SAS Macro for Engle-Granger Co-integration Test

In the coursework of time series analysis, we’ve been taught that a time series regression of Y on X could be valid only when both X and Y are stationary due to the so-call “spurious regression problem”. However, one exception holds that if X and Y, albeit non-stationary, share a common trend such that their trends can be cancelled each other out, then X and Y are co-integrated and the regression of Y on X is valid. As a result, it is important to test the co-integration between X and Y.

Following the definition of co-integration, it is straightforward to formulate a procedure of the co-integration test. First of all, construct a linear combination between Y and X such that e = Y – (a + b * X). Secondly, test if e is stationary with ADF test. If e is stationary, then X and Y are co-integrated. This two-stage procedure is also called Engle-Granger co-integration test.

Below is a SAS macro implementing Engle-Granger co-integration test to show the long-term relationship between GDP and other macro-economic variables, e.g. Personal Consumption and Personal Disposable Income.

SAS Macro

%macro eg_coint(data = , y = , xs = );
*********************************************************************;
* THIS SAS MACRO IMPLEMENTATION ENGLE-GRANGER COINTEGRATION TEST IN *;
* A BATCH MODE TO PROCESS MANY TIME SERIES                          *;
*********************************************************************;
* INPUT PARAMETERS:                                                 *;
*   DATA: A INPUT SAS DATASET                                       *;
*   Y   : A DEPENDENT VARIABLE IN THE COINTEGRATION REGRESSION      *;
*   X   : A LIST OF INDEPENDENT VARIABLE IN THE COINTEGRATION       *;
*         REGRESSION                                                *;
*********************************************************************;
* AUTHOR: WENSUI.LIU@53.COM                                         *;
*********************************************************************;

options nocenter nonumber nodate mprint mlogic symbolgen
        orientation = landscape ls = 150 formchar = "|----|+|---+=|-/\<>*";

%local sig loop;

%let sig = 0.1;

%let loop = 1;

%do %while (%scan(&xs, &loop) ne %str());

  %let x = %scan(&xs, &loop);

  ods listing close;
  ods output FitStatistics = _fit;
  proc reg data = &data;
    model &y = &x;
    output out = _1 residual = r;
  run;
  quit;

  proc sql noprint;
    select cvalue2 into :r2 from _fit where upcase(label2) = "R-SQUARE";
  quit;

  proc arima data = _1;
    ods output stationaritytests = _adf1 (where = (upcase(type) = "ZERO MEAN" and lags = 1) drop = rho probrho fvalue probf);
    identify var = r stationarity = (adf = 1);
  run;
  quit;
  ods listing;

  %if &loop = 1 %then %do;
    data _adf;
      format vars $32. lterm_r2 best12. flg_coint $3.;
      set _adf1 (drop = type lags);
      vars = upcase("&x");
      lterm_r2 = &r2;
      if probtau < &sig then flg_coint = "YES";
      else flg_coint = "NO";
    run;
  %end;
  %else %do;
    data _adf;
      set _adf _adf1 (in = new drop = type lags);
      if new then do;
        vars = upcase("&x");
        lterm_r2 = &r2;
        if probtau < &sig then flg_coint = "YES";
          else flg_coint = "NO";
      end;
    run;
  %end;
    
  %let loop = %eval(&loop + 1);
%end;

proc sort data = _last_;
  by descending flg_coint probtau;
run;

proc report data = _last_ box spacing = 1 split = "/" nowd;
  COLUMN("ENGLE-GRANGER COINTEGRATION TEST BETWEEN %UPCASE(&Y) AND EACH VARIABLE BELOW/ "
         vars lterm_r2 flg_coint tau probtau);
  define vars      / "VARIABLES"                      width = 35;
  define lterm_r2  / "LONG-RUN/R-SQUARED"             width = 15 format =  9.4 center;
  define flg_coint / "COINTEGRATION/FLAG"             width = 15 center;
  define tau       / "TAU STATISTIC/FOR ADF TEST"     width = 20 format = 15.4;
  define probtau   / "P-VALUE FOR/ADF TEST"           width = 15 format =  9.4 center;
run;

%mend eg_coint;

%eg_coint(data = sashelp.citiqtr, y = gdp, xs = gyd gc);

SAS Output

----------------------------------------------------------------------------------------------------------
|                  ENGLE-GRANGER COINTEGRATION TEST BETWEEN GDP AND EACH VARIABLE BELOW                  |
|                                                                                                        |
|                                       LONG-RUN      COINTEGRATION         TAU STATISTIC   P-VALUE FOR  |
|VARIABLES                              R-SQUARED         FLAG               FOR ADF TEST    ADF TEST    |
|--------------------------------------------------------------------------------------------------------|
|GC                                 |      0.9985   |      YES      |             -2.8651|      0.0051   |
|-----------------------------------+---------------+---------------+--------------------+---------------|
|GYD                                |      0.9976   |      YES      |             -1.7793|      0.0715   |
----------------------------------------------------------------------------------------------------------

From the output, it is interesting to see that GDP in U.S. is driven more by Personal Consumption than by Personal Disposable Income.