Types of Data I

Types of Data I

• Panel data: combines these dimensions: compare all individual i’s over all time t’s

Panel Data I

Panel Data II

Panel Data II

Panel Data: Our Motivating Example

The Data I

glimpse(phones)

## Rows: 306
## Columns: 8
## $ year          <fct> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 20…
## $ state         <fct> Alabama, Alaska, Arizona, Arkansas, California, Colorado…
## $ urban_percent <dbl> 30, 55, 45, 21, 54, 34, 84, 31, 100, 53, 39, 45, 11, 56,…
## $ cell_plans    <dbl> 8135.525, 6730.282, 7572.465, 8071.125, 8821.933, 8162.0…
## $ cell_ban      <fct> 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ text_ban      <fct> 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ deaths        <dbl> 18.075232, 16.301184, 16.930578, 19.595430, 12.104340, 1…
## $ year_num      <dbl> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 20…

The Data II

phones %>%
  count(state)

phones %>%
  count(year)

The Data III

phones %>%
  distinct(state)

phones %>%
  distinct(year)

The Data IV

phones %>%
  summarize(States = n_distinct(state),
            Years = n_distinct(year))

Pooled Regression I

• What if we just ran a standard regression:

^Yit=β0+β1Xit+uit

• N number of i groups (e.g. U.S. States)
• T number of t periods (e.g. years)

• This is a pooled regression model: treats all observations as independent

Pooled Regression II

pooled <- lm(deaths ~ cell_plans, data = phones)
pooled %>% tidy()

Pooled Regression III

ggplot(data = phones)+
  aes(x = cell_plans,
      y = deaths)+
  geom_point()+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven")+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=14)

Pooled Regression III

ggplot(data = phones)+
  aes(x = cell_plans,
      y = deaths)+
  geom_point()+
  geom_smooth(method = "lm", color = "red")+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven")+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=14)

Recap: Assumptions about Errors

Biases of Pooled Regression

^Yit=β0+β1Xit+ϵit

• Assumption 3: cor(ui,uj)=0∀i≠j

• Pooled regression model is biased because it ignores:

  • Multiple observations from same group i
  • Multiple observations from same time t

• Thus, errors are serially or auto-correlated; cor(ui,uj)≠0 within same i and within same t

Biases of Pooled Regression: Our Example

^Deathsit=β0+β1Cell Phonesit+uit

• Multiple observations from same state i

  • Probably similarities among u for obs in same state
  • Residuals on observations from same state are likely correlated

• Multiple observations from same year t

  • Probably similarities among u for obs in same year
  • Residuals on observations from same year are likely correlated

Example: Consider Just 5 States

phones %>%
  filter(state %in% c("District of Columbia",
                      "Maryland", "Texas",
                      "California", "Kansas")) %>%
ggplot(data = .)+
  aes(x = cell_plans,
      y = deaths,
      color = state)+
  geom_point()+ 
  geom_smooth(method = "lm")+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=14)+
  theme(legend.position = "top")

Example: Consider Just 5 States

phones %>%
  filter(state %in% c("District of Columbia",
                      "Maryland", "Texas",
                      "California", "Kansas")) %>%
ggplot(data = .)+
  aes(x = cell_plans,
      y = deaths,
      color = state)+ 
  geom_point()+ 
  geom_smooth(method = "lm")+ 
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=14)+
  theme(legend.position = "none")+
  facet_wrap(~state, ncol=3)

Look at All States

ggplot(data = phones)+
  aes(x = cell_plans,
      y = deaths,
      color = state)+ 
  geom_point()+ 
  geom_smooth(method = "lm")+ 
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed")+
  theme(legend.position = "none")+
  facet_wrap(~state, ncol=7)

The Bias in our Pooled Regression

Fixed Effects: DAG

Fixed Effects: DAG

Fixed Effects: Decomposing uit

• Much of the endogeneity in Xit can be explained by systematic differences across i (groups)

• Exploit the systematic variation across groups with a fixed effects model

• Decompose the model error term into two parts:

uit=αi+ϵit

Fixed Effects: αi

• Decompose the model error term into two parts:

uit=αi+ϵit

• αi are group-specific fixed effects

  • group i tends to have higher or lower ˆY than other groups given regressor(s) Xit
  • estimate a separate αi for each group i
  • essentially, estimate a separate constant (intercept) for each group
  • notice this is stable over time within each group (subscript only i, no t)

• This includes all factors that do not change within group i over time

Fixed Effects: ϵit

• Decompose the model error term into two parts:

uit=αi+ϵit

• ϵit is the remaining random error

  • As usual in OLS, assume the 4 typical assumptions about this error:
  • E[ϵit]=0, var[ϵit]=σ2ϵ, cor(ϵit,ϵjt)=0, cor(ϵit,Xit)=0

