Correlogram

Study guide: Exam MAS-II

A correlogram is the plot of the sample autocorrelations $r_k$ of a time series against the lag $k$ . It is drawn with bounds that mark how large a correlation must be before it can be told apart from zero. It is the picture through which the ACF (and, drawn the same way, the PACF) is read, both to identify a model and to check its residuals.

$r_k = \frac{c_k}{c_0}$

$c_k = \frac{1}{n}\sum_{t=1}^{n-k}(x_t - \bar{x})(x_{t+k} - \bar{x})$

$\text{bounds} = -\frac{1}{n} \pm \frac{2}{\sqrt{n}}$

Lag $k$ runs along the horizontal axis and $r_k$ up the vertical. The first bar is always $r_0 = 1$ . When the true $\rho_k = 0$ , $r_k$ is approximately normal with mean $-1/n$ and variance $1/n$ . That gives the bounds Cowpertwait draws. The simpler $\pm 1.96/\sqrt{n}$ used elsewhere in this vault differs from them by about $0.01$ when $n$ is around $100$ .
Patterns to recognise:
- slow, near-linear decay from close to $1$ means a trend or other non-stationarity; difference before reading anything else
- peaks at lags $s, 2s, \ldots$ ( $s = 12$ for monthly data, $s = 4$ for quarterly) mean seasonality
- geometric decay, alternating in sign if $\phi < 0$ , suggests AR behaviour; spikes up to lag $q$ and nothing after suggest MA( $q$ )
- everything inside the bounds looks like white noise
Residual check. After fitting, the correlogram of the residuals should look like white noise. If it does not, the lag of the surviving spike says what is missing. A spike at lag $1$ or $2$ suggests an extra AR or MA term. One at lag $12$ (monthly) or $4$ (quarterly) suggests a seasonal term. Read the residual PACF the same way.
Chance breaches. The bounds are pointwise $95\%$ limits, so about $1$ lag in $20$ crosses them by chance. A marginal breach at an unremarkable lag is weak evidence. A clear breach at lag $1$ or at a seasonal lag is not.
The bounds assume stationarity. On a trending series nearly every bar is significant, and the plot says nothing about the ARMA orders until the trend is removed. See Deterministic and Stochastic Trend.
The cross-correlation of two series is plotted the same way, but over negative as well as positive lags.

Reading the Raw Correlogram of Monthly Claim Counts A series of $n = 96$ monthly reported claim counts has $r_1 = 0.91$ . The autocorrelations decline slowly to $0.55$ at lag $24$ , with local peaks at lags $12$ and $24$ . What does the correlogram show, and what should be done next?
Answer
 $\begin{align*} \text{bounds} &= -\frac{1}{96} \pm \frac{2}{\sqrt{96}} \\ &= -0.010 \pm 0.204 \\ &= (-0.215,\ 0.194) \end{align*}$ 
Every bar is far outside. The slow decay says the series is not stationary (a trend in claim volume), and the peaks at 12 and 24 say there is an annual cycle.

No AR or MA order can be read from this plot. Remove the trend and the seasonal pattern first (a first difference and a lag-12 seasonal difference, or a decomposition), then re-plot the ACF and PACF of what remains.

Residual Correlogram After an AR(1) Fit An AR(1) is fitted to $120$ months of claim frequency. The residual correlogram, plotted to lag $24$ , shows $r_1 = 0.04$ , $r_2 = -0.07$ , $r_5 = 0.18$ and $r_{12} = 0.31$ . All other lags lie within $\pm 0.10$ . Is the model adequate?
Answer
 $\begin{align*} \text{bounds} &= -\frac{1}{120} \pm \frac{2}{\sqrt{120}} \\ &= -0.008 \pm 0.183 \\ &= (-0.191,\ 0.174) \end{align*}$ 
$r_5 = 0.18$ is only just outside, at an unremarkable lag. With $24$ lags plotted, about $1.2$ breaches are expected by chance, so it is not evidence on its own.

$r_{12} = 0.31$ is well outside, and at the seasonal lag. The AR(1) has left the annual pattern in the residuals. Not adequate: add a seasonal term (a seasonal AR or MA at lag $12$ , or a seasonal difference), refit, and check the correlogram again.