PACF Analysis Results
Run the calculator to generate partial autocorrelation diagnostics.
Spikes outside the confidence region are highlighted as statistically notable.
| Absolute | Lower bound | Upper bound | Status | Direction | Interpretation |
|---|
Formula Used
Regression definition: PACF at lag k is the coefficient of yt−k after regressing yt on lags 1 through k.
Approximate confidence limits: ±zα/2 / √n.
Regular difference: Δyt = yt − yt−1.
Seasonal difference: Δsyt = yt − yt−s.
The Yule-Walker and Levinson-Durbin methods recover reflection coefficients recursively. OLS estimates each lag order with a separate regression. Burg estimation minimises forward and backward prediction errors.
How to Use
- Paste values, upload a file, load a preset, or generate data.
- Review and edit observations in the data table.
- Choose missing-value, transformation, differencing, and detrending options.
- Select the maximum lag, estimation method, and confidence level.
- Customise plot labels, dimensions, confidence shading, and display options.
- Calculate the PACF and inspect significant lags, diagnostics, and warnings.
- Export the chart, results, transformed data, JSON summary, or PDF report.
Example Interpretation Guide
| Observed PACF pattern | Possible interpretation | Next check |
|---|---|---|
| One strong spike at lag 1 | Possible AR(1) structure | Inspect ACF decay and residuals |
| Strong spikes through lag 2 | Possible AR(2) structure | Compare candidate AR orders |
| Repeated spikes near lag 12 | Possible seasonal AR behaviour | Check seasonal differencing |
| Many slowly declining spikes | Potential trend or nonstationarity | Apply stationarity diagnostics |
| No notable spikes | Series may resemble white noise | Review Ljung-Box result |
Frequently Asked Questions
What does PACF measure?
PACF measures the direct relationship between observations separated by a lag. It removes effects transmitted through shorter lags.
How is PACF different from ACF?
ACF includes direct and indirect lag relationships. PACF isolates the remaining relationship after controlling intermediate lags.
How can PACF suggest an AR order?
An autoregressive process often shows notable PACF spikes through its order. Later spikes may fall inside confidence limits.
Why should the series be stationary?
Trend and changing variance can create misleading lag patterns. Differencing or transformation may improve interpretability.
What do confidence bands mean?
They provide approximate reference limits under weak serial dependence. Spikes outside them deserve further investigation.
Which PACF method should I use?
Yule-Walker is efficient for many stationary series. OLS and Burg provide useful sensitivity comparisons.
Can isolated significant spikes be trusted?
Multiple lags increase false-positive opportunities. Interpret isolated spikes with theory, seasonality, and residual checks.
What causes seasonal PACF spikes?
Repeated cycles can produce spikes at seasonal intervals. Seasonal AR terms or seasonal differencing may help.
Does PACF confirm an ARIMA model?
No. PACF only supports model identification. Compare alternatives using diagnostics and out-of-sample performance.
Why do methods give different values?
Methods use different finite-sample assumptions and estimators. Differences grow with short, noisy, or unstable series.