PACFCI - Partial Autocorrelation Confidence Interval

Returns the confidence interval limits (upper/lower) for the partial autocorrelation function (PACF).


PACFCI(X, Order, K, Alpha, UL)
is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
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).
is the lag order (e.g., k = 0 (no lag), k = 1 (1st lag), etc.). If missing, the default of k = 1 is assumed.
is the statistical significance level (i.e., alpha). If missing, the default value of 5% is assumed.
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:
    $$\hat \rho_k - Z_{\alpha/2}\times \frac{1}{\sqrt{T}} \leq \rho_{k} \leq \hat{\rho_k}+ Z_{\alpha/2}\times \frac{1}{\sqrt{T}} $$
    • $\rho_k$ 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 ($\phi_k$).
    • $\left[y_{t}\right]$ 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)).

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