AR Order Identification Worked Problems

Using Sample ACF and PACF Tables

When a time series has serial dependence, the order of a plausible autoregressive model is often suggested by the pattern across its sample autocorrelation function (ACF) and partial autocorrelation function (PACF). The decision is not made by counting every sample correlation outside a band. Instead, look for a PACF that is appreciable through an initial set of lags and then mostly small, while the ACF generally tails off. This is evidence for a candidate AR order, not proof that the model is correct.

Definition: For observations $x_1,\ldots,x_n$, the sample ACF at lag $k$ is the sample correlation between observations $k$ time units apart. One common definition is $$ r_k= \frac{\sum_{t=k+1}^{n}(x_t-\bar{x})(x_{t-k}-\bar{x})} {\sum_{t=1}^{n}(x_t-\bar{x})^2}, $$ where $\bar{x}$ is the sample mean. The sample PACF at lag $k$, written $\widehat{\phi}_{kk}$, is the estimated coefficient on $x_{t-k}$ in a linear regression of $x_t$ on its $k$ immediately preceding values, with the mean handled appropriately. It measures the remaining linear association at lag $k$ after accounting for lags $1,\ldots,k-1$.

For a stationary AR($p$) process, the population PACF is zero at lags greater than $p$, while the population ACF typically tails off rather than cutting off at $p$. The observed values are sample estimates, so they do not usually show a perfectly sharp cutoff. A practical identification exercise asks whether the table is broadly consistent with that pattern.

Approximate Significance Bands

A commonly used screening band for the sample ACF and PACF is approximately $\pm 1.96/\sqrt{n}$. It is based on the approximate standard error $1/\sqrt{n}$ under a white-noise null model. Values outside the band are often called individually significant at an approximate 5% level; values inside are not clearly distinguishable from zero by this screen.

\text{approximate band}=\pm\frac{1.96}{\sqrt{n}}

For instance, if $n=121$, then $\sqrt{n}=11$, so the band is $\pm 1.96/11=\pm 0.1782$, rounded to four decimal places. Compare each lag using its absolute value: a value of $-0.20$ is outside this band just as $+0.20$ is. The band is an approximate guide, not an exact cutoff for every process or lag. In particular, it does not account for the fact that many lags are inspected at once; a few isolated spikes can occur by chance.

The simple band is most directly motivated by a white-noise null. When considering a possibly serially correlated series, use it as a convenient visual screen rather than a precise uncertainty calculation for every ACF estimate. Give more weight to the overall lag pattern and to agreement between the ACF and PACF than to one marginal spike.

Why an AR Order Appears in the PACF

The cutoff idea follows from the AR model itself. Suppose a stationary series follows an AR($p$) model with mean $\mu$:

X_t-\mu=\phi_1(X_{t-1}-\mu)+\cdots+\phi_p(X_{t-p}-\mu)+\varepsilon_t,

where the innovations $\varepsilon_t$ have mean zero and are uncorrelated with past values. The PACF at lag $k$ is the coefficient on lag $k$ in the linear projection of $X_t-\mu$ onto its $k$ previous values. For $k>p$, the displayed model already writes $X_t-\mu$ as a linear combination of the first $p$ past values plus an innovation orthogonal to all past values. Adding lag $k$ to the projection therefore does not require a nonzero coefficient on that additional lag. Thus the population PACF at $k>p$ is zero, provided the relevant lag covariance matrix is nonsingular.

This argument explains the theoretical pattern, not an exact sample rule. Sample PACF values beyond $p$ fluctuate around zero, and one may exceed an approximate band. Also, if the highest-order AR coefficient is zero, the effective order is lower than the order used to write the equation.

Calculate the band.
Use $1.96/\sqrt{n}$ as an approximate screen and retain enough precision to compare the entries.

Read the PACF in order.
Look for an initial run of notable lags followed by values that are mostly small. A gap followed by another large spike weakens a simple cutoff interpretation.

