/*  This is a combination forecasting exercise.  */

/*  The data used here is taken from J.A. Brandt and D.A. Bessler 
    "Price Forecasting and Evaluation: An Application in Agriculture,"
     Journal of Forecasting (July - Sept. 1983), pp. 237-248.  */

/*  The data consists of actual hog prices and forecasts from (1) an
    econometric model (ECON) and (2) an ARIMA model (ARIMA).  */

/*  The purpose here is to use these forecasts to build combination
    forecasts based on (1) the Nelson regression method (NELSON)
    and (2) the Granger-Ramanathan regression method (GR).  */ 

/*  Here is the in-sample data for building the combination 
    forecasts.  */

Data Hog;
   Input Period Actual Econ Arima;
   datalines;
   7601  47.99  50.66  48.80
   7602  49.19  46.37  49.06
   7603  43.88  44.00  46.73
   7604  34.25  39.96  39.77
   7701  39.08  42.38  35.33
   7702  40.87  41.65  39.42
   7703  43.85  41.24  40.82
   7704  41.38  49.81  45.08
   7801  47.44  47.18  43.76
   7802  47.84  48.67  45.83
   7803  48.52  44.91  47.22
   7804  50.05  52.39  47.21
   7901  51.98  52.15  51.64
   7902  43.04  52.63  50.39
   7903  38.52  42.41  42.17
   7904  36.39  41.30  37.96
   ;

   /*  Let's print out the data to be sure we entered it correctly.  */

    proc print data = hog;

      run;

 
   /*  Now let's form the regression variables for calculating
       the various combination weights.  */

   data work;
     set hog;
       dep = actual - arima;
       explan = econ - arima;

/*  Now to calculate the Nelson combination weights.  This method
    assumes that the two individual forecasts making up the
    combination forecasts are unbiased.  */

    Proc reg data = work;
      model dep = explan / noint;

    run;

/*  As one can see from this output the optimal combination weight
    to be applied to the econometric forecast is 0.15815 and, 
    therefore, the optimal combination weight for the ARIMA model is
    1 - 0.15815 = 0.84185.  */

/*  Now to calculate the Granger-Ramanathan combination weights.
    This method doesn't assume that both forecasts (ECON and ARIMA)
    are unbiased.  One or more of the forecasts can be biased.  */
   
    Proc reg data = work;
      model actual = econ arima;

    run;

/*  As can be seen from this output, the Granger-Ramanathan forecast
    is of the form: (GR Combo) = 2.88917 + 0.25473*(ECON) +
    0.66104*(ARIMA).  */ 

/*  Now let's see how good these combination forecasts methods are
    in an out-of-sample forecasting experiment as compared to the
    Econometric and ARIMA forecasts by themselves.  */


    Data Out;
    input period actual econ arima;
    datalines;
    8001  36.74  43.51  35.17
    8002  31.18  39.72  36.59
    8003  46.23  37.23  34.34
    8004  46.44  44.38  47.80
    8101  41.13  44.85  47.12
    8102  43.62  41.50  40.45
    8103  50.42  44.21  45.95
    8104  43.24  51.06  45.31
    ;

/*  Now let's compute the out-of-sample forecasts for the
    Nelson combination, the Granger-Ramanathan combination,
    and a naive combination (the simple average of ECON and
    ARIMA forecasts.)  */

data out;
  set out;
  Nelson = 0.15815* econ + 0.84185*arima;
  GR = 2.88917 + 0.25473*econ + 0.66104*arima;
  Ave = 0.5*econ + 0.5*arima;

/*  Let's print out the out-of-sample data set and the
    various forecasts.  */

  proc print data = out;

  run;

/*  Now let's compute the mean square errors and mean absolute errors
    of the various methods in the out-of-sample data.  */

  data compute;
    set out;
    eecon = actual - econ;
    earima = actual - arima;
    enelson = actual - nelson;
    egr = actual - gr;
    eave = actual - ave; 
    aecon = abs(eecon);
    aarima = abs(earima);
    anelson = abs(enelson);
    agr = abs(egr);
    aave = abs(eave);
    secon = eecon**2;
    sarima = earima**2;
    snelson = enelson**2;
    sgr = egr**2;
    save = eave**2;

/*  Here we compute the mean absolute errors of the competing methods.  */

proc univariate data = compute noprint;                                                          
       var aecon aarima anelson agr aave;
       output mean = maeecon maearima maenelson maegr maeave out=results1;

       run;

/*  Here we compute the mean square errors of the competing methods.  */

proc univariate data = compute noprint;                                                          
       var secon sarima snelson sgr save;
       output mean = mseecon msearima msenelson msegr mseave out=results2;

       run; 

/*  Now print out the results of the out-of-sample forecasting
    experiment.  */
 
  proc print data = results1;
     var maeecon maearima maenelson maegr maeave;
     title1 ' '; 
     title2 'The Mean Absolute Errors of the Competing Methods';                                                           
  
     run;

  proc print data = results2;
     var mseecon msearima msenelson msegr mseave;
     title1 ' ';
     title2 'The Mean Square Errors of the Competing Methods';

    run;