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 BoxJenkins 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 partialautocorrelation function for lag k.
 $T$ is the number of nonmissing observations in the input time series.
 $\hat{\rho_{k}}$ is the sample partialautocorrelation 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 $z_{tk}$ that is not accounted for by lags 1 to k1, inclusive.
 Equivalently, PACF(k) is the ordinary least square (OLS) multipleregression kth coefficient ($\phi_k$).
$$\left[y_{t}\right]=\phi_{0}+\sum_{j=1}^{k}\phi_{j}\left[y_{tj}\right]$$
Where:
 $\left[y_{t}\right]$ is the input time series.
 $K$ is the lag order.
 $\phi_j$ is the jth coefficient of the multiple regression (i.e. AR(j)).
Examples
Example 1:


Formula  Description (Result) 

=PACF($B$2:$B$30,1,1)  Partial autocorrelation of order 1 (0.236) 
=PACFCI($B$2:$B$30,1,1,5%,1)  Upper confidence interval for PACF of order 1 (0.364) 
=PACFCI($B$2:$B$30,1,1,5%,0)  Lower confidence interval for PACF of order 1 (0.364) 
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: 0471690740
 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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