CHAPTER(9)

Introduction

The following equation models weekly hours of television viewing by a child:

\[ thours^* = \beta_0 + \beta_{\text{1age}} + \beta_{\text{2age}^2} + \beta_{\text{3motheduc}} + \beta_{\text{4fatheduc}} + \beta_{\text{5sibs}} + u \]

where \(thours^*\) represents the true hours of television viewing, and \(u\) is an error term. The concern is that \(thours^*\) is measured with error in the survey, and observe the reported hours \(thours\).

(4) i. Classical Errors-in-Variables (CEV) Assumptions

1. No Measurement Error in the Dependent Variable

 The classical errors-in-variables model assumes that there is no measurement error in the dependent variable (\( thours^* \)). The observed \( thours \) is considered the true value without measurement error.

2. Linear Relationship

 The relationship between the true values of the variables is assumed to be linear.

3. Exogeneity of the Measurement Error

  The measurement error (\( u \)) is not correlated with the true values of the independent variables.

4. Homoscedasticity

  The variance of the measurement error is constant across all levels of the independent variables.

5. No Measurement Error in the Explanatory Variables

  There is no measurement error in the explanatory variables (age, motheduc, fatheduc, ssibs).

ii. Assessment of CEV Assumptions

1. No Measurement Error in the Dependent Variable:

    This assumption may be challenging to ensure. The accuracy of reported hours (\( thours \)) depends on respondents' honesty and recall accuracy.

2. Linear Relationship

     The assumption of a linear relationship is reasonable and often assumed in regression models.

3. Exogeneity of the Measurement Error

    It depends on the circumstances of the survey. If factors influencing both reported and true hours are correlated, the assumption may be violated.

4. Homoscedasticity

    The homoscedasticity assumption may depend on the nature of reporting errors. If reporting errors vary systematically, homoscedasticity may not hold.

5. No Measurement Error in the Explanatory Variables

   This assumption may be more reasonable for some variables (e.g., age) than others.

Assessing the likelihood of CEV assumptions holding requires careful consideration of survey design, reporting behavior, and the nature of the variables involved.

(C2) Wage2

(i) Estimated Return to Education Using KWW

Using KWW as a proxy, the expected return to education is represented by the coefficient for the “educ” variable in this model.

## 
## 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

(ii) Estimated Return to Education Using / and KWW Together

When KWW and a polynomial term are used as proxy variables, the estimated return to education is represented by the “educ” coefficient in this model.

## 
## 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

(iii) Significance Tests

##             Estimate     Pr(>|t|)
## educ      42.0442036 4.255736e-12
## KWW      -30.3569558 8.406367e-03
## I(KWW^2)   0.6198462 1.783083e-04

(C8) TWOYEAR

(i) Sample Mean and Standard Deviation of stotal

## Loading required package: carData

## Sample Mean of stotal: 0.04748291

## Standard Deviation of stotal: 0.8535441

(ii) Simple Regressions of jc and univ on stotal

## 
## 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

(iii) Test Returns to Two- and Four-Year Colleges

## Linear hypothesis test
## 
## Hypothesis:
## jc - univ = 0
## 
## Model 1: restricted model
## Model 2: lwage ~ stotal + jc + univ
## 
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1   6760 1417.9                           
## 2   6759 1417.8  1   0.16883 0.8049 0.3697

