The following post will give a short introduction about the underlying assumptions of the classical linear regression model (OLS assumptions), which we derived in the following post. Given the Gauss-Markov Theorem we know that the least squares estimator and are unbiased and have minimum variance among all unbiased linear estimators. The Gauss-Markov Theorem is telling us that in a regression model, where the expected value of our error terms is zero, and variance of the error terms is constant and finite and and are uncorrelated for all and the least squares estimator and are unbiased and have minimum variance among all unbiased linear estimators. (A detailed proof of the Gauss-Markov Theorem can be found here)
In the following we will summarize the assumptions underlying the Gauss-Markov in greater depth. In order for a least squares estimator to be BLUE (best linear unbiased estimator) the first four of the following five assumptions have to be satisfied:
Assumption 1: Linear Parameter and correct model specification
Assumption 1 requires that the dependent variable is a linear combination of the explanatory variables and the error terms . Additionally we need the model to be fully specified. A extensive discussion of Assumption 1 can be found here.
Assumption 2: Full Rank of Matrix X
Assumption 2 requires the matrix of explanatory variables X to have full rank. This means that in case matrix X is a matrix . A more detailed elaboration of assumption 2 can be found here.
Assumption 3: Explanatory Variables must be exogenous
Assumption 3 requires data of matrix x to be deterministic or at least stochastically independent of for all . In other words, explanatory variables x are not allowed to contain any information on the error terms , i.e. it must not be possible to explain through X. Mathematically is assumption 3 expressed as
Assumption 4: Independent and Identically Distributed Error Terms
Assumption 4 requires error terms to be independent and identically distributed with expected value to be zero and variance to be constant. Mathematically is assumption 4 expressed as
The exact implications of Assumption 4 can be found here.
Assumption 5: Normal Distributed Error Terms in Population
Assumption 5 is often listed as a Gauss-Markov assumption and refers to normally distributed error terms in population. However, assumption 5 is not a Gauss-Markov assumption in that sense that the OLS estimator will still be BLUE even if the assumption is not fulfilled. You can find more information on this assumption and its meaning for the OLS estimator here.