Time Series Autocorrelation Lab

What is this lab about?

In time series analysis, the Theoretical ACF is an exact, fixed mathematical property of a specific model (like AR or MA). However, in practice, we never know the true model. We estimate the autocorrelations using finite data, yielding the Sample ACF.

"In theory, the ACF is fixed. In practice, we estimate it from finite data, so the estimate varies from sample to sample."

Use the controls below to generate thousands of simulated time series. Watch how the length of the series ($n$) and the model parameters affect the accuracy and variance of our estimates.

Laboratory Parameters

Model Type

Phi (φ₁) / AR1 0.70

Series Length (n) Max Lag

Simulations

A. Mean Sample ACF vs Theoretical

Averaged over simulations.

D. One Random Realization

A single dataset can look very "noisy" compared to the average.

B. Distribution of Sample ACF (Lags 1-3)

Shows the spread of estimates. Solid line = Theoretical, Dashed = Sample Mean.

C. Bias & Error Summary

Lag	Theoretical ACF	Mean Sample ACF	Bias	Std Deviation

Take-Home Messages

• Theoretical ACF is an exact mathematical property of the model.
• Sample ACF is an estimate based on finite data.
• Because data is random, the Sample ACF varies from sample to sample.
• Larger samples (higher N) usually give more reliable estimates with smaller variance.
• Even simple time-series models require substantial data for stable estimation.

Try This

Experiment 1: The effect of N
Set model to AR(1) with Phi = 0.7. Run simulations for N=30, then N=200, then N=1000.
Question: How does the spread (width) of the sample ACF histogram at lag 1 change?

Experiment 2: MA Signature
Set model to MA(1) with Theta = 0.8. Look at Lag 2 and beyond.
Question: The theoretical ACF is exactly 0 for lag > 1. Does the sample ACF also perfectly equal 0?

Experiment 3: Law of Large Numbers
Keep parameters fixed. Change simulations from 100 to 10000.
Question: Does the single realization change? Does the mean sample ACF become smoother?