/* This program demonstrations the Frisch-Waugh theorem using the famous
   M-series data set of Box and Jenkins (1970).  */

OPTIONS PAGESIZE =60 LINESIZE=72 NODATE;
data lead;
  input obs x y;
/* obs lead sales */
cards;
  1 10.01 200.1
  2 10.07 199.5
  3 10.32 199.4
  4  9.75 198.9
  5 10.33 199.0
  6 10.13 200.2
  7 10.36 198.6
  8 10.32 200.0
  9 10.13 200.3
 10 10.16 201.2
 11 10.58 201.6
 12 10.62 201.5
 13 10.86 201.5
 14 11.20 203.5
 15 10.74 204.9
 16 10.56 207.1
 17 10.48 210.5
 18 10.77 210.5
 19 11.33 209.8
 20 10.96 208.8
 21 11.16 209.5
 22 11.70 213.2
 23 11.39 213.7
 24 11.42 215.1
 25 11.94 218.7
 26 11.24 219.8
 27 11.59 220.5
 28 10.96 223.8
 29 11.40 222.8
 30 11.02 223.8
 31 11.01 221.7
 32 11.23 222.3
 33 11.33 220.8
 34 10.83 219.4
 35 10.84 220.1
 36 11.14 220.6
 37 10.38 218.9
 38 10.90 217.8
 39 11.05 217.7
 40 11.11 215.0
 41 11.01 215.3
 42 11.22 215.9
 43 11.21 216.7
 44 11.91 216.7
 45 11.69 217.7
 46 10.93 218.7
 47 10.99 222.9
 48 11.01 224.9
 49 10.84 222.2
 50 10.76 220.7
 51 10.77 220.0
 52 10.88 218.7
 53 10.49 217.0
 54 10.50 215.9
 55 11.00 215.8
 56 10.98 214.1
 57 10.61 212.3
 58 10.48 213.9
 59 10.53 214.6
 60 11.07 213.6
 61 10.61 212.1
 62 10.86 211.4
 63 10.34 213.1
 64 10.78 212.9
 65 10.80 213.3
 66 10.33 211.5
 67 10.44 212.3
 68 10.50 213.0
 69 10.75 211.0
 70 10.40 210.7
 71 10.40 210.1
 72 10.34 211.4
 73 10.55 210.0
 74 10.46 209.7
 75 10.82 208.8
 76 10.91 208.8
 77 10.87 208.8
 78 10.67 210.6
 79 11.11 211.9
 80 10.88 212.8
 81 11.28 212.5
 82 11.27 214.8
 83 11.44 215.3
 84 11.52 217.5
 85 12.10 218.8
 86 11.83 220.7
 87 12.62 222.2
 88 12.41 226.7
 89 12.43 228.4
 90 12.73 233.2
 91 13.01 235.7
 92 12.74 237.1
 93 12.73 240.6
 94 12.76 243.8
 95 12.92 245.3
 96 12.64 246.0
 97 12.79 246.3
 98 13.05 247.7
 99 12.69 247.6
100 13.01 247.8
101 12.90 249.4
102 13.12 249.0
103 12.47 249.9
104 12.47 250.5
105 12.94 251.5
106 13.10 249.0
107 12.91 247.6
108 13.39 248.8
109 13.13 250.4
110 13.34 250.7
111 13.34 253.0
112 13.14 253.7
113 13.49 255.0
114 13.87 256.2
115 13.39 256.0
116 13.59 257.4
117 13.27 260.4
118 13.70 260.0
119 13.20 261.3
120 13.32 260.4
121 13.15 261.6
122 13.30 260.8
123 12.94 259.8
124 13.29 259.0
125 13.26 258.9
126 13.08 257.4
127 13.24 257.7
128 13.31 257.9
129 13.52 257.4
130 13.02 257.3
131 13.25 257.6
132 13.12 258.9
133 13.26 257.8
134 13.11 257.7
135 13.30 257.2
136 13.06 257.5
137 13.32 256.8
138 13.10 257.5
139 13.27 257.0
140 13.64 257.6
141 13.58 257.3
142 13.87 257.5
143 13.53 259.6
144 13.41 261.1
145 13.25 262.9
146 13.50 263.3
147 13.58 262.8
148 13.51 261.8
149 13.77 262.2
150 13.40 262.7
;
 
/*  Create the time trend variable.  */

data lead;
  set lead;
  time = _n_;

run;

/*  Run the "old-fashion" time series regression with trend
    variable (t) included.  */

proc reg data = lead;
   model y = x time;
   title 'Time Series Regression'; 
run;

/*  Now get the coefficient on x in the above equation by using
    the Frisch-Waugh theorem result.  */

proc reg data = lead;
  model y = time;
   output out=result1 r=residyt;
   title 'Regression Producing Residyt';
run;

proc reg data = lead;
   model x = time;
     output out=result2 r=residxt;
	 title 'Regression Producing Residxt';
run;

data lead;
  merge lead result1 result2;

run;

proc reg data = lead;
   model residyt = residxt/noint;
   title 'Final Frisch-Waugh Theorem Regression';
run; 

/*  Here is the Spurious Regression you should NEVER run.  */

proc reg data = lead;
    model y = x;
    title 'Spurious Regression - NEVER EVER Run this!';

run;