What do the dashed confidence bands on an ACF correlogram represent?

## What the bands are testing A correlogram plots the sample autocorrelation `r_k` against lag k. Every one of those values is an *estimate* computed from a finite sample, so every one of them carries sampling error. Even if the underlying process is pure noise with no memory whatsoever, the estimates will not come out at exactly zero — they will scatter around zero. The bands quantify that scatter. The null hypothesis behind them is **white noise**: the observations are uncorrelated at every lag, with constant variance. Under that hypothesis, for reasonably large n, each sample autocorrelation is approximately normally distributed with mean 0 and standard error `1/sqrt(n)`. Multiplying by the 1.96 critical value from the standard normal distribution gives the familiar 95% band: ``` band = +/- 1.96 / sqrt(n) ``` Some n values worked through: | n | band | |---|---| | 50 | about +/- 0.277 | | 100 | about +/- 0.196 | | 400 | about +/- 0.098 | | 1000 | about +/- 0.062 | Two consequences fall straight out of the formula. First, the band **narrows with the square root of the sample size** — quadrupling the data halves the band. Second, on a short series the band is wide, so genuinely real but modest autocorrelation may sit inside it and go undetected. Absence of a crossing on 40 observations is weak evidence of absence. ## The multiplicity trap Each band is a separate 5%-level test at a single lag. A correlogram typically shows 20, 30 or 40 lags. If the series really is white noise, the expected number of bars crossing the band is 5% of however many you plot — about one bar in twenty, about two in forty. Those crossings are guaranteed by the arithmetic, not by the data. So the sensible reading rule is not "any crossing means autocorrelation". It is: - **A spike at an interpretable lag counts.** A crossing at lag 1, or at the seasonal period, is a specific claim you can check against what you know about how the data were generated. - **A pattern counts.** Several consecutive bars outside the band, or a decaying series of them, is not something noise produces by accident. - **One isolated crossing at an arbitrary lag, with everything else inside, is noise** until something else corroborates it. - **The magnitude of the excess matters.** A bar just kissing the band and a bar at three times the band height are very different pieces of evidence. If you want a single verdict over many lags at once rather than eyeballing a plot, a portmanteau test that pools the first h autocorrelations into one statistic is the right instrument; that is what the Ljung-Box test does. ## A refinement on the band width The flat `+/- 1.96/sqrt(n)` band assumes white noise at *every* lag, which is the right assumption when the correlogram is being used to test that null. Some plotting conventions instead widen the band as the lag increases, using Bartlett's formula: when testing whether the process cuts off after lag q, the standard error of `r_k` for `k > q` accounts for the nonzero autocorrelations at lags 1 through q, and is larger than `1/sqrt(n)`. That produces a funnel-shaped band rather than two flat lines. Both conventions exist; know which one you are looking at, because a bar can be significant against the flat band and not against the widening one. The flat band is the more common default and the one interview answers usually mean. ## Assumptions worth stating The approximation relies on n being reasonably large — the normal approximation to the distribution of `r_k` is poor on very short series — and on the estimate being computed from a stationary series. On a trending series the bands are not meaningful in the intended sense, because the null being tested is already obviously false: the plot will show large positive bars decaying slowly across most of the lag range, and the correct conclusion is "there is a trend here", not "there is structure at lag 13". Finally, the bands say nothing about *practical* importance. On a very long series a lag-1 autocorrelation of 0.04 will comfortably clear the band and still be far too small to change any forecast worth making. Statistical significance on a correlogram is a statement about whether the value is distinguishable from zero, not about whether it is big enough to act on.