15 Lecture 6: The Multiple Regression Model I

Slides

7 The Multiple Regression Model (link)

15.1 Introduction

We continue studying the simple regression model.

Figure 15.1: Slides for 7 The Multiple Regression Model.

15.2 Vignette 6.1

Once again, let’s simulate some data. Maybe we are interested in urban and rural towns (70% are urban) :

df <- tibble(urban = sample(c(0,1),500,replace=T,prob=c(.3,.7))) %>%
  ## Urban towns spend, on average, $3 million more on wages than rural towns
  mutate(expen_wages = 3*urban+runif(500,min=0,max=4)) %>%
  ## Urban towns are also have greater incomes (e.g., from taxes), but these are reduced by their high wage expenditures:
  mutate(log_income = 1 + 2*urban - .3*expen_wages + rnorm(500,mean=2)) ## <- Population Eq.

Now we can estimate the effect of wage expenditure on income:

model_a <- lm(log_income ~ expen_wages, data = df) 
summary(model_a)

## 
## Call:
## lm(formula = log_income ~ expen_wages, data = df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.5228 -0.7246  0.0009  0.7553  2.9893 
## 
## Coefficients:
##             Estimate Std. Error t value
## (Intercept)  2.66707    0.12125  21.996
## expen_wages  0.11798    0.02746   4.296
##             Pr(>|t|)    
## (Intercept)  < 2e-16 ***
## expen_wages 2.09e-05 ***
## ---
## Signif. codes:  
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.128 on 498 degrees of freedom
## Multiple R-squared:  0.03574,    Adjusted R-squared:  0.0338 
## F-statistic: 18.46 on 1 and 498 DF,  p-value: 2.09e-05

Wait what? (Interpret a log ~ level)

15.3 Vignette 6.2

Let’s see… How can we remove everything from wages that is explained by urban? How can we remove everything from income that is explained by urban?

df %>% group_by(urban) %>%
  summarise(income_urb= mean(log_income))

## summarise: now 2 rows and 2 columns, ungrouped

## # A tibble: 2 × 2
##   urban income_urb
##   <dbl>      <dbl>
## 1     0       2.43
## 2     1       3.49

df %>% group_by(urban) %>% 
  summarise(expen_wages_urb = mean(expen_wages))

## summarise: now 2 rows and 2 columns, ungrouped