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ommision_irrelevant

ommision_irrelevant - 1 Omission of an important variable...

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1 Omission of an important variable What happens if you omit an important explanatory variable that should have been included in a regression? Let’s assume that the true model is a regression with two explanatory variables Y = β 1 + β 2 X 2 + β 3 X 3 + u and that the usual Gauss-Markov assumptions hold. Suppose that you mistakenly omit the second explanatory variable and instead work with Y = β 1 + β 2 X 2 + u It follows that the OLS estimate will be e β 2 = x 2 y x 2 and we label this estimator of β 2 as e β 2 to distinguish it from the correct OLS estimate. It turns out that e β 2 is biased. In order to show this, let’s begin with a few preliminary calculations. Firstly, Y = Y n = β 1 + β 2 X 2 + β 3 X 3 + u n = β 1 + β 2 X 2 + β 3 X 3 + u Secondly, we can use this equation to write y = Y - Y = ( β 1 + β 2 X 2 + β 3 X 3 ) - ( β 1 + β 2 X 2 + β 3 X 3 + u ) = β 2 x 2 + β 3 x 3 + u - u Thirdly, we can substitute y from above into the formula for e β 2 as follows: e β 2 = x 2 ( β 2 x 2 + β 3 x 3 + u - u ) x 2 2 = β 2 x 2 2 x 2 2 + β 3 x 2 x 3 x 2 2 + x 2 ( u - u ) x 2 2 = β 2 + β 3 x 2 x 3 x 2 2 + x 2 ( u - u ) x 2 2 Finally, we can establish that e β 2 is biased by taking the expected values of both sides of the equation: E e β 2 · = E β 2 + β 3 x 2 x 3 x 2 2 + x 2 ( u - u ) x 2 2 = β 2 + β 3 x 2 x 3 x 2 2 where the derivation above uses the fact that the expected value of a constant is a constant (the

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ommision_irrelevant - 1 Omission of an important variable...

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