Econ 399 Chapter3e - 3.4 The Components of the OLS...

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3.4 The Components of the OLS Variances: Multicollinearity We see in (3.51) that the variance of B j hat depends on three factors: σ 2 , SST j and R j 2 : 1)The error variance, σ 2 Larger error variance = Larger OLS variance -more “noise” in the equation makes it more difficult to accurately estimate partial effects of the variables -one can reduce the error variance by adding (valid) variables to the equation
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3.4 The Components of the OLS Variances: Multicollinearity 2) The Total Sample Variation in x j , SST j Larger x j variance – Smaller OLS j variance -increasing sample size keeps increasing SST j since 2 ) ( j ij j x x SST -This still assumes that we have a random sample
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3.4 The Components of the OLS Variances: Multicollinearity 3) Linear relationships among x variables: R j 2 Larger correlation in x’s – Bigger OLS j variance -R j 2 is the most difficult component to understand - R j 2 differs from the typical R 2 in that it measures the goodness of fit of: ik k i i ij x x x x ... ˆ 2 2 1 1 0 -Where x j itself is not considered an explanatory variable
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3.4 The Components of the OLS Variances: Multicollinearity 3) Linear relationships among x variables: R j 2 -In general, R j 2 is the total variation in x j that is explained by the other independent variables -If R j 2 =1, MLR.3 (and OLS) fails due to perfect multicollinearity (x j is a perfect linear combination of the other x’s) Note that: 2 j R as ) ˆ ( j Var -High (but not perfect) correlation between independent variables is MULTICOLLINEARITY
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3.4 Multicollinearity -Note that an R j 2 close to 1 DOES NOT violate MLR. 3 -unfortunately, the “problem” of multicollinearity is hard to define -No R j 2 is accepted as being too high -A high R j 2 can always be offset by a high SST j or a low σ 2 -Ultimately, how big is B j hat relative to its standard error?
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3.4 Multicollinearity -Ceteris Paribus, it is best to have little correlation between x j and all other independent variables -Dropping independent variables will reduce multicollinearity -But if these variables are valid, we have created bias -Multicollinearity can always be fought by collecting more data -Sometimes multicollinearity is due to over specifying independent variables:
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3.4 Multicollinearity Example -In a study of heart disease, our economic model is: Heart disease=f(fast food, junk food, other)
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