2.2.4 Cross-Covariance Function

Cross-Covariance Function

The cross-covariance function between two time series $\{X_t\}$ and $\{Y_t\}$ measures the linear dependence between $X_s$ and $Y_t$ at possibly different time points, defined as
${\gamma }_{XY}(s,t)=\text{Cov}(X_s,Y_t)=\mathbb{E}[(X_s-{\mu }_X(s))(Y_t-{\mu }_Y(t))],$
where ${\mu }_X(s)=\mathbb{E}[X_s]$ and ${\mu }_Y(t)=\mathbb{E}[Y_t]$ are the respective mean functions. This extends the autocovariance function from a single series to a pair of series, and is the time series analogue of covariance between two random variables.

When both processes are jointly stationary, the cross-covariance depends only on the lag $h=s-t$:
${\gamma }_{XY}(h)=\text{Cov}(X_{t+h},Y_t)\text{for all }t.$

Unlike autocovariance, the cross-covariance is not symmetric in the lag: ${\gamma }_{XY}(h)\ne {\gamma }_{XY}(-h)$ in general. Instead, swapping the roles of $X$ and $Y$ reverses the lag: ${\gamma }_{XY}(h)={\gamma }_{YX}(-h)$.

Direction Matters $h$ encodes lead-lag structure. If ${\gamma }_{XY}(h)$ is large at $h>0$, then $X$ leads $Y$ (values of $X$ covary with future values of $Y$). If ${\gamma }_{XY}(h)$ is large at $h<0$, then $Y$ leads $X$. This asymmetry is what makes cross-covariance useful for identifying causal or predictive relationships between series.

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Intuition

The cross-covariance function answers: does knowing the value of series $X$ at time $s$ tell you anything about series $Y$ at time $t$? While autocovariance looks at memory within a single series, cross-covariance looks at coupling between two series across time.

This is essential in multivariate time series analysis. For example, does a change in interest rates ($X$) predict a future change in stock prices ($Y$)? If ${\gamma }_{XY}(h)$ peaks at a positive lag $h$, it suggests that $X$ leads $Y$ by $h$ periods. The cross-covariance quantifies the strength and timing of such lead-lag relationships, though like all covariance measures, it captures only linear associations.

Properties

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

This reflects the fact that ”$X$ leads $Y$” is a different statement from ”$Y$ leads $X$.”

(Autocovariance as Special Case) When $Y_t=X_t$, the cross-covariance reduces to the autocovariance: ${\gamma }_{XX}(h)={\gamma }_X(h)$.

(Linearity) If $Z_t=a{X}_t+b{Y}_t$, then

${\gamma }_{ZW}(h)=a{\gamma }_{XW}(h)+b{\gamma }_{YW}(h)$
for any third series $\{W_t\}$.

(Scale Dependence) Like autocovariance, ${\gamma }_{XY}(h)$ has units of $(\text{units of }X)\times (\text{units of }Y)$ and is not directly comparable across different pairs of series. The cross-correlation function ${\rho }_{XY}(h)$ normalizes this.

(Independence) If $\{X_t\}$ and $\{Y_t\}$ are independent, then ${\gamma }_{XY}(h)=0$ for all $h$. The converse does not hold in general (zero cross-covariance implies only linear uncorrelatedness).

Examples

Lagged Signal

Let $Y_t=X_{t-d}+W_t$, where $\{X_t\}$ is a stationary signal with autocovariance ${\gamma }_X(h)$, $\{W_t\}\sim \text{WN}(0,{\sigma }_W^2)$ is independent of $\{X_t\}$, and $d>0$ is a fixed delay.

Derive the cross-covariance ${\gamma }_{XY}(h)$. We have ${\gamma }_{XY}(h)=\text{Cov}(X_{t+h},Y_t)=\text{Cov}(X_{t+h},X_{t-d}+W_t)=\text{Cov}(X_{t+h},X_{t-d})={\gamma }_X(h+d),$ since $W_t$ is independent of $X$. The cross-covariance is maximized when $h=-d$, because ${\gamma }_X(h+d)$ is largest at $h+d=0$. This correctly identifies that $Y$ lags $X$ by $d$ periods: the peak at negative $h$ means values of $Y$ are most strongly associated with past values of $X$.

Uncorrelated but Dependent Series

Let $X_t\sim \text{WN}(0,1)$ and $Y_t=X_t^2$.

Show that ${\gamma }_{XY}(0)=0$ despite $Y_t$ being a deterministic function of $X_t$. We have ${\gamma }_{XY}(0)=\text{Cov}(X_t,X_t^2)=\mathbb{E}[X_t^3]-\mathbb{E}[X_t]\mathbb{E}[X_t^2].$ Since $X_t$ has a symmetric distribution around $0$, $\mathbb{E}[X_t^3]=0$ and $\mathbb{E}[X_t]=0$, so ${\gamma }_{XY}(0)=0$. The two series are linearly uncorrelated at every lag, yet $Y_t$ is entirely determined by $X_t$. This illustrates that cross-covariance captures only linear relationships.