Lecture 8 Fixed Effects

Recap

• Last time we talked about how controlling is a common way of blocking back doors to identify an effect
• We can control for a variable W by using our method of using W to explain our other variables, then take the residuals
• Another form of controlling is using a sample that has only observations with similar values of W
• Some variables you want to be careful NOT to control for - you don’t want to close front doors, or open back doors by controlling for colliders

Today

• Today we’ll be starting on our path for the rest of the class, where we’ll be talking about standard methods for performing causal inference
• Different ways of getting identification once we have a diagram!
• Our goal here will be to understand these methods conceptually and to also figure out some good statistical practices for their use
• Our goal is to understand these methods and be able to apply a straightforward version of them

Today

• In particular we’ll be talking about a method that is commonly used to identify causal effects, called fixed effects
• We’ll be discussing the kind of causal diagram that fixed effects can identify
• All of the methods we’ll be discussing are like this - they’ll only apply to particular diagrams
• And so knowing our diagrams will be key to knowing when to use a given method

The Problem

• One problem we ran into last time is that we can’t really control for things if we can’t measure them
• And there are lots of things we can’t measure or don’t have data for!
• So what can we do?

The Solution

• If we observe each person/firm/country multiple times, then we can forget about controlling for the actual back-door variable we’re interested in
• And just control for person/firm/country identity instead!
• This will control for EVERYTHING unique to that individual, whether we can measure it or not!

In Practice

• Let’s do this on the data from the “gapminder” package
• This data tracks life expectancy and GDP per capita in many countries over time

library(gapminder)
data(gapminder)
cor(gapminder$lifeExp,log(gapminder$gdpPercap))

## [1] 0.8076179

gapminder <- gapminder %>% group_by(country) %>%
  mutate(lifeExp.r = lifeExp - mean(lifeExp),
         logGDP.r = log(gdpPercap) - mean(log(gdpPercap))) %>% ungroup()
cor(gapminder$lifeExp.r,gapminder$logGDP.r)

## [1] 0.6404051

So What?

• This isn’t any different, mechanically, from any other time we’ve controlled for something
• So what’s different here?
• Let’s think about what we’re doing conceptually

What’s the Diagram?

• Why are we controlling for things in this gapminder analysis?
• Because there are LOTS of things that might be back doors between GDP per capita and life expectancy
• War, disease, political institutions, trade relationships, health of the population, economic institutions…

What’s the Diagram?

What’s the Diagram?

• There’s no way we can identify this
• The list of back doors is very long
• And likely includes some things we can’t measure!

What’s the Diagram?

• HOWEVER! If we think that these things are likely to be constant within country…
• Then we don’t really have a big long list of back doors, we just have one: “country”

What We Get

• So what we get out of this is that we can identify our effect even if some of our back doors include variables that we can’t actually measure
• When we do this, we’re basically comparing countries to themselves at different time periods!
• Pretty good way to do an apples-to-apples comparison!

Graphically

Graphically

• The post-fixed-effects dots are basically a bunch of “Raw Country X” pasted together.
• Imagine taking “Raw Pakistan” and moving it to the center, then taking “Raw Britain” and moving it to the center, etc.
• Ignoring the baseline differences between Pakistan, Britain, China, etc., in their GDP per capita and life expectancy, and just looking within each country.
• We are ignoring all differences between countries (since that way back doors lie!) and looking only at differences within countries.
• Fixed Effects is sometimes also referred to as the “within” estimator

In Action

Notably

• This does assume, of course, that all those back door variables CAN be described by country
• In other words, that these back doors operate by things that are fixed within country
• If something is a back door and changes over time in that country, fixed effects won’t help!

Varying Over Time

• For example, earlier we mentioned war… that’s not fixed within country! A given country is at war sometimes and not other times.

