Why Use Both the ACF and PACF?
A pure autoregressive process leaves two related but different signatures in its correlograms. Its autocorrelation function (ACF) generally tails off, while its partial autocorrelation function (PACF) has a theoretical cutoff after the autoregressive order. These patterns help propose a model order, but an observed sample correlogram is noisy: it rarely displays a perfect cutoff.
The distinction is useful because the ACF records overall association at each lag, including association passed along through intermediate lags. The PACF at lag \(k\) measures the association between \(X_t\) and \(X_{t-k}\) after accounting linearly for the observations at lags \(1,\ldots,k-1\). For a pure AR process, the PACF pattern is the more direct clue to the order.
The AR-order rules in this tutorial apply to a pure AR(p) process, not automatically to every time-series model. Assume a stationary AR(p) process with a constant mean and uncorrelated, mean-zero innovations of constant variance:
Here \(p\) is the AR order. Stationarity is important: the theoretical ACF and PACF patterns discussed below presume a stable process whose dependence does not change over time. The earlier material on AR stationarity gives conditions for assessing that assumption.
The Theoretical Patterns for a Pure AR Process
For an AR(p) process, the PACF is zero at every lag greater than \(p\). In contrast, the ACF usually continues beyond lag \(p\) and gradually decays. The word usually matters: special parameter values can produce unusual autocorrelation values, so an ACF value of zero at one particular lag is not by itself an AR-order rule.
The ACF's tail-off can take different forms. If the dependence is dominated by a positive real root, autocorrelations may decrease smoothly while staying positive. A negative AR(1) coefficient gives alternating signs as the ACF decays. For some AR(2) processes, the ACF can alternate or oscillate in a damped pattern. These shapes describe how dependence declines; they do not change the rule that the PACF cuts off after the true AR order.
A useful way to understand why the PACF cuts off is to consider what happens when the regression includes the process's full history through lag \(p\). Once lags \(1,\ldots,p\) are included, the AR equation accounts for the predictable part of \(X_t\). Earlier observations beyond that history do not add a new direct lag coefficient in the population regression.
Why the PACF Cuts Off: A Short Justification
For a pure AR(p), the model equation expresses \(X_t-\mu\) as a linear combination of the previous \(p\) centered observations plus the current innovation. The innovation is uncorrelated with past observations. Therefore, after the first \(p\) lags are included in the linear projection, an additional lag \(X_{t-k}\), with \(k>p\), has no remaining direct contribution to \(X_t\). Its partial regression coefficient is zero.
This is a population statement, not a claim that an estimated sample PACF will be exactly zero. A finite sample has random variation, and observations at lags beyond \(p\) can have small, nonzero estimated partial autocorrelations even when the model is truly AR(p).
Look for a tail-off rather than expecting the ACF to stop at the AR order. Note whether the decline is smooth, alternating, or oscillatory.
For a pure AR candidate, identify the last lag with a notable partial autocorrelation and check whether later estimates are small.
Use approximate reference bands as a guide, not a mechanical order-selection rule. A spike near a band or an isolated later spike makes the evidence ambiguous.
Describe what order the pattern suggests and acknowledge alternatives when the sample correlogram does not show a clear cutoff.
Worked Examples
Worked Example: Recognizing an AR(1) Pattern
Suppose a stationary AR(1) process has mean \(\mu\), coefficient \(\phi_1=0.7\), and uncorrelated innovations with mean zero and variance \(\sigma_\epsilon^2\). Find its first four theoretical ACF values and describe the PACF pattern. Then identify the order suggested by the theoretical correlograms.
Method. For a stationary AR(1), the theoretical ACF satisfies \(\rho_k=\phi_1^k\) for \(k\geq 0\). The AR(1) PACF is \(\alpha_{11}=\phi_1\), with zero values at lags greater than 1.
At lag 1, \(\rho_1=0.7^1=0.7\). At lag 2, \(\rho_2=0.7^2=0.49\). At lag 3, \(\rho_3=0.7^3=0.343\). At lag 4, \(\rho_4=0.7^4=0.2401\).
The ACF is positive and gradually decreases: \(0.7, 0.49, 0.343, 0.2401\). The PACF equals \(0.7\) at lag 1 and is zero at lags 2, 3, and every higher lag in the population. Thus, the theoretical pattern suggests AR(1).
