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Interpreting ACF and PACF Confidence Bounds for Model Residuals

Article Quant Q&A · Author: Bazman
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Summary

The document discusses how to interpret autocorrelation and partial autocorrelation plots for errors from an AR(1) model. It addresses residual bars that touch or cross nominal 95% confidence bounds, the number of apparent exceedances across many lags, and whether one-sided breaches imply a pattern. The response treats a bar outside the interval as significant at the chosen level, while emphasizing that borderline readings depend on the exact value and significance threshold.

It cautions that normal-based confidence intervals may be questionable when the data distribution departs from normality, and that a finite sample can show more exceedances than a simple expectation suggests. The sample in the question has 116 observations and the author uses this to illustrate that sampling variation can produce several outlying lags. The exchange offers qualitative guidance rather than a formal multiple-testing correction or a definitive diagnosis; the plots and underlying data are not provided, so the residual autocorrelation cannot be independently assessed.

Key ideas

  • A residual bar outside a confidence bound is significant under the interval’s stated threshold.
  • Nominal confidence bounds rely on assumptions about the sampling distribution.
  • Finite samples can produce more outlying lags than a simple expected count suggests.
  • The number and direction of breaches alone do not establish a definitive residual pattern.

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StatisticsBacktesting
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# How to interpret ACF and PACF plots


# How to interpret ACF and PACF plots












I just want to check that I am interpreting the ACF and PACF plots correctly:

The data corresponds to the errors generated between the actual data points and the estimates generated using an AR(1) model.

I've looked at the answer here:

https://stats.stackexchange.com/questions/106038/estimate-arma-coefficients-through-acf-and-pacf-inspection/

After reading that it seems that the errors are not autocorrelated but I just want to be sure, my concerns are:

1.) The first error is right on the boundary (when this is the case should I accept or reject that there is significant auto-correlation at lag 1)?

2.) The lines represent the 95% confidence interval and given that there are 116 lags I would expect no more than (0.95*116=5.8 which I round up to 6) 6 lags to be exceed the boundary. For the ACF this is the case but for the PACF there are about 10 exceptions. If you include those on the border it's more like 14? Does this still indicate no auto-correlation?

3.) Should I read anything into the fact that all the breaches of the 95% confidence interval occur to the downside?

Thanks

Baz

## Answer by Hawk (score 2)

https://quant.stackexchange.com/a/16795

I can offer my opinion in response to your first two questions:

1.) Unfortunately, this is one of the problems with numbers; the answer is that if the observation is outside of the confidence interval by even a millionth of a percent, it is significant. If it is below by even the smallest amount, it is not significant. Changing your significance level or increasing your sample size might help.

2.) Remember first that, unless you specified otherwise, the confidence interval refers to a standard normal Gaussian distribution (no skew and normal kurtosis and so on); this may or may not be a valid approach if the distribution is perceptibly different from normal in some respect (your information suggests a slightly negative skew...?). Secondly, you only have 116 observations - one moderately-sized sample out of an indefinite set of samples in the observable population: this means that, even if the population is standard normal, there is a definite possibility that any given sample can deviate perceptibly from the population's distribution (such as having 14 points outside of your confidence interval when there should only be 6).

I hope that helps at least a little!

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.