CLRM – Assumption 3: Explanatory Variables must be exogenous

Assumption 3, exogeneity of explanatory variables requires that the explanatory variables X in the model do not explain variation in the error terms, formally we express assumption 3 as

E(\epsilon_i|X)=0

This means that all data in the X matrix are stochastically independent of e_i for all i. Note, the law of iterated expectations ensure that

E(\epsilon)=0 as E(\epsilon) = E[E(\epsilon|X)]=E(0) = 0.

Although having stochastic independence is necessary to fulfill assumption 3, textbooks sometimes mention deterministic X 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 X 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 X_{j,k} additionally to the independence of observation specific characteristics X_{i,k}. For cross sectional data this means that the error term of observation i has to stochastically independent of all explanatory variables of observation i. Furthermore, the error term of observation i has to be independent of the explanatory variables of all other observations j. For time series data the assumption requires inter-temporal independence. This means that the error term of period t must be stochastically independent of all explanatory variables X 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
Assumption 1
Assumption 2
Assumption 3
Assumption 4
Assumption 5

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5 Responses to CLRM – Assumption 3: Explanatory Variables must be exogenous

  1. Pingback: CLRM – Assumption 2: Full Rank of Matrix X | Economic Theory Blog

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