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7.3.3 Applications with Examples - III

Introduction

  • Objective: This session expands on multiple linear regression (MLR) applications by incorporating three independent variables—first income, monthly payment, and utilities—to analyze household debt.
  • Context: Manjula Naik continues her analysis using a more complex MLR model to understand how various financial factors contribute to household debt.

Enhancing the Model

  • Previous Findings: Previous models with two variables provided adjusted R-squared values of 0.67 and 0.45, indicating significant but varying explanations of debt variability.
  • New Approach: Incorporating first income, monthly payment, and utilities into one comprehensive model to capture a broader spectrum of influencing factors.

Methodology

  • Data Preparation:
  • Variables: Debt (Y), first income (X1), monthly payment (X2), and utilities (X3).
  • Location in Spreadsheet: Debt in column K, first income in column H, monthly payments in column I, and utilities in column J.

Regression Analysis

  • Tool Used: Analysis ToolPak in Excel for regression computation.
  • Model Specification: A full MLR model with debt as a function of first income, monthly payment, and utilities.
  • Statistical Outputs:
  • R-squared: Improved to 0.708, indicating that 70.8% of the variability in household debt is now explained by the model.
  • Adjusted R-squared: Slightly lower than R-squared, around 0.70, indicating minimal overfitting despite the increased number of predictors.
  • Standard Error: Reduced to 18699, suggesting increased precision in the model's estimates.

Model Assessment

  • ANOVA Results:
  • SSR: Increased significantly, indicating better explanation of variance by the model.
  • SSE: Decreased, which aligns with the lower standard error and higher explanatory power.
  • F-Statistic: 401.6, with a very low p-value, strongly supporting the model's overall significance.

Coefficient Analysis

  • Interpretation of Coefficients:
  • B1 (First Income): Each unit increase in first income increases debt by 0.017 units.
  • B2 (Monthly Payment): Each unit increase in monthly payment increases debt by 0.96 units.
  • B3 (Utilities): Each unit increase in utilities increases debt by 157.95 units.
  • Significance: All coefficients are significant, indicating a meaningful contribution to the model.

Residuals and Diagnostics

  • Residuals: Standardized residuals mostly within the range of \(\pm3\), suggesting well-behaved errors.
  • Residual Plots: Random patterns across all independent variables and predicted values, affirming the model's adequacy.

Conclusion

  • Model Validation: The enhanced MLR model significantly improves understanding and prediction of household debt dynamics.
  • Future Directions: Encouraged to explore additional datasets and further refine modeling techniques to handle complex business scenarios.