Why ACF and PACF Patterns Help
After deciding whether a series needs differencing, the next question is how its remaining dependence may be represented. The autocorrelation function (ACF) and partial autocorrelation function (PACF) summarize that dependence at successive lags. Their patterns can suggest whether a low-order autoregressive (AR) term, moving-average (MA) term, or a combination is a reasonable candidate.
These patterns are diagnostic clues, not a mechanical order-selection rule. A sample ACF or PACF is noisy, and different models can produce similar-looking plots. The goal is to propose defensible candidates for a suitably differenced series, not to claim that a plot proves the model.
The ACF measures overall linear association between observations \(k\) time steps apart. The PACF measures the association at lag \(k\) after the linear contributions of the intervening lags have been removed. This distinction explains why their patterns can point toward different model components.
Theoretical Patterns for Low-Order Models
Suppose the series being examined is already stationary. Under a pure AR(\(p\)) model, the theoretical ACF generally tails off, while the theoretical PACF is zero at lags greater than \(p\). Under a pure MA(\(q\)) model, the theoretical ACF is zero at lags greater than \(q\), while the PACF generally tails off. For an ARMA model with both components, neither function generally cuts off at a finite lag.
“Cuts off” means that the theoretical correlations are exactly zero beyond a lag, not that every sample value beyond that lag must be zero. The usual pattern guide assumes a correctly specified, nondegenerate, minimal model: for example, the AR representation is stationary, and there is no exact cancellation between AR and MA factors. Special parameter values can produce exceptions.
| Candidate for the stationary series | Theoretical ACF pattern | Theoretical PACF pattern |
|---|---|---|
| White noise | Zero at every positive lag | Zero at every positive lag |
| AR(\(p\)) | Generally tails off | Cuts off after lag \(p\) |
| MA(\(q\)) | Cuts off after lag \(q\) | Generally tails off |
| ARMA(\(p,q\)) | Generally tails off | Generally tails off |
A tail can decay smoothly, alternate in sign, or combine damped oscillations, depending on the model parameters. It need not be a simple positive geometric decline. Thus, the useful contrast is usually a clear cutoff in one function versus a gradual pattern in the other, not an exact visual shape.
Why an AR(1) Has a PACF Cutoff
A short derivation shows why the PACF is especially informative for AR order. Suppose the centered stationary series satisfies \(w_t=\phi w_{t-1}+\varepsilon_t\), where \(|\phi|<1\), and \(\varepsilon_t\) is a zero-mean white-noise innovation with positive variance that is uncorrelated with past values of \(w_t\). Write \(\gamma_k=\operatorname{Cov}(w_t,w_{t-k})\). For \(k\geq 1\), taking covariance with \(w_{t-k}\) gives
The innovation contributes zero covariance because it is uncorrelated with the past. The resulting ACF decays by powers of \(\phi\). For the PACF, \(w_t\) is already represented by \(w_{t-1}\) and a new innovation. Once \(w_{t-1}\) is included in the linear projection, adding older lags supplies no further linear contribution. Therefore the population PACF is \(\phi\) at lag 1 and zero at every lag above 1. This is the AR(1) case of the AR(\(p\)) cutoff pattern.
Sample Values Are Evidence, Not Proof
In data, the ACF and PACF are estimated. One common sample ACF convention first subtracts the sample mean \(\bar w\), then computes
The sample PACF at lag \(k\) can be estimated as the coefficient on \(w_{t-k}\) in a regression of \(w_t\) on \(w_{t-1},\ldots,w_{t-k}\), with an intercept if the series mean is estimated rather than known to be zero. Estimation conventions can differ slightly, so a stated plot or software output should be interpreted according to its convention.
For a rough check under a white-noise null, an approximate pointwise 95% band is \(\pm 1.96/\sqrt{n}\), where \(n\) is the number of observations in the series being analyzed. A bar outside the band is evidence against zero correlation at that lag under the approximation; a bar inside is not proof that the true correlation is zero. The band is not a simultaneous guarantee across all lags, and differencing, parameter estimation, and dependence can affect its accuracy. A few isolated bars outside the band can occur by chance.
Worked Example: A Gradually Decaying ACF and a PACF Cutoff
A hypothetical insurer studies a transformed monthly claim-frequency index \(x_t\). After first differencing, \(w_t=x_t-x_{t-1}\) has 100 observations. The estimated ACF at lags 1 through 4 is \(0.62, 0.39, 0.24, 0.15\), and the estimated PACF is \(0.62, 0.04, -0.03, 0.02\). Propose a low-order candidate for \(x_t\).
