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)).
Examples
Example 1:
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| 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: 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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