Chap2. Use the data in BWGHT to answer this question.

#install.packages("wooldridge")
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(wooldridge)
data<-wooldridge::bwght

6. Using data from 1988 for houses sold in Andover, Massachusetts, from Kiel and McClain (1995), the following equation relates housing price (price) to the distance from a recently built garbage incinerator (dist): log(price)=9.40+0.312 log(dist) n=135, R^2=0.162.

1. Interpret the coefficient on log(dist). Is the sign of this estimate what you expect it to be? Answer: There is a positive relation between log(dist) and log(price) it means the farther you are from a garbage incinerator the more your house will sell for.
2. Do you think simple regression provides an unbiased estimator of the ceteris paribus elasticity of price with respect to dist? (Think about the city’s decision on where to put the incinerator.) Answer: A city wouldn’t put a garbage incinerator next to a neighborhood thats already built because they know that the houses in the surrounding area would lose value. Therefore, the distance of a house to a garbage incinerator is more biased.
3. What other factors about a house affect its price? Might these be correlated with distance from the incinerator? Answer: Other factors about price of a house besides location are how old the house is, if it’s newly updated, and how many sqft it is. Chap4. Use the data in WAGE2 to estimate a simple regression explaining monthly salary (wage) in terms of IQ score (IQ).
4. Find the average salary and average IQ in the sample. What is the sample standard deviation of IQ? (IQ scores are standardized so that the average in the population is 100 with a standard deviation equal to 15.)

library(wooldridge)
Datanew<-wooldridge::wage2
attach(Datanew)
mean(wage)

## [1] 957.9455

mean(IQ)

## [1] 101.2824

sd(IQ)

## [1] 15.05264

modelnew<-lm(wage~IQ)
summary(Datanew)

##       wage            hours             IQ             KWW       
##  Min.   : 115.0   Min.   :20.00   Min.   : 50.0   Min.   :12.00  
##  1st Qu.: 669.0   1st Qu.:40.00   1st Qu.: 92.0   1st Qu.:31.00  
##  Median : 905.0   Median :40.00   Median :102.0   Median :37.00  
##  Mean   : 957.9   Mean   :43.93   Mean   :101.3   Mean   :35.74  
##  3rd Qu.:1160.0   3rd Qu.:48.00   3rd Qu.:112.0   3rd Qu.:41.00  
##  Max.   :3078.0   Max.   :80.00   Max.   :145.0   Max.   :56.00  
##                                                                  
##       educ           exper           tenure            age       
##  Min.   : 9.00   Min.   : 1.00   Min.   : 0.000   Min.   :28.00  
##  1st Qu.:12.00   1st Qu.: 8.00   1st Qu.: 3.000   1st Qu.:30.00  
##  Median :12.00   Median :11.00   Median : 7.000   Median :33.00  
##  Mean   :13.47   Mean   :11.56   Mean   : 7.234   Mean   :33.08  
##  3rd Qu.:16.00   3rd Qu.:15.00   3rd Qu.:11.000   3rd Qu.:36.00  
##  Max.   :18.00   Max.   :23.00   Max.   :22.000   Max.   :38.00  
##                                                                  
##     married          black            south            urban       
##  Min.   :0.000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
##  1st Qu.:1.000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000  
##  Median :1.000   Median :0.0000   Median :0.0000   Median :1.0000  
##  Mean   :0.893   Mean   :0.1283   Mean   :0.3412   Mean   :0.7176  
##  3rd Qu.:1.000   3rd Qu.:0.0000   3rd Qu.:1.0000   3rd Qu.:1.0000  
##  Max.   :1.000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  
##                                                                    
##       sibs           brthord           meduc           feduc      
##  Min.   : 0.000   Min.   : 1.000   Min.   : 0.00   Min.   : 0.00  
##  1st Qu.: 1.000   1st Qu.: 1.000   1st Qu.: 8.00   1st Qu.: 8.00  
##  Median : 2.000   Median : 2.000   Median :12.00   Median :10.00  
##  Mean   : 2.941   Mean   : 2.277   Mean   :10.68   Mean   :10.22  
##  3rd Qu.: 4.000   3rd Qu.: 3.000   3rd Qu.:12.00   3rd Qu.:12.00  
##  Max.   :14.000   Max.   :10.000   Max.   :18.00   Max.   :18.00  
##                   NA's   :83       NA's   :78      NA's   :194    
##      lwage      
##  Min.   :4.745  
##  1st Qu.:6.506  
##  Median :6.808  
##  Mean   :6.779  
##  3rd Qu.:7.056  
##  Max.   :8.032  
## 

rm(Datanew, model1, modelnew)

## Warning in rm(Datanew, model1, modelnew): object 'model1' not found

2. Estimate a simple regression model where a one-point increase in IQ changes wage by a constant dollar amount. Use this model to find the predicted increase in wage for an increase in IQ of 15 points. Does IQ explain most of the variation in wage?

library(wooldridge)
Data4<-wooldridge::wage2
attach(Data4)

## The following objects are masked from Datanew:
## 
##     age, black, brthord, educ, exper, feduc, hours, IQ, KWW, lwage,
##     married, meduc, sibs, south, tenure, urban, wage

mean("wage")

