Problem Set 4

Problem 1

1a. Define omitted variable bias in your own words.

1b. Consider the linear models: \[y = \beta_0 + \beta_1x_1 + \beta_2x_2 + u\] \[y = \beta^*_0 + \beta^*_1x_1 + u^*\]

Derive the formula for \(\beta^*_1\) in terms of \(\beta_1\), \(\beta_2\), and the relationship between \(x_1\) and \(x_2\). Show all steps.

1c. Interpret each term in your formula from 1b. Under what conditions does omitted variable bias occur? When is it zero?

Problem 2

The lecture slides show a nonlinear CEF for GDP and turnout, with different linear approximations (BLPs) for different ranges of GDP. In your own words, explain:

  1. What does it mean for the BLP to be the “best linear approximation” of a nonlinear CEF?
  2. How can the BLP coefficient change sign depending on which range of the data we focus on?
  3. What are the implications for interpreting regression coefficients when the true CEF is nonlinear?

Problem 3

Return to your voter turnout analysis from Problem Set 3.

3a. Re-examine the difference between your bivariate model (turnout ~ income) and multivariate model (turnout ~ income + religiosity + age). Set up a version of the multivariate model that uses only income and religiosity and does not use age (turnout ~ income + religiosity). Calculate the omitted variable bias for income comparing this new multivariate model to the bivariate model using the formula from Problem 1.

# Calculate the components needed for OVB formula
# You'll need:
# 1. β2 from multivariate model (coefficient for religiosity)
# 2. Covariance between income and religiosity
# 3. Variance of income

# Then compute: OVB = β2 * Cov(x1, x2) / Var(x1)

3b. Does the OVB formula correctly predict the difference between the bivariate and multivariate income coefficients? Show your calculations.

3c. Based on the lecture slides’ discussion of model specification: 1. Could adding more variables ever increase bias? Under what conditions? 2. When might it be better to use a bivariate model even if you suspect omitted variables?

Problem 4

Simulation Study of OVB

We’ll study two scenarios of omitted variable bias through simulation.

Scenario A: Confounding (Both X1 and X2 cause Y)

set.seed(789)
n <- 1000
x1 <- rnorm(n)
x2 <- 0.7*x1 + rnorm(n)  # x2 correlated with x1
y <- 2 + 1.5*x1 + 2*x2 + rnorm(n, sd = 0.5)

# Run regressions
model_bivariate <- lm(y ~ x1)
model_multivariate <- lm(y ~ x1 + x2)

summary(model_bivariate)
summary(model_multivariate)

Questions for Scenario A: 1. What is the true value of \(\beta_{1}\)? 2. What is the estimated \(beta_{1}\) in the bivariate model? How biased is it? 3. Use the OVB formula to calculate the expected bias. Does it match the actual bias?

Scenario B: Collider Bias (X2 is a common effect)

set.seed(789)
n <- 1000

# Correct setup for collider scenario
x1 <- rnorm(n)
y <- 2 + 1.5*x1 + rnorm(n, sd = 0.5)  # y depends only on x1
x2 <- 0.7*x1 - 1.5*y + rnorm(n, sd = 0.5)  # x2 is a collider

# Run regressions
model_correct <- lm(y ~ x1)  # Correct specification
model_collider <- lm(y ~ x1 + x2)  # Including collider

summary(model_correct)
summary(model_collider)

Questions for Scenario B: 1. What is the true value of \(\beta_1\)? 2. What happens when we include x2 in the regression? Why? 3. This demonstrates “bad control” or collider bias. Explain in your own words why including x2 creates bias even though x2 is correlated with both x1 and y.

4c. Simulation Synthesis: 1. Create a table comparing both scenarios. 2. Under what circumstances does adding a control variable reduce bias? When might it increase bias? 3. How can researchers decide which variables to include in a regression model?

Problem 5

The lecture slides derive OLS using the plug-in principle and matrix algebra.

5a. Plug-in Principle: 1. Define the plug-in principle in your own words.

5b. Matrix Derivation: The OLS estimator in matrix form is: \[\hat{\beta} = (X^TX)^{-1}X^Ty\]

Using the turnout data with GDP and Temperature:

# Load and clean data
turnout <- read_csv("https://raw.githubusercontent.com/jnseawright/ps405/refs/heads/main/Data/turnout.csv")
turnout_clean <- turnout[13:nrow(turnout), ]  # Remove NA rows as in slides

# Create X matrix with intercept, GDP, Temperature
X <- as.matrix(cbind(1, turnout_clean$GDP, turnout_clean$Temperature))
colnames(X) <- c("Intercept", "GDP", "Temperature")

# Create y vector
y <- turnout_clean$Turnout

# Calculate OLS coefficients using matrix algebra


# Compare with lm() output
  1. Verify that your matrix calculation matches the lm() output.