Basics of Panel Data

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# Basics of Panel Data
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### Ian McCarthy | Emory University
### Workshop on Causal Inference with Panel Data

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# Table of contents

1. [What are Panel Data](#panel)
2. [Estimation](#estimation)
3. [In Practice](#handson)
4. [Semantics](#semantics)

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name: panel

# What are Panel Data?

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# Nature of the Data

- Repeated observations of the same units over time

--
**Notation**
- Unit `\(i=1,...,N\)` over several periods `\(t=1,...,T\)`, which we denote `\(y_{it}\)`
- Treatment status `\(D_{it}\)`
- Regression model, <br>
`\(y_{it} = \delta D_{it} + u_{i} + \epsilon_{it}\)` for `\(t=1,...,T\)` and `\(i=1,...,N\)`

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# Benefits of Panel Data

- *May* overcome certain forms of omitted variable bias
- Allows for unobserved but time-invariant factor, `\(u_{i}\)`, that affects both treatment and outcomes

--
**Still assumes**
- No time-varying confounders 
- Past outcomes do not directly affect current outcomes
- Past outcomes do not affect treatment (reverse causality)

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# Some textbook settings

- Unobserved "ability" when studying schooling and wages
- Unobserved "quality" when studying physicians or hospitals

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# Estimating Regressions with Panel Data

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# Regression model

`\(y_{it} = \delta D_{it} + u_{i} + \epsilon_{it}\)` for `\(t=1,...,T\)` and `\(i=1,...,N\)`

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# Fixed Effects

`\(y_{it} = \delta D_{it} + u_{i} + \epsilon_{it}\)` for `\(t=1,...,T\)` and `\(i=1,...,N\)`

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- Allows correlation between `\(u_{i}\)` and `\(D_{it}\)`
- Physically estimate `\(u_{i}\)` in some cases via set of dummy variables
- More generally, "remove" `\(u_{i}\)` via:
  - "within" estimator
  - first-difference estimator
  
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# Within Estimator
`\(y_{it} = \delta D_{it} + u_{i} + \epsilon_{it}\)` for `\(t=1,...,T\)` and `\(i=1,...,N\)`

--
- Most common approach (default in most statistical software)
- Equivalent to demeaned model,<br>
`\(y_{it} - \bar{y}_{i} = \delta (D_{it} - \bar{D}_{i}) + (u_{i} - \bar{u}_{i}) + (\epsilon_{it} - \bar{\epsilon}_{i})\)`
- `\(u_{i} - \bar{u}_{i} = 0\)` since `\(u_{i}\)` is time-invariant
- Requires *strict exogeneity* assumption (error is uncorrelated with `\(D_{it}\)` for all time periods)

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# First-difference
`\(y_{it} = \delta D_{it} + u_{i} + \epsilon_{it}\)` for `\(t=1,...,T\)` and `\(i=1,...,N\)`

--
- Instead of subtracting the mean, subtract the prior period values<br>
`\(y_{it} - y_{i,t-1} = \delta(D_{it} - D_{i,t-1}) + (u_{i} - u_{i}) + (\epsilon_{it} - \epsilon_{i,t-1})\)`
- Requires exogeneity of `\(\epsilon_{it}\)` and `\(D_{it}\)` only for time `\(t\)` and `\(t-1\)` (weaker assumption than within estimator)
- Sometimes useful to estimate both FE and FD just as a check

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# Keep in mind...

- Discussion only applies to linear case or very specific nonlinear models
- Fixed effects can't solve reverse causality
- Fixed effects doesn't address unobserved, time-varying confounders
- Can't estimate effects on time-invariant variables
- May "absorb" a lot of the variation for variables that don't change much over time

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# Seeing things in action

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# Within Estimator (Default)

.pull-left[
**Stata**<br>

```stata
ssc install bcuse
bcuse wagepan
tsset nr year
xtreg lwage exper expersq, fe
```
]

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**R**<br>

```r
library(readstata13)
library(fixest)
wagepan <- read.dta13("http://fmwww.bc.edu/ec-p/data/wooldridge/wagepan.dta")
feols(lwage~exper + expersq | nr, data=wagepan)
```
]

---
# Within Estimator (Manually Demean)

.pull-left[
**Stata**<br>

```stata
ssc install bcuse
bcuse wagepan
foreach x of varlist lwage exper expersq {
  egen mean_`x'=mean(`x')
  egen demean_`x'=`x'-mean_`x'
}
reg demean_lwage demean_exper demean_expersq
```
]

.pull-right[
**R**<br>

```r
library(readstata13)
wagepan <- read.dta13("http://fmwww.bc.edu/ec-p/data/wooldridge/wagepan.dta")
wagepan <- wagepan %>%
  group_by(nr) %>%
  mutate(demean_lwage=lwage - mean(lwage),
         demean_exper=exper - mean(exper),
         demean_expersq=expersq - mean(expersq))
summary(lm(demean_lwage~demean_exper + demean_expersq, data=wagepan))
```
]

---
# First differencing

.pull-left[
**Stata**<br>

```stata
ssc install bcuse
bcuse wagepan
reg d.lwage d.exper d.expersq, noconstant
```
]

.pull-right[
**R**<br>

```r
library(readstata13)
wagepan <- read.dta13("http://fmwww.bc.edu/ec-p/data/wooldridge/wagepan.dta")
wagepan <- wagepan %>%
  group_by(nr) %>%
  arrange(year) %>%
  mutate(fd_lwage=lwage - lag(lwage),
         fd_exper=exper - lag(exper),
         fd_expersq=expersq - lag(expersq)) %>%
  na.omit()
summary(lm(fd_lwage~0 + fd_exper + fd_expersq, data=wagepan))
```
]