The Pros and Cons of Panel Data

The 3 Panel Data Estimators

1. Pooled OLS: The simplest estimator, it pools all the data and estimates a single OLS regression.

2. Random Effects: Assumes that the unobserved heterogeneity is uncorrelated with the observed variables.

3. Fixed Effects: Controls for unobserved heterogeneity by including a dummy variable for each entity.

Pooled OLS

Simple idea: ignore the panel structure and estimate a single OLS regression.

\[ Y_{it} = \delta D_{it} + \eta_{it} \quad t=1,2,\dots, T\]

Inference in Pooled OLS

The standard errors in Pooled OLS are too small because they do not account for the panel structure.

• Serial correlation: Observations within the same entity are likely to be correlated, but to be consistent POLS assumes: \(\text{Cov}(\varepsilon_{it}, \varepsilon_{it+1}) = 0\).

• Heteroskedasticity: The variance of the error term may vary across entities and time periods.

• Solution \(\Longrightarrow\) Clustered Standard Errors.

Clustered Standard Errors

• Assume that clusters are randomly drawn from a larger population

• Assume that the error term is correlated within the same entity \(i\).

• Available in most statistical software.

Fixed Effects

• Fixed effects accounts for all variables, observed or not, as long as they stay constant within some larger category.

• You can estimate Fixed effects by:

  1. Controlling for units (i.e., person, company, school, country, year, etc.) by including a dummy variable for each unit.
  2. transforming the model by demeaning it (i.e., subtracting the mean of each unit from the variable).

• Fixed effects uses the within-unit variation to estimate the effect.

• All between-unit variation is removed.

Fixed Effects Simple Example

Fixed Effects Simple Example

Visualizing Fixed Effects with Dummies

Fixed Effects Estimation with Dummies

Our panel data model with fixed effects can be written as: \[ Y_{it} = \delta D_{it} + \sum_{j=2}^{N}\gamma_j d_{j} + \eta_{it} \]

• Here \(d_{j}\) is a dummy variable for entity \(j\) (i.e., person, company, school, country, year, etc.).

• Feasible for a small number of entities.

• Cannot include variables that do not vary within the entity.

Estimating the Within Model in R

library(tidyverse); library(modelsummary)
gm <- causaldata::gapminder

gm <- gm %>%
    # Put GDP per capita in log format since it's very skewed
    mutate(log_GDPperCap = log(gdpPercap)) %>%
    # Perform each calculation by group
    group_by(country) %>%
    # Get within variation by subtracting out the mean
    mutate(lifeExp_within = lifeExp - mean(lifeExp),
    log_GDPperCap_within = log_GDPperCap - mean(log_GDPperCap)) %>%
    # We no longer need the grouping
    ungroup()

# Analyze the within variation
m1 <- lm(lifeExp_within ~ log_GDPperCap_within, data = gm)
msummary(
  m1,
  stars = c('*' = .1, '**' = .05, '***' = .01),
  gof_map = list(
    list(raw = "nobs",      clean = "Num. Obs.", fmt = 0),
    list(raw = "r.squared", clean = "R2",      fmt = 3)
  )
)

Fixed Effects in Practice: feols

library(tidyverse); library(modelsummary); library(fixest)
gm <- causaldata::gapminder

# feols: absorb country FE, cluster SEs at country level
m2 <- feols(lifeExp ~ log(gdpPercap) | country,
            cluster = ~country, data = gm)

# Compare within (OLS, for illustration) vs feols (clustered SEs)
msummary(
  list("Within (illustrative)" = m1, "feols (clustered SE)" = m2),
  stars = c('*' = .1, '**' = .05, '***' = .01),
  coef_map = c(
    "log_GDPperCap_within" = "log GDP per Capita",
    "log(gdpPercap)" = "log GDP per Capita"
  ),
  gof_map = list(
    list(raw = "nobs",      clean = "Num. Obs.", fmt = 0),
    list(raw = "r.squared", clean = "R2",      fmt = 3)
  )
)

FE Example: The Returns to Marriage

What is the causal effect of marriage on earnings?

• Married men earn more than single men.

• Cornwell and Rupert (1997) use panel data to estimate the effect of marriage on earnings.

• They estimate the following model: \[ \log(wage_{it}) = \alpha + \delta M_{it} + \beta X_{it} + A_i + \gamma_t + \varepsilon_{it} \]

FE Example: The Returns to Marriage

Table 8.2: Estimated Wage Regressions

Standard errors in parentheses.

Multiple Fixed Effects

• If we observe individuals over time we can include multiple fixed effects in the model. E.g.:
  • Individual fixed effects -> variation within individual
  • Time fixed effects -> variation within year
• This is called a two-way fixed effects model: \[ Y_{it} = \delta D_{it} + u_i + \lambda_t + \eta_{it} \]

What variation remains?

Choosing Between FE and RE: The Hausman Test

• Both FE and RE are consistent when \(\text{Cov}(D_{it}, u_i) = 0\).
• Only FE is consistent when \(\text{Cov}(D_{it}, u_i) \neq 0\).
• The Hausman test (Hausman, 1978) compares FE and RE estimates:
  • \(H_0\): \(\text{Cov}(D_{it}, u_i) = 0\) — RE is consistent and efficient.
  • \(H_1\): \(\text{Cov}(D_{it}, u_i) \neq 0\) — RE is inconsistent; use FE.
• If \(H_0\) is rejected, use FE. If not rejected, RE is preferred (more efficient).

library(plm)
phtest(fe_model, re_model)