On Effective Sample Size

The Problem with Autocorrelated Data

Standard inference assumes independent observations. For \(n\) i.i.d. draws:

\[\text{SE}(\bar{X}) = \frac{\sigma}{\sqrt{n}}\]

Time series data violates this. When observations are positively autocorrelated, each new value carries less new information than an independent draw would.

Treating \(n\) autocorrelated observations as \(n\) independent ones leads to:

• Standard errors that are too small
• Confidence intervals that are too narrow
• \(p\)-values that are too optimistic

Intuition

Consider a slowly meandering series — say, daily average temperature.

High autocorrelation

If \(\rho_1 = 0.95\), tomorrow is nearly predictable from today.
Collecting 365 observations gives far less than 365 “bits” of information.

Low autocorrelation

If \(\rho_1 \approx 0\), each observation is essentially fresh.
\(n_\text{eff} \approx n\).

The effective sample size (ESS) quantifies how many independent observations your autocorrelated series is actually worth.

Finite-Sample Correction

The exact variance of \(\bar{X}\) for a stationary series of length \(n\) is:

\[\text{Var}(\bar{X}) = \frac{\sigma^2}{n}\underbrace{\left(1 + 2\sum_{k=1}^{n-1}\left(1 - \frac{k}{n}\right)\rho_k\right)}_{\tau_n}\]

The weights \(\bigl(1 - k/n\bigr)\) are Bartlett weights — they arise directly from the overlapping structure of the sample mean and shrink toward zero as lag \(k\) approaches \(n\).

Setting \(\text{Var}(\bar{X}) = \sigma^2 / n_{\text{eff}}\) gives the finite-sample ESS:

\[\boxed{n_{\text{eff}}^{(n)} = \frac{n}{\tau_n}, \qquad \tau_n = 1 + 2\sum_{k=1}^{n-1}\!\left(1-\frac{k}{n}\right)\rho_k}\]

This is exact under stationarity — no approximation yet.

Asymptotic Correction

As \(n \to \infty\), the Bartlett weights all \(\to 1\) and the upper limit of the sum extends to \(\infty\), giving the integrated autocorrelation time:

\[\tau_{\text{int}} = \lim_{n\to\infty}\tau_n = 1 + 2\sum_{k=1}^{\infty}\rho_k\]

The asymptotic ESS is then:

\[\boxed{n_{\text{eff}}^{(\infty)} = \frac{n}{\tau_{\text{int}}}}\]

Note

Summability condition. This limit exists and is finite if and only if \(\sum_{k=1}^{\infty}|\rho_k| < \infty\) — satisfied by ARMA processes and short-memory series generally, but not by long-memory (fractionally integrated) processes, where \(\tau_{\text{int}} = \infty\).

This is the version most software implements and most textbooks quote.

How the Two Relate

For \(\phi > 0\) the Bartlett weights downweight the higher lags, so:

\[\tau_n \;<\; \tau_{\text{int}} \qquad \Longrightarrow \qquad n_{\text{eff}}^{(n)} \;>\; n_{\text{eff}}^{(\infty)}\]

The asymptotic formula is the more conservative (pessimistic) estimate. The gap closes as \(n\) grows or \(\phi\) shrinks.

AR(1) closed forms:

\[\tau_{\text{int}} = \frac{1+\phi}{1-\phi}, \qquad \tau_n = \frac{1+\phi}{1-\phi} - \frac{2\phi\!\left(1-\phi^n\right)}{n(1-\phi)^2}\]

The correction term \(\frac{2\phi(1-\phi^n)}{n(1-\phi)^2}\) is the gap; it is \(O(1/n)\) for fixed \(\phi\) and \(O\!\bigl(n^{-1}(1-\phi)^{-2}\bigr)\) near the unit root — largest precisely when \(\phi \approx 1\) and \(n\) is small.

Comparing the Two: AR(1) Table

\(n_{\text{eff}}^{(n)}\) vs. \(n_{\text{eff}}^{(\infty)}\) for selected \(\phi\) and \(n\)

\(\phi\)	\(n = 30\)		\(n = 200\)		\(n = 1000\)
	finite	asymp	finite	asymp	finite	asymp
0.5	10.5	10.0	67.1	66.7	333.8	333.3
0.8	3.9	3.3	22.7	22.2	111.6	111.1
0.9	2.3	1.6	11.0	10.5	53.1	52.6
0.95	1.6	0.8	5.7	5.1	26.2	25.6

Note

The two estimates are close in most practical settings — the gap is only appreciable when \(n\) is small and \(\phi\) is near 1. The asymptotic formula consistently understates \(n_{\text{eff}}\) (is more conservative). The absolute values are genuinely small: \(\phi = 0.9\) with \(n = 200\) really does yield only ~10–11 effective observations.

