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

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.

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).

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.

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.

ARIMA Modeling

2003/2

 
READ NEGIZ4.DAT X1 X2 Y
ARMA Y 2 1 0
LET NUMLAG = 25
SET WRITE DECIMALS 4
LJUNG-BOX TEST RES
    

 
             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