in class exercise 4

Main Model Variables:

• log(psoda): Log of soda prices (dependent variable)

• prpblck: Proportion of Black residents in the zip code

• log(income): Log of median income in the zip code

• prppov: Proportion of population in poverty

• log(hseval): Log of median housing value (added later)

Load required libraries

library(wooldridge) 
library(car)         

## Loading required package: carData

data("discrim")

(i) First regression model

model1 <- lm(log(psoda) ~ prpblck + log(income) + prppov, data = discrim)
summary(model1)

## 
## Call:
## lm(formula = log(psoda) ~ prpblck + log(income) + prppov, data = discrim)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.32218 -0.04648  0.00651  0.04272  0.35622 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.46333    0.29371  -4.982  9.4e-07 ***
## prpblck      0.07281    0.03068   2.373   0.0181 *  
## log(income)  0.13696    0.02676   5.119  4.8e-07 ***
## prppov       0.38036    0.13279   2.864   0.0044 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.08137 on 397 degrees of freedom
##   (9 observations deleted due to missingness)
## Multiple R-squared:  0.08696,    Adjusted R-squared:  0.08006 
## F-statistic:  12.6 on 3 and 397 DF,  p-value: 6.917e-08

• This runs a log-linear regression

• The log transformation on psoda and income helps handle non-linear relationships

• Coefficients will represent percentage changes

(ii) Correlation analysis

cor.test(log(discrim$income), discrim$prppov)

## 
##  Pearson's product-moment correlation
## 
## data:  log(discrim$income) and discrim$prppov
## t = -31.04, df = 407, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.8650980 -0.8071224
## sample estimates:
##       cor 
## -0.838467

• Tests the relationship between income and poverty

• Important for checking potential multicollinearity

(iii) Extended regression model with hseval

model2 <- lm(log(psoda) ~ prpblck + log(income) + prppov + log(hseval), data = discrim)
summary(model2)

## 
## Call:
## lm(formula = log(psoda) ~ prpblck + log(income) + prppov + log(hseval), 
##     data = discrim)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.30652 -0.04380  0.00701  0.04332  0.35272 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.84151    0.29243  -2.878 0.004224 ** 
## prpblck      0.09755    0.02926   3.334 0.000937 ***
## log(income) -0.05299    0.03753  -1.412 0.158706    
## prppov       0.05212    0.13450   0.388 0.698571    
## log(hseval)  0.12131    0.01768   6.860 2.67e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.07702 on 396 degrees of freedom
##   (9 observations deleted due to missingness)
## Multiple R-squared:  0.1839, Adjusted R-squared:  0.1757 
## F-statistic: 22.31 on 4 and 396 DF,  p-value: < 2.2e-16

• Adds housing values as a control variable

• Helps account for general cost of living differences

(iv) Joint significance test for income and prppov

model3 <- lm(log(psoda) ~ prpblck + log(hseval), data = discrim)
anova(model3, model2)

## Analysis of Variance Table
## 
## Model 1: log(psoda) ~ prpblck + log(hseval)
## Model 2: log(psoda) ~ prpblck + log(income) + prppov + log(hseval)
##   Res.Df    RSS Df Sum of Sq      F  Pr(>F)  
## 1    398 2.3911                              
## 2    396 2.3493  2  0.041797 3.5227 0.03045 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Tests whether income and poverty together improve the model

Useful for dealing with potentially correlated variables

vif(model2)

##     prpblck log(income)      prppov log(hseval) 
##    1.951820    7.724903    5.627505    3.193821

par(mfrow=c(2,2))
plot(model2)

• If we still find significant effects of racial composition after controlling for all these factors, it provides stronger evidence of potential discrimination

• It’s less likely to produce false positives in identifying discrimination

v. The most reliable regression would be model2,

which includes log(hseval): log(psoda) ~ prpblck + log(income) + prppov + log(hseval)

Here’s why this model is most reliable:

1. Completeness of Controls:

   • It includes housing values (hseval) which controls for general cost of living and neighborhood characteristics
   • This helps isolate the true effect of racial composition (prpblck) by accounting for economic differences between areas

2. Reduced Omitted Variable Bias:

   • Without housing values, we might incorrectly attribute price differences to racial composition when they’re actually due to overall neighborhood cost structures
   • Housing values serve as a proxy for unobserved neighborhood characteristics

3. Better Model Specification:

   • Including both economic variables (income, poverty) and housing values provides a more complete picture of neighborhood characteristics
   • This helps distinguish between racial discrimination and legitimate cost-based price differences

4. More Conservative Estimate:

   • If we still find significant effects of racial composition after controlling for all these factors, it provides stronger evidence of potential discrimination
   • It’s less likely to produce false positives in identifying discrimination