Why Compare the ACF and PACF?
For a weakly stationary time series, the autocorrelation function (ACF) and partial autocorrelation function (PACF) describe dependence across lags in different ways. Their theoretical patterns are useful for distinguishing candidate pure autoregressive (AR) and moving-average (MA) models: one function cuts off after a finite lag, while the other generally tails off.
The distinction is a guide to model classification, not a guarantee that a finite sample correlogram will show a perfect pattern. Sample autocorrelations vary, and a few small estimated values do not establish a theoretical cutoff. The focus here is the behavior implied by the model itself.
The PACF is not simply the correlation between \(X_t\) and \(X_{t-k}\). That correlation is \(\rho_k\), which includes dependence transmitted through intermediate observations. The PACF measures the remaining linear association at lag \(k\) after those intermediate lags are included.
The Theoretical Pattern
For a pure MA(\(q\)) process, write \(X_t=\varepsilon_t+\theta_1\varepsilon_{t-1}+\cdots+\theta_q\varepsilon_{t-q}\), where the innovations \(\varepsilon_t\) have mean zero, variance \(\sigma^2\), and are uncorrelated across time. For a pure AR(\(p\)) process, write \(X_t=\phi_1X_{t-1}+\cdots+\phi_pX_{t-p}+\varepsilon_t\). Assume the AR process is stationary and causal, so its innovation is uncorrelated with its past. These assumptions support the theoretical patterns below.
| Model | ACF pattern | PACF pattern |
|---|---|---|
| Pure AR(\(p\)) | Generally tails off; it does not cut off at \(p\). | Cuts off after lag \(p\). |
| Pure MA(\(q\)) | Cuts off after lag \(q\). | Generally tails off rather than cutting off at \(q\). |
“Cuts off after lag \(q\)” means that every theoretical autocorrelation at a lag greater than \(q\) is zero. “Tails off” means that correlations may remain nonzero at arbitrarily large lags while tending toward zero. Depending on the model parameters, a tail may decay smoothly, alternate in sign, or have a damped oscillating pattern.
Why the Cutoffs Occur
A finite MA process uses only a finite block of innovations. For MA(\(q\)), \(X_t\) depends on innovations \(\varepsilon_t,\ldots,\varepsilon_{t-q}\), while \(X_{t-k}\) depends on \(\varepsilon_{t-k},\ldots,\varepsilon_{t-k-q}\). If \(k>q\), the two innovation blocks do not overlap. Since innovations at different times are uncorrelated, the two process values are uncorrelated as well.
The AR PACF cutoff follows from the projection defining the PACF. In a causal AR(\(p\)), the equation \(X_t=\phi_1X_{t-1}+\cdots+\phi_pX_{t-p}+\varepsilon_t\) gives a linear predictor based on the most recent \(p\) observations, with residual \(\varepsilon_t\). That residual is orthogonal to all past observations. If the projection includes \(k>p\) past observations, the same predictor, with zero coefficients on the extra lags, still has an orthogonal residual. Uniqueness of the least-squares projection therefore gives zero coefficients on lags beyond \(p\), including the coefficient at lag \(k\). Thus \(\alpha_{kk}=0\) for \(k>p\).
In the other direction, an invertible finite MA process has an infinite AR representation, so its PACF generally tails off rather than cutting off at the MA order. In practice, use the ACF cutoff as the more direct theoretical signature of a pure MA model; do not infer a finite MA order merely because a few PACF values look small.
Computing PACF Values from the ACF
PACF values can be calculated recursively from the autocorrelations. For a nonsingular stationary process, begin with \(\alpha_{11}=\rho_1\). At order \(k\), the PACF is the coefficient on the most distant lag in the order-\(k\) linear projection. The recursion below lets you compute it without solving a new projection from scratch.
Here \(\alpha_{k,j}\) is the coefficient on lag \(j\) in the order-\(k\) projection, and \(\alpha_{kk}\) is its final-lag coefficient. The denominator is the residual variance at the preceding projection order divided by the series variance, so it is positive when the projection covariance matrix is nonsingular.
Worked Example: Recognizing the Cutoff of an AR(1)
A stationary AR(1) model is \(X_t=0.6X_{t-1}+\varepsilon_t\), with mean zero and white-noise innovations of variance \(4\). Find the first three ACF values and determine the PACF cutoff. Use the theoretical patterns to classify the process.
Method. For an AR(1) with coefficient \(\phi=0.6\), the stationary ACF is \(\rho_k=\phi^k\). The PACF has a cutoff after the AR order, provided the process is stationary and the stated order is nonzero.
