5.5 Failure to Meet Assumptions

It is worth thinking about the consequences of not meeting these assumptions.

5.5.1 Failure of Assumption 1.

Assumption 1 states \(\mathbb{E}(\epsilon_{i}) = 0\). This assumption states that, on average, the error for the \(ith\) observation is zero. If instead, \(\mathbb{E}(\epsilon_{i}) = c\) and \(c \neq 0\), and all other assumptions hold, then only the intercept term is biased. Other coefficients OK.

5.5.2 Failure of Assumption 2 or 3.

Assumptions 2 and 3 are the homoskedasticity and no autocorrelation assumption, respectively. If we violate (2) or (3), but all other assumptions hold, (1) variance of \(\hat{\beta}\) is no longer dependable, (2) SEs possibly inaccurate, and (3) significance tests are possibly inaccurate. However, importantly, \(\hat{\beta}\) is still an unbiased and consistent estimator.

5.5.3 Failure of Assumption 5.

Assumption (5) states that that the error of our equation is uncorrelated with all the \(\mathbf{X}\)s. If this assumption fails, while others hold, OLS is no longer a consistent estimator.