• ϵit includes all other factors affecting Yit not contained in group effect αi

  • i.e. differences within each group that change over time
  • Be careful: Xit can still be endogenous due to other factors!
    • cor(Xit,ϵit)≠0

Fixed Effects: New Regression Equation

ˆYit=β0+β1Xit+αi+ϵit

• We've pulled αi out of the original error term into the regression

• Essentially we’ll estimate an intercept for each group (minus one, which is β0)

  • avoiding the dummy variable trap

• Must have multiple observations (over time) for each group (i.e. panel data)

Fixed Effects: Our Example

^Deathsit=β0+β1Cell phonesit+αi+ϵit

• αi is the State fixed effect

  • Captures everything unique about each state i that does not change over time
  • culture, institutions, history, geography, climate, etc!

• There could still be factors in ϵit that are correlated with Cell phonesit!

  • things that do change over time within States
  • perhaps individual States have cell phone bans for some years in our data

Estimating Fixed Effects Models

ˆYit=β0+β1Xit+αi+ϵit

• Two methods to estimate fixed effects models:

1. Least Squares Dummy Variable (LSDV) approach

2. De-meaned data approach

Least Squares Dummy Variable Approach

Least Squares Dummy Variable Approach: Our Example

• Let Alabama be the reference category (β0), include all other States

Our Example in R I

^Deathsit=β0+β1Cell Phonesit+Alaskai+⋯+Wyomingi

• If state is a factor variable, just include it in the regression

• R automatically creates N−1 dummy variables and includes them in the regression

  • Keeps intercept and leaves out first group dummy

Our Example in R II

fe_reg_1 <- lm(deaths ~ cell_plans + state, data = phones)
fe_reg_1 %>% tidy()

De-meaned Approach I

• Alternatively, we can control our regression for group fixed effects without directly estimating them

• We simply de-mean the data for each group to remove the group fixed-effect

• For each group i, find the means (over time, t): ˉYi=β0+β1ˉXi+ˉαi+ˉϵit

• Where:
  • ˉYi: average value of Yit for group i
  • ˉXi: average value of Xit for group i
  • ˉαi: average value of αi for group i (=αi)
  • ˉϵit=0, by assumption 1 about errors

De-meaned Approach II

^† Recall Rule 4 from the 2.3 class notes on the Summation Operator: ∑(Xi−ˉX)=0

De-meaned Approach III

˜Yit=β1˜Xit+˜ϵit

• Yields identical results to dummy variable approach

• More useful when we have many groups (would be many dummies)

• Demonstrates intuition behind fixed effects:

  • Converts all data to deviations from the mean of each group
  • All groups are “centered” at 0, no variation across groups
  • Fixed effects are often called the “within” estimators, they exploit variation within groups, not across groups

De-meaned Approach IV

• We are basically comparing groups to themselves over time

  • apples to apples comparison
  • e.g. Maryland in 2000 vs. Maryland in 2005

• Ignore all differences between groups, only look at differences within groups over time

De-Meaning the Data in R I

# get means of Y and X by state
means_state <- phones %>%
  group_by(state) %>%
  summarize(avg_deaths = mean(deaths),
            avg_phones = mean(cell_plans))
# look at it
means_state

De-Meaning the Data in R II

ggplot(data = means_state)+
  aes(x = fct_reorder(state, avg_deaths),
      y = avg_deaths,
      color = state)+
  geom_point()+
  geom_segment(aes(y = 0,
                   yend = avg_deaths,
                   x = state,
                   xend = state))+
  coord_flip()+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=10)+
  theme(legend.position = "none")

Visualizing “Within Group” Estimates for the 5 States

Visualizing “Within Group” Estimates for All 51 States

De-meaned Approach in R I

• The fixest package is designed for running regressions with fixed effects

• feols() function is just like lm(), with some additional arguments:

#install.packages("fixest")
library(fixest)
fe_reg_1_alt <- feols(deaths ~ cell_plans | state,
                      data = phones)

De-meaned Approach in R II

fe_reg_1_alt %>% summary()

## OLS estimation, Dep. Var.: deaths
## Observations: 306 
## Fixed-effects: state: 51
## Standard-errors: Clustered (state) 
##             Estimate Std. Error  t value  Pr(>|t|)    
## cell_plans -0.001204   0.000143 -8.41708 3.792e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 1.05007     Adj. R2: 0.886524
##                 Within R2: 0.357238