ACF calculations. At lag 1, \(\rho_1=0.6\). At lag 2, \(\rho_2=(0.6)^2=0.36\). At lag 3, \(\rho_3=(0.6)^3=0.216\). The innovation variance does not change these normalized autocorrelations.
PACF check at lag 2. The recursion gives \(\alpha_{22}=(\rho_2-\rho_1^2)/(1-\rho_1^2)\). Substituting the values gives \((0.36-0.36)/(1-0.36)=0/0.64=0\). The AR(1) PACF is therefore \(0.6\) at lag 1 and zero at every lag greater than 1.
Conclusion and check. The ACF continues as \(0.6^k\), while the PACF cuts off after lag 1: this is the theoretical AR(1) signature. The common mistake is to treat the ACF value at lag 2 as evidence that the AR order is at least 2. The AR order is identified by the PACF cutoff here, not by the nonzero ACF at lag 2.
Worked Example: An MA(2) with a Nonzero PACF Tail
Let \(X_t=\varepsilon_t+0.5\varepsilon_{t-1}+0.25\varepsilon_{t-2}\), where the innovations are uncorrelated with mean zero and variance \(1\). Compute the ACF through lag 3 and the PACF through lag 3. Use the pattern to classify the candidate model.
Method. For an MA(2), calculate autocovariances by multiplying coefficients on innovations shared by the two time points. Divide by \(\gamma_0\) to get autocorrelations, then apply the PACF recursion.
Autocovariances and ACF. The coefficient list is \(1,0.5,0.25\). Thus \(\gamma_0=1^2+0.5^2+0.25^2=1+0.25+0.0625=1.3125=21/16\). At lag 1, the shared innovations give \(\gamma_1=0.5(1)+0.25(0.5)=0.625=10/16\), so \(\rho_1=(10/16)/(21/16)=10/21\). At lag 2, \(\gamma_2=0.25(1)=4/16\), so \(\rho_2=(4/16)/(21/16)=4/21\). At lag 3 there are no shared innovations, so \(\gamma_3=0\) and \(\rho_3=0\). The same reasoning gives \(\rho_k=0\) for all \(k>2\).
PACF at lag 1. The first partial autocorrelation is \(\alpha_{11}=\rho_1=10/21\).
PACF at lag 2. The numerator is \(\rho_2-\rho_1^2=4/21-100/441=84/441-100/441=-16/441\). The denominator is \(1-\rho_1^2=1-100/441=341/441\). Therefore \(\alpha_{22}=(-16/441)/(341/441)=-16/341\). The order-2 coefficient on lag 1 is \(\alpha_{2,1}=\alpha_{1,1}-\alpha_{2,2}\alpha_{1,1} =(10/21)(1+16/341)=170/341\).
PACF at lag 3. The numerator is \(\rho_3-\alpha_{2,1}\rho_2-\alpha_{2,2}\rho_1 =0-(170/341)(4/21)-(-16/341)(10/21) =-680/7161+160/7161=-520/7161\). The denominator is \(1-\alpha_{2,1}\rho_1-\alpha_{2,2}\rho_2 =1-1700/7161+64/7161=5525/7161\). Thus \(\alpha_{33}=(-520/7161)/(5525/7161)=-520/5525=-104/1105\approx-0.0941\). The PACF values through lag 3 are \(10/21\), \(-16/341\), and approximately \(-0.0941\).
Conclusion and check. The ACF cuts off after lag 2, as required for this MA(2), while the PACF has not cut off at lag 2. The common arithmetic trap is to calculate the lag-3 denominator as \(4041/7161\). Substitution gives \(7161-1700+64=5525\), not \(4041\); carrying the incorrect denominator would also give an incorrect PACF.
Worked Example: Confirming an AR(2) PACF Cutoff
Consider the zero-mean AR(2) model \(X_t=0.5X_{t-1}-0.2X_{t-2}+\varepsilon_t\), where the innovations are white noise with variance \(1\). The stationarity conditions for an AR(2), expressed as \(|\phi_2|<1\), \(\phi_1+\phi_2<1\), and \(\phi_2-\phi_1<1\), are satisfied: \(0.2<1\), \(0.3<1\), and \(-0.7<1\). Find the ACF through lag 3 and the PACF through lag 3.
Method. Use the AR(2) autocorrelation equations \(\rho_1=\phi_1/(1-\phi_2)\) and \(\rho_k=\phi_1\rho_{k-1}+\phi_2\rho_{k-2}\) for \(k\geq2\), with \(\rho_0=1\). Then use the PACF recursion.
