2.2.3 Autocorrelation Function

Autocorrelation Function (ACF)

The autocorrelation function of a stationary time series $\{X_t\}$ is the normalized autocovariance, measuring the linear dependence between $X_t$ and $X_{t+h}$ on a unitless scale:
${\rho }_X(h)=\frac{{\gamma }_X(h)}{{\gamma }_X(0)}=\frac{\text{Cov}(X_t,X_{t+h})}{\text{Var}(X_t)},$
where ${\gamma }_X(h)$ is the autocovariance at lag $h$ and ${\gamma }_X(0)=\text{Var}(X_t)$. This is the time series analogue of the correlation coefficient ${\rho }_{X,Y}$, applied to a single series at two different times.

For non-stationary processes, the general form uses two time indices:
${\rho }_X(s,t)=\frac{{\gamma }_X(s,t)}{\sqrt{{\gamma }_X(s,s){\gamma }_X(t,t)}}.$

The ACF is one of the most important diagnostic tools in time series analysis: its shape reveals the memory structure and guides model selection (e.g., AR vs. MA vs. ARMA).

ACF Measures Linear Dependence Only correlation, ${\rho }_X(h)=0$ means $X_t$ and $X_{t+h}$ are linearly uncorrelated, but they may still be dependent in nonlinear ways. For Gaussian processes, uncorrelatedness implies independence; in general, it does not.

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Intuition

The ACF answers: how strongly does the value at time $t$ predict the value $h$ steps into the future, on a scale from $-1$ to $1$? A slowly decaying ACF signals persistence: the series drifts for long stretches above or below its mean. A sharply truncating ACF (dropping to zero after lag $q$) is the signature of a moving average MA$(q)$ process. An ACF that decays geometrically suggests an autoregressive AR process.

In practice, the sample ACF $\hat{\rho }(h)=\hat{\gamma }(h)/\hat{\gamma }(0)$ is plotted as a correlogram, with confidence bands at $\pm 1.96/\sqrt{T}$ (under the null of white noise). Lags where $\hat{\rho }(h)$ exceeds these bands indicate statistically significant serial dependence.

Properties

(Normalization) ${\rho }_X(0)=1$ always.

(Boundedness) $-1\le {\rho }_X(h)\le 1$ for all $h$, by the Cauchy–Schwarz bound.

(Symmetry) ${\rho }_X(h)={\rho }_X(-h)$.

(Positive Semi-Definiteness) The function ${\rho }_X(h)$ is positive semi-definite: for any $n$, times $t_1,\dots ,t_n$, and constants $a_1,\dots ,a_n$, ${\sum }_{i,j}{a}_i{a}_j{\rho }_X(t_i-t_j)\ge 0$.

(White Noise) If $\{W_t\}\sim \text{WN}(0,{\sigma }^2)$, then ${\rho }_W(h)=\{\begin{array}{ll}1& h=0,\\ 0& h\ne 0.\end{array}$

(MA(q) Cutoff) If $X_t={\sum }_{j=0}^q{\theta }_j{W}_{t-j}$, then ${\rho }_X(h)=0$ for all $|h|>q$. This sharp cutoff is the defining ACF signature of MA processes.

(AR(1) Decay) If $X_t=\varphi X_{t-1}+W_t$ with $|\varphi |<1$, then ${\rho }_X(h)={\varphi }^{|h|}$, which decays geometrically.

Examples

ACF of MA(1)

Let $X_t=W_t+\theta W_{t-1}$ with $\{W_t\}\sim \text{WN}(0,{\sigma }^2)$.

Derive the ACF. autocovariance of MA(1): ${\rho }_X(h)=\{\begin{array}{ll}1& h=0,\\ \frac{\theta }{1+{\theta }^2}& |h|=1,\\ 0& |h|\ge 2.\end{array}$ Note that $|{\rho }_X(1)|\le 1/2$ for all $\theta$, since $\frac{\theta }{1+{\theta }^2}$ is maximized at $\theta =1$ giving $1/2$. This means an MA(1) process can never have lag-1 correlation exceeding $0.5$ in absolute value.

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ACF of AR(1)

Let $X_t=\varphi X_{t-1}+W_t$ with $|\varphi |<1$ and $\{W_t\}\sim \text{WN}(0,{\sigma }^2)$.

Derive the ACF and describe its behavior. ${\gamma }_X(0)=\frac{{\sigma }^2}{1-{\varphi }^2}$ and ${\gamma }_X(h)={\varphi }^{|h|}\cdot {\gamma }_X(0)$ (by recursive substitution), the ACF is ${\rho }_X(h)={\varphi }^{|h|}.$ If $\varphi >0$, the ACF decays monotonically toward zero, indicating positive persistence. If $\varphi <0$, the ACF alternates in sign while decaying in magnitude, indicating oscillatory mean-reversion. The closer $|\varphi |$ is to $1$, the slower the decay and the longer the memory.

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