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

- The time series is homogeneous or equally spaced.
- The sample ACF and PACF plots (i.e., correlograms) are tools commonly used for model identification in Box-Jenkins models.
- 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 $k$.
- $P(Z \geq Z_\frac{\alpha}{2}) = \frac{\alpha}{2}$.
- $Z\sim N(0,1)$.

- PACF is the autocorrelation between $z_t$ and $z_{t-k}$ that is not accounted for by lags 1 to k-1, inclusive.
- 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)).

## Files Examples

## Related Links

## References

- D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906.
- James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
- Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.
- Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848.
- Walter Enders; Applied Econometric Time Series; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568.

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