Heteroskedastic and Serially Correlated Residuals

The Classical Linear Regression Model (CLRM) rests on seven main assumptions: (A1) linearity in parameters, (A2) random sampling, (A3) no perfect multicollinearity, (A4) strict exogeneity, (A5) homoskedasticity, (A6) no autocorrelation, and (A7) normality of the error term. According to the Gauss-Markov theorem, if A1–A5 are satisfied, the Ordinary Least Squares (OLS) estimators are Best Linear Unbiased Estimators (BLUE). These assumptions ensure that the estimators are unbiased, have the smallest variance among the class of linear unbiased estimators, and allow valid statistical inference through t and F tests.

The Gauss-Markov theorem states that if the classical linear regression assumptions (A1-A5) hold, the OLS estimators are Best Linear Unbiased Estimators (BLUE). This means they are linear in parameters, unbiased (on average, they hit the true parameter), and have the smallest variance among all linear unbiased estimators.

Strict exogeneity means that the error term has zero conditional mean given all explanatory variables. Mathematically, E(u|X) = 0. It ensures no correlation between the errors and any explanatory variable, preventing biased coefficient estimates.

Autocorrelation refers to the correlation between error terms across different observations, often in time series data. It violates the assumption that Cov(u_i, u_j) = 0 for i ≠ j, leading to inefficient estimates and invalid inference.

Homoskedasticity assumes constant variance of the error term across observations: Var(u|X) = σ^2. This condition is crucial for deriving valid standard errors, confidence intervals, and hypothesis tests.

Heteroskedasticity occurs when the variance of the error term varies across observations. It violates the homoskedasticity assumption and affects the reliability of standard errors and test statistics.

No, OLS estimators remain unbiased under heteroskedasticity. However, their standard errors become biased, making t-tests and F-tests invalid. OLS also becomes inefficient.

No, R-squared is not affected by heteroskedasticity. It still measures the proportion of the variance in the dependent variable explained by the model.

Efficiency is compromised. The estimators no longer have the smallest possible variance among linear unbiased estimators.

It tests the null hypothesis of homoskedasticity, i.e., constant error variance across observations. The alternative is that variance depends on explanatory variables.

The squared residuals from the original model are regressed on the original explanatory variables.

White test includes not only the explanatory variables but also their squares and cross-product terms to capture more general forms of heteroskedasticity.

It uses only the fitted values and their squares from the original regression, reducing the number of variables in the auxiliary regression.

Solutions include using robust standard errors, Weighted Least Squares (WLS), Feasible Generalized Least Squares (FGLS), variable transformation (e.g., log transformation), or adopting non-linear models.

Log-linear models are useful when the dependent variable has an exponential or multiplicative relationship with the predictors. They also help in correcting heteroskedasticity by reducing variance in higher values.

Autocorrelation in time series data implies that residuals from one period are correlated with those from another. This violates OLS assumptions and leads to inefficient estimates and misleading inference.

The dwtest() function conducts the Durbin-Watson test and is part of the lmtest package in R.

Autocorrelation may result from omitted variables, incorrect functional form, measurement errors, or the inherent time-related nature of the data (e.g., inertia, seasonality).

Autocorrelation distorts standard error estimates, leading to invalid t- and F-tests. This results in misleading conclusions about the significance of explanatory variables.

The Durbin-Watson test examines the presence of first-order autocorrelation by comparing the difference in residuals across consecutive observations. Values near 2 indicate no autocorrelation; values significantly different suggest its presence.

It is not suitable when lagged dependent variables are included as regressors, as it can produce misleading results in such cases.

Alternative tests include the Breusch-Godfrey LM test, which allows testing for higher-order serial correlation and is applicable even when lagged dependent variables are included.

First, estimate the original regression model and obtain residuals. Then regress the residuals on all original regressors plus their lagged values. The R² from this auxiliary regression is used to compute the LM statistic.

The test statistic is calculated as LM = n × R² from the auxiliary regression. It follows a chi-square distribution. A significant value indicates autocorrelation.

It suggests model misspecification or data issues. Correcting for it may involve including lagged variables, using GLS, or applying robust standard errors.

The Breusch-Godfrey test, implemented as bgtest() in R (also from the lmtest package), allows for testing higher-order autocorrelation and accommodates more flexible model structures.