenges when constructing predictions.

Marginal Effects

• Understanding the marginal effect of experience using quadratic functional forms.

• Example: Wage equation analysis for determining returns to experience.

• It is critical to analyze the sample around turning point outcomes (e.g., return to experience).

Interaction Effects

• Challenges in interpretation due to the presence of interaction effects in models.

• Models with interaction terms require careful interpretation of parameter estimates.

Average Partial Effects (APE)

• In models featuring quadratics and interactions, partial effects depend on one or more explanatory variables.

• APE serves as a summary measure to illustrate the relationship between the dependent variable and each explanatory variable.

• Method: Compute partial effects, estimate parameters, and average them across the sample.

Goodness-of-Fit and R-squared

• General notes on R-squared:

  • A high value does not imply causation; a low value does not rule out precise estimation of effects.

  • The Adjusted R-squared penalizes additional regressors.

    • It increases only if the t-statistic of the newly added regressor exceeds one in absolute value.

• Caution against comparing R-squared across models with different numbers of parameters.

• Adjusted R-squared can aid in selecting between nonnested models.

Model Comparison and Limitations

• Avoid comparing models differing in their definitions of the dependent variable.

• For example, comparing CEO compensation against firm performance metrics.

• Controlling too many variables might distort regression results (e.g., beer consumption’s impact on vehicle fatalities or health expenditures).

• The effectiveness of regressors: Ensure that added variables genuinely contribute to reducing error variance without exacerbating multicollinearity.

Conclusion on Regressors

• Adding new regressors can reduce error variance but may also intensify multicollinearity.

• Look for uncorrelated regressors to minimize issues while enhancing estimation precision, as in beer price elasticity estimates.

• Predicting y when log(y) is the dependent variable relies on the independence assumption of error terms from predictor variables: $u \perp x1, \ldots, xk$.