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Specification Bias Explanation (2)

Specification Bias Explanation (2) - expected value of Y...

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Specification Bias a. Specification mistake - suppose an important variable, X 2 , is left out of the regression model. The true model is: 0 1 1 2 2 i i i i Y X X u β β β = + + + But, you assume: 0 1 1 i i i Y X v α α = + + (What CRM assumptions have been violated? Assumption #1 and Assumption #3.) b. What happens - Verbally. Your model assumes only X 1 causes Y to change, but in truth, the variable X 2 also causes Y to change. The effects of X 2 on Y are not accounted for in your model. As a result, the effect of X 2 on Y gets tangled up with the effect of X 1 on Y . We can't get a clear picture of how changes in X 1 affect changes in Y . c. What happens - Mathematically. The estimator that you use is: 1 2 1 ˆ i i 1 i x Y = x α This will be biased . To show this take the expected value of the estimator and use the
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Unformatted text preview: expected value of Y when evaluating: First, insert the true expected value of Y i ˆ 1 i 2 i 1 i 1 2 1 2 1 i ( + + ) x X X E [ ] = x ∑ ∑ , and : ˆ 1 i 2 i 2 i 1 i 1 i 1 i 1 1 2 1 2 2 2 2 1 i 1 i 1 i x x x X X X E [ ] = + = + x x x ∑ ∑ ∑ ∑ ∑ ∑ . which says that the expected value of 1 ˆ equals the true effect of X 1 on Y , 1 , plus the bias due to model misspecification . The bias due to model misspecification is made up of two parts: (1) 2- the true effect of X 2 on Y ; and (2) the relationship between X 2 and X 1 . Moral of this story: Leaving out an important independent variable can lead to biased parameter estimates....
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