parcorr

Sample partial autocorrelation

Syntax

[pacf,lags] = parcorr(y)

PACFTbl = parcorr(Tbl)

[___,bounds] = parcorr(___)

[___] = parcorr(___,Name=Value)

parcorr(___)

parcorr(ax,___)

[___,h] = parcorr(___)

Description

[pacf,lags] = parcorr(y) returns the sample partial autocorrelation function (PACF) pacf and associated lags lags of the univariate time series y.

PACFTbl = parcorr(Tbl) returns the table PACFTbl containing variables for the sample PACF and associated lags of the last variable in the input table or timetable Tbl. To select a different variable in Tbl, for which to compute the PACF, use the DataVariable name-value argument.

[___,bounds] = parcorr(___) uses any input-argument combination in the previous syntaxes, and returns the output-argument combination for the corresponding input arguments and the approximate upper and lower confidence bounds bounds on the PACF.

[___] = parcorr(___,Name=Value) uses additional options specified by one or more name-value arguments. For example, parcorr(Tbl,DataVariable="RGDP",NumLags=10,NumSTD=1.96) returns 10 lags of the sample PACF of the table variable "RGDP" in Tbl and 95% confidence bounds.

parcorr(___) plots the sample PACF of the input series with confidence bounds.

parcorr(ax,___) plots on the axes specified by ax instead of the current axes (gca). ax can precede any of the input argument combinations in the previous syntaxes.

[___,h] = parcorr(___) plots the sample PACF of the input series and additionally returns handles to plotted graphics objects. Use elements of h to modify properties of the plot after you create it.

Examples

Compute PACF from Vector of Time Series Data

Compute the PACF of a univariate time series. Input the time series data as a numeric vector.

Load the quarterly real GDP series in Data_GDP.mat. Plot the series, which is stored in the numeric vector Data.

load Data_GDP
plot(Data)

The series exhibits exponential growth.

Compute the returns of the series.

ret = price2ret(Data);

ret is a series of real GDP returns; it has one less observation than the real GDP series.

Compute the PACF of the real GDP returns, and return the associated lags.

[pacf,lags] = parcorr(ret);
[pacf lags]

ans = 21×2

    1.0000         0
    0.3329    1.0000
    0.0828    2.0000
   -0.1205    3.0000
   -0.1080    4.0000
   -0.0869    5.0000
    0.0226    6.0000
   -0.0254    7.0000
   -0.0243    8.0000
    0.0699    9.0000
      ⋮

Let ${\mathit{y}}_{\mathit{t}}$ be the real GDP return at time t. pacf(3) = 0.0828 means that the correlation between ${\mathit{y}}_{\mathit{t}}$ and ${\mathit{y}}_{\mathit{t}-2}$, after adjusting for the linear effects of ${\mathit{y}}_{\mathit{t}-1}$ on ${\mathit{y}}_{\mathit{t}}$, is 0.0828.

Compute PACF of Table Variable

Compute the PACF of a time series, which is one variable in a table.

Load the electricity spot price data set Data_ElectricityPrices.mat, which contains the daily spot prices in the timetable DataTimeTable.

load Data_ElectricityPrices.mat
DataTimeTable.Properties.VariableNames

ans = 1x1 cell array
    {'SpotPrice'}

Plot the series.

plot(DataTimeTable.SpotPrice)

The time series plot does not clearly indicate an exponential trend or unit root.

Compute the PACF of the raw spot price series.

PACFTbl = parcorr(DataTimeTable)

PACFTbl=21×2 table
    Lags      PACF  
    ____    ________

      0            1
      1       0.5541
      2      0.10938
      3     0.099833
      4     0.029511
      5     0.038836
      6     0.065892
      7     0.029965
      8     0.034951
      9     0.050091
     10     0.051031
     11     0.033994
     12     0.051877
     13     0.028973
     14     0.047456
     15      0.11895
      ⋮

parcorr returns the results in the table PACFTbl, where variables correspond to the PACF (PACF) and associated lags (Lags).

By default, parcorr computes the PACF of the last variable in the table. To select a variable from an input table, set the DataVariable option.

Return PACF Confidence Bounds

Consider the electricity spot prices in Compute PACF of Table Variable.

Load the electricity spot price data set Data_ElectricityPrices.mat. Compute the PACF and return the PACF confidence bounds.

load Data_ElectricityPrices
[PACFTbl,bounds] = parcorr(DataTimeTable)

PACFTbl=21×2 table
    Lags      PACF  
    ____    ________

      0            1
      1       0.5541
      2      0.10938
      3     0.099833
      4     0.029511
      5     0.038836
      6     0.065892
      7     0.029965
      8     0.034951
      9     0.050091
     10     0.051031
     11     0.033994
     12     0.051877
     13     0.028973
     14     0.047456
     15      0.11895
      ⋮

bounds = 2×1

    0.0532
   -0.0532

Assuming the spot prices follow a Gaussian white noise series, an approximate 95.4% confidence interval on the PACF is (-0.0532, 0.0532).

