🎳Intro to Econometrics Unit 4 – Gauss-Markov Assumptions & OLS Properties

Key Concepts and Definitions

• Gauss-Markov assumptions a set of conditions that ensure OLS estimators are BLUE (Best Linear Unbiased Estimators)
• Linear regression a statistical method for modeling the linear relationship between a dependent variable and one or more independent variables
• Ordinary Least Squares (OLS) a method for estimating the parameters of a linear regression model by minimizing the sum of squared residuals
• Estimator a rule or formula used to estimate the value of an unknown parameter based on sample data
• Unbiasedness an estimator is unbiased if its expected value is equal to the true value of the parameter being estimated
• Efficiency an estimator is efficient if it has the smallest variance among all unbiased estimators
• Homoscedasticity the assumption that the variance of the error term is constant across all observations
• Multicollinearity the presence of high correlation among independent variables in a regression model

Linear Regression Basics

• Linear regression models the relationship between a dependent variable and one or more independent variables
• The goal is to find the line of best fit that minimizes the sum of squared residuals (differences between observed and predicted values)
• Simple linear regression involves one independent variable, while multiple linear regression involves two or more independent variables
• The regression equation is represented as $Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \beta_kX_k + \varepsilon$, where:
  • $Y$ is the dependent variable
  • $\beta_0$ is the intercept
  • $\beta_1, \beta_2, ..., \beta_k$ are the slope coefficients for each independent variable
  • $X_1, X_2, ..., X_k$ are the independent variables
  • $\varepsilon$ is the error term
• The slope coefficients represent the change in the dependent variable for a one-unit change in the corresponding independent variable, holding other variables constant
• The intercept represents the value of the dependent variable when all independent variables are zero

The Gauss-Markov Assumptions Explained

• The Gauss-Markov assumptions ensure that OLS estimators are BLUE (Best Linear Unbiased Estimators)
• Assumption 1: Linearity the relationship between the dependent variable and independent variables is linear
• Assumption 2: Random sampling observations are randomly sampled from the population
• Assumption 3: No perfect collinearity no independent variable is a perfect linear combination of other independent variables
• Assumption 4: Zero conditional mean the expected value of the error term is zero given any values of the independent variables ($E[\varepsilon|X] = 0$)
• Assumption 5: Homoscedasticity the variance of the error term is constant across all levels of the independent variables ($Var[\varepsilon|X] = \sigma^2$)
  • Violation of this assumption is called heteroscedasticity
• When these assumptions hold, OLS estimators are BLUE, meaning they are unbiased and have the smallest variance among all linear unbiased estimators

Ordinary Least Squares (OLS) Method

• OLS is a method for estimating the parameters of a linear regression model by minimizing the sum of squared residuals
• The OLS estimators for the slope coefficients and intercept are obtained by solving the minimization problem: $\min_{\beta_0, \beta_1, ..., \beta_k} \sum_{i=1}^n (Y_i - \beta_0 - \beta_1X_{1i} - ... - \beta_kX_{ki})^2$
• The OLS estimators for the slope coefficients are given by: $\hat{\beta}_j = \frac{\sum_{i=1}^n (X_{ji} - \bar{X}_j)(Y_i - \bar{Y})}{\sum_{i=1}^n (X_{ji} - \bar{X}_j)^2}$, where $\bar{X}_j$ and $\bar{Y}$ are the sample means of $X_j$ and $Y$, respectively
• The OLS estimator for the intercept is given by: $\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1\bar{X}_1 - ... - \hat{\beta}_k\bar{X}_k$
• OLS estimators are consistent and asymptotically normal under the Gauss-Markov assumptions

Properties of OLS Estimators

• Under the Gauss-Markov assumptions, OLS estimators have several desirable properties:
  • Unbiasedness the expected value of the OLS estimator is equal to the true value of the parameter ($E[\hat{\beta}_j] = \beta_j$)
  • Efficiency OLS estimators have the smallest variance among all linear unbiased estimators (Gauss-Markov Theorem)
  • Consistency as the sample size increases, the OLS estimators converge in probability to the true parameter values
  • Asymptotic normality as the sample size increases, the OLS estimators follow a normal distribution
• The variance of the OLS estimator for $\beta_j$ is given by: $Var[\hat{\beta}_j] = \frac{\sigma^2}{\sum_{i=1}^n (X_{ji} - \bar{X}_j)^2}$, where $\sigma^2$ is the variance of the error term
• The standard errors of the OLS estimators are the square roots of their variances and are used for hypothesis testing and constructing confidence intervals

Violations and Consequences

• Violation of the Gauss-Markov assumptions can lead to biased, inefficient, or inconsistent OLS estimators
• Omitted variable bias occurs when a relevant variable is excluded from the model, leading to biased and inconsistent estimators
• Measurement error in the independent variables can cause attenuation bias, where the OLS estimators are biased towards zero
• Heteroscedasticity (non-constant variance of the error term) leads to inefficient estimators and invalid standard errors
  • Heteroscedasticity can be detected using tests like the Breusch-Pagan test or White test
  • Robust standard errors (e.g., White's heteroscedasticity-consistent standard errors) can be used to obtain valid inference in the presence of heteroscedasticity
• Autocorrelation (correlation between error terms across observations) leads to inefficient estimators and invalid standard errors
  • Autocorrelation can be detected using tests like the Durbin-Watson test or Breusch-Godfrey test
  • Generalized Least Squares (GLS) or robust standard errors can be used to address autocorrelation

Real-World Applications

• Linear regression is widely used in various fields, such as economics, finance, social sciences, and natural sciences
• Examples of applications include:
  • Estimating the impact of education on earnings (Mincer equation)
  • Analyzing the relationship between advertising expenditure and sales
  • Examining the effect of price and income on consumer demand
  • Predicting housing prices based on property characteristics (hedonic pricing models)
  • Assessing the impact of government policies on economic outcomes
• In practice, researchers must carefully consider the assumptions underlying the linear regression model and address any violations to obtain reliable and meaningful results

Common Pitfalls and How to Avoid Them

• Misspecification of the functional form assuming a linear relationship when the true relationship is nonlinear can lead to biased estimates
  • Use scatterplots and residual plots to assess the linearity assumption
  • Consider transformations (e.g., logarithmic, polynomial) or nonlinear regression methods if necessary
• Omitting relevant variables can cause omitted variable bias and lead to incorrect conclusions
  • Carefully consider the theoretical foundations and previous literature to identify important variables
  • Use techniques like stepwise regression or Lasso to select relevant variables
• Ignoring multicollinearity can lead to unstable and difficult-to-interpret estimates
  • Check for high correlations among independent variables using a correlation matrix
  • Use variance inflation factors (VIF) to detect multicollinearity
  • Consider dropping one of the highly correlated variables or using techniques like principal component analysis (PCA) to combine them
• Failing to check for heteroscedasticity or autocorrelation can result in invalid inference
  • Always perform diagnostic tests for heteroscedasticity and autocorrelation
  • Use robust standard errors or appropriate estimation methods (e.g., GLS) to address these issues
• Overfitting the model by including too many independent variables can lead to poor out-of-sample performance
  • Use model selection criteria like adjusted R-squared, AIC, or BIC to balance goodness-of-fit and model complexity
  • Perform cross-validation to assess the model's performance on unseen data