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3.3 — Omitted Variable Bias

ECON 480 • Econometrics • Fall 2020

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/metricsF20
metricsF20.classes.ryansafner.com

Review: u

Yi=β0+β1Xi+ui

  • Error term, ui includes all other variables that affect Y

  • Every regression model always has omitted variables assumed in the error

    • Most unobservable (hence "u") or hard to measure
    • Examples: innate ability, weather at the time, etc

  • Again, we assume u is random, with E[u|X]=0 and var(u)=σ2u

  • Sometimes, omission of variables can bias OLS estimators (^β0 and ^β1)

Omitted Variable Bias I

  • Omitted variable bias (OVB) for some omitted variable Z exists if two conditionsa are met:

1. Z is a determinant of Y

  • i.e. Z is in the error term, ui

Omitted Variable Bias I

  • Omitted variable bias (OVB) for some omitted variable Z exists if two conditionsa are met:

Omitted Variable Bias I

  • Omitted variable bias (OVB) for some omitted variable Z exists if two conditionsa are met:

1. Z is a determinant of Y

  • i.e. Z is in the error term, ui

2. Z is correlated with the regressor X

  • i.e. cor(X,Z)≠0
  • implies cor(X,U)≠0
  • implies X is endogenous

Omitted Variable Bias II

  • Omitted variable bias makes X endogenous
    • E(ui|Xi)≠0⟹ knowing X tells you something about ui
    • Knowing X tells you something about Y not by way of X!

Omitted Variable Bias III

  • ^β1 is biased: E[^β1]≠β1

  • ^β1 systematically over- or under-estimates the true relationship (β1)

  • ^β1 “picks up” both:

    • X→Y
    • X←Z→Y

Omited Variable Bias: Class Size Example

Example: Consider our recurring class size and test score example: Test scorei=β0+β1STRi+ui

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

Omited Variable Bias: Class Size Example

Example: Consider our recurring class size and test score example: Test scorei=β0+β1STRi+ui

  • Which of the following possible variables would cause a bias if omitted?
  1. Zi: time of day of the test

Omited Variable Bias: Class Size Example

Example: Consider our recurring class size and test score example: Test scorei=β0+β1STRi+ui

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

  1. Zi: time of day of the test

  2. Zi: parking space per student

Omited Variable Bias: Class Size Example

Example: Consider our recurring class size and test score example: Test scorei=β0+β1STRi+ui

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

  1. Zi: time of day of the test

  2. Zi: parking space per student

  3. Zi: percent of ESL students

Recall: Endogeneity and Bias

  • The true expected value of ^β1 is actually:†

E[^β1]=β1+cor(X,u)σuσX

Recall: Endogeneity and Bias

  • The true expected value of ^β1 is actually:†

E[^β1]=β1+cor(X,u)σuσX

1) If X is exogenous: cor(X,u)=0, we're just left with β1

Recall: Endogeneity and Bias

  • The true expected value of ^β1 is actually:†

E[^β1]=β1+cor(X,u)σuσX

1) If X is exogenous: cor(X,u)=0, we're just left with β1

2) The larger cor(X,u) is, larger bias: (E[^β1]−β1)

Recall: Endogeneity and Bias

  • The true expected value of ^β1 is actually:†

E[^β1]=β1+cor(X,u)σuσX

1) If X is exogenous: cor(X,u)=0, we're just left with β1

2) The larger cor(X,u) is, larger bias: (E[^β1]−β1)

3) We can “sign” the direction of the bias based on cor(X,u)

  • Positive cor(X,u) overestimates the true β1 (^β1 is too high)
  • Negative cor(X,u) underestimates the true β1 (^β1 is too low)

† See 2.4 class notes for proof.

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
corr<-cor(CAcorr)
corr
##                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

  • el_pct is strongly (negatively) correlated with testscr (Condition 1)

  • el_pct is reasonably (positively) correlated with str (Condition 2)

Endogeneity and Bias: Correlations II

  • Here is where checking correlations between variables helps: 
# Make a correlation plot
library(corrplot)
corrplot(corr, type="upper", 
         method = "number", # number for showing correlation coefficient
         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))
ABCDEFGHIJ0123456789
ESL
<chr>
Average_test_score
<dbl>
High ESL643.9591
Low ESL664.3540
2 rows

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

E[^β1]=β1+bias

E[^β1]= β1 + cor(X,u) σuσX

  • cor(STR,u) is positive (via %EL)

  • cor(u,Test score) is negative (via %EL)

  • β1 is negative (between Test score and STR)

  • Bias is positive

    • But since β1 is negative, it’s made to be a larger negative number than it truly is
    • Implies that β1 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 ...