Method. Check the approximate band, then compare the two patterns. Because the series being analyzed is the first difference, a stationary-series candidate AR(\(p\)) becomes an ARIMA(\(p,1,0\)) candidate for \(x_t\).
Step 1: Calculate the approximate band. Here \(n=100\), so \(\sqrt{n}=10\) and \(1.96/\sqrt{n}=1.96/10=0.196\). The approximate bounds are \(-0.196\) and \(0.196\).
Step 2: Assess the ACF. The first three values, \(0.62, 0.39,\) and \(0.24\), exceed \(0.196\); the fourth, \(0.15\), does not. The values decline rather than ending abruptly at lag 1. Their first two ratios are \(0.39/0.62\approx0.629\) and \(0.24/0.39\approx0.615\), consistent with a roughly geometric decline.
Step 3: Assess the PACF. The lag-1 value \(0.62\) exceeds the band. The absolute values at lags 2 through 4, \(0.04, 0.03,\) and \(0.02\), are all below it. This is consistent with a PACF cutoff after lag 1.
Candidate and check. The combined pattern suggests an AR(1) model for \(w_t\), hence an ARIMA(1,1,0) candidate for \(x_t\). The conclusion is a candidate, not proof: the sample cutoff may be affected by noise, and other models may also fit. The common mistake avoided is calling the ACF “cut off” just because its later bars become small; here it declines gradually while the PACF shows the clearer cutoff.
Worked Example: An ACF Cutoff Suggesting an MA Term
A hypothetical quarterly reserve-change series has been differenced once, producing \(n=64\) observations \(w_t\). Its estimated ACF at lags 1 through 4 is \(0.48,-0.08,0.03,-0.11\), and its estimated PACF is \(0.48,-0.23,0.12,-0.07\). Assess an MA(1) candidate for the differenced series and translate it to the original series.
Step 1: Calculate the band. Since \(\sqrt{64}=8\), the approximate bound is \(1.96/8=0.245\), giving bounds of \(-0.245\) and \(0.245\).
Step 2: Read the ACF. The lag-1 value \(0.48\) is outside the band. The remaining ACF values have absolute values \(0.08, 0.03,\) and \(0.11\), each below \(0.245\). This supports an ACF cutoff after lag 1.
Step 3: Read the PACF. The lag-1 PACF is \(0.48\), while the lag-2 value \(-0.23\) is just inside the band because \(0.23<0.245\); the values at lags 3 and 4 are also inside. This does not establish that the PACF truly tails off, but it is compatible with the MA(1) pattern.
Optional parameter check. Under the convention \(w_t=\varepsilon_t+\theta\varepsilon_{t-1}\), a stationary MA(1) with innovation variance \(\sigma^2>0\) has \(\rho_1=\theta/(1+\theta^2)\), and ACF zero for lags above 1. Setting the observed \(0.48\) equal to the theoretical lag-1 expression gives \(0.48\theta^2-\theta+0.48=0\). The discriminant is \(1-4(0.48)(0.48)=0.0784\), whose square root is \(0.28\). The roots are \((1-0.28)/0.96=0.75\) and \((1+0.28)/0.96=1.333\). The invertible choice is \(\theta=0.75\), since \(|\theta|<1\). This rough check is not a fitted estimate; sample ACF values are uncertain.
Candidate and check. The suggested original-series model is ARIMA(0,1,1). The MA(1) theoretical lag-1 correlation at \(\theta=0.75\) is \(0.75/(1+0.75^2)=0.75/1.5625=0.48\), matching the illustrative input. The common mistake avoided is treating \(-0.23\) as outside the band: its magnitude is smaller than \(0.245\).
Worked Example: Both Functions Tail Off
A hypothetical monthly premium-adjustment series \(x_t\) is first-differenced. For the resulting \(n=225\) values, the sample ACF at lags 1 through 5 is \(0.60,0.36,0.22,0.13,0.08\), and the sample PACF is \(0.60,0.20,0.13,0.09,0.06\). What can be concluded about low-order candidates?
Step 1: Calculate the approximate band. Since \(\sqrt{225}=15\), the bound is \(1.96/15=0.1307\) to four decimal places. Values with magnitude greater than about \(0.1307\) are outside this approximate pointwise band.
