A More Flexible Ljung-Box Test in SAS

This article is originally published at https://statcompute.wordpress.com

Ljung-Box test is an important diagnostic to check if residuals from the time series model are independently distributed. In SAS / ETS module, it is easy to perform Ljung-Box with ARIMA procedure. However, test outputs are only provided for Lag 6, 12, 18, and so on, which cannot be changed by any option.

data one;
  do i = 1 to 100;
    x = uniform(1);
	output;
  end;
run;

proc arima data = one;
  identify var = x whitenoise = ignoremiss;
run;
quit;
/*
                            Autocorrelation Check for White Noise

 To        Chi-             Pr >
Lag      Square     DF     ChiSq    --------------------Autocorrelations--------------------
  6        5.49      6    0.4832     0.051    -0.132     0.076    -0.024    -0.146     0.064
 12        6.78     12    0.8719     0.050     0.076    -0.046    -0.025    -0.016    -0.018
 18       10.43     18    0.9169     0.104    -0.053     0.063     0.038    -0.085    -0.065
 24       21.51     24    0.6083     0.007     0.178     0.113    -0.046     0.180     0.079
*/

The SAS macro below is a more flexible way to perform Ljung-Box test for any number of lags. As shown in the output, test results for Lag 6 and 12 are identical to the one directly from ARIMA procedure.

%macro LBtest(data = , var = , lags = 4);
***********************************************************;
* SAS MACRO PERFORMING LJUNG-BOX TEST FOR INDEPENDENCE    *;
* ======================================================= *;
* INPUT PAREMETERS:                                       *;
*  DATA : INPUT SAS DATA TABLE                            *;
*  VAR  : THE TIME SERIES TO TEST FOR INDEPENDENCE        *;
*  LAGS : THE NUMBER OF LAGS BEING TESTED                 *;
* ======================================================= *;
* AUTHOR: [email protected]                               *;
***********************************************************;

%local nlag; 

data _1 (keep = &var);
  set &data end = eof;
  if eof then do;
    call execute('%let nlag = '||put(_n_ - 1, 8.)||';');
  end;
run;

proc arima data = _last_;
  identify var = &var nlag = &nlag outcov = _2 noprint;
run;
quit;

%do i = 1 %to &lags;
  data _3;
    set _2;
	where lag > 0 and lag <= &i;
  run;

  proc sql noprint;
    create table
	  _4 as
	select
      sum(corr * corr / n) * (&nlag + 1) * (&nlag + 3) as _chisq,
	  1 - probchi(calculated _chisq, &i.)              as _p_chisq,
	  &i                                               as _df
	from
	  _last_;
  quit;

  %if &i = 1 %then %do;
  data _5;
    set _4;
  run;
  %end;
  %else %do;
  data _5;
    set _5 _4;
  run;
  %end;
%end;

title;
proc report data = _5 spacing = 1 headline nowindows split = "*";
  column(" * LJUNG-BOX TEST FOR WHITE NOISE *
           * H0: RESIDUALS ARE INDEPENDENTLY DISTRIBUTED UPTO LAG &lags * "
          _chisq _df _p_chisq);
  define _chisq   / "CHI-SQUARE" width = 20 format = 15.10;
  define _df      / "DF"         width = 10 order;
  define _p_chisq / "P-VALUE"    width = 20 format = 15.10;
run;

%mend LBtest;

%LBtest(data = one, var = x, lags = 12);

/*
             LJUNG-BOX TEST FOR WHITE NOISE
 H0: RESIDUALS ARE INDEPENDENTLY DISTRIBUTED UPTO LAG 12

           CHI-SQUARE         DF              P-VALUE
 ------------------------------------------------------
         0.2644425904          1         0.6070843322
         2.0812769288          2         0.3532290858
         2.6839655476          3         0.4429590625
         2.7428168168          4         0.6017432831
         5.0425834917          5         0.4107053939
         5.4851972398          6         0.4832476224
         5.7586229652          7         0.5681994829
         6.4067856029          8         0.6017645131
         6.6410385135          9         0.6744356312
         6.7142471241         10         0.7521182318
         6.7427585395         11         0.8195164211
         6.7783018413         12         0.8719097622
*/

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