## Warning in mean.default("wage"): argument is not numeric or logical: returning
## NA

## [1] NA

mean("IQ")

## Warning in mean.default("IQ"): argument is not numeric or logical: returning NA

## [1] NA

sd("IQ")

## Warning in var(if (is.vector(x) || is.factor(x)) x else as.double(x), na.rm =
## na.rm): NAs introduced by coercion

## [1] NA

model<-lm(wage~IQ)
summary(model)

## 
## Call:
## lm(formula = wage ~ IQ)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -898.7 -256.5  -47.3  201.1 2072.6 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 116.9916    85.6415   1.366    0.172    
## IQ            8.3031     0.8364   9.927   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 384.8 on 933 degrees of freedom
## Multiple R-squared:  0.09554,    Adjusted R-squared:  0.09457 
## F-statistic: 98.55 on 1 and 933 DF,  p-value: < 2.2e-16

3. Now, estimate a model where each one-point increase in IQ has the same percentage effect on wage. If IQ increases by 15 points, what is the approximate percentage increase in predicted wage?

Answer:An increase of IQ of 15 points will increase Wage by 124.5 points.Also, after running the command for linear regression in r-studio, we get R2=0.094. Variation in IQ explains only 9.4% of variation in Wage. Therefore, variation in IQ doesn’t explain most of the variations in Wage.

Chapter3. 6 i.IQ~ educ The coef is 3.53, meaning that 1 year increase in education resulting in 3.53 point increase in IQ

model <- lm(IQ~educ, data= data)
summary(model)

## 
## Call:
## lm(formula = IQ ~ educ, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -50.228  -7.262   0.907   8.772  37.373 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  53.6872     2.6229   20.47   <2e-16 ***
## educ          3.5338     0.1922   18.39   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.9 on 933 degrees of freedom
## Multiple R-squared:  0.2659, Adjusted R-squared:  0.2652 
## F-statistic:   338 on 1 and 933 DF,  p-value: < 2.2e-16

theta1= model$coefficients[2]

ii.wage ~ educ The coef is 0.059839, meaning that 1 year increase in education resulting in 0.06 USD increase in log(wage)

model <- lm(log(wage)~educ, data= data)
summary(model)

## 
## Call:
## lm(formula = log(wage) ~ educ, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.94620 -0.24832  0.03507  0.27440  1.28106 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 5.973062   0.081374   73.40   <2e-16 ***
## educ        0.059839   0.005963   10.04   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4003 on 933 degrees of freedom
## Multiple R-squared:  0.09742,    Adjusted R-squared:  0.09645 
## F-statistic: 100.7 on 1 and 933 DF,  p-value: < 2.2e-16

beta1= model$coefficients[2]

3. wage ~ educ+iq The coef are 0.0391199 and 0.0058631 for educ and IQ respectively. It indicates that if the years of education do not change, 1 point increase in IQ resulting in 0.006 USD increase in the log(wage). Besidesm if the IQ level is fixed, 1 year increase in education resulting in 0.039 USD increase in the log(wage).

model <- lm(log(wage)~educ+IQ, data= data)
summary(model)

## 
## Call:
## lm(formula = log(wage) ~ educ + IQ, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.01601 -0.24367  0.03359  0.27960  1.23783 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 5.6582876  0.0962408  58.793  < 2e-16 ***
## educ        0.0391199  0.0068382   5.721 1.43e-08 ***
## IQ          0.0058631  0.0009979   5.875 5.87e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3933 on 932 degrees of freedom
## Multiple R-squared:  0.1297, Adjusted R-squared:  0.1278 
## F-statistic: 69.42 on 2 and 932 DF,  p-value: < 2.2e-16

beta1_2= model$coefficients[2]
beta2= model$coefficients[3]

iv.The verification Due to the error term we don’t have exactly substraction equal to 0, the result near 0 at -0.02

beta1_2+beta2*theta1 - beta1

## educ 
##    0

C8 i) Mean and standard deviation The unit of measurement in income is USD while that for prpblck is the percentage

data8 <- discrim
print(paste("The mean of income will be:", mean(data8$income, na.rm=TRUE)))

## [1] "The mean of income will be: 47053.7848410758"

print(paste("The standard deviation of income will be:", sd(data8$income, na.rm=TRUE)))

## [1] "The standard deviation of income will be: 13179.2860689389"

print(paste("The mean of people black will be:", mean(data8$prpblck,na.rm=TRUE)))

## [1] "The mean of people black will be: 0.113486396497833"

print(paste("The standard deviation of people black will be:", sd(data8$prpblck, na.rm=TRUE)))

## [1] "The standard deviation of people black will be: 0.182416467486231"

2. psoda~ prpblck+income The coef of prpblck is 0.0115, meaning that if we increase black people by 1% in zipcode, the price of medium soda will increase by 0.0115 USD. It is not economically lag.

sample_size <- data8$psoda %>% na.omit %>% as.data.frame()
sample_size <- nrow(sample_size)
model <- lm(psoda~ prpblck+income, data=data8)
summary <- summary(model)
summary

