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SAS/ETS Examples
Estimating an Almost Ideal Demand System Model
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SAS Program
/*-----------------------------------------------------------------
Example: Estimating an Almost Ideal Demand System Model
Requires: SAS/ETS
Version: 9.0
------------------------------------------------------------------*/
data aids_;
input year qtr pop beef_q pork_q chick_q turkey_q beef_p pork_p chick_p
turkey_p cpi pc_exp;
date = yyq(year,qtr);
format date yyq6.;
datalines;
1975 1 215.132 22.0991 11.8074 9.2631 1.0724 134.8333 120.8 58.9 72.13 52.43 1143.95149
1975 2 215.646 21.0304 11.0085 10.1645 1.3916 152.6333 129.8 58.97 70.83 53.23 1175.192619
1975 3 216.294 22.2717 9.6583 10.2047 1.8937 163.2 157.4 68.93 73.87 54.37 1210.389562
1975 4 216.851 22.7502 10.4302 9.7381 3.9068 158.1667 161.8 66.3 78.27 55.23 1240.483097
1976 1 217.315 23.8753 10.939 10.3294 1.145 148.7 149.4 61.93 78.1 55.77 1278.213653
1976 2 217.773 22.8964 10.3597 10.9598 1.5151 148.2 146.3 60.7 77 56.47 1298.48512
1976 3 218.337 24.4652 11.0404 11.031 2.0091 142.8 145.1 60.9 76.87 57.37 1329.141028
1976 4 218.92 23.1085 13.1403 10.1654 4.2247 142.9333 126.6 55.13 78.23 58.03 1365.909921
1977 1 219.424 22.9976 11.9593 10.3395 1.2198 142.1667 127.2 58.27 76.07 59.03 1403.219338
1977 2 219.953 22.6131 11.4488 11.2841 1.3872 143.9333 128.8 60.8 74.1 60.33 1432.468374
1977 3 220.573 23.4054 11.2324 11.1785 2.109 146.5 138.6 61.9 76.83 61.2 1464.254465
1977 4 221.206 22.7401 12.3959 10.4996 4.0197 150.8 135.7 59.3 80.42 61.87 1503.010768
1978 1 221.711 22.0441 11.6947 10.8966 1.2211 160 144.9 61.4 74.53 62.93 1533.30582
1978 2 222.278 21.7602 11.4791 11.8033 1.589 182.5333 150.7 66.93 79.6 64.53 1595.978
1978 3 222.929 21.6064 11.4611 11.6097 2.0336 186.2 153.1 70.5 84.37 66.07 1628.433268
1978 4 223.574 21.8814 12.3615 11.1396 3.88 186.4333 158.8 67.07 87.73 67.4 1666.789669
1979 1 224.144 20.5086 12.1608 11.4686 1.3533 211.7 165.1 69.6 90.33 69.07 1708.165287
1979 2 224.738 19.0408 13.1374 12.6605 1.7558 231.5 156.8 70.33 91.03 71.47 1743.033539
1979 3 225.421 19.1457 13.5239 12.582 2.1241 222.7 146 65.93 89.83 73.83 1796.749194
1979 4 226.135 19.3989 14.8575 11.6184 4.0155 223.8333 142.1 64.87 88.17 75.93 1842.372919
1980 1 226.753 18.947 14.5643 12.1178 1.7761 231.1667 141.7 67.57 92.77 78.93 1891.930629
1980 2 227.387 18.898 14.8637 12.6824 1.9508 227.5 132.5 63.53 90.2 81.83 1890.283724
1980 3 228.07 19.2127 13.5274 11.7869 2.5774 237.5333 152.6 75.07 93.63 83.33 1947.980769
1980 4 228.696 19.4988 14.3512 11.4689 3.9459 238.1667 163.2 77.27 98.17 85.53 2010.533657
1981 1 229.148 19.3429 14.0974 12.0277 1.6042 233.5 157.3 75.43 98.13 87.8 2065.37274
1981 2 229.668 18.9816 13.4164 12.6866 1.9006 230.7 153.1 71.9 98.47 89.83 2097.266489
