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

Non-Normal Errors (Centered Gamma)

Errors = (Gamma(2,3.7672) ) Y t = X 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= , SD= Average s 2 (b1) =

Non-Constant Error Variance (Heteroscedasticity) Mean: E(Y|X) =     X = X Standard Deviation:  Y|X = X Distribution: Normal

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

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

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

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  1 contained 10 in 76.72% of the samples. Average= , SD= Average s 2 (b1) = >> ( ) 2