MATH7970 Chap.4 Autocorrelation, PACF and White Noise
Autocorrelation, PACF and White Noise
Dependence by lag
The ACF measures standardised association across lags, while the PACF removes intermediate linear routes. Read the whole shape rather than selecting one convenient bar.
Residual noise as a target
White-noise residuals show no material serial correlation under the diagnostics used.
This is a failure-to-reject statement, not proof that the fitted model is the true data-generating process.
Build correlation from lagged pairs
Sample autocorrelation starts with deviations from the mean and products of values separated by a lag. Distant lags use fewer pairs, so their bars are more variable. One extreme observation can enter several products and create apparent structure.
Record the available pair count and inspect the series itself before treating an isolated late spike as evidence for another parameter. Repeated seasonal positions or a coherent decay carry much stronger support.
Describe shapes before labels
Geometric positive decay often accompanies positive autoregression. Alternating decay may indicate negative feedback or effects of differencing.
An ACF cutoff with a PACF tail supports finite MA memory, while the reverse supports an AR starting point. Mixed models can tail in both.
Write cutoff, tail-off, alternation and seasonal repetition first, because that evidence statement remains auditable even if later diagnostics reject the proposed order.
Reference bands are not an order selector
Approximate white-noise bands screen for sample correlations that are individually unusual.
Inspecting many lags makes occasional exceedances likely, and adjacent bars are not independent pieces of evidence. Prefer structures that have a mechanism and appear across neighbouring or seasonal lags.
A lone marginal bar should rarely justify raising model order when the resulting coefficient is unstable and held-out forecasting does not improve.
Repeat the analysis after fitting
Raw ACF and PACF identify candidates; residual plots judge whether those candidates removed the dependence. If residual ACF retains slow decay, seasonal spikes or alternation, name the pattern and make one targeted repair.
Squared residuals answer a different question about variance. Report a clean result as no material structure at inspected lags, not proof of independence. A later holdout or break can still reveal failure outside the diagnostic window.
Connect lag evidence to a mechanism
A correlation pattern becomes more persuasive when the lag has a process interpretation.
Daily carryover, weekly scheduling or annual seasonality gives a reason for dependence at particular positions. Without such a link, a high-order parameter can merely reproduce sampling noise. For every proposed lag, write what information is transmitted across that interval and how a changed data frequency would move the feature. Then check that the fitted coefficient, impulse response and residual plot tell the same story.
This prevents the correlation functions from becoming a visual codebook detached from the series being analysed.
What this chapter covers
- 01
autocovariance
- 02
sample ACF
- 03
partial autocorrelation
- 04
cutoff and tail-off
- 05
significance bands
- 06
white-noise residuals
Worked example · free
Read an AR signature
- 3Describe both patterns.
- 2Name and qualify the model.
Key terms
- Autocovariance
- Covariance between observations separated by a specified lag.
- Autocorrelation Function
- The sequence of standardised lag covariances.
- Partial Autocorrelation
- Lag association after intermediate lags are linearly removed.
- Cutoff
- A pattern becoming negligible after a particular lag.
- White Noise
- An uncorrelated constant-variance innovation process with zero mean.
Autocorrelation, PACF and White Noise FAQ
Why use both ACF and PACF?
They separate persistence visible through all routes from the direct relation remaining after intermediate lags are removed. Their contrasting shapes narrow the set of parsimonious starting models.
Does one bar outside a band prove dependence?
No. Inspecting many lags can produce occasional exceedances, so coherent patterns matter more than one marginal bar. Repeated or smoothly decaying structure provides a much stronger modelling signal.
What if residual ACF still has structure?
The fitted model has not captured all linear dependence, and the pattern should guide a targeted specification repair. Refit the candidate and check that the named pattern actually disappears.
Exam move
Sketch idealised AR and MA correlation shapes from memory. For every empirical plot, describe cutoff, decay, alternation and seasonal spikes before selecting an order. Draw paired ACF and PACF sketches for positive AR, negative AR, finite MA and seasonal dependence. Cover the labels and describe every sketch using cutoff, tail-off, alternation and repeated seasonal lags.
Then select one parsimonious candidate and write the evidence supporting it. After fitting, revisit the residual versions and demand that the named feature disappear. Mark how many pairs support distant lags so isolated spikes receive suitable caution, and finish by distinguishing proposed identification from demonstrated adequacy. Calculate one lag correlation from deviations by hand.
The arithmetic reveals the shrinking pair count and explains why distant isolated bars deserve less confidence than a repeated structured sequence. Recheck the same patterns after transformation because differencing can change their interpretation.
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