parcorr - Partial autocorrelation

Syntax

parcorr(Series,nLags,R,nSTDs) [PartialACF,Lags,Bounds] = ... parcorr(Series,nLags,R,nSTDs)

Description

parcorr(Series,nLags,R,nSTDs) computes and plots the sample partial autocorrelation function (partial ACF) of a univariate, stochastic time series. parcorr computes the partial ACF by fitting successive autoregressive models of orders 1, 2, ... by ordinary least squares, retaining the last coefficient of each regression. To plot the partial ACF sequence without the confidence bounds, set nSTDs = 0.
[PartialACF,Lags,Bounds] = ... parcorr(Series,nLags,R,nSTDs) computes and returns the partial ACF sequence.

Input Arguments

`Series`	Vector of observations of a univariate time series for which `parcorr` returns or plots the sample partial autocorrelation function (partial ACF). The last element of `Series` contains the most recent observation of the stochastic sequence.
`nLags`	Positive scalar integer indicating the number of lags of the partial ACF to compute. If `nLags = []` or is unspecified, `parcorr` computes the partial ACF sequence at lags 0, 1, 2, ..., T, where T `=` `min([20,length(Series)-1])`.
`R`	Nonnegative integer scalar indicating the number of lags beyond which `parcorr` assumes the theoretical partial ACF is zero. Assuming that `Series` is an AR(R) process, the estimated partial ACF coefficients at lags greater than `R` are approximately zero-mean, independently distributed Gaussian variates. In this case, the standard error of the estimated partial ACF coefficients of a fitted `Series` with N observations is approximately for lags greater than `R`. If `R = []` or is unspecified, the default is `0`. The value of `R` must be less than `nLags`.
`nSTDs`	Positive scalar indicating the number of standard deviations of the sample partial ACF estimation error to display, assuming that `Series` is an AR(R) process. If the `R`th regression coefficient (the last ordinary least squares (OLS) regression coefficient of `Series` regressed on a constant and `R` of its lags) includes N observations, specifying `nSTDs` results in confidence bounds at . If `nSTDs = []` or is unspecified, the default is `2` (approximate 95 percent confidence interval).

Output Arguments

`PartialACF`	Sample partial ACF of `Series`. `PartialACF` is a vector of length `nLags` + 1 corresponding to lags 0, 1, 2, ..., `nLags`. The first element of `PartialACF` is unity, that is, `PartialACF(1)` `= 1 =` OLS regression coefficient of `Series` regressed upon itself. `parcorr` includes this element as a reference.
`Lags`	Vector of lags, of length `nLags` + 1. The elements correspond to the elements of `PartialACF`.
`Bounds`	Two-element vector indicating the approximate upper and lower confidence bounds, assuming that `Series` is an AR(R) process. `Bounds` is approximate for lags greater than `R` only.

Examples

Example 1

Create a stationary AR(2) process from a sequence of 1000 Gaussian deviates:

strm = RandStream('mt19937ar'); % reproducible
RandStream.setDefaultStream(strm);
x = randn(1000, 1);
y = filter(1, [1 -0.6 0.08], x);
[PartialACF, Lags, Bounds] = parcorr(y, [], 2); 
[Lags, PartialACF]
ans =
         0    1.0000
    1.0000    0.5854
    2.0000   -0.0931
    3.0000   -0.0597
    4.0000   -0.0049
    5.0000    0.0346
    6.0000    0.0302
    7.0000   -0.0359
    8.0000    0.0190
    9.0000   -0.0347
   10.0000    0.0342
   11.0000    0.0344
   12.0000    0.0217
   13.0000   -0.0273
   14.0000    0.0252
   15.0000   -0.0231
   16.0000   -0.0158
   17.0000    0.0065
   18.0000   -0.0036
   19.0000    0.0379
   20.0000   -0.0142

Bounds
Bounds =
    0.0633
   -0.0633

Visually assess whether the partial ACF is zero for lags greater than 2:

parcorr(y, [], 2) % Use the same example, but plot
                  % the partial ACF sequence with
                  % confidence bounds.

Example 2

See Pre-Estimation Analysis.

References

[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 1994.

[2] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.