Autocorrelation ACF and PACF

Memory in a series

The autocorrelation at lag $h$ is the correlation between the series and itself shifted by $h$ steps: $\rho(h)=\gamma(h)/\gamma(0)$. The sample version sums the products of deviations from the mean and divides by the total squared deviation. For pure noise with $T$ observations the sample autocorrelations are roughly normal with standard deviation $1/\sqrt{T}$, so a bar outside $\pm1.96/\sqrt{T}$ is unusual at the 5% level. Daily stock returns show almost no autocorrelation, which is why simple return-prediction rules fail. Their squares do: big moves are followed by big moves, the signature of volatility clustering.

The partial autocorrelation at lag $h$ is the correlation between $x_t$ and $x_{t+h}$ after the effect of the lags in between has been removed. The two plots identify a model. An autoregression of order $p$ has an ACF that fades gradually and a PACF that cuts off after lag $p$. A moving average of order $q$ is the mirror image: an ACF that cuts off after lag $q$ and a PACF that fades. After fitting a model, the Ljung-Box statistic $Q=T(T+2)\sum_{k=1}^{m}\hat\rho(k)^{2}/(T-k)$ tests the first $m$ residual autocorrelations together. If they are not significant, the model has captured the memory.

• AR(p): the ACF fades and the PACF cuts off after lag $p$.
• MA(q): the ACF cuts off after lag $q$ and the PACF fades.
• White noise: every autocorrelation stays inside $\pm1.96/\sqrt{T}$ about 95% of the time.

Lab · what's graded

• The 95% band15 pts
• A sample autocorrelation15 pts
• An AR(1) autocorrelation15 pts
• Reading the plots15 pts
• Match the vocabulary15 pts
• Identifying a model10 pts
• Returns versus squared returns15 pts