Econ 399 Chapter2c

# Econ 399 Chapter2c - 2.5 Variances of the OLS Estimators-We...

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2.5 Variances of the OLS Estimators -We have proven that the sampling distribution of OLS estimates (B 0 hat and B 1 hat) are centered around the true value -How FAR are they distributed around the true values? -The best estimator will be most narrowly distributed around the true values -To determine the best estimator, we must calculate OLS variance or standard deviation -recall that standard deviation is the square root of the variance.

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Gauss-Markov Assumption SLR.5 (Homoskedasticity) The error u has the same variance given any value of the explanatory variable. In other words, 2 x) | ar(u V
Gauss-Markov Assumption SLR.5 (Homoskedasticity) -While variance can be calculated via assumptions SLR.1 to SLR.4, this is very complicated -The traditional simplifying assumption is homoskedasticity, or that the unobservable error, u, has a CONSTANT VARIANCE -Note that SLR.5 has no impact on unbiasness -SLR.5 simply simplifies variance calculations and gives OLS certain efficiency properties

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Gauss-Markov Assumption SLR.5 (Homoskedasticity) -While assuming x and u are independent will also simplify matters, independence is too strong an assumption -Note that: ) | ( E 0 ) | ( E )] | ( [ ) | ( x) | ar(u 2 2 2 2 2 x u x u x u E x u E V
Gauss-Markov Assumption SLR.5 (Homoskedasticity) -if the variance is constant given x, it is always constant -Therefore: ) ( ) ( E 2 2 u Var u 2 is also called the ERROR VARIANCE or DISTURBANCE VARIANCE

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Gauss-Markov Assumption SLR.5 (Homoskedasticity) -SLR.4 and SLR.5 can be rewritten as conditions of y (using the fact we expect the error to be zero and y only varies due to the error) (2.56) ) | ( (2.55) ) | ( 2 1 0 x y Var x x y E
Heteroskedastistic Example Consider the following model: (ie) u 065 . 0 130 i i i income weight -here it is assumed that weight is a function of income -SLR.5 requires that:

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