# A question about Omitted Variables in a regression

I've been working through some econometrics worksheets recently. I came to a problem that I was hoping to get some help with. Assume we have a model,

(1)     yi = βo + β1chemi + β2disti + εi
and we know that
E[ εi | chemi, disti ] = 0.
Further, assume that we only have data to estimate the regression using the following model:
(2)     yi = βo + β1chemi + γi.

Then, on average, what will be the estimate of β̂1 if we exclude disti from the regression, in terms of the parameters from model (1), and why?

Many thanks.

• Google "omitted variable bias". – Dimitriy V. Masterov Sep 29 '17 at 3:03

Since you only have one explanatory variable in the model you estimate, you have $\hat{\beta}_1=\hat{c}ov(y,chem)/\hat{v}ar(chem)$
According to the correct model this gives $\hat{\beta}_1=\hat{c}ov(\beta_1 chem,chem)/\hat{v}ar(chem)+\hat{c}ov(\beta_2 dist,chem)/\hat{v}ar(chem)$
$\hat{\beta}_1-\beta_1=\beta_2\hat{c}ov(dist,chem)/\hat{v}ar(chem)$
It is zero only if dist and chem are uncorrelated or $\beta_2=0.$