2.2.5 Cross-Correlation Function

Cross-Correlation Function (CCF)

The cross-correlation function between two jointly stationary time series $\{X_t\}$ and $\{Y_t\}$ is the normalized cross-covariance, measuring the linear dependence between $X_{t+h}$ and $Y_t$ on a unitless scale:
${\rho }_{XY}(h)=\frac{{\gamma }_{XY}(h)}{\sqrt{{\gamma }_X(0){\gamma }_Y(0)}}=\frac{\text{Cov}(X_{t+h},Y_t)}{\sqrt{\text{Var}(X_t)\text{Var}(Y_t)}},$
where ${\gamma }_{XY}(h)$ is the cross-covariance at lag $h$, and ${\gamma }_X(0)$, ${\gamma }_Y(0)$ are the variances of each series. This extends the autocorrelation function to a pair of series, and is the time series analogue of the correlation coefficient ${\rho }_{X,Y}$.

Like cross-covariance, the CCF is not symmetric: ${\rho }_{XY}(h)\ne {\rho }_{XY}(-h)$ in general. The relationship ${\rho }_{XY}(h)={\rho }_{YX}(-h)$ holds, so reading the CCF from one direction gives the lead-lag structure from $X$‘s perspective, and reversing it gives $Y$‘s.

Spurious Cross-Correlation $\{X_t\}$ and $\{Y_t\}$ share a common trend or seasonal component, the CCF can show large values at many lags even when the series have no genuine dynamic relationship. It is standard practice to pre-whiten both series (remove autocorrelation and trend) before interpreting the CCF causally.

If both

Intuition

The CCF answers: on a scale from $-1$ to $1$, how strongly is the value of series $X$ at time $t+h$ linearly associated with the value of series $Y$ at time $t$? It is the primary tool for detecting lead-lag relationships between two series.

A peak in ${\rho }_{XY}(h)$ at $h>0$ means that $X$ leads $Y$ by $h$ periods (future $X$ covaries with current $Y$, equivalently current $X$ covaries with future $Y$). A peak at $h<0$ means $Y$ leads $X$. A peak at $h=0$ indicates contemporaneous association. In practice, the sample CCF is plotted and compared against confidence bands at $\pm 1.96/\sqrt{T}$ to identify statistically significant lead-lag structure.

Properties

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

(Asymmetry) ${\rho }_{XY}(h)={\rho }_{YX}(-h)$, but ${\rho }_{XY}(h)\ne {\rho }_{XY}(-h)$ in general.

(Autocorrelation as Special Case) When $Y_t=X_t$, the CCF reduces to the ACF: ${\rho }_{XX}(h)={\rho }_X(h)$.

(Normalization) ${\rho }_{XY}(h)={\gamma }_{XY}(h)/{\sigma }_X{\sigma }_Y$, where ${\sigma }_X=\sqrt{{\gamma }_X(0)}$ and ${\sigma }_Y=\sqrt{{\gamma }_Y(0)}$.

(Independence) If $\{X_t\}$ and $\{Y_t\}$ are independent, then ${\rho }_{XY}(h)=0$ for all $h$.

(Invariance to Location-Scale) If ${\tilde{X}}_t=a{X}_t+b$ and ${\tilde{Y}}_t=c{Y}_t+d$ with $a,c>0$, then ${\rho }_{\tilde{X}\tilde{Y}}(h)={\rho }_{XY}(h)$. The CCF is unaffected by shifts and positive rescalings.

Examples

Delayed Response

An economist models monthly advertising spend $\{X_t\}$ and monthly revenue $\{Y_t\}$ for a company. Both series are stationary after differencing. The sample CCF shows a significant peak at $h=2$.

Interpret the peak. $h=2$ in ${\rho }_{XY}(h)$ means that $X_{t+2}$ is correlated with $Y_t$, or equivalently, $X_t$ is correlated with $Y_{t+2}$. This indicates that advertising spend leads revenue by roughly $2$ months: an increase in ad spending today is associated with a revenue increase two months later. The magnitude of ${\rho }_{XY}(2)$ quantifies the strength of this association on a $[-1,1]$ scale.

A peak at

CCF of Lagged Signal

Let $Y_t=\beta X_{t-d}+W_t$, where $\{X_t\}$ is stationary white noise with variance ${\sigma }_X^2$, $\{W_t\}\sim \text{WN}(0,{\sigma }_W^2)$ is independent of $\{X_t\}$, and $d>0$.

Derive the CCF. cross-covariance: ${\gamma }_{XY}(h)=\text{Cov}(X_{t+h},\beta X_{t-d}+W_t)=\beta \text{Cov}(X_{t+h},X_{t-d})=\{\begin{array}{ll}\beta {\sigma }_X^2& h=-d,\\ 0& h\ne -d,\end{array}$ since $\{X_t\}$ is white noise and $\{W_t\}$ is independent. The variances are ${\gamma }_X(0)={\sigma }_X^2$ and ${\gamma }_Y(0)={\beta }^2{\sigma }_X^2+{\sigma }_W^2$. So ${\rho }_{XY}(h)=\{\begin{array}{ll}\frac{\beta {\sigma }_X^2}{{\sigma }_X\sqrt{{\beta }^2{\sigma }_X^2+{\sigma }_W^2}}=\frac{\beta {\sigma }_X}{\sqrt{{\beta }^2{\sigma }_X^2+{\sigma }_W^2}}& h=-d,\\ 0& h\ne -d.\end{array}$ The CCF is a single spike at $h=-d$, correctly identifying that $Y$ lags $X$ by $d$ periods. The height of the spike depends on the signal-to-noise ratio ${\beta }^2{\sigma }_X^2/{\sigma }_W^2$.

From the