Lecture 6: Heteroskedasticity and Time Series Modelling

Introduction

• Lecture title: ECMT2150 Lecture 6 Intermediate Econometrics
• Semester: 1, 2025
• Coordinator: Felipe Pelaio
  • Office: Room 512, Social Sciences Building
  • Email: felipe.queirozpelaio@sydney.edu.au
  • Office Hours: Tuesdays 5-6pm

Topics Today

• Specification Errors
  • Heteroskedasticity
• Regression Analysis with Time Series – Part I

Heteroskedasticity

Overview

• Definition and implications for OLS estimators
• Detection methods
• Handling heteroskedasticity

What is Heteroskedasticity?

• Assumption MLR.5: Constant variance of error term
  • Homoskedasticity: Var(u|x1, …, xK) = σ²
  • If this assumption fails: Heteroskedasticity occurs, where Var(u|x1, …, xK) = σi²
• Example: Variance in wages dependent on education

Implications for OLS Estimators

• OLS coefficients remain unbiased and consistent even with heteroskedastic errors
• However, standard errors become biased under heteroskedasticity leading to:
  • Invalid inference tests (t and F statistics)
  • OLS may not be the best estimator
• Adjusting Statistics
  • We can adjust t and F statistics to be robust to heteroskedasticity

Standard Errors with Heteroskedasticity

• OLS estimator: β₁̂ = β₁ + [ σ(xi - x̄)ûi / σ(xi - x̄)² ]
• Variance of β₁̂ needs modification:
  • Var(β₁̂) = [ σi² / SSTx² ]
• For consistency, White (1980) provides parameters that apply regardless of homoskedasticity.

Consistent Standard Errors in MLR

• Also known as White/Huber/Eicker standard errors:
  • Var(β̂_j) = [ σi² r̂ij² ûi² / SSRj² ]
• Corrects for degrees of freedom: multiply by n/(n-k-1)

Summary of Heteroskedasticity

• In large samples, report and use heteroskedasticity-robust standard errors
• With small samples and homoskedasticity assumption satisfied:
  • Standard t statistics retain their properties
  • t statistics may differ significantly from robust ones if there is heteroskedasticity present

Detection of Heteroskedasticity

Strategies

1. Economic Theory: Consider whether heteroskedasticity is expected based on the topic.
   • Example: firm-size or income relations
2. Visual Analysis: Plot residuals to check for patterns.
3. Formal Statistical Tests:
   • Breusch-Pagan test
   • White test

Example of Detection: Modeling wages against education

• Wages expected to rise with education
• Residual plots can indicate heteroskedasticity

Formal Statistical Tests for Heteroskedasticity

Breusch-Pagan Test

• Assesses if variance of errors relates linearly to explanatory variables:
  • H₀: Var(u|x) = σ²
  • Regress squared residuals on explanatory variables
  • High R² indicates rejection of homoskedasticity

White Test

• Tests for the presence of heteroskedasticity of unknown form:
  • More general than Breusch-Pagan including squares and cross-products
  • Can lead to many regressors, complicating analysis

Remedies for Heteroskedasticity

1. Robust Standard Errors: Calculate after OLS.
2. Alternative Estimators: Weighted Least Squares (WLS)
3. Log Transforms: Sometimes reduces heteroskedasticity but not guaranteed

Regression Analysis with Time Series

Nature of Time Series Data

• Temporal ordering: past affects future
• Defined as stochastic process

Static Models

• Example: Relationship between y and z at the same time:
  • yt = β₀ + β₁zt + u_t
  • Change in z impacts y immediately

Finite Distributed Lag (FDL) Models

• Allow for explanatory variables to affect dependent variable with a lag:
  • Example: gfrt = α₀ + δ₀pet + δ₁pet-1 + δ₂pet-2 + ut
• Impact Propensity: Immediate effect; Long-Run Propensity: Total effect over time

Properties of OLS for Time Series

• Unbiasedness under specific assumptions (TS-1 to TS-6) including:
  • No perfect collinearity
  • Zero conditional mean
• Assumptions Impact Downstream Results: E(β̂j) = βj

Conclusion

• Time series analysis requires attention to structural dynamics and temporal relationships.
• Understanding variations and statistical properties ensures precision in econometric modeling.