3.5 Cross-Covariance and Cross-Correlation

Cross-Covariance Definition¶

When examining the relationship between two time series, it is often useful to see how they move in tandem. One powerful tool for this is the cross-covariance function defined as

for time series $x_s$ and $y_t$. Analogous to stationarity for autocovariance, the two series are said to be jointly stationary if both:

1. $x_s$ and $y_t$ are individually stationary.

2. The cross-covariance is only a function of the lag $h=s-t$.

In such a case, Eq. (1) can be simplified to

Since we are dealing with two distinct time series, it is generally the case that $\gamma_{x,y}(h)\neq\gamma_{x,y}(-h)$ for $h\neq0$. This is due to the fact that $\gamma_{x,y}(h)$ represents the covariance between $y$ and $x$ at $h$ lags in the future, whereas $\gamma_{x,y}(-h)$ represents the covariance between $y$ and $x$ at $h$ lags in the past. Put differently, there is no reason to assume the covariance between employment and the following month’s inflation is the same as the covariance between employment and the preceding month’s inflation.

Cross-Correlation Definition¶

The cross-correlation function is defined as

For jointly stationary time series, Eq. (3) simplifies to

Cross-Covariance and Cross-Correlation in statsmodels¶

In statsmodels, the cross-covariance and cross-correlation are accessed by statsmodels.tsa.stattools.ccovf and statsmodels.tsa.stattools.ccf, respectively. The only arguments we need to worry about at this point are x and y, the two time series. Note that these functions only calculate positive $h$ values, to get both positive and negative values (i.e. both x and y leading) you must run both ccf(x,y) and ccf(y,x).