ACF calculations. With \(\phi_1=0.5\) and \(\phi_2=-0.2\), \(\rho_1=0.5/(1-(-0.2))=0.5/1.2=5/12\). Next, \(\rho_2=0.5(5/12)-0.2(1)=5/24-1/5=(25-24)/120=1/120\). Then \(\rho_3=0.5(1/120)-0.2(5/12)=1/240-1/12=-19/240\). These values show that the ACF need not vanish after lag 2.
PACF at lags 1 and 2. We have \(\alpha_{11}=5/12\). For lag 2, the numerator is \(\rho_2-\rho_1^2=1/120-25/144=(6-125)/720=-119/720\). The denominator is \(1-\rho_1^2=1-25/144=119/144\). Therefore \(\alpha_{22}=(-119/720)/(119/144)=-1/5\). The updated lag-1 coefficient is \(\alpha_{2,1}=5/12-(-1/5)(5/12)=1/2\).
PACF at lag 3. Its numerator is \(\rho_3-\alpha_{2,1}\rho_2-\alpha_{2,2}\rho_1 =-19/240-(1/2)(1/120)-(-1/5)(5/12) =-19/240-1/240+1/12=0\). The denominator is positive because the order-2 projection covariance matrix is nonsingular; explicitly it is \(1-\alpha_{2,1}\rho_1-\alpha_{2,2}\rho_2 =1-5/24+1/600=119/150>0\). Hence \(\alpha_{33}=0\). For this AR(2), the PACF is \(5/12\) at lag 1, \(-1/5\) at lag 2, and zero from lag 3 onward.
Conclusion and check. The PACF cutoff after lag 2 matches the AR(2) order; the nonzero ACF at lag 3 is not a contradiction. A common mistake is to assume that both functions must cut off at the model order. For a pure AR model, it is the PACF that cuts off, while the ACF generally tails off.
Using the Patterns Carefully
A practical classification starts by asking which theoretical pattern is most consistent with the candidate: a PACF cutoff with a tailing ACF suggests a pure AR model; an ACF cutoff with a tailing PACF suggests a pure MA model. The cutoff lag suggests the order, subject to checking that the last included coefficient is not zero.
- Do not reverse the functions. For pure AR(\(p\)), the PACF cuts off; for pure MA(\(q\)), the ACF cuts off.
- Do not confuse a small value with a zero. Theoretical cutoff means exactly zero at all higher lags, not merely values that look small.
- Do not treat a single zero as a cutoff. ACF or PACF values at selected lags can equal zero even when the function does not cut off there.
- Keep theory separate from sample evidence. Sample correlograms are estimates and can fluctuate around the theoretical values.
These patterns are aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus. On a classification problem, first state the model assumptions, then identify the function and lag at which a genuine theoretical cutoff occurs. That makes the conclusion clear and avoids mistaking a tail-off for a finite-order cutoff.
Check Your Understanding
For each item, use the stated theoretical model or correlogram pattern; distinguish a true cutoff from a value that is merely small.
- A stationary pure MA(2) process has \(\rho_1=0.30\), \(\rho_2=-0.10\), and \(\rho_k=0\) for every \(k\geq3\). Which function has the theoretical cutoff, and what order does it suggest? (Answer: The ACF cuts off after lag 2, suggesting MA(2).)
- A stationary AR(1) process has coefficient \(\phi=-0.4\). Find \(\rho_1,\rho_2,\rho_3\), and state the PACF cutoff. (Answer: \(\rho_1=-0.4\), \(\rho_2=0.16\), \(\rho_3=-0.064\); the PACF cuts off after lag 1.)
- A stationary AR(2) process has \(\phi_1=0.3\) and \(\phi_2=0.4\). Find \(\rho_1\) and \(\alpha_{22}\). (Answer: \(\rho_1=0.3/(1-0.4)=0.5\), and \(\alpha_{22}=0.4\).)
- A pure MA(1) process is \(X_t=\varepsilon_t-0.5\varepsilon_{t-1}\), with uncorrelated innovations of variance \(2\). Find \(\rho_1\) and \(\rho_2\), and identify which function cuts off after lag 1. (Answer: \(\rho_1=-0.5/(1+0.5^2)=-0.4\), \(\rho_2=0\); the ACF cuts off after lag 1.)
- A candidate process has a theoretical PACF that is nonzero at lags 1, 2, and 3 and zero at every lag greater than 3; its ACF remains nonzero at some higher lags. Which pure model classification is suggested, and what is the suggested order? (Answer: Pure AR(3), because the PACF cuts off after lag 3 while the ACF tails off.)