Violations of Assumptions In Least Squares Regression

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

Non-Normality Errors (t3 errors)

Errors = 3*t3 E(et) = E(3*t3)=3E(t3)=3*0=0 V(et) = V(3*t3)=9V(t3)=9*(3/(3-2))=27 Based on 100,000 simulations, the 95% CI for b1 contained 10 in 95.31% of the samples. Average=9.999271, SD=0.3478 Average s2(b1) = 0.1216903

Non-Constant Error Variance (Heteroscedasticity) Mean: E(Y) = b0 + b1 X = 50 + 10X Standard Deviation: sX = 0.1(X+5) Distribution: Normal

Non-Constant Error Variance (Heteroscedasticity) Based on 100,000 simulations, the 95% CI for b1 contained 10 in 92.62% of the samples. Average=9.999724, SD=0.4667337 Average s2(b1) = 0.1813961 < 0.2184

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

Correlated Errors Based on 100,000 simulations, the 95% CI for b1 contained 10 in 84.46% of the samples. Average=9.99853, SD=0.3037228 Average s2(b1) = 0.0474835 < 0.092157

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

Measurement Error in X Based on 100,000 simulations, the 95% CI for b1 contained 10 in 76.56% of the samples. Average=9.197172, SD=0.6112944 Average s2(b1) = 0.4290028 >> (0.20226)2