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Published byLenard Hancock Modified over 4 years ago

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Violations of Assumptions In Least Squares Regression

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Standard Assumptions in Regression Errors are Normally Distributed with mean 0 Errors have constant variance Errors are independent X is Measured without error

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Example X s and OLS Estimators “t” is used to imply time ordering

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Non-Normal Errors (Centered Gamma)

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Errors = (Gamma(2,3.7672)-7.3485) Y t = 50 + 10X t + ( t -7.35) = 0 * + 1 X t + t * E( t * ) = 0 V( t * ) = 27 Based on 100,000 simulations, the 95% CI for 1 contained 10 in 95.05% of the samples. Average=9.99887, SD=0.3502 Average s 2 (b1) = 0.1224804

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Non-Constant Error Variance (Heteroscedasticity) Mean: E(Y|X) = X = 50 + 10X Standard Deviation: Y|X = X + 0.5 Distribution: Normal

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Non-Constant Error Variance (Heteroscedasticity) Based on 100,000 simulations, the 95% CI for 1 contained 10 in 92.62% of the samples. Average=9.998828, SD = 0.467182 Average s 2 (b1) = 0.1813113 < 0.2184

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Correlated Errors Example: 2 = 9, =0.5, n=22

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Correlated Errors Based on 100,000 simulations, the 95% CI for 1 contained 10 in 84.56% of the samples. Average=10.00216, SD=0.3039251 Average s 2 (b1) = 0.0476444 < 0.092157

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Measurement Error in X Z=True Value of Independent Variable (Unobserved) X=Observed Value of Independent Variable X=Z+U (Z can be fixed or random) Z, U independent (assumed) when Z is random V(U) is independent of Z when Z is fixed

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Measurement Error in X Based on 100,000 simulations, the 95% CI for 1 contained 10 in 76.72% of the samples. Average=9.197568, SD=0.6076758 Average s 2 (b1) = 0.4283653 >> (0.20226) 2

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