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PS9591: Regressions and Causal Inference

12 Lecture 5: The Simple Regression Model II

Slides

  • 5 The Simple Regression Model (link)

12.1 Introduction

We continue studying the simple regression model.

Figure 12.1: Slides for 4 The Simple Regression Model II.

12.2 Vignette 5.1

Let’s create some data:

## This is my population regression:
df <- tibble(x=rnorm(1000000)) %>%
  mutate(y = 1 + 2*x + rnorm(1000000))

df %>%
  ggplot(aes(x=x,y=y)) +
  geom_point(color="royalblue3",alpha = .05) + 
  theme_minimal() 
model_pop <- lm(y~x,data = df)
summary(model_pop)
## 
## Call:
## lm(formula = y ~ x, data = df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.6014 -0.6730  0.0006  0.6733  5.0313 
## 
## Coefficients:
##              Estimate Std. Error t value
## (Intercept) 0.9980409  0.0009995   998.5
## x           2.0003711  0.0009996  2001.1
##             Pr(>|t|)    
## (Intercept)   <2e-16 ***
## x             <2e-16 ***
## ---
## Signif. codes:  
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9995 on 999998 degrees of freedom
## Multiple R-squared:  0.8002, Adjusted R-squared:  0.8002 
## F-statistic: 4.004e+06 on 1 and 999998 DF,  p-value: < 2.2e-16

12.2.1 Theorem SLR.1: Unbiasedness of OLS:

Let’s say that I take n RANDOM samples of size N from the population:

n_samples <- 200
N_size <- 1000

beta_coefs <- NULL

for(i in 1:n_samples){
  df_sample <- df %>%
    slice_sample(n=N_size)
  model_sam <- lm(y~x,data = df_sample)
  
  beta_coefs <- cbind(beta_coefs,coef(model_sam)[2])
}
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Let’s see the mean of these betas:

mean(beta_coefs)
## [1] 2.00359
hist(beta_coefs)

12.3 Vignette 5.2

12.3.1 a. Increased variation in \(\mu\) 

## This is my population regression a:
df_a <- tibble(x=rnorm(100000)) %>%
  mutate(y = 1 + 2*x + rnorm(100000,sd=1))

### Let's say that I take n RANDOM samples of size N from the population:
n_samples <- 200
N_size <- 1000

beta_coefs_a <- NULL

for(i in 1:n_samples){
  df_sample <- df_a %>%
    slice_sample(n=N_size)
  model_sam <- lm(y~x,data = df_sample)
  
  beta_coefs_a <- rbind.data.frame(beta_coefs_a,coef(model_sam)[2])
}
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colnames(beta_coefs_a) <- "betas"
beta_coefs_a$model <- "df_a"

## This is my population regression b:
df_b <- tibble(x=rnorm(100000)) %>%
  mutate(y = 1 + 2*x + rnorm(100000,sd=5))

### Let's say that I take n RANDOM samples of size N from the population:
n_samples <- 200
N_size <- 1000

beta_coefs_b <- NULL

for(i in 1:n_samples){
  df_sample <- df_b %>%
    slice_sample(n=N_size)
  model_sam <- lm(y~x,data = df_sample)
  
  beta_coefs_b <- rbind.data.frame(beta_coefs_b,coef(model_sam)[2])
}
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colnames(beta_coefs_b) <- "betas"
beta_coefs_b$model <- "df_b"

## Now we compare:
beta_coefs_comp <- rbind.data.frame(beta_coefs_a,beta_coefs_b)

beta_coefs_comp %>%
  group_by(model) %>%
  mutate(mean_beta=mean(betas)) %>%
  ungroup() %>%
  ggplot(aes(x=betas,fill=model,color=model)) +
  geom_density(alpha=0.5) + 
  geom_vline(aes(xintercept=mean_beta,color=model),linetype="dashed") +
  theme_minimal() 
## ungroup: no grouping variables remain

12.3.2 b. Increased variation in x is good: 

## This is my population regression a:
df_a <- tibble(x=rnorm(100000,sd=1)) %>%
  mutate(y = 1 + 2*x + rnorm(100000))

### Let's say that I take n RANDOM samples of size N from the population:
n_samples <- 200
N_size <- 1000

beta_coefs_a <- NULL

for(i in 1:n_samples){
  df_sample <- df_a %>%
    slice_sample(n=N_size)
  model_sam <- lm(y~x,data = df_sample)
  
  beta_coefs_a <- rbind.data.frame(beta_coefs_a,coef(model_sam)[2])
}
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colnames(beta_coefs_a) <- "betas"
beta_coefs_a$model <- "df_a"

## This is my population regression b:
df_b <- tibble(x=rnorm(100000,sd=5)) %>%
  mutate(y = 1 + 2*x + rnorm(100000))

### Let's say that I take n RANDOM samples of size N from the population:
n_samples <- 200
N_size <- 1000

beta_coefs_b <- NULL

for(i in 1:n_samples){
  df_sample <- df_b %>%
    slice_sample(n=N_size)
  model_sam <- lm(y~x,data = df_sample)
  
  beta_coefs_b <- rbind.data.frame(beta_coefs_b,coef(model_sam)[2])
}
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colnames(beta_coefs_b) <- "betas"
beta_coefs_b$model <- "df_b"

## Now we compare:
beta_coefs_comp <- rbind.data.frame(beta_coefs_a,beta_coefs_b)

beta_coefs_comp %>%
  group_by(model) %>%
  mutate(mean_beta=mean(betas)) %>%
  ungroup() %>%
  ggplot(aes(x=betas,fill=model,color=model)) +
  geom_density(alpha=0.5) + 
  geom_vline(aes(xintercept=mean_beta,color=model),linetype="dashed") +
  theme_minimal() 
## ungroup: no grouping variables remain

12.3.3 c. Increased sample size is good: 

## This is my population regression a:
df_a <- tibble(x=rnorm(100000)) %>%
  mutate(y = 1 + 2*x + rnorm(100000))

### Let's say that I take n RANDOM samples of size N from the population:
n_samples <- 200
N_size <- 20

beta_coefs_a <- NULL

for(i in 1:n_samples){
  df_sample <- df_a %>%
    slice_sample(n=N_size)
  model_sam <- lm(y~x,data = df_sample)
  
  beta_coefs_a <- rbind.data.frame(beta_coefs_a,coef(model_sam)[2])
}
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colnames(beta_coefs_a) <- "betas"
beta_coefs_a$model <- "df_a"

## This is my population regression b:
df_b <- tibble(x=rnorm(100000)) %>%
  mutate(y = 1 + 2*x + rnorm(100000))

### Let's say that I take n RANDOM samples of size N from the population:
n_samples <- 200
N_size <- 2000

beta_coefs_b <- NULL

for(i in 1:n_samples){
  df_sample <- df_b %>%
    slice_sample(n=N_size)
  model_sam <- lm(y~x,data = df_sample)
  
  beta_coefs_b <- rbind.data.frame(beta_coefs_b,coef(model_sam)[2])
}
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colnames(beta_coefs_b) <- "betas"
beta_coefs_b$model <- "df_b"

## Now we compare:
beta_coefs_comp <- rbind.data.frame(beta_coefs_a,beta_coefs_b)

beta_coefs_comp %>%
  group_by(model) %>%
  mutate(mean_beta=mean(betas)) %>%
  ungroup() %>%
  ggplot(aes(x=betas,fill=model,color=model)) +
  geom_density(alpha=0.5) + 
  geom_vline(aes(xintercept=mean_beta,color=model),linetype="dashed") +
  theme_minimal() 
## ungroup: no grouping variables remain
### NOTE: beta is unbiased!!