Autocorrelation, Stationarity, and Effective Sample Size

200 dependent observations can be worth 16 independent ones — see the ACF, the PACF, and the standard error you should have reported.

Ch 12 12.2

Explorer

Temporal & Longitudinal

18 mindifficulty 3/5

Controls

AR(1) coefficientφ0.850

Stationary only while |φ| < 1.

Linear drift0

A deterministic trend added to the process.

Seasonal amplitude0

Seasonal periods12

Innovation SDσ1

Series lengthT200

First difference

Plot and analyse Δxₜ = xₜ − xₜ₋₁ instead.

Random seed

Every result on this page is a deterministic function of the seed and the controls.

The series

Successive values are dependent — the observations are not a random sample.

ACF

Correlation with the series k steps earlier.

sample ACFtheoretical φᵏwhite-noise band

Partial ACF

AR(1) shows one spike, then nothing.

Lag plot

xₜ against xₜ₋₁ with the fitted AR(1) line — dependence made visible.

φ̂ from the lag regression

0.800

set to 0.85

r₁

0.795

white-noise band ±0.139

Half-life of a shock

3.1 steps

log 0.5 / log φ

Effective sample size

T = 200; T(1−φ)/(1+φ)

Naive SE of the mean

0.1200

assumes independence

Dependence-corrected SE

0.3598

inflation ×3

Sample size is not observation count

With φ = 0.85, 200 observations carry about as much information about the mean as 16 independent ones. Every standard error, confidence interval, and p-value computed as if the data were i.i.d. is too small by the inflation factor shown above.

Predict before you look

You collect 200 daily measurements from a process with lag-one correlation 0.85 and compute a 95% interval for the mean as x̄ ± 1.96 s/√200. How does that interval compare with the correct one?

Scenarios

Teacher-ready starting points.

Documentation

What this shows

An AR(1) process with optional deterministic drift and seasonality, shown together with its sample ACF against the theoretical φᵏ curve, its PACF, and a lag-one scatter plot with the fitted autoregression. Two standard errors for the mean are reported side by side: the naive one and the one corrected for dependence.

Dependence and the information it costs

For xₜ = φxₜ₋₁ + εₜ with |φ| < 1 the process is stationary with autocorrelation ρₖ = φᵏ, so the ACF decays geometrically and the PACF is zero beyond lag 1. That signature is how AR order is identified: the ACF tails off while the PACF cuts off.

At |φ| = 1 the process is a random walk. There is no fixed mean or variance to estimate, the sample ACF decays roughly linearly rather than geometrically, and inference on levels is meaningless. First differencing recovers white noise, which is the “I” in ARIMA.

Dependence reduces information. The variance of the sample mean of an AR(1) series is inflated by roughly (1 + φ)/(1 − φ), so the effective sample size is about T(1 − φ)/(1 + φ). At φ = 0.9 that is 5% of the nominal T, and every interval computed under independence is about 4.4 times too narrow.

A deterministic trend or a seasonal component induces large autocorrelations of its own. An ACF computed before detrending therefore cannot be read as evidence of an autoregressive mechanism.

xₜ = φxₜ₋₁ + εₜ,  ρₖ = φᵏ,  Var(x) = σ²/(1 − φ²)

Var(x̄) ≈ (σ²/T) · (1 + φ)/(1 − φ)

T_eff ≈ T (1 − φ)/(1 + φ)

half-life of a shock = log(0.5)/log φ

Try this

Set φ = 0 and confirm every ACF bar lies inside the ±1.96/√T band.
Set φ = 0.85: read r₁, compare the ACF bars with the dashed φᵏ curve, and count PACF spikes.
Compare the naive and corrected standard errors at φ = 0.5, 0.85, and 0.95.
Set φ = 1.0 and describe the ACF; then switch on first differencing.
Set drift 0.05 with φ = 0.3 and explain why the ACF looks persistent anyway.
Add seasonal amplitude 3 with period 12 and find the spikes at lags 12 and 24.

Common misconception

“I have 200 measurements, so my standard error is s/√200.”

That formula assumes independence. With AR(1) dependence the variance of the mean is inflated by (1 + φ)/(1 − φ): at φ = 0.85 the 200 observations carry the information of about 16 independent ones, and the honest standard error is roughly 3.5 times the naive one. Confidence intervals and p-values computed as if time-ordered data were a random sample are systematically overconfident — this is one of the most common errors in applied longitudinal work.

DSPA3 Chapter 12 — Time series and longitudinal analysis