Problem Set 8

Due Date: March 6, 2026
Submission: https://canvas.northwestern.edu/courses/245562/assignments/1687753

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Problem 1

1a.

Compare and contrast the following transformations: - \(\log(X)\) vs. \(\sqrt{X}\) - \(\text{asinh}(X)\) vs. \(\log(X + 1)\)

1b.

Using the kidiq dataset from the slides:

library(rosdata)
data(kidiq)

# Fit three models:
# 1. Without centering: kid_score ~ mom_hs + mom_iq + mom_hs:mom_iq
# 2. With centering mom_iq at its mean
# 3. With standardization (z-scores for mom_iq)

# Your tasks:
# 1. Fit all three models
# 2. Create a table comparing coefficients
# 3. Interpret the interaction term in each model
# 4. Explain why centering matters for interpretation

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Problem 2

Using QOG data:

library(rqog)
library(dplyr)
library(ggplot2)

# Load and prepare data
qog_data <- read_qog(which_data = "standard", data_type = "time-series")

democracy_data <- qog_data %>%
  filter(year == 2020) %>%
  select(
    country = cname,
    democracy = vdem_libdem,
    gdp_pc = wdi_gdpcappppcon2017,
    population = wdi_pop,
    corruption = ti_cpi
  ) %>%
  filter(!is.na(democracy), !is.na(gdp_pc), gdp_pc > 0) %>%
  mutate(
    log_gdp = log(gdp_pc),
    log_pop = log(population)
  )

# Your tasks:
# 1. Fit four different models of democracy:
#    a. Linear: democracy ~ gdp_pc
#    b. Log-X: democracy ~ log_gdp
#    c. Log-Y: lm(log(democracy) ~ gdp_pc)  # Note: democracy is 0-1
#    d. Log-log: lm(log(democracy + 0.01) ~ log_gdp)  # Handle zeros

# 2. For each model:
#    a. Create residual plots
#    b. Interpret coefficients substantively

# 3. Create a visualization comparing predicted vs. actual values
#    for all four models on the original scale

# 4. Test polynomial specifications:
#    democracy ~ poly(log_gdp, 2)
#    democracy ~ poly(log_gdp, 3)

Questions: 1. Which transformation provides the best fit? Justify using both statistical and substantive criteria.

Problem 3

Using the democracy model from Problem 2:

# Choose the best model from Problem 2
best_model <- lm(democracy ~ log_gdp, data = democracy_data)

# Create comprehensive diagnostics:
library(ggplot2)
library(patchwork)

# 1. Create all four diagnostic plots (residuals vs. fitted values, Q-Q plot of residuals, scale-location plot, and residuals vs. leverage plot) 
# 2. Calculate and plot relevant influence statistics (Cook's distance and/or DFFITS at your discretion)
# 3. Identify influential observations
# 4. Test for heteroskedasticity (Breusch-Pagan test)

Questions: 1. Which countries are influential observations in your model? Why? 2. Is there evidence of heteroskedasticity? What are the implications? 3. Are the residuals normally distributed? Why does this matter?