ACF, PACF, ADF and Ljung–Box Tests

Suppose you are analyzing the number of inquiries an online customer service center receives each week. If there were many inquiries last week, will there be many this week too? If inquiry volume has risen steadily since a service overhaul, can you use past patterns as they are to predict the future? And once you've built a forecasting model, you also need to check that no time-related patterns the model missed are left over.

Tools often used together here are the autocorrelation function (ACF), the partial autocorrelation function (PACF), the ADF unit root test and the Ljung–Box test. The four look like a list of similarly named time series diagnostics, but the questions they actually answer differ. ACF and PACF look at the relationship between past and current values, the ADF test examines whether a unit root is present, and the Ljung–Box test assesses whether the autocorrelations at several lags are jointly 0.

In this article we'll use a customer service center's weekly inquiry counts over 156 weeks (3 years) as an example to go through how the four methods work, the order of analysis, how to interpret the numbers and common misunderstandings.

This week's inquiry count may be related to last week's, and scheduled events such as month-end billing or product updates can create recurring fluctuations. If you ignore time, it's easy to miss dependence that actually exists in the data.

The example uses 156 weekly inquiry counts compiled with the same definition from the first week of 2023 through the last week of 2025. Before the analysis, check that no weeks are missing, that the weekly counting rule didn't change and whether there were exceptional events such as a major outage. Don't conclude that rising inquiry counts are a trend right away; also consider the possibility of seasonality or a structural change.

For example, the lag-1 autocorrelation is the relationship between this week's and last week's inquiry counts, and the lag-4 autocorrelation is the relationship between this week's count and the count 4 weeks earlier. Conceptually, the population autocorrelation at lag 1 can be written as:

A positive autocorrelation means that when past values were high, current values also tend to be high. A negative one means high values tend to be followed by low values. But autocorrelation doesn't imply causation, and it can be large simply because of a trend or seasonality.

It is usually examined with an ACF plot, a bar chart with lags on the horizontal axis and correlation coefficients on the vertical axis. Suppose the original inquiry count series gave the following sample ACF.

There is a strong positive correlation between inquiry volumes in adjacent weeks, and the correlation decays relatively slowly even as the lag grows. This pattern can appear in a highly persistent stationary series, but it can also be seen in a nonstationary series with a trend or a unit root. A slowly decaying ACF alone can't confirm a unit root.

When looking at an ACF, check not only the height of the bars but also the confidence band and the state of the series used in the analysis (original series, differenced series or residuals).

Lags not listed in the table are drawn by smoothly connecting the tabulated values.

For example, the lag-3 PACF assesses the relationship between the current value and the value 3 weeks earlier after controlling for the linear effects of lags 1 and 2.

For a stationary pure AR(1) process, the PACF is theoretically 0 after lag 1 while the ACF decays gradually. For a pure MA(1) process it's the other way around: the ACF is theoretically 0 after lag 1 and the PACF decays gradually.

These are idealized textbook patterns, though. In real samples, noise, mixed ARMA structures, seasonality and estimation error can keep a clear cutoff from appearing.

The ACF and PACF are especially useful for screening candidates for the AR order p and the MA order q in a series that has been made stationary.

In practice, use these patterns as clues for finding initial candidates, not as rules that settle the model. The final model is best chosen by comparing AIC and BIC, residual diagnostics and forecasting performance from time series cross-validation or a holdout sample, all together.

Not every time series model requires the same stationarity assumption, but it is particularly important in traditional ARMA analysis.

If customer service inquiries rise over a long period as the service grows, the mean level can change over time. Interpreting the autocorrelation of the original series as is in that situation can mostly reflect the trend rather than the actual short-term dependence. So review the plot and the data-generating process, and if needed consider detrending, seasonal adjustment or differencing.

Differencing isn't always necessary, though. For a trend-stationary process or a series with a structural change, differencing automatically can actually throw away information.

The test is run on a regression that includes lagged difference terms to account for autocorrelation in the errors. A simplified ADF regression is:

• Null hypothesis H₀: a unit root is present (γ = 0).
• Alternative hypothesis H₁: the process has no unit root, given the deterministic terms specified (constant, trend, etc.).

