Track 12 · Time Series and Forecasting · Lesson 9 of 11

Autocorrelation

Calculate and read autocorrelation (ACF) and partial autocorrelation (PACF) plots, recognise the signatures of trend, seasonality and white noise, and test for white noise with the Ljung–Box test.

35 minAdvanced7 graded exercisesMembers

If today's call volume tells you something about next Sunday's, a forecaster should use it; if yesterday's fund return tells you nothing about today's, no clever model will find a pattern. The autocorrelation function measures how much a series remembers its own past.

The autocorrelation at lag $k$ , $r_k$ , is the correlation between the series and itself shifted by $k$ periods: how similar is $y_t$ to $y_{t-k}$ across the whole series? Plotting $r_1, r_2, \ldots$ against $k$ gives the ACF plot (correlogram).

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It has 7 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Calculate a sample autocorrelation and the ±1.96/√n white-noise band
Recognise ACF signatures of trend, seasonality, AR behaviour and white noise
Explain what the PACF adds to the ACF
Use and correctly interpret the Ljung–Box test

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Formula sheet

Sample autocorrelation (the convention of R and statsmodels: both sums divided by n, which cancels)
$r_k = \frac{\sum_{t=1}^{n-k} (y_t - \bar y)(y_{t+k} - \bar y)}{\sum_{t=1}^{n} (y_t - \bar y)^2}, \qquad \text{white-noise band: } \pm \frac{1.96}{\sqrt n}$
Ljung–Box portmanteau test of the first h autocorrelations together
$Q(h) = n(n+2) \sum_{k=1}^{h} \frac{r_k^2}{n-k} \ \sim\ \chi^2_{h} \text{ under white noise}$

Reference

Autocorrelation (ACF) and partial autocorrelation (PACF)
The correlation of a series with its own past at each lag (ACF), and the same after removing the effect of shorter lags (PACF), used to detect dependence, seasonality and AR order.
Stationarity and autoregressive (AR) models
An AR(p) model predicts a stationary series from a linear combination of its own last p values; differencing first turns many trending series into stationary ones.

Working method

Read the question twice: what is being asked, about which group or variable?
Name the method before calculating.
Keep full precision; round only the final answer.
Ask whether the answer is plausible in context.