How can both patterns be true at the same time?

USA: aggregate income grew, but median incomes stagnated due to rising inequality. Most people did not experience the income growth the average implied.

Richer countries also have:

• Stronger institutions and rule of law
• Better public health systems
• Higher social trust
• More stable political histories

These all correlate with both income and happiness — but they are not income.

• A country’s culture and institutions do not change year to year
• Firm “DNA” is largely stable across survey waves
• A person’s innate ability does not fluctuate

The payoff: estimates that use only within-unit change over time, holding stable characteristics constant by construction.

This is what today’s session is about.

The WHR panel is unbalanced — not every country is surveyed in every year. This is typical and not problematic for FE estimation.

The problem: observations from the same country across years are not independent. The error \(\epsilon_{it}\) is not truly random — it has two components:

\[\epsilon_{it} = \underbrace{\eta_i}_{\text{stable, country-specific}} + \underbrace{v_{it}}_{\text{idiosyncratic}}\]

• \(\eta_i\): a country’s culture, institutions, social trust — stable but unobserved
• \(v_{it}\): genuine year-to-year noise

Scandinavian countries have high happiness and high GDP and strong institutions. Those institutions are not caused by income — they are \(\eta_i\).

The fix: \(\eta_i\) is stable over time, so subtract each country’s time-average:

\[\tilde{y}_{it} = y_{it} - \bar{y}_i \qquad \tilde{x}_{it} = x_{it} - \bar{x}_i\]

Because \(\eta_i - \bar{\eta}_i = 0\), the unobserved country effect vanishes entirely.

Strategy: if \(\eta_i\) is time-invariant, subtract each unit’s time-average from its observations.

For unit \(i\), compute \(\bar{y}_i = T^{-1}\sum_t y_{it}\) and \(\bar{x}_i = T^{-1}\sum_t x_{it}\), then:

\[\tilde{y}_{it} = y_{it} - \bar{y}_i \qquad \tilde{x}_{it} = x_{it} - \bar{x}_i\]

Because \(\eta_i - \bar{\eta}_i = 0\), the individual effect vanishes entirely.

OLS on the demeaned data gives the Fixed Effects (FE) estimator, also called the within estimator.

The question it answers:

When a country’s income rises above its own historical average, does its happiness tend to rise too?

This is much more credible for causal interpretation than pooled OLS, because all stable country-level confounders — observed or not — are controlled for by construction.

Variables that do not change over time for a given unit are differenced out entirely:

• Geographic region
• Colonial history
• Language group
• Any characteristic constant across the panel

Their coefficients cannot be estimated with FE.

But what about shocks that hit all countries in a given year?

• A global financial crisis (2008–2009)
• A worldwide pandemic (2020)
• A global commodity price surge

These affect all countries in the same period, regardless of their individual characteristics.

If such shocks correlate with our explanatory variable, unit FE alone does not protect us.

This controls for any factor that shifts the outcome equally across all units in a given year — observed or not.

The two-way fixed effects model:

\[y_{it} = \underbrace{\alpha_i}_{\text{country FE}} + \underbrace{\gamma_t}_{\text{year FE}} + \beta \, x_{it} + v_{it}\]

\(\hat\beta\) is now identified from variation that is neither explained by which country it is nor which year it is.

Standard OLS standard errors assume independence and are therefore too small, leading to overconfident inference.

The solution: cluster standard errors at the unit level.

This is the meaningful fit statistic for FE models.

Classical \(R^2\) can be misleadingly high: unit means are removed before estimation, so part of the total variation is never in play.

fixest reports:

• Within R² — fit on demeaned data ← report this
• R² — overall (includes between variation) ← less informative

Pooled OLS has no such argument. It mixes within- and between-country variation; stable country differences (institutions, culture, geography) contaminate \(\hat\beta\).

Fixed effects provides one:

“By comparing each country to itself over time, I remove all stable between-country differences — observed or not. The remaining income variation is not contaminated by time-invariant confounders.”

1. Time-varying confounders

Variables that change over time and independently affect both income and happiness — e.g. a political reform that simultaneously boosts growth and improves governance. Partial remedies: year FE (common shocks), explicit controls (Gini, unemployment).

2. No feedback loops

FE requires strict exogeneity: past happiness must not affect future income. If dissatisfied workers reduce effort → lower income → lower happiness, the within-country coefficient is biased even after demeaning.

3. FE is one step on a longer ladder

Already controlled for:

• Country FE: stable culture, institutions, geography — all time-invariant confounders
• Year FE: global shocks affecting all countries simultaneously

Not yet controlled for: time-varying factors that change differently across countries and independently affect both income and happiness.

Most WHR variables — social support, healthy life expectancy, freedom to choose — are likely mediators, not confounders.

Better time-varying controls for this analysis:

• Gini coefficient: directly tests whether rising inequality explains the US paradox
• Unemployment rate: macroeconomic conditions independent of income levels

Why pooled OLS fails: \(\eta_i\) correlated with \(x_{it}\) → OVB.

The FE estimator: within-transformation removes \(\eta_i\); identifies within-unit variation only; cannot estimate time-invariant covariates.

Between vs. within: two different questions, two different answers — be deliberate about which variation you use.

Two-way FE: adds year fixed effects \(\gamma_t\) to control for common period shocks (recessions, pandemics).

In R: fixest::feols(), cluster at country level, report within-\(R^2\), compare models with modelsummary.