ADF critical values and p-values are computed differently from those of an ordinary t test. Results can also change depending on whether you include a constant or a time trend and how many lags you use. And ADF is not an all-purpose tool that tests every kind of nonstationarity; structural changes or seasonal unit roots need separate examination.

The figures below were set up to illustrate the diagnostic workflow alongside the separate ACF example.

A p-value of .573 for the original series means there isn't enough evidence to conclude there is no unit root; it is not proof that a unit root definitely exists. For the first-differenced series, on the other hand, the unit root null can be rejected under that test specification.

Accordingly, you can consider an ARIMA differencing order of d = 1 as a candidate. But you still need to review the differenced plot, the ACF and PACF and the model fit, and you shouldn't ignore structural changes or seasonality.

Even if the original series rises every week, if the size of the increase is stable, the differenced series can show a more constant mean. But differencing costs one observation, and the level information of the original series is no longer directly visible. Overdifferencing can also create unnecessary negative autocorrelation.

In this example, suppose that after differencing the ACF weakens quickly at short lags and a clear spike appears at lag 1 of the PACF. In that case ARIMA(1,1,0) can serve as an initial candidate, but you need to compare its actual fit with ARIMA(0,1,1) and other candidates.

Unlike the ACF, which looks at one lag at a time, it combines the correlations up to a specified maximum lag. The test statistic is:

Here n is the number of observations, h the maximum lag included in the test and ρ̂_k the sample autocorrelation at lag k. The null hypothesis is usually that the autocorrelations from lag 1 through lag h are all 0.

The Ljung–Box test can also be used to explore dependence in the original series, but it is used most often to check whether autocorrelation remains in the residuals of a forecasting model. When applying it to ARIMA residuals, it's common to adjust the test's degrees of freedom for the number of estimated AR and MA parameters.

The tests in the table below are assumed to use lags up to 12, with degrees of freedom adjusted for the number of estimated AR and MA parameters.

For candidate A the null hypothesis is rejected, so you should consider whether the model fails to capture the time dependence adequately. For candidate B the null is not rejected, but that doesn't prove the residuals are perfect white noise. There are limits related to the test's power and the maximum lag chosen.

Also, Ljung–Box mainly examines linear autocorrelation left over from the mean structure. To check for volatility clustering or conditional heteroscedasticity, you can add checks on the pattern of squared residuals or ARCH-related diagnostics.

In practice, the following order is useful.

1. Plot the original series: check for trend, seasonality, outliers and structural change
2. ACF and PACF of the original series: understand the form of time dependence
3. ADF test: check for a unit root under the specified trend and constant conditions
4. Needed transformations: apply differencing, seasonal differencing, detrending, etc.
5. ACF and PACF of the transformed series: choose candidate AR and MA orders
6. Model fitting: compare the fit and forecasting performance of several candidates
7. Residual ACF and Ljung–Box: check for unexplained autocorrelation

This order is a starting point, not a fixed procedure for every time series. For example, if seasonality is pronounced, you may need to look at the ACF at seasonal lags and seasonal differencing first; if there is a clear structural change, it's best to start by looking at when that event happened and how to reflect it in the model.

For both tests, failing to reject the null hypothesis should not be read as evidence that the hypothesis is true.

Also, settling the ARIMA orders from the shapes of the ACF and PACF bars alone can lead to overfitting or the wrong amount of differencing. Weekly data can have an annual cycle (about 52 weeks) and holiday or event effects, and if you estimate 52-week seasonality from the 156 weeks of data in this example, keep in mind that the cycle repeats only three times.

In an actual report, state the deterministic terms in the ADF test and how the lag order was chosen, the maximum lag and degrees-of-freedom adjustment for Ljung–Box, the ARIMA model used and the conditions of the residual diagnostics.

ADF tests for the presence of a unit root, and Ljung–Box tests whether the autocorrelations at several lags are jointly 0.

In time series analysis, rather than settling on a model from just one of the four tools, it's important to connect the characteristics of the original series, the needed transformations, the comparison of candidate models and residual diagnostics into a single workflow. Remembering that ACF and PACF explore structure, ADF checks for unit roots and Ljung–Box looks for leftover autocorrelation makes it easy to keep each analysis's role straight.