Check the ACF pattern.
For an AR candidate, the ACF generally tails off, possibly with alternating signs or small sample fluctuations, rather than showing a clear finite cutoff at the same lag.

State a candidate, not a verdict.
Describe the order the table supports and note ambiguity. ACF/PACF patterns alone do not establish that the fitted model is adequate or stationary.

Worked Identification Problems

Worked Example: A Plausible AR(1) Candidate

A hypothetical insurer tracks a weekly index of incoming claim notices for $n=121$ weeks. The sample ACF values at lags 1 through 6 are $0.56, 0.46, 0.13, 0.06, -0.10, -0.08$. The sample PACF values at those lags are $0.56, 0.04, -0.09, 0.06, -0.11, 0.03$. Identify a plausible order using the approximate bands.

Method. Compute the band, compare absolute values, and then check whether the two functions have the expected broad pattern.

Step 1: Band. Since $\sqrt{121}=11$, the band is $1.96/11=0.1782$, rounded to four decimal places.

Step 2: PACF. At lag 1, $|0.56|>0.1782$. At lags 2 through 6, the absolute values are $0.04, 0.09, 0.06, 0.11, 0.03$, all below $0.1782$. The sample PACF therefore has one clear initial spike and then small values.

Step 3: ACF. The absolute values at lags 1 and 2 are $0.56$ and $0.46$, both beyond the band. Lag 3, $0.13$, is inside, as are the remaining displayed values in absolute magnitude. The ACF generally tails off, with small fluctuations; it does not show a PACF-like cutoff at lag 3.

Conclusion and check. These patterns support AR(1) as a plausible candidate. The common mistake to avoid is saying the ACF decreases at every lag: it does not, since the signed value rises from $-0.10$ to $-0.08$, and its absolute magnitude rises from $0.06$ to $0.10$. “Generally tails off, with small fluctuations” describes the observed pattern more accurately.

Worked Example: A Plausible AR(2) Candidate

A hypothetical monthly count series records maintenance requests for a group of facilities. There are $n=100$ observations. The sample PACF values at lags 1 through 6 are $0.49, -0.28, 0.07, -0.12, 0.15, -0.04$, and the ACF values are $0.52, 0.23, 0.06, -0.09, -0.13, 0.05$. What order is suggested?

Method. Use the approximate band as a screen, remembering that a negative partial autocorrelation can be just as informative as a positive one.

Step 1: Band. Here $\sqrt{100}=10$, so $1.96/\sqrt{100}=1.96/10=0.196$.

Step 2: PACF. The first two absolute values are $0.49$ and $0.28$, both greater than $0.196$. The remaining absolute values are $0.07, 0.12, 0.15, 0.04$, all less than $0.196$. This is consistent with an initial run through lag 2 followed by small PACF values.

Step 3: ACF. The first two ACF values, $0.52$ and $0.23$, exceed the band; the remaining displayed values have absolute magnitudes $0.06, 0.09, 0.13, 0.05$, all inside it. The ACF generally tails off rather than showing a separate clear cutoff at lag 2.

Conclusion and check. AR(2) is a plausible candidate from these tables. Do not discard the second PACF spike merely because it is negative: the comparison is based on absolute magnitude, and $|-0.28|=0.28>0.196$. The tables suggest an order; they do not prove that the process follows AR(2).

Worked Example: Considering a Higher-Order Candidate

A hypothetical energy cooperative studies a daily index of peak demand, with $n=144$ observations. The PACF at lags 1 through 7 is $0.43, 0.25, -0.20, 0.09, 0.13, -0.05, 0.02$. The ACF is $0.50, 0.39, 0.20, 0.08, -0.06, -0.11, 0.04$. Identify the most plausible order among the short AR candidates.

Method. Find the band and determine where the PACF changes from a notable initial run to values that are mostly small. Then check whether the ACF broadly tails off.

