Assign5

Q1.(i) How many single-person households are there in the data set?

k401ksubs <- read_excel("/Users/jingwu/Desktop/401ksubs.xls")
sum(k401ksubs$fsize==1)

## [1] 2017

2. Use OLS to estimate the model

single_house <- subset(k401ksubs, fsize==1)
fit_ksubs <- lm(nettfa~inc+age, data = `single_house`)
summary(fit_ksubs)

## 
## Call:
## lm(formula = nettfa ~ inc + age, data = single_house)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -179.95  -14.16   -3.42    6.03 1113.94 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -43.03981    4.08039 -10.548   <2e-16 ***
## inc           0.79932    0.05973  13.382   <2e-16 ***
## age           0.84266    0.09202   9.158   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 44.68 on 2014 degrees of freedom
## Multiple R-squared:  0.1193, Adjusted R-squared:  0.1185 
## F-statistic: 136.5 on 2 and 2014 DF,  p-value: < 2.2e-16

The coefficient is positive but not that large as what I would expected.

3. Does the intercept from the regression in part (ii) have an interesting meaning? Explain.

Considering we are taking age as one control, 0 age may not be really meaningful in this case.

4. Find the p-value for the test H0 :β2 =1 against Ha :β2 <1. Do you reject H0 at the 1% significance level? (0.84266 - 1 )/0.09202=-1.709846

pnorm(-1.709846)

## [1] 0.04364718

thus it’s not enough to reject at 1% significance level.

22. If you do a simple regression of nettfa on inc, is the estimated coefficient on inc much different from the estimate in part (ii)? Why or why not?

single_house <- subset(k401ksubs, fsize==1)
fit_ksubs <- lm(nettfa~inc, data = `single_house`)
summary(fit_ksubs)

## 
## Call:
## lm(formula = nettfa ~ inc, data = single_house)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -185.12  -12.85   -4.85    1.78 1112.66 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -10.5709     2.0607   -5.13 3.18e-07 ***
## inc           0.8207     0.0609   13.48  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 45.59 on 2015 degrees of freedom
## Multiple R-squared:  0.08267,    Adjusted R-squared:  0.08222 
## F-statistic: 181.6 on 1 and 2015 DF,  p-value: < 2.2e-16

The coefficient didn’t change a lot though the intercept changed.

Q2. In R, use a seed of 123 and generate n = 1000 samples, if we want to test β2 − β1 = 0, we are trying to see β1 = β2, and β1x1 + β2x2 = β1*(x1+x2), that’s why we create w = x1 + x2 and regress y on w and x1.

set.seed(123)
n <- 1000
X1 <- rnorm(n,mean=5,sd=sqrt(10))
X2 <- 2.5+3*(X1)+rnorm(n,mean=0,sd=1)
Y <- 10+2*(X1)+2*(X2)+rnorm(n,mean=0,sd=1)
w <- (X1)+(X2)
data <- data.frame(Y,X1,X2,w)
model_1 <- lm(Y~X1+X2, data=data)
model_2 <- lm(Y~X1+w, data=data)
summary(model_1)

## 
## Call:
## lm(formula = Y ~ X1 + X2, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.8360 -0.6277 -0.0370  0.6538  3.3787 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  9.94428    0.09441  105.33   <2e-16 ***
## X1           1.91068    0.09373   20.38   <2e-16 ***
## X2           2.02751    0.03079   65.86   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9788 on 997 degrees of freedom
## Multiple R-squared:  0.9985, Adjusted R-squared:  0.9985 
## F-statistic: 3.344e+05 on 2 and 997 DF,  p-value: < 2.2e-16

summary(model_2)

## 
## Call:
## lm(formula = Y ~ X1 + w, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.8360 -0.6277 -0.0370  0.6538  3.3787 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  9.94428    0.09441 105.333   <2e-16 ***
## X1          -0.11682    0.12439  -0.939    0.348    
## w            2.02751    0.03079  65.860   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9788 on 997 degrees of freedom
## Multiple R-squared:  0.9985, Adjusted R-squared:  0.9985 
## F-statistic: 3.344e+05 on 2 and 997 DF,  p-value: < 2.2e-16

Q3. See attached picture.