Check. Each ACF value is \(0.7\) times the preceding value, as required by \(\rho_k=\phi_1^k\). The conclusion uses the PACF cutoff, not an imagined ACF cutoff. A common mistake is to call this AR(4) because four ACF values were calculated; the number of listed autocorrelations does not determine the order.
Worked Example: Reading an AR(2) Pattern From Its Coefficients
A stationary AR(2) process has \(\phi_1=0.5\) and \(\phi_2=-0.2\), with mean \(\mu\) and uncorrelated, mean-zero innovations of constant variance. Calculate \(\rho_1,\ldots,\rho_4\), calculate the PACF at lags 1 and 2, and identify the theoretical order.
Method. The Yule–Walker relations for this AR(2) give \(\rho_1=\phi_1/(1-\phi_2)\), \(\rho_2=\phi_1\rho_1+\phi_2\), and \(\rho_k=\phi_1\rho_{k-1}+\phi_2\rho_{k-2}\) for \(k\geq2\). The lag-2 PACF is \(\alpha_{22}=(\rho_2-\rho_1^2)/(1-\rho_1^2)\).
First, \(\rho_1=0.5/(1-(-0.2))=0.5/1.2=0.4167\), to four decimal places. Then \(\rho_2=0.5(0.4167)-0.2=0.2083-0.2=0.0083\). Using the unrounded value \(\rho_1=5/12\), the exact value is \(\rho_2=1/120=0.0083\).
Next, \(\rho_3=0.5(1/120)-0.2(5/12)=1/240-1/12=-19/240=-0.0792\). Also, \(\rho_4=0.5(-19/240)-0.2(1/120)=-19/480-1/600=-99/2400=-0.0413\).
At lag 1, the PACF is \(\alpha_{11}=\rho_1=5/12=0.4167\). At lag 2, \[ \alpha_{22} = \frac{1/120-(5/12)^2}{1-(5/12)^2} = \frac{1/120-25/144}{1-25/144} = \frac{-119/720}{119/144} =-\frac{1}{5} =-0.2. \] For a pure AR(2), \(\alpha_{kk}=0\) for every \(k>2\). The theoretical PACF therefore cuts off after lag 2, so the candidate order is 2, even though the ACF values do not stop at lag 2.
Check. The recurrence gives \(\rho_3\) and \(\rho_4\) from the two preceding ACF values, and the lag-2 PACF recovers the second AR coefficient. The small value \(\rho_2=0.0083\) does not make the process AR(1); order is indicated by the PACF cutoff, not by whether one ACF value happens to be small.
Worked Example: Handling a Borderline Sample PACF
A hypothetical analyst fits no model yet and calculates correlograms from \(n=100\) consecutive observations. The sample ACF at lags 1 through 6 is \(0.58, 0.36, 0.20, 0.09, 0.03, -0.04\). The sample PACF at lags 1 through 5 is \(0.50, 0.24, 0.21, -0.08, 0.05\). Use approximate 95% white-noise reference bands to assess what AR order is suggested, and explain the uncertainty.
Method. A common rough reference band for an individual sample autocorrelation under a white-noise reference is \(\pm1.96/\sqrt{n}\). Here \(\sqrt{100}=10\), so the approximate limits are \(1.96/10=0.196\), or \(-0.196\) and \(0.196\). These limits are a screening guide for this exercise, not a definitive test of AR order.
The PACF estimates at lags 1 and 2, \(0.50\) and \(0.24\), exceed \(0.196\) in absolute value. The lag-3 estimate, \(0.21\), also exceeds the limit, but by only \(0.21-0.196=0.014\). The estimates at lags 4 and 5, \(-0.08\) and \(0.05\), lie inside the band. The sample ACF decreases gradually from \(0.58\) to values near zero, which is consistent with tailing off.
A reasonable reading is that AR(2) is a candidate, but the lag-3 PACF spike makes the cutoff ambiguous; an AR(3) candidate should not be ruled out solely by this plot. The lag-3 value is only slightly outside the rough band, and sample estimates vary. A single borderline spike is not conclusive evidence for a higher order.
Check. The ACF does not cut off after lag 2, and the PACF has small later values except for a borderline lag-3 value. The defensible conclusion is “AR(2) is suggested, with AR(3) a plausible alternative,” rather than a certain order. A common mistake is to treat the approximate band as an exact rule and declare AR(3) automatically; the correction is to consider the full pattern and the uncertainty of the sample.