1981 3 230.3 19.6459 13.0621 12.8079 2.4992 239.0333 166.6 74.9 101.83 92.37 2139.062395
1981 4 230.906 19.2868 14.0918 11.9454 4.5618 235.4333 167.9 70.47 92.97 93.7 2150.663367
1982 1 231.392 18.8193 12.7405 11.9425 1.7318 233.3 169.4 71.77 92.47 94.47 2185.689621
1982 2 231.901 18.7968 12.2607 12.834 2.0328 242.9667 179.2 72.13 91.3 95.9 2208.705889
1982 3 232.496 19.9882 11.5797 12.9951 2.5874 244.0333 195.7 72.33 95.47 97.7 2251.333897
1982 4 233.077 19.4242 12.5368 11.8858 4.2141 233.1333 198 69.2 92.9 97.93 2307.933232
1983 1 233.543 19.2252 12.114 12.4026 2.0296 233.8667 193.6 69.53 92.07 97.87 2342.609284
1983 2 234.02 19.3361 12.7472 13.129 2.1634 240.9333 181 68.73 92.57 99.1 2414.323562
1983 3 234.601 20.4952 12.8496 12.5883 2.4668 234.3667 175 74.43 92.7 100.27 2471.643343
1983 4 235.157 19.5995 14.0587 11.6903 4.3635 227.1667 169.1 77.17 91.33 101.17 2527.997312
1984 1 235.601 19.3191 12.8106 12.3375 1.9057 238.4667 170.9 85.13 94.33 102.3 2575.434739
1984 2 236.077 19.3744 12.6312 13.3105 2.155 238 168.7 82.4 97.2 103.4 2627.749903
1984 3 236.655 19.9738 12.3802 13.1955 2.5966 232.2 173.5 80.23 101.77 104.53 2659.045148
1984 4 237.238 19.7597 13.6804 12.6693 4.3788 233.2333 172.7 76.27 102.37 105.3 2706.148901
1985 1 237.667 19.0586 12.7537 12.7081 1.9896 234.9 175 77.13 107.23 105.97 2769.52627
1985 2 238.172 20.0757 12.855 13.7851 2.1634 230.4333 167.8 75.67 103.43 107.27 2815.297348
1985 3 238.789 20.8175 12.8035 13.5748 2.7299 222.7333 170.4 75.73 105.23 108.03 2878.907485
1985 4 239.392 19.2488 13.4707 13.0119 4.7012 226.4333 172.4 76.77 104.93 109 2909.036225
1986 1 239.858 19.0212 12.4462 13.0699 2.3152 229.2667 177.4 76.8 106.3 109.23 2944.554695
1986 2 240.365 20.1607 12.3281 14.1113 2.3758 222.9 173.2 77.2 103.3 109 2971.522476
1986 3 240.961 20.7108 11.5663 13.813 3.0149 225.6333 200.3 91.9 109 109.8 3038.458506
1986 4 241.544 18.9416 12.6462 13.2814 5.1658 229.3 204.1 88.1 107.7 110.4 3073.980463
1987 1 242.005 18.2746 12.2342 14.0183 2.5318 230.6 195.7 81.9 103.27 111.63 3110.892566
1987 2 242.516 18.5266 11.7023 14.5088 2.8807 239.1 194 78.17 102.77 113.1 3176.601083
1987 3 243.118 19.175 11.7285 14.5456 3.4995 242.1667 206.9 77.77 104.73 114.4 3234.547081
1987 4 243.724 17.8807 13.513 14.2982 5.8184 241.6667 200.7 76.1 93.97 115.37 3264.861072
1988 1 244.205 18.1678 12.8332 14.3959 3.035 241.7333 194.6 74.6 92.33 116.07 3337.155259
1988 2 244.708 18.4639 12.5507 14.7447 3.4076 250.1 195.5 80.8 91.73 117.53 3391.286758
1988 3 245.352 18.7622 12.9249 14.2711 3.7854 254.5333 196.7 95.77 98.7 119.1 3451.164042
1988 4 245.97 17.2552 14.1567 14.0575 5.4503 255 189.4 90.3 100.1 120.33 3516.790666
1989 1 246.454 16.9765 12.8213 14.3563 3.1189 260.7 190.5 90.57 97.27 121.67 3562.328061
1989 2 247.01 17.5192 13.0187 15.0116 3.2763 266.9667 188.9 95.83 99.9 123.67 3616.149144
1989 3 247.695 17.5975 12.6332 15.0852 4.1933 268.0333 194.6 95.33 103.57 124.67 3660.658594
