Partial Autocorrelation Tool Calculator

The partial autocorrelation at lag k measures the correlation between xₜ and xₜ₋ₖ after removing the linear effects of the intermediate lags xₜ₋₁ … xₜ₋ₖ₊₁.

• OLS method: for each lag k, fit xₜ = c + Σⱼ₌₁ᵏ φⱼ xₜ₋ⱼ + εₜ. The PACF at lag k is the coefficient φₖ.
• Durbin–Levinson method: compute autocorrelations rₖ and solve the Yule–Walker recursion. The PACF is the last coefficient φₖₖ produced by the recursion at lag k.

Significance band uses the common approximation: ± z / √N. It is a quick guide, not a strict hypothesis test.

1. Paste your observations into the values box.
2. Set the maximum lag you want to inspect.
3. Pick a method and confidence level for bands.
4. Optionally difference or center the series.
5. Press Calculate to view the table above the form.

For AR order selection, look for a sharp PACF cutoff.

Partial autocorrelation at lag k is the correlation between xₜ and xₜ₋ₖ after controlling for lags 1…k−1. In regression form, PACF(k) is the last coefficient φₖ in xₜ = c + Σⱼ₌₁ᵏ φⱼ xₜ₋ⱼ + εₜ. Spikes that remain after conditioning often point to the autoregressive order; an AR(2) series typically shows strong lags 1–2 and small values after. Negative values indicate inverse dependence after nearer lags are removed.

The calculator reports an approximate band of ±z/√N. At 95% confidence, z≈1.96, so N=100 gives ±0.196 and N=400 gives ±0.098; N=25 gives about ±0.392. At 90% use z≈1.645; at 99% use z≈2.576. Values outside the band are flagged “Yes” to highlight lags unlikely under pure randomness. When many lags are inspected, treat the flag as screening, then confirm with residual diagnostics.

Durbin–Levinson computes PACF via Yule–Walker recursion from autocorrelations rₖ (with r₀=1), yielding φₖₖ efficiently for each lag. OLS estimates PACF by regressing xₜ on an intercept and k lagged values, then reading the last coefficient. Both are standard in ARIMA identification and should align on well-behaved series. With short series, reduce K so n−k remains comfortably large and coefficients stay stable.

Centering removes the mean so correlations reflect fluctuations, not level. First differencing, xₜ−xₜ₋₁, helps when trend inflates low‑lag dependence and can reveal a simpler structure. Differencing shortens the sample by one point, so confidence bands widen slightly. If you see a repeating spike at a fixed lag, consider seasonal terms or seasonal differencing.

A practical starting point is K≈10·log10(N): N=50 suggests K≈17 and N=200 suggests K≈23, but you should cap K well below N to avoid spurious spikes. Use the table to pick candidate AR(p) orders, fit models, and recheck residual ACF/PACF for remaining pattern. If outliers dominate, clean or robustify first. Export CSV or PDF to document inputs, bands, and decisions for quick sharing.