## 
## Call:
## lm(formula = psoda ~ prpblck + income, data = data8)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.29401 -0.05242  0.00333  0.04231  0.44322 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 9.563e-01  1.899e-02  50.354  < 2e-16 ***
## prpblck     1.150e-01  2.600e-02   4.423 1.26e-05 ***
## income      1.603e-06  3.618e-07   4.430 1.22e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.08611 on 398 degrees of freedom
##   (9 observations deleted due to missingness)
## Multiple R-squared:  0.06422,    Adjusted R-squared:  0.05952 
## F-statistic: 13.66 on 2 and 398 DF,  p-value: 1.835e-06

print(paste("The sample size is:",sample_size))

## [1] "The sample size is: 402"

print(paste("The Rsquared is:",summary$r.squared))

## [1] "The Rsquared is: 0.0642203910903628"

3. psoda ~ prpblck The coef is 0.0649. Hence, the effect is smaller when we control for income

model <- lm(psoda~prpblck, data=data8)
summary(model)

## 
## Call:
## lm(formula = psoda ~ prpblck, data = data8)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.30884 -0.05963  0.01135  0.03206  0.44840 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.03740    0.00519  199.87  < 2e-16 ***
## prpblck      0.06493    0.02396    2.71  0.00702 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0881 on 399 degrees of freedom
##   (9 observations deleted due to missingness)
## Multiple R-squared:  0.01808,    Adjusted R-squared:  0.01561 
## F-statistic: 7.345 on 1 and 399 DF,  p-value: 0.007015

4. log(psoda) ~ prpblck ~ log(income) The coef is 0.1216. Hence, the effect is smaller when we control for income.

model <- lm(log(psoda)~prpblck+ log(income), data=data8)
summary(model)

## 
## Call:
## lm(formula = log(psoda) ~ prpblck + log(income), data = data8)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.33563 -0.04695  0.00658  0.04334  0.35413 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.79377    0.17943  -4.424 1.25e-05 ***
## prpblck      0.12158    0.02575   4.722 3.24e-06 ***
## log(income)  0.07651    0.01660   4.610 5.43e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0821 on 398 degrees of freedom
##   (9 observations deleted due to missingness)
## Multiple R-squared:  0.06809,    Adjusted R-squared:  0.06341 
## F-statistic: 14.54 on 2 and 398 DF,  p-value: 8.039e-07

22. log(psoda) ~ prpblck ~ log(income)+ prppov The coef decrease to 0.07281 and the significant level also decreases

model <- lm(log(psoda)~prpblck+ log(income)+prppov, data=data8)
summary(model)

## 
## Call:
## lm(formula = log(psoda) ~ prpblck + log(income) + prppov, data = data8)
## 
## 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

6. correlation The correlation is not as my expectation. It tells that proportion of poverty and price of soda is only 0.0259, meaning no relation. I expected the poorer the population, the less price we obtain. # Remove missing values from the vectors

data8_6 <- data8 %>% select(psoda,prppov) %>% na.omit 
cor(data8_6)

##             psoda     prppov
## psoda  1.00000000 0.02598077
## prppov 0.02598077 1.00000000

7. statement evaluation Because the correaltion is very high at -0.83. It means that the added value of prppove is very low when consider information of income already. Besides, 2 correlated variables in the model also can cause the multicolinearity. Hence, the statement is true. # Remove missing values from the vectors

data8_7 <- data8 %>% select(income,prppov) %>% na.omit %>% mutate(income= log(income))
cor(data8_7)

##           income    prppov
## income  1.000000 -0.838467
## prppov -0.838467  1.000000

C9 i) model report

model <- lm(log(psoda)~prpblck+ log(income)+prppov, data=data8)
summary(model)

## 
## Call:
## lm(formula = log(psoda) ~ prpblck + log(income) + prppov, data = data8)
## 
## 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

2. correlation The correlation is -0.838 Both variables are statistically significant with p(log(income)~0, p(prppove)= 0.0044

data9_2 <- data8 %>% select(income,prppov) %>% na.omit %>% mutate(income= log(income))
cor(data9_2)

##           income    prppov
## income  1.000000 -0.838467
## prppov -0.838467  1.000000

3. Model modification The coef is 0.121, meaning that an increase 1 USD in housing value will result in 0.121 USD in price of soda. And the null hypothesis is strongly reject with p-value~0

model <- lm(log(psoda)~prpblck+ log(income)+prppov+log(hseval), data=data8)
summary(model)

## 
## Call:
## lm(formula = log(psoda) ~ prpblck + log(income) + prppov + log(hseval), 
##     data = data8)
## 
## 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

4. Insignificant value At 5% significance level, the 2 variables have joint effect

modelur <- lm(log(psoda)~prpblck+ log(income)+prppov+log(hseval), data=data8)
modelr <- lm(log(psoda)~prpblck+log(hseval), data=data8)
ftest <- anova(modelur, modelr)
print(ftest)

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

v)Model Intepretation Based on the model in part 9.3, we conclude that prpblck and log(hseval) are two most reliable variables because of their significance value in both coefficients and small p-value