1989 4 248.38 17.2416 13.5038 14.8018 6.0078 266.9333 199.8 89.07 96.8 125.87 3699.069973
1990 1 248.92 16.5516 12.5572 14.9451 3.5354 272.6333 207.6 90.2 98.87 128.03 3771.098689
1990 2 249.562 17.3763 11.8816 15.4012 3.6762 281.2 220.5 90.9 98.9 129.33 3812.887805
1990 3 250.301 17.4207 12.1079 15.5829 4.2115 280.3667 235.5 91.2 101.83 131.57 3866.944199
1990 4 251.04 16.4315 13.216 15.5973 6.1275 289.8667 236 87.37 97.57 133.7 3877.278277
1991 1 251.649 15.9842 12.2575 15.1466 3.6626 294.2667 227.6 89.6 99.47 134.8 3879.014024
1991 2 252.299 17.0449 12.0445 16.4942 3.9074 295.2 225.6 88.2 101.03 135.6 3922.536202
1991 3 253.041 17.5591 12.2515 16.5271 4.0353 284.6333 227 87.7 103.1 136.67 3950.158176
1991 4 253.756 16.2093 13.8041 15.8341 6.2823 279.2 216.4 86.63 95.67 137.7 3964.0442
1992 1 254.337 16.3952 13.1929 16.8084 3.4475 282.2667 210.4 86.2 95.37 138.67 4052.800129
1992 2 255.04 16.9538 12.697 17.1634 3.7342 286.8333 207.3 85.87 98.47 139.8 4089.06446
1992 3 255.832 17.227 13.2128 17.3147 4.2 282.6667 211.8 88.03 100.1 140.9 4129.378126
1992 4 256.561 15.9093 13.9768 16.4821 6.4864 286.6667 208.5 87.63 93.97 141.9 4207.876895
1993 1 257.157 15.8915 13.018 17.3378 3.5527 292.1333 205.9 87.5 99.07 143.1 4229.517377
1993 2 257.785 16.2058 12.6251 17.9871 3.7316 300.4 205.5 88.43 101.37 144.2 4287.786892
1993 3 258.511 17.043 12.8477 17.9543 3.9801 292 211.8 89.43 102.43 144.77 4340.434256
1993 4 259.177 15.9513 13.8657 16.9636 6.4533 289.2333 213.2 90.7 97.5 145.77 4397.292964
1994 1 259.65 16.2494 12.5018 17.516 3.5479 286.6667 212.5 89.27 98.5 146.7 4442.326208
1994 2 260.261 16.8868 12.9018 18.0614 3.7688 286.1667 210.4 90.63 98.83 147.63 4493.085787
1994 3 260.948 17.3775 13.1741 18.6015 4.414 279.5 210.5 90.87 102.73 148.93 4553.589221
1994 4 261.59 16.4723 14.4918 17.3363 6.1044 279.1667 204.8 89.6 100.07 149.63 4607.69641
1995 1 262.123 16.3097 13.1044 17.9111 3.5628 283.8667 202.7 90.23 99.77 150.87 4643.430756
1995 2 262.707 17.0895 12.9167 18.5493 3.9053 283.1 201.2 90.33 102.9 152.2 4704.584774
1995 3 263.396 17.647 12.6978 17.343 4.1648 285.1 206.9 92.2 106.53 152.87 4750.641619
1995 4 264.055 16.3476 13.7183 17.0398 6.173 285.2667 213.5 93.9 100.27 153.6 4789.162843
1996 1 264.535 17.0783 12.6109 18.2436 3.7235 278.7 218.3 93.8 105.03 155 4848.611429
1996 2 265.13 17.6763 11.5758 18.6502 3.9 277.4 227.4 95.47 103.27 156.53 4920.237092
1996 3 265.836 17.1606 12.0383 18.5709 4.6382 279.8 243.8 98.93 106.5 157.37 4950.138431
1996 4 266.515 16.2764 12.8608 17.486 6.206 285.0333 245.4 100.87 102.5 158.5 5007.138448
1997 1 267.032 16.2921 11.7752 17.559 3.4603 278.8 244.4 101.1 105.9 159.57 5084.418879
1997 2 267.668 17.2994 11.5617 18.7398 3.9826 278.9667 243.1 100.07 105.17 160.2 5105.49469
1997 3 268.402 17.0513 12.0187 18.4364 4.1989 281.0667 248.1 99.5 108.5 160.83 5187.275058