(iv) Test for Quadratic Term in stotal

##   female phsrank BA AA black hispanic  id exper        jc     univ    lwage
## 1      1      65  0  0     0        0  19   161 0.0000000 0.000000 1.925291
## 2      1      97  0  0     0        0  93   119 0.0000000 7.033333 2.796494
## 3      1      44  0  0     0        0  96    81 0.0000000 0.000000 1.625600
## 4      1      34  0  0     0        1 119    39 0.2666667 0.000000 2.223312
## 5      1      80  0  0     0        0 132   141 0.0000000 0.000000 1.642083
## 6      0      59  0  0     0        0 156   165 0.0000000 0.000000 2.079442
##       stotal smcity medcity submed lgcity sublg vlgcity subvlg ne nc south
## 1 -0.4417497      0       0      0      0     1       0      0  1  0     0
## 2  0.0000000      1       0      0      0     0       0      0  0  1     0
## 3 -1.3570027      0       0      0      0     1       0      0  1  0     0
## 4 -0.1900551      1       0      0      0     0       0      0  0  0     0
## 5  0.0000000      0       0      0      0     0       0      0  0  0     1
## 6  1.3887565      1       0      0      0     0       0      0  0  0     1
##     totcoll
## 1 0.0000000
## 2 7.0333333
## 3 0.0000000
## 4 0.2666667
## 5 0.0000000
## 6 0.0000000

##  [1] "female"   "phsrank"  "BA"       "AA"       "black"    "hispanic"
##  [7] "id"       "exper"    "jc"       "univ"     "lwage"    "stotal"  
## [13] "smcity"   "medcity"  "submed"   "lgcity"   "sublg"    "vlgcity" 
## [19] "subvlg"   "ne"       "nc"       "south"    "totcoll"

## Linear hypothesis test
## 
## Hypothesis:
## I(stotal^2) = 0
## 
## Model 1: restricted model
## Model 2: lwage ~ stotal + I(stotal^2) + jc + univ
## 
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1   6759 1417.8                           
## 2   6758 1417.8  1  0.013251 0.0632 0.8016

(v) Test Joint Significance of Interaction Terms

## Linear hypothesis test
## 
## Hypothesis:
## stotal:jc = 0
## stotal:univ = 0
## 
## Model 1: restricted model
## Model 2: lwage ~ stotal + jc + univ + stotal:jc + stotal:univ
## 
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1   6759 1417.8                           
## 2   6757 1417.5  2   0.28045 0.6684 0.5125

(vi) Final Model Controlling for Ability

## 
## Call:
## lm(formula = lwage ~ stotal + I(stotal^2) + jc + univ + stotal:jc + 
##     stotal:univ, data = twoyear)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.87058 -0.30239  0.01589  0.32046  1.80138 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.107884   0.008862 237.861  < 2e-16 ***
## stotal       0.074176   0.011169   6.641 3.35e-11 ***
## I(stotal^2)  0.005856   0.006196   0.945    0.345    
## jc           0.065377   0.007374   8.866  < 2e-16 ***
## univ         0.059737   0.002928  20.405  < 2e-16 ***
## stotal:jc   -0.008969   0.009949  -0.901    0.367    
## stotal:univ -0.004873   0.003899  -1.250    0.211    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.458 on 6756 degrees of freedom
## Multiple R-squared:  0.1188, Adjusted R-squared:  0.118 
## F-statistic: 151.7 on 6 and 6756 DF,  p-value: < 2.2e-16

Exercise_7

Maral

2023-12-08

CHAPTER(9)

Introduction

(4) i. Classical Errors-in-Variables (CEV) Assumptions

1. No Measurement Error in the Dependent Variable

2. Linear Relationship

3. Exogeneity of the Measurement Error

4. Homoscedasticity

5. No Measurement Error in the Explanatory Variables

ii. Assessment of CEV Assumptions

1. No Measurement Error in the Dependent Variable:

2. Linear Relationship

3. Exogeneity of the Measurement Error

4. Homoscedasticity

5. No Measurement Error in the Explanatory Variables

Assessing the likelihood of CEV assumptions holding requires careful consideration of survey design, reporting behavior, and the nature of the variables involved.

(C2) Wage2

(i) Estimated Return to Education Using KWW

(ii) Estimated Return to Education Using / and KWW Together

(iii) Significance Tests

(C8) TWOYEAR

(i) Sample Mean and Standard Deviation of stotal

(ii) Simple Regressions of jc and univ on stotal

(iii) Test Returns to Two- and Four-Year Colleges

(iv) Test for Quadratic Term in stotal

(v) Test Joint Significance of Interaction Terms

(vi) Final Model Controlling for Ability