Module 6

Multiple Regression: Further Issues

Extend multiple regression beyond the linear-in-levels baseline by learning how scaling, logs, quadratics, interactions, adjusted R-squared, prediction, and careful controls change interpretation.

20 lessons8 to 12 hoursIntermediate

Skills

Convert coefficient interpretations when variables are measured in different units.Interpret standardized coefficients without treating them as causal importance rankings.Choose between level, log-level, level-log, and log-log specifications.Use exact percentage changes for log-dependent-variable models.Explain when logs, quadratics, and interactions help and when they can mislead.Compare adjusted R-squared across nonnested models while respecting economic logic.Distinguish bad controls from controls that improve precision.Build predictions and prediction intervals without fabricating certainty.Correctly convert predictions from log outcomes back to the original outcome scale.Use bootstrap standard errors as an applied robustness check.
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Module resources

20 quiz checks6 datasets11 notebooks
M6_GPA_INTERACTIONS6 variablesM6_HOUSING_LOGS7 variablesM6_POLICY_CONTROLS7 variablesM6_SALES_ADVERTISING7 variablesM6_STARTUP_PREDICTION6 variablesM6_WAGE_SCALING8 variables

Lessons

1Why Units Matter in Multiple RegressionWhy does the same regression relationship look different when units change?Open lesson2Coefficients under RescalingWhat happens to a slope when x or y is divided by 100 or 1,000?Open lesson3Standardized CoefficientsWhen can standardized coefficients help compare variables measured in different units?Open lesson4Level-Log and Log-Level ModelsHow does interpretation change when only one side of the model is logged?Open lesson5Log-Log ElasticitiesWhy is the slope in a log-log model called an elasticity?Open lesson6Exact Percent Changes in Log ModelsWhen does the usual 100 times beta shortcut become too rough?Open lesson7When Logs Are Not AppropriateWhat should you check before taking logs?Open lesson8Quadratic Terms and Turning PointsHow can a regression slope change as x changes?Open lesson9Marginal Effects in Quadratic ModelsWhy is there no single slope in a quadratic model?Open lesson10Interactions between Continuous VariablesHow can the effect of one variable depend on another variable?Open lesson11Interactions with Dummy VariablesHow do slopes or intercepts differ across groups?Open lesson12Centering Variables before InteractionsWhy can centering make interaction models easier to read?Open lesson13Adjusted R-Squared and Model SizeWhy can R-squared rise when a variable adds little value?Open lesson14Comparing Nonnested ModelsHow should students compare models when one is not just a larger version of the other?Open lesson15Over-Control and Bad ControlsWhen can adding more controls make interpretation worse?Open lesson16Controls for PrecisionWhen can a control improve precision without changing the target coefficient?Open lesson17Prediction with Multiple RegressionHow do we build and explain a fitted prediction?Open lesson18Prediction Intervals and UncertaintyWhy is a prediction not just one number?Open lesson19Predictions when the Dependent Variable Is LoggedWhy is exp(predicted log y) usually not the best prediction of y?Open lesson20Module 6 Applied Forecasting ProjectHow can we build a careful model that uses transformations, interactions, and prediction without overclaiming?Open lesson

Notebook labs

M6_WAGE_SCALING

Scaling and Coefficient Interpretation

Convert slopes across wage units and confirm that fitted relationships do not change when units are handled correctly.

Open interactive notebookDownload .ipynb