One reason why the omitted variable leads to biased estimates is that omitting a relevant variable violates assumption 3 of the necessary assumptions of the classical regression model that states that all explanatory variables must be exogenous, i.e.

From this post, we know that omitting a relevant variable from the regression causes the error term and the explanatory variables to be correlated.

Suppose that the data generating process in the population is as follows:

However, we omit a variable, in this case and estimate the following regression model:

,

where and

Note that, if it implies that . Hence, we violate assumption three. That is, the error term will be correlated with another explanatory variable.

Now, how can we mathematically prove that omitting indeed causes endogeneity. To prove this, lets start from the limit of the OLS estimator. Let denote the full matrix of explanatory variables, in our case , and let be the error term containing , that is . Additionally, let be the vector of parameters that we want to estimate, i.e. .

Immediately, we see that . This is always the case whenever , i.e. whenever we have a correlation between and .

In case we specify our model correctly, the second term in the third row would be \mathbb{E}(\left[ (X’X)^{-1}X’u \epsilon]) and collaps to zero. That is . This happens because , since , which holds because the original assumption that each of the explanatory variables are uncorrelated with the error term .

###### Omitted Variable Bias

- Overview
- Introduction
- Understanding the Bias
- Explanation and Example
- Consequences
- What can we do about it?
- Concluding Remarks

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