Calculates the cross-correlation function between two time series.

## Syntax

**XCF**(**Y**, **X**, **K**, **Method**, **Return_type**)

- Y
- is the first univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
- X
- is the second univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
- K
- is the lag order (e.g., 0 = no lag, 1 = 1st lag, etc.) to use with second-time series input (X). If missing, the default lag order of zero (i.e., no-lag) is assumed.
- Method
- is a switch to select the calculation method (1 = Pearson (default), 2 = Spearman, 3 = Kendall).
Order Description 1 Pearson (default). 2 Spearman. 3 Kendall. - Return_type
- is a switch to select the return output (1 = correlation value (default), 2 = Std Error).
Method Description 1 Correlation Value. 2 Standard Error.

## Remarks

- The time series is homogeneous or equally spaced.
- The two time series must be identical in size.
- The Pearson correlation, $r_{xy}$, is defined as follows: $$r_{xy}= \frac{\sum_{i=1}^N(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^N(x_i-\bar{x})^2\times\sum_{i=1}^N(y_i-\bar{y})^2}}$$ Where:
- $\bar{x}$ is the sample average of time series X.
- $\bar{y}$ is the sample average of time series Y.
- $x_i \in X$ is a value from the first input time series data.
- $y_i \in Y$ is a value from the second input time series data.
- $N$ is the number of pairs $\left ( x_i,y_i \right )$ that do not contain a missing observation.

- The Spearman rank correlation, $h$, is defined as follows: $$r =1-\frac{6\sum ( x_i-y_i )^2}{N\times(N^2-1}$$ Where:
- $x_i \in X$ (e.g., $x_i$ is in the first input time series data).
- $y_i \in Y$ (e.g., $y_i$ is in the second input time series data).
- $N$ is the number of pairs $\left ( x_i,y_i \right )$ that do not contain a missing observation.

- The Kendall tau ($\tau$) rank correlation is defined as follows: $$\tau =\frac{N_C-N_D}{\frac{1}{2}N(N-1)}$$ Where:
- $N_C$ is the number of concordant pairs of observations, $(x_i, y_i)$ and $(x_j, y_j)$, defined such that the ranks of the pairs of elements are in agreement. That is, if $x_i \gt x_j$ then $y_i \gt y_j$, and if $x_i \lt x_j$ then $y_i \lt y_j$.
- $N_D$ is the number of discordant pairs of observations, $(x_i, y_i$) and $(x_j, y_j)$, defined such that the ranks of the pairs of elements are not in agreement.
- $N$ is the number of pairs $\left ( x_i,y_i \right )$ that do not contain a missing observation.

## Files Examples

## Related Links

- Wikipedia - Correlation.
- Wikipedia - Cross-Correlation.
- Wikipedia - Pearson product-moment correlation coefficient.
- Wikipedia - Spearman's rank correlation coefficient.
- Wikipedia - Kendall tau rank correlation coefficient.

## References

- Hamilton, J.D.; Time Series Analysis, Princeton University Press (1994), ISBN 0-691-04289-6.
- Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740.

## Comments

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