Step 1: Band. Since $\sqrt{144}=12$, the band is $1.96/12=0.1633$, rounded to four decimal places.

Step 2: PACF. Lags 1, 2, and 3 are beyond the band because $0.43>0.1633$, $0.25>0.1633$, and $|-0.20|=0.20>0.1633$. At lags 4 through 7, the absolute values $0.09, 0.13, 0.05, 0.02$ are all below the band. This gives an initial run through lag 3.

Step 3: ACF. The first three ACF values are beyond the band, while subsequent displayed values are inside. The ACF generally tails off across these lags, with small fluctuations, rather than exhibiting a clear cutoff after lag 3.

Conclusion and check. AR(3) is a reasonable candidate based on this pattern. The PACF value at lag 3 is negative but still beyond the band in absolute value. It would be incorrect to call it insignificant solely because its sign is negative. As in the earlier examples, the proposed order remains provisional; additional model assessment is needed before treating it as an adequate model.

Ambiguous Patterns and Practical Limits

Not every table has a clean sequence of PACF spikes. If a later lag is outside the band after several small lags, the pattern may reflect sampling variation, a more complicated dependence structure, or an order that is difficult to distinguish from the available data. Avoid selecting the order mechanically from the last isolated spike. Likewise, a PACF value just beyond the approximate band is weaker evidence than a clear initial run of spikes followed by consistently small values.

The ACF and PACF should be considered together. A gradual or oscillating ACF tail is compatible with an AR pattern; it need not decrease monotonically. Small up-and-down movements, including changes in sign, are expected in finite samples. What matters is the broad contrast: a PACF with an apparent initial cutoff and an ACF that generally tails off. If that contrast is absent or unclear, report the evidence as ambiguous rather than forcing an order.

Key takeaway: Calculate the approximate band, compare PACF values by absolute magnitude, and use the ACF as a pattern check. A clear PACF cutoff can suggest an AR order, but sample tables provide evidence for a candidate, not proof of the model.

Check Your Understanding

For each problem, use the approximate band as a screening tool and state what the patterns suggest.

A hypothetical weekly staffing index has $n=64$ observations. Its PACF at lags 1 through 4 is $0.41, 0.12, -0.08, 0.06$, and its ACF generally tails off. What order is suggested? (Answer: The band is $\pm 1.96/8=\pm 0.2450$; only PACF lag 1 exceeds it, so AR(1) is a plausible candidate.)
A hypothetical monthly service series has $n=100$. Its PACF through lag 5 is $0.46, -0.24, 0.10, 0.18, -0.03$. Which order is suggested by the PACF? (Answer: The band is $\pm 0.196$; only lags 1 and 2 exceed it in absolute value, so AR(2) is a plausible candidate.)
A hypothetical daily temperature-adjustment index has $n=169$. Its PACF through lag 5 is $0.39, 0.22, -0.17, 0.05, -0.11$. What cutoff is suggested? (Answer: The band is $\pm 1.96/13=\pm 0.1508$; the first three PACF values exceed it in absolute value and the next two do not, suggesting AR(3).)
A hypothetical quarterly utilization series has $n=81$. Its PACF at lags 1 through 4 is $0.31, 0.20, 0.24, 0.06$. Is there a clean cutoff? (Answer: The band is $\pm 1.96/9=\pm 0.2178$; lags 1 and 3 are outside, but lag 2 is inside, so the pattern does not show a clean initial cutoff and the order is ambiguous.)
A hypothetical claims-processing index has $n=121$, PACF values $0.52, 0.11, -0.07, 0.04$, and ACF values $0.52, 0.33, 0.20, 0.12, -0.03$. Which pattern should guide the candidate order? (Answer: The band is $\pm 0.1782$; only the first PACF value is outside, supporting AR(1). The ACF has later sample values outside the band, but its general tail-off pattern does not by itself imply a higher AR order.)

What You'll Learn