# PACFCI - Partial Autocorrelation Confidence Interval

Returns the confidence interval limits (upper/lower) for the partial autocorrelation function (PACF).

## Syntax

PACFCI(X, Order, K, Alpha, UL)
X
is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
Order
is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).
Order Description
1 Ascending (the first data point corresponds to the earliest date) (default).
0 Descending (the first data point corresponds to the latest date).
K
is the lag order (e.g., k = 0 (no lag), k = 1 (1st lag), etc.). If missing, the default of k = 1 is assumed.
Alpha
is the statistical significance level (i.e., alpha). If missing, the default value of 5% is assumed.
UL
is a flag to specify whether an upper (ul = 1), or lower (ul = 0) confidence interval bound is desired.

## Remarks

1. The time series is homogeneous or equally spaced.
2. The sample ACF and PACF plots (i.e., correlograms) are tools commonly used for model identification in Box-Jenkins models.
3. The PACFCI function is calculated as:
$$\hat \rho_k - Z_{\alpha/2}\times \frac{1}{\sqrt{T}} \leq \rho_{k} \leq \hat{\rho_k}+ Z_{\alpha/2}\times \frac{1}{\sqrt{T}}$$
Where:
• $\rho_k$ is the population partial-autocorrelation function for lag $k$.
• $T$ is the number of non-missing observations in the input time series.
• $\hat{\rho_{k}}$ is the sample partial-autocorrelation function for lag $k$.
• $P(Z \geq Z_\frac{\alpha}{2}) = \frac{\alpha}{2}$.
• $Z\sim N(0,1)$.
4. PACF is the autocorrelation between $z_t$ and $z_{t-k}$ that is not accounted for by lags 1 to k-1, inclusive.
5. 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 multiple regression (i.e., AR(j)).