ECON 523

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photo: World Bank/Peter Kapuscinski (2015)

Instructor:
Pamela Jakiela

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Empirical Exercise 2

This exercise makes use of the data set E2-BanerjeeEtAl-data.dta, a subset of the data used in the paper A multifaceted program causes lasting progress for the very poor: Evidence from six countries by Abhijit Banerjee, Esther Duflo, Nathanael Goldberg, Dean Karlan, William Pariente, Jeremy Shapiro, Bram Thuysbaert, and Chris Udry, published in the Science in 2015.

The authors examine the impacts of a “graduation” program first designed by the Bangladeshi NGO BRAC. The program offers extremely poor households an asset transfer, temporary consumption support, skills training, home visits, and access to savings technologies. The program was evaluated through a randomized trial in six countries.

In this exercise, we use data on the program’s impacts on food security to explore the mechanics of fixed effects.


Getting Started

Create an R script that contains the following preliminaries:

# preliminaries -----------------------------------------

## libraries

#install.packages("tidyverse")
#install.packages("haven") # load dta files
#install.packages("fixest") # OLS w/ robust SEs

library(tidyverse)
library(haven)
library(fixest)

## load data 

urlfile <- 'https://pjakiela.github.io/ECON523/exercises/E2-BanerjeeEtAl-data.dta'
e2data <- read_dta(urlfile)


In-Class Activity

Extend your script as you answer the following questions, so that you can run the code from start to finish and re-generate all your answers.

Question 1

Familiarize yourself with the data set. How many countries are included in the study, and how many observations are there in each country? What fraction of the observations from each country were treated?

Hint: use count() to count the number of observations from each country, use table() to do a cross-tabulation of two variables, and use group_by() followed by summarize() to calculate the mean value of treatment in each each country.

Question 2

Take a look at the outcome variable e_foodsec. What is the mean value in each country? What is the mean value in the treatment group in each country? What does a histogram of the food security index look like?

Hint: use hist() to make a simple histogram.

Question 3

Regress food security on treatment. What do you find? How should we interpret this coefficient?

Hint: use feols(), as we did in Exercise 1. The general syntax is:

feols(y ~ x, data = df, vcov = 'hc1')

Question 4

Now regress food security on treatment controlling for country fixed effects. How do the results change?

Hint: to add fixed effects for categorical variable z to your feols regression equation, use the syntax y ~ x | z.

Question 5

What if we regress food security on treatment separately for each country? In how many of the six countries do we see a positive and statistically significant treatment effect?

Hint: e2data[e2data$country == 1] will identify the subset of the rows of e2data with country equal to 1.

Question 6

The regression including country fixed effects is equivalent to a regression where we first subtract off country-specific means and then regress de-meaned (or normalized) food security on normalized treatment. Show that this is the case: generate variables mean_t and mean_fs capturing the within-country means of treatment and food security, and then generate norm_t and norm_fs capturing the normalized values (calculated by subtracting the country-specific mean). Regress norm_fs on norm_t without country fixed effects to confirm that the regression coefficient from Question 4.

Hint: use group_by(), then mutate() to generate the variables mean_t and mean_fs, and then ungroup() to return the data frame to one where the unit of observation is an individual rather than a country.

Question 7

The regression including country fixed effects is also equivalent to a regression of residualized food security (predicted from a regression of food security on country fixed effects) on residualized treatment (predicted the same way). Show that this is the case by generating new variables fs_resid and t_resid that capture the residuals from regressions of food security and treatment on the country fixed effects. Regress fs_resid on t_resid and compare your results to the coefficients from Questions 4 and 6.

Hint: the example below illustrates how to capture the residuals from a regression:

model <- feols(y ~ x, data = df, vcov = 'hc1')
y_resid <- model$residuals

Question 8

The regression including country fixed effects is also equivalent to a weighted average of the country-specific treatment effects. The weights are proportional to N*p*(1-p) where N is the number of observations in a country and p is the proportion treated in that country. The weights are normalized by dividing by the sum of all the weights. Extend the program below to calculate the treatment effect that you would get from a regression controlling for fixed effects.

## check: what values should your new variables take on?
e2data %>% group_by(country, treatment) %>% 
  summarize(mean = mean(e_foodsec))

## calculate the weights and the regression coefficient
e2data %>% group_by(country) %>% 
  summarize(mean_fs_t = mean(e_foodsec[treatment == 1], na.rm = TRUE)) %>% 
  mutate(weight = n*mean_t*(1-mean_t) / sum(n*mean_t*(1-mean_t)))

Question 9

Show that you can also recover the coefficient and standard error from the fixed effects regression by regressing e_foodsec on treatment while controlling for the propensity score which is probability that an observation is treated, proxied for by the proportion treated within each country.


Empirical Exercise

For this part of the exercise, we’re going to drop all the observations in the treatment group, and then simulate alternative scenarios to better understand how fixed effects work. Create a new R script that begins with the code below, and then extend your program as you answer the questions. Make sure you understand what the code below does before proceeding.