Compare OLS and Yule-Walker PACFs

Load the US quarterly macroeconomic series data Data_USEconModel.mat. Remove all missing values from the timetable of data DataTimeTable by using listwise deletion.

load Data_USEconModel
DataTimeTable = rmmissing(DataTimeTable);

Compute the PACF of the raw effective federal funds rate FEDFUNDS by using the OLS method (the default method when the data does not contain any missing values). Change the name of the PACF variable of the output table to ols.

PACFTbl = parcorr(DataTimeTable,DataVariable="FEDFUNDS",Method="ols");
PACFTbl = renamevars(PACFTbl,"PACF","ols");

Compute the PACF of the raw effective federal funds rate FEDFUNDS by solving the Yule-Walker equations. Store the result as the variable yw in PACFTbl.

PACFTbl.yw = parcorr(DataTimeTable.FEDFUNDS,Method="yule-walker");

Compare the PACFs between the methods.

PACFTbl

PACFTbl=21×3 table
    Lags       ols          yw    
    ____    _________    _________

      0             1            1
      1       0.92881      0.91502
      2      0.074551     0.061337
      3       0.14949      0.15031
      4      -0.23745     -0.20808
      5      0.073346     0.073911
      6      -0.25854     -0.21128
      7      0.021783     0.018625
      8       0.10817     0.092159
      9       0.16147      0.10987
     10     -0.081344    -0.077452
     11      0.050378     0.034769
     12      0.075574     0.070003
     13      0.026074      0.02403
     14     -0.036989    -0.025924
     15      0.074656     0.048657
      ⋮

Plot the PACFs in the same stem plot.

stem(repmat(PACFTbl.Lags,1,2),PACFTbl{:,["ols" "yw"]},"filled")
title("PACF of Effective Federal Funds Rate")
legend(["OLS" "Yule-Walker"])

Plot PACF of Simulated Time Series

Specify the AR(2) model:

$$y_t=0.6y_{t-1}-0.5y_{t-2}+{\epsilon }_t,$$

where $${\epsilon }_t$$ is Gaussian with mean 0 and variance 1.

rng(1); % For reproducibility
Mdl = arima(AR={0.6 -0.5},Constant=0,Variance=1)

Mdl = 
  arima with properties:

     Description: "ARIMA(2,0,0) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 2
               D: 0
               Q: 0
        Constant: 0
              AR: {0.6 -0.5} at lags [1 2]
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: 1

Simulate 1000 observations from Mdl.

y = simulate(Mdl,1000);

Plot the PACF. Specify that the series is an AR(2) process.

parcorr(y,NumAR=2)

The PACF cuts off after the second lag. This behavior indicates an AR(2) process.

Specify Additional Lags for PACF Plot

Specify the multiplicative seasonal ARMA $$(2,0,1)\times (3,0,0{)}_{12}$$ model:

$$(1-0.75L-0.15L^2)(1-0.9L^{12}+0.75L^{24}-0.5L^{36})y_t=2+{\epsilon }_t-0.5{\epsilon }_{t-1},$$

where $${\epsilon }_t$$ is Gaussian with mean 0 and variance 1.

Mdl = arima(AR={0.75 0.15},SAR={0.9 -0.75 0.5}, ...
    SARLags=[12 24 36],MA=-0.5,Constant=2, ...
    Variance=1);

Simulate data from Mdl.

rng(1);
y = simulate(Mdl,1000); 

Plot the default partial autocorrelation function (PACF).

figure
parcorr(y)

The default correlogram does not display the dependence structure for higher lags.

Plot the PACF for 40 lags.

figure
parcorr(y,NumLags=40)

The correlogram shows the larger correlations at lags 12, 24, and 36.

Input Arguments

y — Observed univariate time series

numeric vector

Observed univariate time series for which parcorr computes or plots the PACF, specified as a numeric vector.

Data Types: double

Tbl — Time series data

table | timetable

Time series data, specified as a table or timetable. Each row of Tbl contains contemporaneous observations of all variables.

Specify a single series (variable) by using the DataVariable argument. The selected variable must be numeric.

ax — Axes on which to plot

Axes object

Axes on which to plot, specified as an Axes object.

By default, parcorr plots to the current axes (gca).

