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Specification Bias

# Specification Bias - Specification Bias We discussed...

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Specification Bias We discussed estimators in class for models that were misspecified, either by omitting important variables or including irrelevant variables. We found that the OLS estimators for the model with omitted variables were biased , but had smaller variances than the estimators for a properly specified model. You might feel that this doesn’t make sense. If you leave out important variables, it must make the estimator of 2 s biased as well. If the estimator for 2 s is biased upwards, then the estimators would not be efficient. This question is quite different from the issues that we focused on in class. Let me clarify by providing notes for the topics in class and address the question of evaluating the OLS estimators and the issue of estimating 2 s . Let us consider two possible models: (1) 1 1 1 Y = X + u b , and (2) 1 1 2 2 2 Y = X + X + u b b b b . There are two possibilities: Model (1) is correct, which means that the set of independent variables in X 2 are truly irrelevant (we might say 2 = 0 b ). Model (2) is correct, which means that the set of independent variables in X 2 are truly important and should have been included ( 2 0 b ). There are also two ways that we can make a specification error. We specify model (1), but model (2) is the truth. We are therefore guilty of omitting a set of independent variables that should have been included. We assume model (2), but model (1) is the truth. In this case, our mistake is including a set of independent variables that are irrelevant. Omitted Regressors: Let’s evaluate the properties of the estimators under each of the two specification errors. When estimating we apply OLS so all the estimators will be linear estimators. Let’s start with the first form of specification error, omitting relevant independent variables . The OLS estimator is: (3) -1 1 1 1 1 b = (X X ) X Y .

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Specification Bias - Specification Bias We discussed...

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