# or using broom's tidy()
fe_reg_1_alt %>% tidy()

Two-Way Fixed Effects

Two-Way Fixed Effects

Two-Way Fixed Effects

• A one-way fixed effects model estimates a fixed effect for groups

• Two-way fixed effects model estimates fixed effects for both groups and time periods ^Yit=β0+β1Xit+αi+θt+νit

• αi: group fixed effects

  • accounts for time-invariant differences across groups

• θt: time fixed effects

  • accounts for group-invariant differences over time

• νit remaining random error

  • all remaining factors that affect Yit that vary by state and change over time

Two-Way Fixed Effects: Our Example

^Deathsit=β0+β1Cell phonesit+αi+θt+νit

• αi: State fixed effects

  • differences across states that are stable over time (note subscript i only)
  • e.g. geography, culture, (unchanging) state laws

• θt: Year fixed effects

  • differences over time that are stable across states (note subscript t only)
  • e.g. economy-wide macroeconomic changes, federal laws passed

Visualizing Year Effects I

# find averages for years
means_year <- phones %>%
  group_by(year) %>%
  summarize(avg_deaths = mean(deaths),
            avg_phones = mean(cell_plans))
means_year

Visualizing Year Effects II

ggplot(data = phones)+
  aes(x = year,
      y = deaths)+
  geom_point(aes(color = year))+
  # Add the yearly means as black points
  geom_point(data = means_year,
             aes(x = year,
                 y = avg_deaths),
             size = 3,
             color = "black")+
  # connect the means with a line
  geom_line(data = means_year,
            aes(x = as.numeric(year),
                y = avg_deaths),
            color = "black",
            size = 1)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size = 14)+
  theme(legend.position = "none")

Estimating Two-Way Fixed Effects

ˆYit=β0+β1Xit+αi+θt+νit

• As before, several equivalent ways to estimate two-way fixed effects models:

1) Least Squares Dummy Variable (LSDV) Approach: add dummies for both groups and time periods (separate intercepts for groups and times)

2) Fully De-meaned data: ˜Yit=β1˜Xit+˜νit

where for each variable: ~varit=varit−¯vart−¯vari

3) Hybrid: de-mean for one effect (groups or years) and add dummies for the other effect (years or groups)

LSDV Method

fe2_reg_1 <- lm(deaths ~ cell_plans + state + year,
                data = phones)
fe2_reg_1 %>% tidy()

With fixest

fe2_reg_2 <- feols(deaths ~ cell_plans | state + year,
                 data = phones)
fe2_reg_2 %>% summary()

## OLS estimation, Dep. Var.: deaths
## Observations: 306 
## Fixed-effects: state: 51,  year: 6
## Standard-errors: Clustered (state) 
##            Estimate Std. Error   t value Pr(>|t|) 
## cell_plans   -3e-04   0.000305 -0.980739  0.33144 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.930036     Adj. R2: 0.909197
##                  Within R2: 0.011989

fe2_reg_2 %>% tidy()

Adding Covariates

Adding Covariates I

^Deathsit=β1Cell Phonesit+αi+θt+urban pctit+cell banit+text banit

• Can still add covariates to remove endogeneity not soaked up by fixed effects
  • factors that change within groups over time
  • e.g. some states pass bans over the time period in data (some years before, some years after)

Adding Covariates II

fe2_controls_reg <- feols(deaths ~ cell_plans + text_ban + urban_percent + cell_ban | state + year,
                          data = phones) 
fe2_controls_reg %>% summary()

## OLS estimation, Dep. Var.: deaths
## Observations: 306 
## Fixed-effects: state: 51,  year: 6
## Standard-errors: Clustered (state) 
##                Estimate Std. Error  t value Pr(>|t|)    
## cell_plans    -0.000340   0.000277 -1.22780 0.225269    
## text_ban1      0.255926   0.243444  1.05127 0.298188    
## urban_percent  0.013135   0.009815  1.33822 0.186878    
## cell_ban1     -0.679796   0.335655 -2.02528 0.048194 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.920123     Adj. R2: 0.910039
##                  Within R2: 0.032939

fe2_controls_reg %>% tidy()

Comparing Models

library(huxtable)
huxreg("Pooled" = pooled,
       "State Effects" = fe_reg_1,
       "State & Year Effects" = fe2_reg_1,
       "With Controls" = fe2_controls_reg,
       coefs = c("Intercept" = "(Intercept)",
                 "Cell phones" = "cell_plans",
                 "Cell Ban" = "cell_ban1",
                 "Texting Ban" = "text_ban1",
                 "Urbanization Rate" = "urban_percent"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 4)