Note

Specify missing observations using NaN. The parcorr function treats missing values as missing completely at random.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: parcorr(Tbl,DataVariable="RGDP",NumLags=10,NumSTD=3) plots 10 lags of the sample PACF of the variable "RGDP" in Tbl, and displays confidence bounds consisting of 3 standard errors away from 0.

NumLags — Number of lags

positive integer

Number of lags in the sample PACF, specified as a positive integer. parcorr uses lags 0:NumLags to estimate the PACF.

The default is min([20,T – 1]), where T is the effective sample size of the input time series.

Example: parcorr(y,Numlags=10) plots the sample PACF of y for lags 0 through 10.

Data Types: double

NumAR — Number of lags in theoretical AR model

0 (default) | nonnegative integer

Number of lags in a theoretical AR model of the input time series, specified as a nonnegative integer less than NumLags.

parcorr uses NumAR to estimate confidence bounds. For lags > NumAR, parcorr assumes that the input times series is a Gaussian white noise process. Consequently, the standard error is approximately $$1/\sqrt{T},$$ where T is the effective sample size of the input time series.

Example: parcorr(y,NumAR=10) specifies that y is an AR(10) process and plots confidence bounds for all lags greater than 10.

Data Types: double

NumSTD — Number of standard errors in confidence bounds

2 (default) | nonnegative scalar

Number of standard errors in the confidence bounds, specified as a nonnegative scalar. For all lags greater than NumAR, the confidence bounds are 0 ± NumSTD*$$\widehat{\sigma }$$, where $$\widehat{\sigma }$$ is the estimated standard error of the sample partial autocorrelation.

The default yields approximate 95% confidence bounds.

Example: parcorr(y,NumSTD=1.5) plots the PACF of y with confidence bounds 1.5 standard errors away from 0.

Data Types: double

Method — PACF estimation method

"ols" | "yule-walker" | character vector

PACF estimation method, specified as a value in this table.

Value	Description	Restrictions
"ols"	Ordinary least squares (OLS)	The input times series must be fully observed (it cannot contain any NaN values).
"yule-walker"	Yule-Walker equations	None.

If the input time series is fully observed, the default is "ols". Otherwise, the default is "yule-walker".

Example: parcorr(y,Method="yule-walker") computes the PACF of y using the Yule-Walker equations.

Data Types: char | string

DataVariable — Variable in Tbl

last variable (default) | string scalar | character vector | integer | logical vector

Variable in Tbl for which parcorr computes the PACF, specified as a string scalar or character vector containing a variable name in Tbl.Properties.VariableNames, or an integer or logical vector representing the index of a name. The selected variable must be numeric.

Example: DataVariable="GDP"

Example: DataVariable=[false true false false] or DataVariable=2 selects the second table variable.

Data Types: double | logical | char | string

Output Arguments

pacf — Sample PACF

numeric vector

Sample PACF, returned as a numeric vector of length NumLags + 1. parcorr returns pacf only when you supply the input y.

The elements of pacf correspond to lags 0, 1, 2, ..., NumLags (that is, elements of lags). For all time series, the lag 0 partial autocorrelation pacf(1) = 1.

lags — PACF lags

numeric vector

PACF lags, returned as a numeric vector with elements 0:NumLags. parcorr returns lags only when you supply the input y.

PACFTbl — Sample PACF

table

Sample PACF, returned as a table with variables for the outputs pacf and lags. parcorr returns PACFTbl only when you supply the input Tbl.

bounds — Approximate upper and lower confidence bounds

numeric vector

Approximate upper and lower confidence bounds assuming the input series is an AR(NumAR) process, returned as a two-element numeric vector. The NumSTD option specifies the number of standard errors from 0 in the confidence bounds.

h — Handles to plotted graphics objects

graphics array

Handles to plotted graphics objects, returned as a graphics array. h contains unique plot identifiers, which you can use to query or modify properties of the plot.

More About

Partial Autocorrelation Function

The partial autocorrelation function measures the correlation between y_t and y_{t + k} after adjusting for the linear effects of y_{t + 1},...,y_{t + k – 1}.

The estimation of the PACF involves solving the Yule-Walker equations with respect to the autocorrelations. However, if the time series is fully observed, then the PACF can be estimated by fitting successive autoregressive models of orders 1, 2, ... using ordinary least squares. For details, see [1], Chapter 3.

Missing Completely at Random

Observations of a random variable are missing completely at random if the tendency of an observation to be missing is independent of both the random variable and the tendency of all other observations to be missing.

Tips

• To plot the PACF without confidence bounds, set NumSTD=0.

Algorithms

parcorr plots the PACF when you do not request any output or when you request the fourth output h.

References

[1] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.

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

Version History

Introduced before R2006a