6.2 Regression analysis for impact estimation

Regression analysis is a powerful tool for estimating impact in evaluation studies. It models relationships between variables, allowing researchers to isolate the effects of interventions while controlling for other factors. This technique helps uncover causal links and quantify program impacts.

Key considerations include selecting appropriate variables, addressing assumptions, and interpreting results correctly. Advanced methods like fixed effects and difference-in-differences can strengthen causal inference. Proper analysis and interpretation of regression coefficients provide valuable insights into program effectiveness.

Principles of Regression Analysis

Fundamentals of Regression

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• Regression analysis models relationship between dependent variable and one or more independent variables
• Ordinary least squares (OLS) method estimates regression coefficients by minimizing sum of squared residuals
• Linear regression assumes linear relationship between variables represented by straight line (simple regression) or hyperplane (multiple regression)
• Homoscedasticity requires constant variance of residuals across all levels of independent variables
• Independence of observations means each data point not influenced by other observations in dataset

Key Assumptions and Considerations

• Normality of residuals allows for valid hypothesis testing and confidence interval estimation
• Multicollinearity (high correlation between independent variables) can lead to unreliable coefficient estimates
  • Assess using variance inflation factor (VIF)
  • Address by removing correlated variables or using regularization techniques (ridge regression)
• Outliers can significantly impact regression results
  • Identify using scatter plots or Cook's distance
  • Consider robust regression techniques (M-estimators, least trimmed squares)
• Non-linear relationships may require transformation of variables or non-linear regression models
  • Polynomial regression for quadratic relationships
  • Log transformation for exponential relationships

Regression Models for Impact Evaluation

Treatment and Control Variables

• Impact evaluation models include treatment variable to capture effect of intervention on outcome
• Control variables account for other factors influencing outcome, reducing omitted variable bias
  • Demographic characteristics (age, gender, education)
  • Economic indicators (income, employment status)
• Interaction terms examine how treatment impact varies across subgroups or conditions
  • Treatment * Gender to assess differential effects by gender
  • Treatment * Income to explore impact variation by socioeconomic status

Advanced Modeling Techniques

• Fixed effects models control for time-invariant unobserved heterogeneity in panel data analysis
  • Entity fixed effects (individual, firm, country)
  • Time fixed effects (year, month, season)
• Difference-in-differences estimates treatment effects by comparing outcome changes between groups over time
  • Pre-treatment period establishes baseline differences
  • Post-treatment period captures intervention effect
• Instrumental variable regression addresses endogeneity issues when treatment assignment not random
  • Requires valid instrument correlated with treatment but not directly with outcome
  • Two-stage least squares (2SLS) estimation method commonly used
• Consider robust or clustered standard errors for heteroscedasticity or within-group correlation
  • Huber-White robust standard errors for general heteroscedasticity
  • Cluster-robust standard errors for grouped data structures

Interpretation of Regression Coefficients

Understanding Coefficient Meaning

• Regression coefficient represents change in dependent variable associated with one-unit change in independent variable, holding others constant
• Sign of coefficient indicates direction of relationship (positive or negative)
• Magnitude quantifies strength of relationship, considering scale of variables
  • For continuous variables: "A one-unit increase in X is associated with a β-unit change in Y"
  • For binary variables: "The presence of X is associated with a β-unit difference in Y compared to its absence"
• Standardized coefficients (beta coefficients) compare relative importance of variables measured on different scales
  • Expressed in standard deviation units for both dependent and independent variables

Statistical Significance and Inference

• Assess statistical significance using t-tests or p-values (common threshold p < 0.05)
• Confidence intervals provide range of plausible values for true population parameter
  • 95% CI interpretation: "We are 95% confident that the true population parameter lies between [lower bound] and [upper bound]"
• Interpret interaction terms by considering combined effects of interacting variables
  • Main effects represent effects when other interacting variable is zero
  • Interaction coefficient represents how effect of one variable changes as other variable changes

Goodness of Fit and Predictive Power

Model Fit Measures

• Coefficient of determination (R-squared) measures proportion of variance in dependent variable explained by independent variables
  • Ranges from 0 to 1, higher values indicate better fit
  • Interpretation: "X% of the variation in Y is explained by the model"
• Adjusted R-squared accounts for number of predictors, useful for comparing models with different variables
• F-statistic tests overall significance of regression model
  • Null hypothesis: All coefficients are zero
  • Large F-statistic with low p-value indicates model is statistically significant

Prediction Accuracy and Model Validation

• Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) measure average prediction error in original units of dependent variable
  • RMSE more sensitive to large errors due to squaring
  • MAE less affected by outliers, easier to interpret
• Cross-validation assesses model's predictive performance on unseen data
  • K-fold cross-validation divides data into k subsets, trains on k-1 and tests on remaining subset
  • Leave-one-out cross-validation for small datasets
• Residual analysis helps identify potential violations of regression assumptions
  • Residuals vs. fitted values plot checks linearity and homoscedasticity
  • Q-Q plot assesses normality of residuals
• Information criteria (AIC, BIC) used for model selection, balancing fit and complexity
  • Lower values indicate better model
  • AIC tends to select more complex models compared to BIC