🥖Linear Modeling Theory Unit 9 – Diagnostics & Remedies for Linear Regression

Key Concepts

• Linear regression models the relationship between a dependent variable and one or more independent variables
• Ordinary least squares (OLS) estimation minimizes the sum of squared residuals to find the best-fitting line
• Residuals represent the difference between observed and predicted values of the dependent variable
• Coefficient of determination (R-squared) measures the proportion of variance in the dependent variable explained by the model
  • Adjusted R-squared accounts for the number of predictors in the model
• Multicollinearity occurs when independent variables are highly correlated with each other
• Heteroscedasticity refers to non-constant variance of the residuals across the range of predicted values
• Outliers are data points that are far from the majority of the data and can heavily influence the regression line

Diagnostic Tools

• Residual plots visualize the relationship between residuals and predicted values to check for patterns or trends
  • Residuals should be randomly scattered around zero with no discernible pattern
• Normal probability plots (Q-Q plots) assess the normality of residuals by comparing their distribution to a theoretical normal distribution
• Cook's distance measures the influence of individual observations on the regression coefficients
• Variance Inflation Factor (VIF) quantifies the severity of multicollinearity for each independent variable
  • VIF values greater than 5 or 10 indicate problematic multicollinearity
• Breusch-Pagan test checks for the presence of heteroscedasticity in the residuals
• Durbin-Watson test detects autocorrelation in the residuals of a time series regression model

Common Issues in Linear Regression

• Omitted variable bias occurs when a relevant predictor is not included in the model, leading to biased coefficient estimates
• Misspecification of the functional form (e.g., assuming a linear relationship when it is non-linear) can lead to poor model fit and biased estimates
• Measurement errors in the variables can introduce bias and reduce the precision of the estimates
• Endogeneity arises when an independent variable is correlated with the error term, violating the assumption of exogeneity
  • Endogeneity can be caused by omitted variables, simultaneous causality, or measurement errors
• Sample selection bias occurs when the sample is not representative of the population of interest, leading to biased estimates
• Overfitting happens when a model is too complex and fits the noise in the data rather than the underlying relationship
  • Overfitted models have poor generalization performance on new data

Assumption Violations

• Linearity assumption: The relationship between the dependent and independent variables is linear
  • Violated when the true relationship is non-linear (e.g., quadratic, exponential)
• Independence of errors assumption: The residuals are uncorrelated with each other
  • Violated when there is autocorrelation in the residuals (common in time series data)
• Homoscedasticity assumption: The variance of the residuals is constant across all levels of the predicted values
  • Violated when there is heteroscedasticity (non-constant variance)
• Normality assumption: The residuals follow a normal distribution with a mean of zero
  • Violated when the residuals are skewed or have heavy tails
• No multicollinearity assumption: The independent variables are not highly correlated with each other
  • Violated when there is significant multicollinearity among predictors

Remedial Measures

• Variable transformation (e.g., logarithmic, square root) can help address non-linearity and heteroscedasticity
• Adding interaction terms or polynomial terms can capture non-linear relationships between variables
• Robust standard errors (e.g., White's heteroscedasticity-consistent standard errors) can be used when heteroscedasticity is present
• Weighted least squares (WLS) estimation assigns different weights to observations based on the variance of the residuals to address heteroscedasticity
• Ridge regression and Lasso regression are regularization techniques that can help mitigate multicollinearity by shrinking the coefficient estimates
  • Ridge regression adds a penalty term to the OLS objective function based on the L2 norm of the coefficients
  • Lasso regression uses the L1 norm penalty, which can also perform variable selection by setting some coefficients to zero
• Instrumental variables (IV) estimation can address endogeneity by using an instrument that is correlated with the endogenous variable but uncorrelated with the error term

Advanced Techniques

• Generalized linear models (GLMs) extend linear regression to handle non-normal response variables (e.g., logistic regression for binary outcomes, Poisson regression for count data)
• Mixed-effects models (also known as hierarchical or multilevel models) account for clustered or nested data structures by incorporating random effects
• Quantile regression estimates the relationship between the independent variables and specific quantiles of the dependent variable, providing a more comprehensive view of the data
• Generalized additive models (GAMs) allow for non-linear relationships between the dependent and independent variables using smooth functions
  • GAMs are more flexible than traditional linear models and can capture complex patterns in the data
• Bayesian linear regression incorporates prior knowledge about the parameters and updates the estimates based on the observed data
  • Bayesian methods provide a probabilistic framework for inference and can handle small sample sizes or high-dimensional data

Real-World Applications

• Predicting house prices based on features such as square footage, number of bedrooms, and location
• Analyzing the factors that influence customer satisfaction in a service industry
• Estimating the effect of advertising expenditure on sales revenue for a company
• Identifying the key drivers of employee turnover in an organization
• Forecasting energy consumption based on historical data and weather variables
• Assessing the impact of socioeconomic factors on health outcomes in a population
• Predicting stock prices using financial indicators and market sentiment data

Key Takeaways

• Diagnostic tools help identify potential issues in linear regression models, such as non-linearity, heteroscedasticity, and multicollinearity
• Assumption violations can lead to biased and inefficient estimates, requiring appropriate remedial measures
• Variable transformations, robust standard errors, and regularization techniques can address common issues in linear regression
• Advanced techniques like GLMs, mixed-effects models, and GAMs extend the capabilities of linear regression to handle more complex data structures and relationships
• Proper model specification, variable selection, and diagnostic checking are crucial for obtaining reliable and interpretable results
• Linear regression has wide-ranging applications in various fields, including finance, marketing, healthcare, and social sciences
• Understanding the assumptions, limitations, and remedial measures of linear regression is essential for effective data analysis and decision-making