Review: u

Omitted Variable Bias I

Omitted Variable Bias I

Omitted Variable Bias I

Omitted Variable Bias II

Omitted Variable Bias III

Omited Variable Bias: Class Size Example

• Which of the following possible variables would cause a bias if omitted?

1. \(Z_i\): time of day of the test

2. \(Z_i\): parking space per student

3. \(Z_i\): percent of ESL students

Recall: Endogeneity and Bias

Endogeneity and Bias: Correlations I

• Here is where checking correlations between variables helps:

# Select only the three variables we want (there are many)
CAcorr <- CASchool %>%
  select("str","testscr","el_pct")
# Make a correlation table
cor_table <- cor(CAcorr)
cor_table # look at it

##                str    testscr     el_pct
## str      1.0000000 -0.2263628  0.1876424
## testscr -0.2263628  1.0000000 -0.6441237
## el_pct   0.1876424 -0.6441237  1.0000000

Endogeneity and Bias: Correlations II

• Here is where checking correlations between variables helps:

# Make a correlation plot
library(corrplot)
corrplot(cor_table, type="upper", 
         method = "circle",
         order="original")

• el_pct is strongly correlated with testscr (Condition 1)
• el_pct is reasonably correlated with str (Condition 2)

Look at Conditional Distributions I

# make a new variable called EL
# = high (if el_pct is above median) or = low (if below median)
CASchool <- CASchool %>% # next we create a new dummy variable called ESL
  mutate(ESL = ifelse(el_pct > median(el_pct), # test if ESL is above median
                     yes = "High ESL", # if yes, call this variable "High ESL"
                     no = "Low ESL")) # if no, call this variable "Low ESL"
# get average test score by high/low EL
CASchool %>%
  group_by(ESL) %>%
  summarize(Average_test_score = mean(testscr))

Look at Conditional Distributions II

ggplot(data = CASchool)+
  aes(x = testscr,
      fill = ESL)+
  geom_density(alpha=0.5)+
  labs(x = "Test Score",
       y = "Density")+
  ggthemes::theme_pander(
    base_family = "Fira Sans Condensed",
    base_size=20
    )+
  theme(legend.position = "bottom")

Look at Conditional Distributions III

esl_scatter <- ggplot(data = CASchool)+
  aes(x = str,
      y = testscr,
      color = ESL)+
  geom_point()+
  geom_smooth(method = "lm")+
  labs(x = "STR",
       y = "Test Score")+
  ggthemes::theme_pander(
    base_family = "Fira Sans Condensed",
    base_size=20
    )+
  theme(legend.position = "bottom")
esl_scatter

Look at Conditional Distributions III

esl_scatter+
  facet_grid(~ESL)+
  guides(color = F)

Omitted Variable Bias in the Class Size Example

• \(cor(STR,u)\) is positive (via \(\%EL\))

• \(cor(u, \text{Test score})\) is negative (via \(\%EL\))

• \(\beta_1\) is negative (between Test score and STR)

• Bias is positive

  • But since \(\color{red}{\beta_1}\) is negative, it’s made to be a larger negative number than it truly is
  • Implies that \(\color{red}{\beta_1}\) overstates the effect of reducing STR on improving Test Scores

Omitted Variable Bias: Messing with Causality I

• If school districts with higher Test Scores happen to have both lower STR AND districts with smaller \(STR\) sizes tend to have less \(\%EL\) ...

• How can we say \(\hat{\beta_1}\) estimates the marginal effect of \(\Delta STR \rightarrow \Delta \text{Test Score}\)?

• (We can’t.)

Omitted Variable Bias: Messing with Causality II

RCTs Neutralize Omitted Variable Bias I

RCTs Neutralize Omitted Variable Bias II

But We Rarely, if Ever, Have RCTs

Another Way to Control for Variables

Another Way to Control for Variables

Controlling for Variables

Controlling for Variables

Multivariate Econometric Models Overview

^† Note Bailey defines k to include both the number of variables plus the constant.