Varying Over Time

• Of course, in this case, we could control for War as well and be good!
• Time-varying things doesn’t mean that fixed effects doesn’t work, it just means you need to control for that stuff too
• It always comes down to thinking carefully about your diagram
• Fixed effects mainly works as a convenient way of combining together lots of different constant-within-country back doors into something that lets us identify the model even if we can’t measure them all

Fixed Effects in Regression

• We can just do fixed effects as we did-subtract out the group means and analyze (perhaps with regression) what’s left
• We can also include dummy variables for each group/individual, which accomplishes the same thing

\[ Y = \beta_0 + \beta_1Group1 + \beta_2Group2 + ... + \]

\[ \beta_NGroupN + \beta_{N+1}X + \varepsilon \]

\[ Y = \beta_i + \beta_1X + \varepsilon \]

Fixed Effects in Regression

• Why does that work?
• We want to “control for group/individual” right? So… just… put in a control for group/individual
• Of course, like all categorical variables as predictors, we leave out a reference group
• But here, unlike with, say, a binary predictor, we’re rarely interested in the FE coefficients themselves. Most software works with the mean-subtraction approach (or a variant) and don’t even report them!

Fixed Effects in Regression: Variation

• Remember we are isolating within variation
• If an individual has no within variation, say their treatment never changes, they basically get washed out entirely!
• A fixed-effects regression wouldn’t represent them. And can’t use FE to study things that are fixed over time
• And in general if there’s not a lot of within variation, FE is going to be very noisy. Make sure there’s variation to study!

Fixed Effects in Regression: Notes

• It’s common to cluster standard errors at the level of the fixed effects, since it seems likely that errors would be correlated over time (autocorrelated errors)
• It’s possible to have more than one set of fixed effects. \(Y = \beta_i + \beta_j + \beta_1X + \varepsilon\)
• But interpretation gets tricky - think through what variation in \(X\) you’re looking at at that point!

Coding up Fixed Effects

• We will use the fixest package
• It’s very fast, and can be easily adjusted to do FE with other regression methods like logit, or combined with instrumental variables
• Clusters at the first listed fixed effect by default

library(fixest)

m1 <- feols(outcome ~ predictors | FEs, data = data)
msummary(m1)

Example: Sentencing

• What effect do sentencing reforms have on crime?
• One purpose of punishment for crime is to deter crime
• If sentences are more clear and less risky, that may reduce a deterrent to crime and so increase crime
• Marvell & Moody study this using data on reforms in US states from 1969-1989

Example: Sentencing

• I’ve omitted code reading in the data
• But in our data we have multiple observations per state

head(mmdata)

## # A tibble: 6 x 6
##   state   year assault robbery pop1000 sentreform
##   <chr>  <dbl>   <dbl>   <dbl>   <dbl>      <dbl>
## 1 "ALA "    70    7413    1731    3450          0
## 2 "ALA "    71    7645    2005    3497          0
## 3 "ALA "    72    7431    2407    3540          0
## 4 "ALA "    73    8362    2809    3581          0
## 5 "ALA "    74    8429    3562    3628          0
## 6 "ALA "    75    8440    4446    3681          0

mmdata <- mmdata %>% mutate(assaultper1000 = assault/pop1000,
         robberyper1000 = robbery/pop1000)

Fixed Effects

• We can see how robbery rates evolve in each state over time as states implement reform

Fixed Effects

• You can tell that states are more or less likely to implement reform in a way that’s correlated with the level of robbery they already had
• So SOMETHING about the state is driving both the level of robberies AND the decision to implement reform
• Who knows what!
• Our diagram has reform -> robberies and reform <- state -> robberies, which is something we can address with fixed effects.

Fixed Effects

sentencing_ols <- lm(robberyper1000 ~ sentreform, data = mmdata)
sentencing_fe <- feols(robberyper1000 ~ sentreform | state, data = mmdata)
msummary(list('OLS' = sentencing_ols, 'FE' = sentencing_fe), stars = TRUE, gof_omit = 'AIC|BIC|F|Lik|Adj|Pseudo')

Example

• The 1.254, 0.352 included the fact that different kinds of states tend to institute reform
• The 0.245 doesn’t!
• Looks like the deterrent effect was real! Although important to consider if there might be time-varying back doors too, we don’t account for those in our analysis
• What things might change within state over time that would be related to robberies and to sentencing reform?

Practice

• We want to know the effect of your teacher on the test scores of high school students
• Some potential back doors might go through: parents' intelligence, age, demographics, school, last year's teacher
• Draw a diagram including all these variables, plus maybe some unobservables where appropriate
• If you used fixed effects for students, what back doors would still be open?
• What would the feols() command for this regression look like?

Practice Answers

• Fixed effects would close your back doors for parents' intelligence, demographics, and school, but leave open age and last year's teacher

m <- feols(TestScore ~ Teacher + Age + LastYearsTeacher | 
             Student, data = data)