PACFCI - Partial Autocorrelation Confidence Interval

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)).

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.

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)).

Example 1:

Files Examples

Related Links

• Wikipedia - Partial-autocorrelation function (PACF)
• Wikipedia - Box-Jenkins methodology

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