Testing Ergodicity in Panel Data (Stata)

Key Point

There is no standard Stata command that directly tests ergodicity. Ergodicity is a property of the underlying stochastic process rather than a finite sample. Instead, researchers test properties that imply or are consistent with ergodicity.

1. Test for Stationarity

Most econometric applications assume a process is ergodic if it is:

Stationary
Weakly dependent (mixing)
Has finite moments

Typical Stata commands:

xtunitroot llc y
xtunitroot ips y
xtunitroot fisher y

* Single time series
dfuller y
pperron y
kpss y

2. Compare Time and Cross-Sectional Averages

A defining property of ergodicity is that long-run time averages converge to the corresponding ensemble (cross-sectional) averages.

Compute both:

bysort year: egen csmean = mean(y)
bysort id: egen timemean = mean(y)

summ csmean
summ timemean

histogram timemean

Evidence supporting ergodicity would be that the distribution of time averages approaches the cross-sectional distribution as the time dimension grows.

3. Examine Heterogeneity

Persistent differences across individuals may indicate non-ergodicity.

bys id: egen mu = mean(y)
bys id: egen sd = sd(y)

Large differences in means or variances suggest heterogeneous underlying processes.

4. Examine Persistence

Ergodic stationary processes generally exhibit declining autocorrelation.

Useful commands:

corrgram y
xtreg y L.y
xtregar

Near-unit-root behavior or highly persistent autocorrelation argues against practical ergodicity.

5. Ergodicity Economics (Ole Peters)

In ergodicity economics, compare:

Ensemble-average growth
Time-average growth

Example:

gen growth = (wealth-L.wealth)/L.wealth
gen loggrowth = ln(1+growth)

summ growth
summ loggrowth

If the arithmetic and logarithmic growth rates differ substantially, the process is non-ergodic.

Practical Workflow

For a panel dataset, a practical assessment of ergodicity consists of:

Testing for stationarity.
Examining heterogeneity across units.
Evaluating autocorrelation and persistence.
Comparing time averages with cross-sectional averages.
Checking parameter stability over time.

This combination provides much stronger evidence than any single “ergodicity test.”