AR(1) Closed Forms

For \(X_t = \phi X_{t-1} + \varepsilon_t\), \(\rho_k = \phi^k\):

Asymptotic (infinite sum, weights \(= 1\)):

\[n_{\text{eff}}^{(\infty)} = n\,\frac{1-\phi}{1+\phi}\]

Finite-sample (Bartlett-weighted, exact):

\[n_{\text{eff}}^{(n)} = \frac{n(1-\phi)^2}{\;(1-\phi^2) - \tfrac{2\phi(1-\phi^n)}{n}\;}\]

Note

A useful rule of thumb: the two agree to within 5 % when \(n \gtrsim 10\,/\,(1-\phi)\), i.e. when the series is roughly ten times longer than the process’s characteristic decorrelation length. Below that threshold, prefer the finite-sample formula.

Estimation in Practice

Population autocorrelations \(\rho_k\) are unknown; substitute \(\hat\rho_k\).

Finite-sample plug-in (preferred for small \(n\)):

\[\hat\tau_n = 1 + 2\sum_{k=1}^{n-1}\!\left(1-\frac{k}{n}\right)\hat\rho_k\]

This is the Newey–West / Bartlett kernel HAC variance estimator, widely used in econometrics (the \(1/n\) correction is built in).

Asymptotic plug-in (truncated at lag \(K\)):

\[\hat\tau_{\text{int}} = 1 + 2\sum_{k=1}^{K}\hat\rho_k\]

Choosing \(K\) still requires care: fixed window, automatic noise-threshold cutoff \(|\hat\rho_k| < c/\sqrt{n}\), batch means, or spectral (\(\tau_{\text{int}} = \pi S(0)/\sigma^2\)).

Warning

The asymptotic estimator without truncation (\(K = n-1\) but weights \(= 1\)) is inconsistent — use Bartlett weights or a principled truncation.

Worked Example in R

set.seed(42)
phi <- 0.90   # true AR parameter (known in simulation)
n   <- 200
x   <- arima.sim(model = list(ar = phi), n = n)

# Asymptotic ESS — population closed form
ess_asymp  <- n * (1 - phi) / (1 + phi)

# Finite-sample ESS — population closed form (exact)
tau_fs     <- (1 + phi)/(1 - phi) - 2*phi*(1 - phi^n) / (n*(1-phi)^2)
ess_finite <- n / tau_fs

# Sample-based ESS — fit AR(1), plug phi_hat into finite-sample formula
phi_hat    <- coef(arima(x, order = c(1, 0, 0)))["ar1"]
tau_hat    <- (1 + phi_hat)/(1 - phi_hat) -
              2*phi_hat*(1 - phi_hat^n) / (n*(1 - phi_hat)^2)
ess_hat    <- n / tau_hat

cat("n                         :", n,                    "\n")

n                         : 200 

cat("phi_hat (AR(1) MLE)       :", round(phi_hat, 4),    "\n")

phi_hat (AR(1) MLE)       : 0.8535 

cat("ESS asymptotic  (pop.)    :", round(ess_asymp,  1),  "\n")

ESS asymptotic  (pop.)    : 10.5 

cat("ESS finite-smpl (pop.)    :", round(ess_finite, 1),  "\n")

ESS finite-smpl (pop.)    : 11 

cat("ESS finite-smpl (plug-in) :", round(ess_hat,    1),  "\n")

ESS finite-smpl (plug-in) : 16.3 

```

Warning

A common mistake: summing Bartlett weights over all \(n-1\) lags without truncation is inconsistent — the accumulated noise in the sample ACF dominates and can produce ESS \(> n\). Always truncate at a principled bandwidth \(K\), or estimate \(\phi\) and use the closed form as above.

Corrected Inference

Once \(n_{\text{eff}}\) is in hand, apply it wherever \(n\) normally appears:

Quantity	Naive (wrong)	Corrected
Standard error of mean	\(\hat\sigma / \sqrt{n}\)	\(\hat\sigma / \sqrt{n_{\text{eff}}}\)
95 % CI half-width	\(1.96\,\hat\sigma / \sqrt{n}\)	\(1.96\,\hat\sigma / \sqrt{n_{\text{eff}}}\)
\(t\)-statistic d.f.	\(n - 1\)	\(n_{\text{eff}} - 1\)
Required sample size	\(n^*\)	\(n^* \cdot \tau\)

Use \(n_{\text{eff}}^{(n)}\) (finite-sample) when \(n\) is small or \(\phi\) is large; \(n_{\text{eff}}^{(\infty)}\) (asymptotic) is a safe default for large \(n\) and is what most software returns.

Key Takeaways

1. Two distinct corrections exist. The finite-sample formula uses Bartlett weights \((1-k/n)\); the asymptotic formula takes \(n \to \infty\) and drops them. They are not the same thing.

2. The asymptotic formula is conservative. It underestimates \(n_{\text{eff}}\) relative to the exact formula, especially at high persistence or small \(n\).

3. The gap is \(O(1/n)\) but can be large when \(\phi \approx 1\), because the correction term scales as \((1-\phi)^{-2}\).

4. Rule of thumb: use the finite-sample correction when \(n \lesssim 10/(1-\phi)\); otherwise the asymptotic version suffices.

5. Both versions use \(n_{\text{eff}}\) for all downstream inference — SEs, CIs, \(t\)-tests, power calculations.