Common mistake 1 with Correlogram (ACF / PACF)

Reading the correlogram of a non-stationary series as if it identified an AR model.

Common mistake 2 with Correlogram (ACF / PACF)

Treating every bar outside the band as real. With 20 lags, one false positive at 5% is expected.

Common mistake 3 with Correlogram (ACF / PACF)

Identifying a model from the correlogram of a seasonal series without looking at the seasonal lags. The interesting spike is at 12, and a chart drawn to 20 lags on monthly data shows it only just.

Common mistake 4 with Correlogram (ACF / PACF)

Computing a correlogram on a series with a strong seasonal pattern and no seasonal differencing. The seasonal spikes dominate every other lag and hide the short-run structure entirely.

EconStatsCraft › Correlogram (ACF / PACF)

Correlogram (ACF / PACF)

Autocorrelation and partial autocorrelation with Ljung–Box tests. Use it to choose the AR and MA orders of an ARIMA model.

What it needs

Series

The intuition

The ACF asks "how much does today resemble k days ago?" - but that includes everything transmitted through the days in between. The PACF strips those out and asks about the DIRECT link only. An AR(1) has an ACF that decays forever (yesterday affects today, which affects tomorrow) but a PACF that cuts off after lag 1, because there is no direct channel beyond one period.

When to use it

Before fitting ARIMA, and after fitting anything to a time series to check the residuals are white noise.

Where the estimator comes from

\rho_k=\gamma_k/\gamma_0 where \gamma_k=\operatorname{Cov}(y_t,y_{t-k}). The partial autocorrelation \phi_{kk} is the coefficient on y_{t-k} in a regression of y_t on all lags up to k, obtained recursively by the Durbin–Levinson algorithm. Ljung–Box aggregates the first m autocorrelations: Q=n(n+2)\sum_{k=1}^{m}\hat{\rho}_k^2/(n-k), asymptotically \chi^2(m).

The model

\rho_k=\frac{\operatorname{Cov}(y_t,y_{t-k})}{\operatorname{Var}(y_t)}\qquad Q=n(n+2)\sum_{k=1}^{m}\frac{\hat{\rho}_k^{2}}{n-k}

What it assumes

The series is stationary. On a non-stationary series the ACF decays slowly and tells you nothing else.
The observations are evenly spaced in time. A correlogram computed on rows a day apart in one stretch and a month apart in another is measuring lags that do not exist.
The sample is long enough for the lags requested. Each successive lag is estimated from fewer pairs, so the right-hand end of a correlogram is always its least reliable part.
Missing periods have not been silently dropped. Removing gaps glues together observations that were never adjacent, which biases every autocorrelation towards zero.

Statistical properties

Under white noise, each ρ̂ₖ is approximately N(0, 1/n) - hence the ±1.96/√n band.
With 20 lags plotted at 5%, roughly one spike outside the band is expected by chance. Do not over-read a single one.
The ACF of a non-stationary series decays linearly and slowly; that pattern identifies a unit root, not an AR process.
Ljung–Box on RESIDUALS should use df reduced by the number of estimated ARMA parameters.
Seasonality shows as spikes at multiples of the period, not at every lag.

How to read the output

ACF cuts off sharply after lag q, PACF decays → MA(q).
PACF cuts off after lag p, ACF decays → AR(p).
Both decay gradually → ARMA.
ACF that decays extremely slowly and stays positive → a unit root. Difference first.
A spike at lag 4 (quarterly) or 12 (monthly) → seasonality.
Ljung–Box on residuals: a large p-value is the good outcome here.
Read the ACF and PACF together, not separately: an AR process decays in the ACF and cuts off in the PACF, an MA process does the reverse. Either one alone is ambiguous.
The significance band is 1.96 over the square root of the sample size, so it narrows with more data. A bar that clears the band on 500 observations would not clear it on 50.
The Ljung-Box column tests the lags jointly up to that point, which is a different question from whether any individual bar is significant - and it is the one that matters for whether a model has left structure behind.
A slowly decaying ACF is a diagnosis, not a failure: it says the series needs differencing before this chart can say anything about its short-run structure.

A worked reading

ACF significant to lag 6, PACF cutting off after lag 2, on a series that the unit-root test called I(0). That is an AR(2) signature: start from ARIMA(2,0,0) and let BIC check whether an MA term improves it.

Symptom → diagnosis → what to do

If you see this	It means	Do this
ACF decays very slowly, stays positive	Unit root	Difference first, then re-read the correlogram
ACF cuts off after q, PACF decays	Moving average	MA(q)
PACF cuts off after p, ACF decays	Autoregressive	AR(p)
Both decay geometrically	Mixed	ARMA - use information criteria to choose orders
Large negative spike at lag 1 after differencing	Over-differenced	Reduce d by one
Spikes at 4, 8, 12 (quarterly)	Seasonality	Add seasonal terms or seasonally difference

Common mistakes

Reading the correlogram of a non-stationary series as if it identified an AR model.
Treating every bar outside the band as real. With 20 lags, one false positive at 5% is expected.
Identifying a model from the correlogram of a seasonal series without looking at the seasonal lags. The interesting spike is at 12, and a chart drawn to 20 lags on monthly data shows it only just.
Computing a correlogram on a series with a strong seasonal pattern and no seasonal differencing. The seasonal spikes dominate every other lag and hide the short-run structure entirely.

Where to read this properly

Box & Jenkins (1970), Time Series Analysis: Forecasting and Control
Ljung & Box (1978)
Hamilton (1994), Time Series Analysis, Princeton University Press, chapter 3

Related Time series - one series

Time-varying coefficients (Kalman) · Unit root tests · ARIMA / SARIMA · Rolling-origin cross-validation · Theta method · Unobserved components · STL forecasting · Forecast combination · Markov regime switching · Change frequency (resample) · ARCH / GARCH volatility · Decompose & filter · Spectral analysis · Exponential smoothing · Forecast accuracy

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Each example is simulated from a known model, so you can check whether the estimator recovers the truth - and nothing here is anyone's private data.

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Correlogram (ACF / PACF)

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

Related Time series - one series

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