Step 2: Count the notable early lags. In the ACF, \(0.60,0.36,\) and \(0.22\) exceed \(0.1307\); \(0.13\) is slightly below it, and \(0.08\) is below it. In the PACF, \(0.60\) and \(0.20\) exceed the bound; \(0.13,0.09,\) and \(0.06\) do not. Both functions show early dependence followed by smaller values, rather than a clear, well-separated cutoff pattern.
Step 3: State what the evidence supports. A mixed ARMA candidate for the differenced series is plausible because neither plot gives a decisive cutoff. However, these values do not identify unique orders: an AR(1), a low-order ARMA model, or another parsimonious candidate could be considered depending on the full plots and other evidence. If using a mixed candidate such as ARMA(1,1) for \(w_t\), its corresponding original-series notation is ARIMA(1,1,1).
Check. The lag-4 ACF value \(0.13\) is not outside the calculated band because \(0.13<0.1307\). The common mistake avoided is rounding the bound to \(0.13\) and then treating equality at that rounded value as evidence beyond the band. Neither a sample bar inside the band nor one just outside it proves a theoretical zero or nonzero correlation.
A Practical Reading Sequence
Use the series after the chosen differencing operation. If \(w_t=\Delta^d x_t\), candidate AR and MA orders describe \(w_t\); the corresponding integrated model for \(x_t\) has differencing order \(d\).
Ask whether the ACF or PACF has a plausible cutoff, or whether both taper. Do not select an order from one isolated bar without considering neighboring lags.
For a white-noise reference, calculate \(1.96/\sqrt{n}\) using the sample size of the analyzed series. Treat this as approximate, pointwise evidence rather than a definitive test across all lags.
Translate a plausible low-order pattern into a candidate model and retain alternatives when the sample does not distinguish them clearly.
Common Traps and Why the Distinction Matters
- Confusing sample bars with theoretical cutoffs. A sample ACF bar rarely equals exactly zero. A cutoff is a model property, while a plotted sample value is an estimate.
- Switching the ACF and PACF roles. A PACF cutoff points toward an AR order; an ACF cutoff points toward an MA order, under the standard nondegenerate model conditions.
- Ignoring the differencing already applied. If the examined series is \(\Delta x_t\), an AR(1) pattern describes a candidate AR term for the differenced series, not an AR(1) model for the original undifferenced series.
- Over-reading significance bands. A bar inside a rough band does not establish a zero correlation, and a handful of bars outside pointwise bands can arise by chance.
- Assuming one pattern has only one explanation. Finite samples can make an AR process resemble an ARMA process or obscure an expected cutoff. ACF and PACF guide candidate construction; they do not by themselves settle model choice.
These distinctions matter because an incorrect order can leave important dependence unrepresented or add unnecessary terms. The disciplined conclusion is calibrated: state the pattern, name the candidate it suggests, and acknowledge when the sample evidence is ambiguous. This is aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus.
Check Your Understanding
For each problem, distinguish the observed sample pattern from what it can support about a candidate model.
- A stationary transformed series has \(n=100\), sample ACF values \(0.50,0.25,0.12\), and PACF values \(0.50,0.02,-0.03\) at lags 1–3. Calculate the approximate band and give a low-order candidate. (Answer: Band \(\pm0.196\); the pattern suggests AR(1).)
- A first-differenced series has \(n=49\). Its ACF is \(0.40,0.05,-0.08\) and its PACF is \(0.40,0.18,0.09\) at lags 1–3. What candidate is suggested for the original series? (Answer: Band \(\pm0.280\); the ACF suggests MA(1) for the differenced series, hence ARIMA(0,1,1) for the original series.)
- For a stationary series with \(n=81\), both ACF and PACF decline gradually, and both have values outside the approximate band at lags 1 and 2 but not at lags 3–5. What is a defensible conclusion? (Answer: The band is \(\pm0.218\); a low-order ARMA candidate is plausible, but the pattern does not identify unique orders.)
- A series \(x_t\) is differenced twice. The differenced series has an ACF that cuts off after lag 2 and a PACF that tails off. State the suggested model class and its notation for \(x_t\). (Answer: An MA(2) candidate for the twice-differenced series, corresponding to ARIMA(0,2,2) for \(x_t\).)
- A white-noise reference band is calculated as \(\pm0.196\). A sample PACF at lag 2 is \(0.197\), with all other early PACF values small. What does this single bar justify? (Answer: It is only marginal evidence beyond the approximate pointwise band; it does not prove a PACF cutoff at lag 2 or establish an AR(2) model.)