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Notes6 - Lecture Notes 6 Econ 410 Introduction to...

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Lecture Notes 6 Econ 410 – Introduction to Econometrics 1 The multiple regression model Omitted Variable Bias In the linear regression model with just one regressor, we focused on the effect of the variable i X on i Y and we included all the other determinants that have an effect on i Y in the error term. Ex.: i Y is the random variable “earnings” and i X is the random variable “years of schooling”. Some omitted variables are: the person IQ, his/her family and regional background, the schools he/she attended, age, working experience, etc. Omitted variables can make the OLS estimator biased (omitted variable bias). In particular, the bias happens when the omitted variable is a determinant of the dependent variable i Y and it is correlated with the included regressor i X . Omitted variable bias means that the first least square assumption doesn’t hold, that is: ( ) 0 | i i X u E . Indeed, if the omitted variable influences i Y , then it must be included in the error term, and if the omitted variable is correlated with i X , then the error term will be correlated with i X . So: If you suspect that a variable you did not include in your regression has an influence on i Y , then check the data. In particular, check the correlation between your regressor and this omitted variable. If you find a nonzero correlation, then the OLS estimator is biased because the mean of the sampling distribution of the OLS estimator might not equal the true effect on i Y of a unit change in i X . Ex.: Consider the omitted variable IQ. It is plausible to think that this omitted variable is correlated with the level of education of a person, and it is also plausible to think that this variable can affect his/her earnings. So, omitting the IQ might introduce omitted variable bias. Important: Why there is no omitted variable bias if the omitted variable is uncorrelated with the included regressor i X ? Why there is no omitted variable bias if the omitted variable does not affect the dependent variable i Y ? Because in both cases the OLS estimator would not incorrectly incorporate the influence of the omitted variable in predicting the effect on i Y of a unit change in i X .
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Lecture Notes 6 Econ 410 – Introduction to Econometrics 2 Measuring the omitted variable bias If the omitted variable bias is present, than it is possible to show that: X u Xu p σ σ ρ β β + → 1 1 ˆ where: ) , ( i i Xu u X corr = ρ This formula implies that, in large samples, 1 ˆ β is close to X u Xu σ σ ρ β + 1 with high probability. It follows that, if the omitted variable bias is present: 1 ˆ β is not a consistent estimator of 1 β . 1 ˆ β is a biased estimator of 1 β ; X u Xu σ σ ρ is the bias that persists even in large samples.
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