Marginal Effects I

$$Y_i= \beta_0+\beta_1 X_{1i} + \beta_2 X_{2i}$$

• Consider changing \(X_1\) by \(\Delta X_1\) while holding \(X_2\) constant:

$$\begin{align*} Y&= \beta_0+\beta_1 X_{1} + \beta_2 X_{2} && \text{Before the change}\\ \end{align*}$$

Marginal Effects I

$$Y_i= \beta_0+\beta_1 X_{1i} + \beta_2 X_{2i}$$

• Consider changing \(X_1\) by \(\Delta X_1\) while holding \(X_2\) constant:

$$\begin{align*} Y&= \beta_0+\beta_1 X_{1} + \beta_2 X_{2} && \text{Before the change}\\ Y+\Delta Y&= \beta_0+\beta_1 (X_{1}+\Delta X_1) + \beta_2 X_{2} && \text{After the change}\\ \end{align*}$$

Marginal Effects I

$$Y_i= \beta_0+\beta_1 X_{1i} + \beta_2 X_{2i}$$

• Consider changing \(X_1\) by \(\Delta X_1\) while holding \(X_2\) constant:

$$\begin{align*} Y&= \beta_0+\beta_1 X_{1} + \beta_2 X_{2} && \text{Before the change}\\ Y+\Delta Y&= \beta_0+\beta_1 (X_{1}+\Delta X_1) + \beta_2 X_{2} && \text{After the change}\\ \Delta Y&= \beta_1 \Delta X_1 && \text{The difference}\\ \end{align*}$$

Marginal Effects I

$$Y_i= \beta_0+\beta_1 X_{1i} + \beta_2 X_{2i}$$

• Consider changing \(X_1\) by \(\Delta X_1\) while holding \(X_2\) constant:

$$\begin{align*} Y&= \beta_0+\beta_1 X_{1} + \beta_2 X_{2} && \text{Before the change}\\ Y+\Delta Y&= \beta_0+\beta_1 (X_{1}+\Delta X_1) + \beta_2 X_{2} && \text{After the change}\\ \Delta Y&= \beta_1 \Delta X_1 && \text{The difference}\\ \frac{\Delta Y}{\Delta X_1} &= \beta_1 && \text{Solving for } \beta_1\\ \end{align*}$$

Marginal Effects II

$$\beta_1 =\frac{\Delta Y}{\Delta X_1}\text{ holding } X_2 \text{ constant}$$

Similarly, for \(\beta_2\):

$$\beta_2 =\frac{\Delta Y}{\Delta X_2}\text{ holding }X_1 \text{ constant}$$

And for the constant, \(\beta_0\):

$$\beta_0 =\text{predicted value of Y when } X_1=0, \; X_2=0$$

You Can Keep Your Intuitions...But They're Wrong Now

The “Constant”

• Alternatively, we can write the population regression equation as: $$Y_i=\beta_0\color{#e64173}{X_{0i}}+\beta_1X_{1i}+\beta_2X_{2i}+u_i$$

• Here, we added \(X_{0i}\) to \(\beta_0\)

• \(X_{0i}\) is a constant regressor, as we define \(X_{0i}=1\) for all \(i\) observations

• Likewise, \(\beta_0\) is more generally called the “constant” term in the regression (instead of the “intercept”)

• This may seem silly and trivial, but this will be useful next class!

The Population Regression Model: Example I

• Let's see what you remember from micro(econ)!

• What measures the price effect? What sign should it have?

• What measures the income effect? What sign should it have? What should inferior or normal (necessities & luxury) goods look like?

• What measures the cross-price effect(s)? What sign should substitutes and complements have?

The Population Regression Model: Example I

• Interpret each \(\hat{\beta}\)

Multivariate OLS in R

# run regression of testscr on str and el_pct
school_reg_2 <- lm(testscr ~ str + el_pct, 
                 data = CASchool)

lm(y ~ x1 + x2, data = df)

Multivariate OLS in R II

# look at reg object
school_reg_2

## 
## Call:
## lm(formula = testscr ~ str + el_pct, data = CASchool)
## 
## Coefficients:
## (Intercept)          str       el_pct  
##    686.0322      -1.1013      -0.6498

Multivariate OLS in R III

summary(school_reg_2) # get full summary

## 
## Call:
## lm(formula = testscr ~ str + el_pct, data = CASchool)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -48.845 -10.240  -0.308   9.815  43.461 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 686.03225    7.41131  92.566  < 2e-16 ***
## str          -1.10130    0.38028  -2.896  0.00398 ** 
## el_pct       -0.64978    0.03934 -16.516  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.46 on 417 degrees of freedom
## Multiple R-squared:  0.4264,    Adjusted R-squared:  0.4237 
## F-statistic:   155 on 2 and 417 DF,  p-value: < 2.2e-16

Multivariate OLS in R IV: broom

# load packages
library(broom)
# tidy regression output
tidy(school_reg_2)

Multivariate Regression Output Table

library(huxtable)
huxreg("Model 1" = school_reg,
       "Model 2" = school_reg_2,
       coefs = c("Intercept" = "(Intercept)",
                 "Class Size" = "str",
                 "%ESL Students" = "el_pct"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 2)