SED navigation bar go to SED home page go to Dataplot home page go to NIST home page SED Home Page SED Staff SED Projects SED Products and Publications Search SED Pages
Dataplot Vol 1 Vol 2

LJUNG-BOX TEST

Name:
    LJUNG-BOX TEST
Type:
    Analysis Command
Purpose:
    Perform a Ljung-Box test for randomness.
Description:
    There are a large number of tests of randomness (e.g., the runs tests). Autocorrelation plots are one common method test for randomness. The Ljung-Box test is based on the autocorrelation plot. However, instead of testing randomness at each distinct lag, it tests the "overall" randomness based on a number of lags. For this reason, it is often referred to as a "portmanteau" test.

    More formally, the Ljung-Box test can be defined as follows.

    H0: The data are random.
    Ha: The data are not random.
    Test
    Statistic:
    The test statistic is:

      \( Q_{LB} = n(n+2) \sum_{j=1}^{h}{\frac{\rho^{2}(j)}{n-j}} \)

    where n is the sample size, \( \rho_{j} \) is the autocorrelation at lag j, and h is the number of lags being tested.

    Significance
    Level:
    \( \alpha \)
    Critical
    Region:
    The hypothesis of randomness is rejected if

      \( Q_{LB} = \chi_{1-\alpha;h}^{2} \)

    where \( \chi^{2} \) is the percent point function of the chi-square distribution.

    The Ljung-Box test is commonly used in ARIMA modeling. Note that it is applied to the residuals of a fitted ARIMA model, not the original series.

Syntax:
    LJUNG-BOX TEST <y>       <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable being tested;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LJUNG-BOX TEST RES
    LJUNG-BOX TEST Y1 SUBSET TAG > 1
Note:
    The number of lags to test can be set by entering any of the following command (before entering the LJUNG-BOX command):

      LET LAGS = <value>
      LET LAG = <value>
      LET NUMLAG = <value>

    By default, Dataplot will use the same number of lags as the autocorrelation plot. Typically, you will want to test fewer lags. Although the choice is somewhat arbitrary, 25 is a reasonable number for many series.

Note:
    The following statistic is also supported:

      LET A = LJUNG BOX TEST Y

    In addition to the above LET command, built-in statistics are supported for about 20+ different commands (enter HELP STATISTICS for details).

Default:
    None
Synonyms:
    LJUNG TEST and LJUNG-BOX are synonyms for LJUNG-BOX TEST.
Related Commands: Reference:
    G. M. Ljung and G. E. P. Box (1978), "On a Measure of a Lack of Fit in Time Series Models," Biometrika, Vol. 65, pp. 297-303.

    Peter Brockwell and Richard Davis 2002, "Introduction to Time Series and Forecasting," 2nd. Ed., Springer, p. 36.

Applications:
    ARIMA Modeling
Implementation Date:
    2003/2
Program:
     
    READ NEGIZ4.DAT X1 X2 Y
    ARMA Y 2 1 0
    LET NUMLAG = 25
    SET WRITE DECIMALS 4
    LJUNG-BOX TEST RES
        
Dataplot generates the following output for the Ljung-Box test:
     
                 Ljung-Box Test for Randomness
      
     Response Variable: RES
      
     H0: The Data Are Random
     Ha: The Data Are Not Random
      
     Summary Statistics:
     Number of Observations:                  559
     Lag Tested:                              24
     Lag 1 Autocorrelation:                   -0.0010
     Lag 2 Autocorrelation:                   0.0062
     Lag 3 Autocorrelation:                   0.0052
      
     Ljung-Box Test Statistic:                31.9107
     CDF Value:                               0.8708
     P-Value:                                 0.1292
      
      
      
                 Conclusions (Upper One-Tailed Test)
      
     ------------------------------------------------------------------------
                                                                         Null
             Null     Confidence           Test       Critical     Hypothesis
       Hypothesis          Level      Statistic      Value (>)     Conclusion
     ------------------------------------------------------------------------
           Random          0.0%         31.9107         0.0000         REJECT
           Random          50.0%        31.9107        23.3367         REJECT
           Random          75.0%        31.9107        28.2412         REJECT
           Random          90.0%        31.9107        33.1962         ACCEPT
           Random          95.0%        31.9107        36.4150         ACCEPT
           Random          97.5%        31.9107        39.3641         ACCEPT
           Random          99.0%        31.9107        42.9798         ACCEPT
           Random          99.9%        31.9107        51.1786         ACCEPT
        

Privacy Policy/Security Notice
Disclaimer | FOIA

NIST is an agency of the U.S. Commerce Department.

Date created: 3/10/2003
Last updated: 10/31/2016

Please email comments on this WWW page to alan.heckert@nist.gov.