ECON 4370/6370 — Homework 2

Author

Your Name (SMU ID)

Published

September 21, 2026

This homework is a coding assignment. Please follow the instructions to complete the tasks.
Use Quarto/Knitr chunks for all work.
Turn in the rendered HTML and your .qmd source via Canvas.

0. Setup

set.seed(123)  # Please don't change this seed.

# You may load packages you plan to use (optional):
library(tidyverse)
library(readxl)
library(zoo)
library(lubridate)
library(forecast)

# Read in the CPS data we used in the lecture
cps <- read_csv("./data/cps09mar.csv")

# Read in the Welch Goyal Data we used in the lecture
data_raw <- read_excel("./data/PredictorData2018.xlsx",
                       sheet = "Monthly",
                       col_names = TRUE,
                       na = "NaN")

# Transform the Welch-Goyal data
data_wg <- data_raw %>% 
    transmute(
        date = zoo::as.yearmon(lubridate::ymd(yyyymm, truncated = 2)),
        ExReturn = log(Index) - log(c(1,Index[-length(Index)])),
        dp = log(D12) - log(Index),
        dy = log(D12) - log( c(1, Index[-length(Index)]) ),
        ep = log(E12) - log(Index),
        tms = lty - tbl,
        dfy = BAA - AAA,
        dfr = corpr - ltr,
        bm = `b/m`,
        tbl = tbl,
        ltr = ltr,
        ntis = ntis,
        svar = svar,
        infl = infl
    ) %>% 
    filter(row_number() > 1)

1. Return to Education for Subgroups

In this exercise, we will estimate the Mincer equation for different subgroups based on gender and education level. The Mincer equation relates log wages to education and experience:

\[\log(w_i) = \alpha + \beta_1 \cdot \text{education}_i + \beta_2 \cdot \text{experience}_i + \beta_3 \cdot \text{experience}_i^2 + \varepsilon_i\]

Step 1: Prepare the data

First, prepare the CPS data for the Mincer regression analysis:

# TODO: Filter and prepare the CPS data for Mincer regression
# - Keep only workers with positive earnings, hours, and weeks
# - Calculate hourly wage and log wage
# - Calculate potential experience as age - education - 6
# - Create education groups: "High School or Less" (≤12 years) vs "College+" (>12 years)
# - Create gender labels

# Step-by-step instructions for data preparation:

# 1. Filter the data to keep only valid workers:
#    - earnings > 0 (positive earnings)
#    - hours > 0 (positive hours worked)
#    - week > 0 (positive weeks worked)
#    - hours <= 80 (reasonable maximum hours per week)
#    - week <= 52 (reasonable maximum weeks per year)

# 2. Calculate derived variables:
#    - wage_hourly = earnings / (hours * week)
#    - log_wage = log(wage_hourly)
#    - experience = pmax(age - education - 6, 0)  # potential experience, non-negative
#    - exp2 = experience^2  # squared experience for Mincer equation

# 3. Create subgroup indicators:
#    - gender_group: "Male" if female == 0, "Female" if female == 1
#    - edu_group: "High School or Less" if education <= 12, "College+" if education > 12
#    - subgroup: combination of gender and education (4 groups total)

# 4. Remove any observations with missing values in key variables

# cps_mincer <- cps %>%
  # TODO: Add your filtering and transformation code here
  # Hint: Follow the pattern from the lecture

Step 2: Estimate the Mincer equation for different subgroups

Estimate the Mincer equation separately for each subgroup:

# TODO: Estimate Mincer equation for each subgroup
# Create a function to estimate Mincer equation for a given subset of data
mincer_est <- function(data) {
  # TODO: Implement the Mincer regression: log_wage ~ education + experience + exp2
  # Use lm() function to estimate the model
  # Return: coefficients, standard errors, R-squared, and other summary statistics
  # Hint: model <- lm(log_wage ~ education + experience + exp2, data = data)
}

# TODO: Estimate for each subgroup and store results
# The four subgroups are:
# 1. Male, High School or Less (gender_group == "Male" & edu_group == "High School or Less")
# 2. Male, College+ (gender_group == "Male" & edu_group == "College+")
# 3. Female, High School or Less (gender_group == "Female" & edu_group == "High School or Less")
# 4. Female, College+ (gender_group == "Female" & edu_group == "College+")

# Create a results data frame to store estimates for all subgroups
# Columns should include: subgroup, alpha, beta_edu, beta_exp, beta_exp2, r_squared, n_obs

# Example structure for results:
# results <- data.frame(
#   subgroup = character(),
#   alpha = numeric(),
#   beta_edu = numeric(),
#   beta_exp = numeric(), 
#   beta_exp2 = numeric(),
#   r_squared = numeric(),
#   n_obs = integer()
# )

# TODO: Loop through each subgroup and estimate the Mincer equation
# Use filter() to subset data for each subgroup
# Store results in the results data frame

