Fundamental Assumptions of Ordinary Least Squares

Solution & Explanation

When discussing the assumptions underlying the Ordinary Least Squares (OLS) method, we are essentially referring to the conditions that need to be satisfied for OLS estimators to be valid and efficient. These assumptions are crucial for ensuring that the results obtained from an OLS regression are reliable and interpretable. Below is a detailed explanation of each assumption:

1. Linearity of Relationship

   • Explanation: The relationship between the dependent variable and the independent variables should be linear. This means that the effect of the independent variables on the dependent variable is additive and can be represented as a straight line in a two-dimensional space.
   • Significance: If the true relationship is not linear, the OLS estimates may be biased, leading to incorrect conclusions.

2. No Multicollinearity

   • Explanation: Multicollinearity occurs when two or more independent variables in a regression model are highly correlated, making it difficult to isolate the individual effect of each variable.
   • Significance: High multicollinearity can inflate the standard errors of the coefficients, leading to less reliable statistical tests and wider confidence intervals.

3. Normality of Residuals

   • Explanation: The residuals (errors) of the regression model should be normally distributed with a mean of zero.
   • Significance: Normality of residuals is important for conducting hypothesis tests and constructing confidence intervals. While OLS estimates remain unbiased without this assumption, the inference (e.g., t-tests) may not be valid if the residuals are not normally distributed.

4. Homoscedasticity

   • Explanation: The variance of the error term should remain constant across all levels of the independent variables. This means that the spread of the residuals should be roughly the same for all fitted values.
   • Significance: Homoscedasticity ensures that the OLS estimates are efficient (i.e., they have the smallest possible variance) and that the standard errors are correctly estimated, which is crucial for hypothesis testing.

5. No Autocorrelation

   • Explanation: The residuals should not be correlated with each other. In other words, the error term for one observation should not be related to the error term of another.
   • Significance: Autocorrelation can lead to inefficient estimates and underestimated standard errors, affecting the validity of hypothesis tests.

6. No Endogeneity

   • Explanation: There should be no correlation between the independent variables and the error term. Each independent variable should be exogenous, meaning it is not influenced by the error term.
   • Significance: Endogeneity can lead to biased and inconsistent estimates, as the OLS estimator assumes that the independent variables are fixed and not correlated with the error term.

Each of these assumptions plays a critical role in ensuring that the OLS regression provides valid, unbiased, and efficient estimates. Understanding these assumptions allows data scientists to diagnose potential issues in their models and apply necessary corrections, such as transforming variables, adding interaction terms, or using alternative estimation methods like Generalized Least Squares (GLS) or Instrumental Variables (IV) regression.