To perform valid regression analysis, especially in time-series data, we must rely on the foundational assumptions of the Classical Linear Regression Model (CLRM). These assumptions ensure that the Ordinary Least Squares (OLS) estimators are BLUE - Best Linear Unbiased Estimators. Any violation can lead to incorrect inferences, which is particularly critical for Researchers, Economists, and UGC-NET aspirants.
Below are the six core assumptions:
Linearity in Parameters
Explanation: The model must be linear in parameters, i.e., the relationship should be expressed as:
Yt = β0 + β1X1t + β2X2t + ... + εt
Here, the parameters (βi) must appear linearly, even if Xt involves logs or polynomials.
Time-Series Focus: Even in lagged dependent variable models like:
Yt = β0 + β1Yt-1 + β2Xt + εt
...linearity must still hold in βs.
No Perfect Collinearity
Explanation: No independent variable should be a perfect linear function of others.
Mathematical condition:
Corr(Xi, Xj) ≠ ±1
Time-Series Note: Common in macroeconomic data where GDP, investment, and consumption might be highly correlated.
Remedy: Drop or combine collinear variables using Principal Component Analysis (PCA) or domain judgment.
Zero Conditional Mean (Exogeneity)
Explanation: The error term εt should have an expected value of zero, conditional on the regressors:
E(εt | X1t, X2t, ..., Xkt) = 0
Time-Series Application: Violated when relevant lagged variables or omitted predictors influence both Yt and εt.
Remedy: Include omitted variables or use instrumental variables (IV) if needed.
Homoskedasticity
Explanation: Constant variance of error terms:
Var(εt | X) = σ² (a constant)
Time-Series Issue: In financial data, variance often changes over time (heteroskedasticity).
Detection: Breusch-Pagan or White tests.
Remedy: Use robust standard errors or weighted least squares (WLS).
No Serial Correlation (Autocorrelation)
Explanation: Errors should be uncorrelated across time:
Cov(εt, εt-1) = 0
Time-Series Insight: Commonly violated in time-dependent data like temperature, stock prices, etc.
Detection: Durbin-Watson test or Breusch-Godfrey LM test.
Remedy: Use Generalized Least Squares (GLS) or add lagged dependent variables.
Normality of Errors
Explanation: The error terms εt should be normally distributed, particularly important for hypothesis testing in small samples.
Detection: Jarque-Bera Test, histogram, or Q-Q plots.
Remedy: Use transformations like log or square root, or bootstrapping techniques.
Consequences of Violating Assumptions
| Violated Assumption | Effect on OLS |
|---|---|
| Linearity | Model mis-specification, invalid estimates |
| No Collinearity | Inflated standard errors, unstable coefficients |
| Exogeneity | Biased and inconsistent estimates |
| Homoskedasticity | Incorrect standard errors, invalid test statistics |
| No Serial Correlation | Underestimates standard errors, false significance |
| Normality | Inefficient testing; invalid p-values in small samples |
Example:
Suppose a time-series model:
Yt = 2 + 0.8Yt-1 + 0.5Xt + εt
Where Y is GDP, X is government expenditure, and εt is the error term. With n = 10 years of data, you calculate residuals and find:
- Durbin-Watson = 1.1 → Suggests positive autocorrelation
- Breusch-Pagan p-value = 0.02 → Indicates heteroskedasticity
Interpretation: The model violates two key assumptions. Remedies involve re-specifying the model or using GLS with corrected standard errors.
Application Cases
- Econometrics: GDP forecasting where accurate model specification is critical to avoid policy misguidance.
- Finance: Stock price regression models with lagged returns must check for autocorrelation and heteroskedasticity.
- Public Policy: Budget expenditure modeling exogeneity and homoskedasticity are vital to valid conclusions.
- Climate Studies: Time-series of temperature or rainfall often show serial correlation and changing variance.
Summary
The Classical Linear Model assumptions serve as the backbone of regression validity. For time-series models, special care must be taken to detect and correct issues like autocorrelation and changing variance. Being aware of these assumptions, their violations, consequences, and remedies is essential for UGC NET aspirants aiming to master Business Statistics and apply it to real-world problems.