Step 3: Visualize the results

Create visualizations to compare the returns to education across subgroups:

# TODO: Create visualizations to compare results across subgroups
# Suggestions:
# 1. Bar plot comparing returns to education (β₁) across subgroups
# 2. Scatter plot of log wages vs education by subgroup
# 3. Experience-wage profiles by subgroup

# 1. Bar plot of returns to education (beta_edu) by subgroup
# Use your results data frame with columns: subgroup, beta_edu
# ggplot(results, aes(x = subgroup, y = beta_edu)) + geom_col()

# 2. Scatter plot of log wages vs education by subgroup
# Use the original cps_mincer data with subgroup information
# ggplot(cps_mincer, aes(x = education, y = log_wage, color = subgroup)) + 
#   geom_point(alpha = 0.3) + 
#   geom_smooth(method = "lm", se = FALSE)

# 3. Experience-wage profiles by subgroup
# Create predicted wage profiles using the estimated coefficients
# For each subgroup, create a sequence of experience values (0 to 40 years)
# Calculate predicted log wages: alpha + beta_edu * education + beta_exp * exp + beta_exp2 * exp^2
# Plot the predicted profiles for each subgroup

# TODO: Bar plot of returns to education
# TODO: Scatter plot of log wages vs education  
# TODO: Experience-wage profiles

Step 4: Interpret the results

Please briefly interpret your results.

2. Rolling Window Estimation of the AR(1) Model

The AR(1) Model

The AR(1) model for inflation is: \[\text{infl}_t = \alpha + \rho \cdot \text{infl}_{t-1} + \varepsilon_t\]

Where: - \(\alpha\) is the intercept (constant term) - \(\rho\) is the persistence parameter (how much current inflation depends on past inflation) - \(\varepsilon_t\) is the error term

In this exercise, we will implement rolling window estimation to examine how the persistence of inflation changes over time.

What is Rolling Window Estimation?

Rolling window estimation is a technique used in time series analysis to examine how model parameters change over time. Instead of estimating a model once using all available data, we estimate the model repeatedly using only a fixed-size “window” of recent observations.

Key concepts:

Window size: The number of observations used in each estimation (we’ll use 120 months = 10 years)
Rolling: The window moves forward one observation at a time
Time-varying parameters: This allows us to see how relationships change over time

Example with our data:

Start with observations 1-120, estimate AR(1) model
Move to observations 2-121, estimate AR(1) model
Move to observations 3-122, estimate AR(1) model
Continue until we reach the end of the data

Visual representation (window size = 120):

Time:     1  2  3  4  ........................... 120 121 ... T
Window 1: [████████████████████████████████████████]
Window 2:    [████████████████████████████████████████]
Window 3:       [████████████████████████████████████████]
...

Each window gives us one estimate of ρ, creating a time series of ρ estimates.

Why use rolling windows? 1. Structural breaks: Economic relationships may change over time 2. Parameter stability: Test if model parameters are constant 3. Forecasting: More recent data may be more relevant for predictions 4. Policy analysis: Understand how economic relationships evolve

Step 1: Prepare the inflation data

# TODO: Prepare the inflation time series
# - Extract the inflation series from data_wg
# - Remove any missing values
# - Create lagged inflation variable

# infl_data <- data_wg %>%
  # TODO: Add your data preparation code here
  # Hint: select relevant variables and create lags

Step 2: Implement rolling window estimation

# TODO: Implement rolling window estimation
# - Fix window size to 120 months (10 years)
# - For each window, estimate AR(1) model: infl_t = α + ρ * infl_{t-1} + ε_t
# - Store the estimated coefficients and their standard errors
# - Store the R-squared for each window

# Function to estimate AR(1) for a given window
ar1_est <- function(data_window) {
  # TODO: Implement AR(1) estimation for a window of data
  # Steps:
  # 1. Create lagged inflation: infl_lag = lag(infl, 1)
  # 2. Remove first observation (due to lag)
  # 3. Estimate: lm(infl ~ infl_lag, data = data_window)
  # 4. Extract coefficients, standard errors, R-squared
  # 5. Return as a list or data frame
}

# TODO: Implement rolling window loop
# Detailed steps:
# 1. Calculate total number of windows: n_windows = nrow(infl_data) - window_size + 1
# 2. Initialize results data frame
# 3. Loop through each window:
#    for (i in 1:n_windows) {
#      # Get window data: infl_data[i:(i + window_size - 1), ]
#      # Estimate AR(1) model
#      # Store results with date = infl_data$date[i + window_size - 1]
#    }

window_size <- 120

Step 3: Visualize the results

# TODO: Create visualizations of the rolling estimates
# 1. Time series plot of the estimated ρ (persistence parameter)

# TODO: Plot rolling estimates of ρ over time

What do you observe?