# preliminaries ---------------------------------

## libraries

#install.packages("tidyverse")
#install.packages("haven") # load dta files
#install.packages("fixest") # OLS w/ robust SEs

library(tidyverse)
library(haven)
library(fixest)

## load data 

urlfile <- 'https://pjakiela.github.io/ECON523/exercises/E2-BanerjeeEtAl-data.dta'
e2dataraw <- read_dta(urlfile)

## drop treatment group, randomly assign observations to four groups
e2data <- e2dataraw %>% 
  filter(treatment != 1) %>% 
  select(!treatment) %>% 
  mutate(randnum = runif(n())) %>%  
  arrange(country, randnum) %>%  
  group_by(country) %>%
  mutate(within_id = row_number(),  
         group = (within_id %% 4)) %>%  
  mutate(group = ifelse(group == 0, 4, group)) %>% 
  ungroup()

Question 1: fixed effects when p is constant across countries

Part (a)

Create a treatment variable t1 and assign observations in groups 1 and 2 to treatment. Then, create a variable impact1 that is equal to 2 for observations in the treatment group and 0 otherwise. This is the treatment effect for the purposes of this (first) simulation. Generate an outcome variable y1 that is endline foodsecurity (e_foodsec) plus impact1. Now regress y1 on t1 with and without country fixed effects. How do the estimated treatments effects and the levels of statistical significance compare across the two specifications?

Part (b)

When the probability of treatment does not vary across countries, including country fixed effects is not necessary - but it may increase statistical power. In the example above, fixed effects did not improve statistical power much because the mean of the outcome variable does not vary across countries (it is normalized to zero in the control group in every country). Change this by increasing y1 by 10 in two countries and decreasing y1 by 20 in two other countries. Now rerun your two regressions (with and without fixed effects). You should see that including fixed effects now changes the standard error on your estimated treatment effect substantially (though it still should not impact your estimated coefficient much).

Part (c)

The estimated coefficient from a regression with fixed effects is a weighted average of the estimated country-specific treatment effects (i.e. the within-country differences in means between treatment and control). The weights are proportional to the sample size within each country. Given this, if you increased the treatment effect in Peru from 2 to 11, what you expect the treatment effect to be? Calculate the expected regression coefficient by hand (using R as a calculator) and then adjust your code and run the fixed effects regression to confirm your result.

Question 2: when are fixed effects necessary?

Fixed effects are needed when treatment probabilities vary across countries and the mean of the outcome variable also varies across countries (because then treatment is correlated with the outcome, even in the absence of a treatment effect). To see this, generate a variable t2 that is equal to 1 for all observations in group 1 plus the observations in group 2 in Ethiopia, Ghana, and Honduras (countries 1, 2, and 3). In this simulation, we are not going to add any treatment effect - so the null hypothesis is true. Generate an outcome variable y2 that is equal to food security, and then add 5 to it in Ethiopia, Ghana, and Honduras (for observations in the treatment and control groups in those countries). How do the results of regressions with and without country fixed effects compare?

Question 3: how observations are weighted?

For the last question, we need to have the same number of observations in each country. The code below does this. You can see that we now have equal numbers of observations from groups 1, 2, 3, and 4 in each country.

q3 <- e2data %>% 
  filter(within_id <= 360)
count(q3, group)
table(q3$country, q3$group)

Generate a treatment variable t3. t3 should be equal to one for observations in group 1 in Ethiopia and Ghana. t3 should be equal to one for observations in groups 1 and 2 in Honduras and India. t3 should be equal to 1 for observations in groups 1, 2, and 3 in Pakistan and Peru. Given this, what is the proportion treated in each country?

Part (a)

First, consider what happens when we only have a treatment effect in the countries with the lowest proportion treated. Create a variable impact3a that is equal to 10 for treated observations in Ethiopia and Ghana, and equal to zero for everybody else. Then, create an outcome variable y3a that is the sum of e_foodsec and impact3a. You can see the average treatment effect across all the treated observations in the sample by summarizing impact3a among treated individuals. How does that compare to the results of regressions with and without fixed effects? Which do you think better captures the average treatment effect, and why (this is a trick question)?

Part (b)

Now replicate the exercise above, but have the treatment effect occur in Honduras and India (where the proportion treated is one half) rather than in Ethiopia and Ghana (where the proportion treated is one quarter). Generate new variables impact3b and y3b and repeat your analysis.

Part (c)

Now replicate the exercise again, but have the treatment effect occur in Pakistan and Peru (where the proportion treated is three quarters) rather than in Honduras and India (where the proportion treated is one half). Generate new variables impact3c and y3c and repeat your analysis.

Part (d)

In the simulations above, the regressions without fixed effects recovered a reasonable estimate of average treatment effect on the treated across all the countries in the analysis. Why was this? How would you modify simulation 3c so that the regression without fixed effects would not recover an unbiased estimate of the average treatment effect across all treated individuals in the sample?

Question 4: Takeaways.

Based on the above, which countries received relatively low weight in the analysis of Banerjee et al. conditional on the number of observations from that country? How do you think that might have impacted their results?


This exercise is part of the module Revisiting Regression.