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 ^β1 estimates the marginal effect of ΔSTR→ΔTest Score?

Omitted Variable Bias: Messing with Causality II

  • Consider an ideal random controlled trial (RCT)

  • Randomly assign experimental units (e.g. people, cities, etc) into two (or more) groups:

    • Treatment group(s): gets a (certain type or level of) treatment
    • Control group(s): gets no treatment(s)

  • Compare results of two groups to get average treatment effect

RCTs Neutralize Omitted Variable Bias I

Example: Imagine an ideal RCT for measuring the effect of STR on Test Score

  • School districts would be randomly assigned a student-teacher ratio

  • With random assignment, all factors in u (family size, parental income, years in the district, day of the week of the test, climate, etc) are distributed independently of class size

RCTs Neutralize Omitted Variable Bias II

Example: Imagine an ideal RCT for measuring the effect of STR on Test Score

  • Thus, cor(STR,u)=0 and E[u|STR]=0, i.e. exogeneity

  • Our ^β1 would be an unbiased estimate of β1, measuring the true causal effect of STR → Test Score

But We Rarely, if Ever, Have RCTs

  • But our data is not an RCT, it is observational data!

  • “Treatment” of having a large or small class size is NOT randomly assigned!

  • %EL: plausibly fits criteria of O.V. bias!

    1. %EL is a determinant of Test Score
    2. %EL is correlated with STR

  • Thus, “control” group and “treatment” group differs systematically!

    • Small STR also tend to have lower %EL; large STR also tend to have higher %EL
    • Selection bias: cor(STR,%EL)≠0, E[ui|STRi]≠0

Treatment Group

Control Group

Another Way to Control for Variables

  • Causal pathways connecting str and test score:
    • str → test score
    • str ← ESL → testscore

Another Way to Control for Variables

  • Causal pathways connecting str and test score:

    • str → test score
    • str ← ESL → testscore

  • DAG rules tell us we need to control for ESL in order to identify the causal effect of

  • So now, how do we control for a variable?

Controlling for Variables

  • Look at effect of STR on Test Score by comparing districts with the same %EL.

    • Eliminates differences in %EL between high and low STR classes
    • “As if” we had a control group! Hold %EL constant

  • The simple fix is just to not omit %EL!

    • Make it another independent variable on the righthand side of the regression

Treatment Group

Control Group

Controlling for Variables

  • Look at effect of STR on Test Score by comparing districts with the same %EL.

    • Eliminates differences in %EL between high and low STR classes
    • “As if” we had a control group! Hold %EL constant

  • The simple fix is just to not omit %EL!

    • Make it another independent variable on the righthand side of the regression

The Multivariate Regression Model

Multivariate Econometric Models Overview

Yi=β0+β1X1i+β2X2i+⋯+βkXki+ui

Multivariate Econometric Models Overview

Yi=β0+β1X1i+β2X2i+⋯+βkXki+ui

  • Y is the dependent variable of interest
    • AKA "response variable," "regressand," "Left-hand side (LHS) variable"

Multivariate Econometric Models Overview

Yi=β0+β1X1i+β2X2i+⋯+βkXki+ui

  • Y is the dependent variable of interest
    • AKA "response variable," "regressand," "Left-hand side (LHS) variable"
  • X1 and X2 are independent variables
    • AKA "explanatory variables", "regressors," "Right-hand side (RHS) variables", "covariates"

Multivariate Econometric Models Overview

Yi=β0+β1X1i+β2X2i+⋯+βkXki+ui

  • Y is the dependent variable of interest
    • AKA "response variable," "regressand," "Left-hand side (LHS) variable"
  • X1 and X2 are independent variables
    • AKA "explanatory variables", "regressors," "Right-hand side (RHS) variables", "covariates"
  • Our data consists of a spreadsheet of observed values of (X1i,X2i,Yi)

Multivariate Econometric Models Overview

Yi=β0+β1X1i+β2X2i+⋯+βkXki+ui

  • Y is the dependent variable of interest
    • AKA "response variable," "regressand," "Left-hand side (LHS) variable"
  • X1 and X2 are independent variables
    • AKA "explanatory variables", "regressors," "Right-hand side (RHS) variables", "covariates"
  • Our data consists of a spreadsheet of observed values of (X1i,X2i,Yi)
  • To model, we "regress Y on X1 and X2"

Multivariate Econometric Models Overview

Yi=β0+β1X1i+β2X2i+⋯+βkXki+ui

  • Y is the dependent variable of interest
    • AKA "response variable," "regressand," "Left-hand side (LHS) variable"
  • X1 and X2 are independent variables
    • AKA "explanatory variables", "regressors," "Right-hand side (RHS) variables", "covariates"
  • Our data consists of a spreadsheet of observed values of (X1i,X2i,Yi)
  • To model, we "regress Y on X1 and X2"
  • β0,β1,⋯,βk are parameters that describe the population relationships between the variables
    • We estimate k+1 parameters (“betas”)†

