Measurement Error Assumption: The observed variable thours thours is measured with error, meaning that thours ∗ thours ∗ is not directly observed and is subject to measurement error. Zero Conditional Mean Assumption: The measurement error is not systematically related to the true values of the explanatory variables. In mathematical terms, this is expressed as E ( u ∣ rage , motheduc , fatheduc , ssibs ) = 0 E(u∣rage,motheduc,fatheduc,ssibs)=0. Exogeneity Assumption: The measurement error in thours thours is not correlated with the unobserved error term u u. Mathematically, C o v ( u , thours ) = 0 Cov(u,thours)=0. Now, regarding whether these assumptions are likely to hold:
Measurement Error Assumption: This would depend on the quality and accuracy of the data collection method used to measure thours thours. If there are inaccuracies or biases in the measurement process, this assumption may not hold. Zero Conditional Mean Assumption: This assumption may be violated if there is a systematic pattern in the measurement errors related to the true values of the explanatory variables. For example, if there is a tendency to underreport television hours for certain groups, this assumption may be violated. Exogeneity Assumption: If the measurement error in thours thours is correlated with the unobserved factors influencing television viewing behavior (captured by u u), this assumption may not hold.
##
## Call:
## lm(formula = wage ~ educ + KWW, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -906.43 -245.00 -34.28 197.40 2298.61
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -71.091 81.574 -0.871 0.384
## educ 43.460 6.019 7.221 1.07e-12 ***
## KWW 12.413 1.731 7.172 1.51e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 372.4 on 932 degrees of freedom
## Multiple R-squared: 0.1537, Adjusted R-squared: 0.1519
## F-statistic: 84.63 on 2 and 932 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = wage ~ educ + KWW + I(KWW^2), data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -953.39 -239.83 -30.71 201.54 2253.05
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 648.6825 207.7162 3.123 0.001846 **
## educ 42.0442 5.9887 7.021 4.26e-12 ***
## KWW -30.3570 11.4948 -2.641 0.008406 **
## I(KWW^2) 0.6198 0.1647 3.763 0.000178 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 369.8 on 931 degrees of freedom
## Multiple R-squared: 0.1664, Adjusted R-squared: 0.1637
## F-statistic: 61.94 on 3 and 931 DF, p-value: < 2.2e-16## Estimate Pr(>|t|)
## educ 42.0442036 4.255736e-12
## KWW -30.3569558 8.406367e-03
## I(KWW^2) 0.6198462 1.783083e-04## Loading required package: carData## Sample Mean of stotal: 0.04748291## Standard Deviation of stotal: 0.8535441##
## Call:
## lm(formula = jc ~ stotal, data = twoyear)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.3633 -0.3424 -0.3384 -0.3113 3.5196
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.338364 0.009403 35.983 <2e-16 ***
## stotal 0.011177 0.011001 1.016 0.31
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7721 on 6761 degrees of freedom
## Multiple R-squared: 0.0001527, Adjusted R-squared: 4.767e-06
## F-statistic: 1.032 on 1 and 6761 DF, p-value: 0.3097##
## Call:
## lm(formula = univ ~ stotal, data = twoyear)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.4319 -1.8707 -0.4968 1.6909 7.8927
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.87073 0.02520 74.25 <2e-16 ***
## stotal 1.16968 0.02948 39.68 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.069 on 6761 degrees of freedom
## Multiple R-squared: 0.1889, Adjusted R-squared: 0.1888
## F-statistic: 1575 on 1 and 6761 DF, p-value: < 2.2e-16