LJUNG-BOX TEST

LJUNG-BOX TEST

Analysis Command

Perform a Ljung-Box test for randomness.

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: where n is the sample size, r(j) is the autocorrelation at lag j, and h is the number of lags being tested.
Significance Level:
Critical Region:	The hypothesis of randomness is rejected if QLB > where 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.

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.

LJUNG-BOX TEST RES
LJUNG-BOX TEST Y1 SUBSET TAG > 1

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.

None

LJUNG TEST and LJUNG-BOX are synonyms for LJUNG-BOX TEST.

AUTOCORRELATION PLOT	= Generate an autocorrelation plot.
RUNS TEST	= Perform a runs test for randomness.
ARMA	= Perform an ARIMA fit.

"On a Measure of a Lack of Fit in Time Series Models", G. M. Ljung and G. E. P. Box, Biometrika, 1978, 65, pp. 297-303.

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

ARIMA Modeling

2003/2

 
READ NEGIZ4.DAT X1 X2 Y
ARMA Y 2 1 0
LET NUMLAG = 25
LJUNG-BOX TEST RES
    

 
      **************************
      **  ljung-box test res  **
      **************************


              LJUNG-BOX TEST FOR RANDOMNESS

1. STATISTICS:
      NUMBER OF OBSERVATIONS      =      559
      LAG TESTED                  =       25
      LAG 1 AUTOCORRELATION       =  -0.1012441E-02
      LAG 2 AUTOCORRELATION       =   0.6160716E-02
      LAG 3 AUTOCORRELATION       =   0.5182213E-02

   LJUNG-BOX TEST STATISTIC       =    31.93575

2. PERCENT POINTS OF THE REFERENCE CHI-SQUARE DISTRIBUTION
   (REJECT HYPOTHESIS OF RANDOMNESS IF TEST STATISTIC VALUE
   IS GREATER THAN PERCENT POINT VALUE)
   FOR LJUNG-BOX TEST STATISTIC
      0          % POINT    =          0.
      50         % POINT    =    24.33659
      75         % POINT    =    29.33885
      90         % POINT    =    34.38158
      95         % POINT    =    37.65248
      99         % POINT    =    44.31411


3. CONCLUSION (AT THE 5% LEVEL):
      THE DATA ARE RANDOM.