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Violations of Assumptions In Least Squares Regression

Violations of Assumptions In Least Squares Regression. Standard Assumptions in Regression. Errors are Normally Distributed with mean 0 Errors have constant variance Errors are independent X is Measured without error. Example X s and OLS Estimators. “t” is used to imply time ordering.

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Violations of Assumptions In Least Squares Regression

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  1. Violations of Assumptions In Least Squares Regression

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

  3. Example Xs and OLS Estimators “t” is used to imply time ordering

  4. Non-Normal Errors (Centered Gamma)

  5. Errors = (Gamma(2,3.7672)-7.3485) • Yt = 50 + 10Xt + (et-7.35) = b0*+b1Xt+et* • E(et*) = 0 V(et*) = 27 Based on 100,000 simulations, the 95% CI for b1 contained 10 in 95.05% of the samples. Average=9.99887, SD=0.3502 Average s2(b1) = 0.1224804

  6. Non-Constant Error Variance (Heteroscedasticity) • Mean: E(Y|X) = b0 + b1X = 50 + 10X • Standard Deviation: sY|X = X + 0.5 • Distribution: Normal

  7. Non-Constant Error Variance (Heteroscedasticity) Based on 100,000 simulations, the 95% CI for b1 contained 10 in 92.62% of the samples. Average=9.998828, SD = 0.467182 Average s2(b1) = 0.1813113 < 0.2184

  8. Correlated Errors Example: s2= 9, r=0.5, n=22

  9. Correlated Errors Based on 100,000 simulations, the 95% CI for b1 contained 10 in 84.56% of the samples. Average=10.00216, SD=0.3039251 Average s2(b1) = 0.0476444 < 0.092157

  10. 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

  11. Measurement Error in X Based on 100,000 simulations, the 95% CI for b1 contained 10 in 76.72% of the samples. Average=9.197568, SD=0.6076758 Average s2(b1) = 0.4283653 >> (0.20226)2

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