# 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)).

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
A B
Date Data
January 10, 2008 -0.30
January 11, 2008 -1.28
January 12, 2008 0.24
January 13, 2008 1.28
January 14, 2008 1.20
January 15, 2008 1.73
January 16, 2008 -2.18
January 17, 2008 -0.23
January 18, 2008 1.10
January 19, 2008 -1.09
January 20, 2008 -0.69
January 21, 2008 -1.69
January 22, 2008 -1.85
January 23, 2008 -0.98
January 24, 2008 -0.77
January 25, 2008 -0.30
January 26, 2008 -1.28
January 27, 2008 0.24
January 28, 2008 1.28
January 29, 2008 1.20
January 30, 2008 1.73
January 31, 2008 -2.18
February 1, 2008 -0.23
February 2, 2008 1.10
February 3, 2008 -1.09
February 4, 2008 -0.69
February 5, 2008 -1.69
February 6, 2008 -1.85
February 7, 2008 -0.98

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)