Autocorrelation Function

Study guides: Exam SRM, Exam MAS-II

The autocorrelation function (ACF) gives the correlation between a Time Series and itself $k$ periods earlier. Plotted against $k$ — the correlogram — it is the primary tool for identifying the MA order of an ARIMA model and for checking residuals.

$\rho_k = \frac{\gamma_k}{\gamma_0} = \frac{\mathrm{Cov}(Y_t, Y_{t+k})}{\mathrm{Var}(Y_t)}$

$\hat{\rho}_k = \frac{\sum_{t=1}^{n-k}(y_t - \bar{y})(y_{t+k} - \bar{y})}{\sum_{t=1}^{n}(y_t - \bar{y})^{2}}$

$\rho_0 = 1$ always, and $\rho_k = \rho_{-k}$ , so only positive lags are plotted
Significance bands are $\pm 1.96/\sqrt{n}$ ; a spike inside them is indistinguishable from zero
The identification rule: an MA( $q$ ) process has $\rho_k = 0$ for $k > q$ — the ACF cuts off after lag $q$ . An AR( $p$ ) process has an ACF that tails off geometrically
Slow, near-linear decay signals non-stationarity and calls for Differencing; a spike at lag $s$ and its multiples signals Seasonality
On residuals it is the White Noise check: all spikes inside the bands means the model has extracted the structure
The companion Partial Autocorrelation Function does the reverse job, cutting off for AR and tailing off for MA
For two series, the Cross-Correlation function correlates one with the other shifted by $k$ periods; with the same series twice it is the ACF

Identifying a Model from the ACF A differenced series of 144 monthly observations has $\hat\rho_1 = -0.52$ , $\hat\rho_2 = 0.06$ , $\hat\rho_3 = -0.04$ , $\hat\rho_4 = 0.09$ , with the PACF tailing off. What model does this suggest?

Answer Bands: $\pm 1.96/\sqrt{144} = \pm 0.163$ . Only $\hat\rho_1$ clears them; lags 2–4 are all inside. ACF cuts off after lag 1, PACF tails off → MA(1) on the differenced series, i.e. ARIMA(0, 1, 1).