Interpreting Ambiguous Sample Correlograms
In practice, the population patterns are not visible. A sample ACF or PACF is an estimate, and even when the underlying process has a clear AR order, random variation can create small spikes after the theoretical cutoff. Conversely, a true nonzero PACF can be estimated close to zero in a particular sample.
For a quick visual screen, approximate bands of \(\pm1.96/\sqrt{n}\) are often drawn around zero. A value outside the band is a reason to investigate, not a guarantee that a population partial autocorrelation is nonzero. The bands are approximate and should not be treated as a collection of independent, definitive decisions. The smaller the sample, the wider these bands become, and the harder it may be to distinguish nearby orders.
| Observed pattern | Reasonable interpretation for a pure AR candidate | Caution |
|---|---|---|
| ACF tails off; PACF has a clear early cutoff after lag 1 | AR(1) is suggested | Sample noise may hide or create small spikes |
| ACF tails off; PACF has a clear early cutoff after lag 2 | AR(2) is suggested | Check whether later PACF estimates are consistently small |
| ACF tails off; PACF has a borderline spike just beyond a possible cutoff | More than one nearby order may be plausible | A single borderline value does not settle the order |
| PACF has several substantial later spikes with no apparent cutoff | The pure low-order AR pattern is not clear | Do not force an order from an unclear correlogram |
The ACF still provides useful context. A steadily declining ACF may fit positive persistence; an alternating ACF may indicate negative or oscillatory dependence. But the ACF alone is not a reliable way to read off the AR order. Likewise, an isolated PACF spike at a later lag can result from sample variability. A careful conclusion distinguishes a candidate suggested by the pattern from a certainty about the underlying process.
Common Exam Traps
- Looking for an ACF cutoff at the AR order. For a pure AR(p), it is the theoretical PACF that cuts off after \(p\); the ACF generally tails off.
- Counting calculated lags as the order. Reporting six ACF values does not imply an AR(6). The order is about the underlying model, not the number of plotted lags.
- Treating sample estimates as population values. A theoretical PACF is zero after \(p\), but sample PACF estimates after \(p\) need not equal zero.
- Promoting one borderline spike to a definite order. Approximate bands are aids to interpretation. Consider the full correlogram and state uncertainty where the cutoff is unclear.
- Applying the AR pattern without checking the model setting. The cutoff rule here is for a pure AR process. Do not assume that every observed time series follows that structure.
Check Your Understanding
For each problem, state the candidate AR order or explain why the sample evidence is ambiguous.
- A stationary AR(1) has \(\phi_1=-0.4\). Calculate \(\rho_1,\rho_2,\rho_3\), describe the sign pattern, and state where its theoretical PACF cuts off. (Answer: \(\rho_1=-0.4,\ \rho_2=0.16,\ \rho_3=-0.064\); the ACF alternates with decreasing magnitude, and the PACF cuts off after lag 1.)
- A stationary pure AR(2) has sample PACF estimates \(0.46,-0.29,0.04,0.10\) at lags 1 through 4, and its sample ACF gradually declines. What order is suggested by the combined pattern? (Answer: AR(2) is suggested; the PACF estimates after lag 2 are small.)
- For \(n=81\), use the approximate limits \(\pm1.96/\sqrt{n}\). A sample PACF has values \(0.41,0.27,0.20,-0.24,0.03\) at lags 1 through 5. Calculate the limits and assess whether a clean cutoff is evident. (Answer: the limits are approximately \(\pm0.2178\); lags 1, 2, and 4 exceed them in absolute value, so there is no clean cutoff and a definite AR order is not justified.)
- For \(n=36\), the approximate 95% limits are \(\pm1.96/\sqrt{36}\). A sample PACF has values \(0.35,0.31,-0.12,0.09\). Identify the likely candidate and explain one reason for caution. (Answer: the limits are approximately \(\pm0.3267\), so only lag 1 exceeds them; AR(1) is a tentative candidate, but the small sample and the lag-2 estimate near the boundary make a firm order uncertain.)
- A hypothetical correlogram has ACF values \(0.64,0.40,0.25,0.14,0.08\) and PACF values \(0.64,0.01,-0.06,0.04\) at the displayed lags. What order does the theoretical AR pattern most closely suggest, and why is the ACF not evidence for that same order? (Answer: AR(1) is suggested by the PACF cutoff after lag 1; the nonzero, declining ACF values are consistent with tail-off, not an ACF cutoff.)