# PACF - Partial Autocorrelation

Calculates the sample partial autocorrelation function (PACF).

## Syntax

PACF ([x], order, k)

[X]
Required. Is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
Order
Optional. Is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).
Value Order
1 Ascending (the first data point corresponds to the earliest date) (default).
0 Descending (the first data point corresponds to the latest date).
K
Optional. Is the lag order (e.g., k = 0 (no lag), k = 1 (1st lag), etc.). If missing, the default of k = 1 is assumed.

## Remarks

1. The time series is homogeneous or equally spaced.
2. ACF and PACF plots (i.e., correlograms) are tools commonly used for model identification in Box-Jenkins models.
3. PACF is the autocorrelation between $z_t$ and $z_{t-k}$ that is not accounted for by lags 1 to k-1, inclusive.
4. Equivalently, PACF(k) is the ordinary least square (OLS) multiple-regression k-th coefficient ($\phi_k$).

$$\left[y_{t}\right]=\phi_{0}+\sum_{j=1}^{k}\phi_{j}\left[y_{t-j}\right]$$

Where:

• $\left[y_{t}\right]$ is the input time series.
• $k$ is the lag order.
• $\phi_j$ is the j-th coefficient of the linear multiple regression (i.e., AR(j)).