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

Marginal Effects I

Yi=β0+β1X1i+β2X2i

  • Consider changing X1 by ΔX1 while holding X2 constant:

Y=β0+β1X1+β2X2Before the change

Marginal Effects I

Yi=β0+β1X1i+β2X2i

  • Consider changing X1 by ΔX1 while holding X2 constant:

Y=β0+β1X1+β2X2Before the changeY+ΔY=β0+β1(X1+ΔX1)+β2X2After the change

Marginal Effects I

Yi=β0+β1X1i+β2X2i

  • Consider changing X1 by ΔX1 while holding X2 constant:

Y=β0+β1X1+β2X2Before the changeY+ΔY=β0+β1(X1+ΔX1)+β2X2After the changeΔY=β1ΔX1The difference

Marginal Effects I

Yi=β0+β1X1i+β2X2i

  • Consider changing X1 by ΔX1 while holding X2 constant:

Y=β0+β1X1+β2X2Before the changeY+ΔY=β0+β1(X1+ΔX1)+β2X2After the changeΔY=β1ΔX1The differenceΔYΔX1=β1Solving for β1

Marginal Effects II

β1=ΔYΔX1 holding X2 constant

Marginal Effects II

β1=ΔYΔX1 holding X2 constant

Similarly, for β2:

β2=ΔYΔX2 holding X1 constant

Marginal Effects II

β1=ΔYΔX1 holding X2 constant

Similarly, for β2:

β2=ΔYΔX2 holding X1 constant

And for the constant, β0:

β0=predicted value of Y when X1=0,X2=0

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

  • We have been envisioning OLS regressions as the equation of a line through a scatterplot of data on two variables, X and Y

    • β0: "intercept"
    • β1: "slope"

  • With 3+ variables, OLS regression is no longer a "line" for us to estimate

The "Constant"

  • Alternatively, we can write the population regression equation as: Yi=β0X0i+β1X1i+β2X2i+ui

  • Here, we added X0i to β0

  • X0i is a constant regressor, as we define X0i=1 for all i observations

  • Likewise, β0 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

Example:

Beer Consumptioni=β0+β1Pricei+β2Incomei+β3Nachos Pricei+β4Wine Price+ui

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

The Population Regression Model: Example I

Example:

Beer Consumptioni=β0+β1Pricei+β2Incomei+β3Nachos Pricei+β4Wine Price+ui

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

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

The Population Regression Model: Example I

Example:

Beer Consumptioni=β0+β1Pricei+β2Incomei+β3Nachos Pricei+β4Wine Price+ui

  • 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?

The Population Regression Model: Example I

Example:

Beer Consumptioni=β0+β1Pricei+β2Incomei+β3Nachos Pricei+β4Wine Price+ui

  • 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

Example:

^Beer Consumptioni=20−1.5Pricei+1.25Incomei−0.75Nachos Pricei+1.3Wine Pricei

  • Interpret each ˆβ

Multivariate OLS in R

# run regression of testscr on str and el_pct
school_reg_2 <- lm(testscr ~ str + el_pct, 
                 data = CASchool)
  • Format for regression is lm(y ~ x1 + x2, data = df)
  • y is dependent variable (listed first!)
  • ~ means “modeled by”
  • x1 and x2 are the independent variable
  • df is the dataframe where the data is stored

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
  • Stored as an lm object called school_reg_2, a list object

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)
ABCDEFGHIJ0123456789
term
<chr>
estimate
<dbl>
std.error
<dbl>
(Intercept)686.03224877.41131248
str-1.10129590.38027832
el_pct-0.64977680.03934255
3 rows | 1-3 of 5 columns

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)
Model 1Model 2
Intercept698.93 ***686.03 ***
(9.47)   (7.41)   
Class Size-2.28 ***-1.10 ** 
(0.48)   (0.38)   
%ESL Students       -0.65 ***
       (0.04)   
N420       420       
R-Squared0.05    0.43    
SER18.58    14.46    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Review: u

Yi=β0+β1Xi+ui

  • Error term, ui includes all other variables that affect Y

  • Every regression model always has omitted variables assumed in the error

    • Most unobservable (hence "u") or hard to measure
    • Examples: innate ability, weather at the time, etc

  • Again, we assume u is random, with E[u|X]=0 and var(u)=σ2u

  • Sometimes, omission of variables can bias OLS estimators (^β0 and ^β1)

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