Assumption 3, exogeneity of explanatory variables requires that the explanatory variables in the model do not explain variation in the error terms, formally we express assumption 3 as
This means that all data in the matrix are stochastically independent of for all . Note, the law of iterated expectations ensure that
Although having stochastic independence is necessary to fulfill assumption 3, textbooks sometimes mention deterministic as a sufficient requirement. I should mention that, we fulfill assumption 3 per definition when applying OLS on deterministic data. However, we do not require the strong demand of deterministic data to meet assumption 3. Ensuring stochastically independence of is sufficient to secure an unbiased and consistent estimator. You will recognize soon enough that even stochastically independent data are not easy to find.
Summarizing assumption 3 of the classical linear regression model (clrm) in mildly different word: In order to fulfill assumption 3 the data generating process of X has to be independent of the data generating process of the error terms.
Impact of assumption 3
Assumption 3 requires the error term to be stochastically independent of all additionally to the independence of observation specific characteristics . For cross sectional data this means that the error term of observation has to stochastically independent of all explanatory variables of observation . Furthermore, the error term of observation has to be independent of the explanatory variables of all other observations . For time series data the assumption requires inter-temporal independence. This means that the error term of period must be stochastically independent of all explanatory variables of the past, the presence and the future.
Violating assumption 3
The OLS estimator is neither consistent nor unbiased in case assumption 3 is violated. Unfortunately, we violate assumption 3 very easily. Common case that violate assumption 3 include omitted variables, measurement error and simultaneity.
Assumptions of Classical Linear Regressionmodels (CLRM)
Overview of all CLRM Assumptions
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