Understanding Multiple Regression Basics

Aug 10, 2024

CFA Level 2: Quantitative Methods - Multiple Regression Basics

Introduction

  • Focus on basics of multiple regression and underlying assumptions.
  • Important for understanding upcoming modules.
  • New module presentation allows for concise material and better linkage.

Learning Outcome Statements (LOS)

  1. Basic Linear Regression

    • Transition from simple to multiple linear regression.
    • Review of previously learned concepts is necessary.

  2. Interpretation of Residual Plots

    • Pay attention as it is heavily featured in the module vignette questions.

Multiple Regression Model Overview

  • Model Structure:
    • Dependent Variable (Y)
    • Intercept (b)
    • Multiple Independent Variables (X1, X2, ..., Xn)
    • Error Term (Disturbance Term)
  • Partial Slope Coefficients:
    • Represents relationship while holding other variables constant.
    • Important for understanding the impact of each independent variable.

Purpose of Multiple Regression

  • Testing Theories:
    • Examine relationships between various economic variables.
  • Identifying Relationships:
    • Helps uncover hidden or latent relationships.
  • Forecasting:
    • Useful for predicting future values (e.g., inflation).

Example Analysis

  • Impact of Inflation and Real Rates on USD Price:
    • Dependent Variable: Price of the USD Index.
    • Independent Variables: Inflation, Real Interest Rate.
    • Data collection challenges: needing consistent time-trends for analysis.

Statistical Significance

  • Coefficients Interpretation:
    • Negative relationship between inflation and USD price.
    • Positive relationship between real interest rates and USD price.
  • P-values and Test Statistics:
    • Use to determine significance (e.g., p-value < 0.05 indicates significance).
    • Example: Inflation has p-value of 0.27 (not significant), Real Interest Rate has p-value of 0.01 (significant).

Key Assumptions of Multiple Regression

  1. Linearity:
    • Relationship between independent and dependent variables is linear.
  2. Independence of Errors:
    • Error terms should not be correlated; tested using Durbin-Watson statistic.
  3. Homoscedasticity:
    • Variance of error terms should be constant across observations.
    • Violations lead to heteroskedasticity.
  4. Normality of Errors:
    • Residuals should be normally distributed for valid inference.

Heteroskedasticity vs Homoscedasticity

  • Heteroskedasticity:
    • Non-constant variance of error terms leading to biased estimates.
  • Homoscedasticity:
    • Constant variance across all observations.

Residual Plot Interpretation

  • Important for checking assumptions of linearity, independence, normality, and homoscedasticity.
  • Look for patterns in residuals to determine model adequacy.

Conclusion

  • Understand the importance of multiple regression and its assumptions.
  • Focus on interpreting residual plots as they relate to LOS.
  • Preparation for upcoming modules requires grasp of these concepts.