1997 4 269.085 16.2354 13.3464 17.635 5.9572 279.3 244.4 100.1 100.67 161.47 5231.896984
1998 1 269.583 16.6884 12.7173 17.464 3.9209 273.4667 245.6 102.03 101.03 161.9 5299.583615
1998 2 270.216 17.1985 12.3142 18.1367 3.9191 278.1 239.8 102.57 97.33 162.77 5381.075475
1998 3 270.949 17.5085 13.0799 18.5167 4.1742 277.3667 244.9 105.53 102.8 163.4 5434.243039
1998 4 271.633 16.6475 14.384 18.7589 6.0081 279.5333 240.4 107.33 97.1 163.97 5497.95128
1999 1 272.136 16.6785 13.5476 19.5127 3.84 278 235.8 106.43 98.47 164.6 5595.374363
1999 2 272.774 17.7635 13.055 19.9553 3.831 284.7667 238.4 104.13 97.2 166.2 5683.103381
1999 3 273.52 17.6689 13.218 19.2361 4.3288 289.2333 246.4 105.6 102.77 167.23 5761.655019
;
run;
proc means data=aids_;
run;
data aids;
set aids_;
if year < 1975 then delete;
t = _n_ ;
co1 = cos(1/2*3.14159*t);
co2 = cos(2/2*3.14159*t);
si1 = sin(1/2*3.14159*t);
si2 = sin(2/2*3.14159*t);
x = beef_p*beef_q + pork_p*pork_q + chick_p*chick_q
+ turkey_p*turkey_q;
w_beef = beef_p*beef_q/x;
w_pork = pork_p*pork_q/x;
w_chick = chick_p*chick_q/x;
w_turkey = turkey_p*turkey_q/x;
run;
data aids ;
set aids;
obs = _n_ ;
run;
proc gplot data=aids;
plot w_beef*date w_pork*date w_chick*date w_turkey*date/
overlay cframe=ligr haxis=axis1 vaxis=axis2;
title 'Budget Shares Plots';
footnote1 c=blue ' * beef '
c=red ' . pork '
c=green ' o chicken '
c=black ' + turkey ';
symbol1 c=blue i=join v=star;
symbol2 c=red i=join v=dot;
symbol3 c=green i = join v = circle ;
symbol4 c=black i = join v = plus ;
axis1 label=('Time') ;
axis2 label=(angle=90 'Budget Share');
run;
quit;
footnote;
proc means data=aids noprint;
var beef_p pork_p chick_p turkey_p;
output out=pm mean=b_m p_m c_m t_m;
run;
data aids ;
if _n_ = 1 then set pm ;
set aids ;
pb_ = (beef_p/b_m);
pp_ = (pork_p/p_m);
pc_ = (chick_p/c_m);
pt_ = (turkey_p/t_m);
lpb = log(pb_);
lpp = log(pp_);
lpc = log(pc_);
lpt = log(pt_);
lx = log(x);
/* Specify the linear price index for the LA-AIDS model */
lp0 = sum(w_beef*lpb + w_pork*lpp + w_chick*lpc + w_turkey*lpt);
lxp = lx-lp0;
run;
proc means noprint;
var w_beef w_pork w_chick w_turkey pb_ pp_ pc_ pt_ x lp0;
output out=meanw mean= w_beef w_pork w_chick w_turkey bm pm cm tm x lp0;
run;
/* Linear Approximate AIDS (LA-AIDS) Model*/
proc model data=aids ;
/*imposing homogeneity and symmetry restrictions on the parameters*/
gbt = 0-gbb-gbp-gbc;
gpt = 0-gbp-gpp-gpc;
gct = 0-gbc-gpc-gcc;
/* delete last equation(turkey) for adding up*/
w_beef = ab + gbb*lpb + gbp*lpp + gbc*lpc + gbt*lpt + bb*lxp;
w_pork = ap + gbp*lpb + gpp*lpp + gpc*lpc + gpt*lpt + bp*lxp;
w_chick = ac + gbc*lpb + gpc*lpp + gcc*lpc + gct*lpt + bc*lxp;
fit w_beef w_pork w_chick / itsur nestit outest=fin0;
parms ab bb gbb gbp gbc
ap bp gpp gpc
ac bc gcc ;
run;
quit ;
proc syslin itsur data = aids outest = fin1;
b: model w_beef = lpb lpp lpc lpt lxp co1 si1 t ;
p: model w_pork = lpb lpp lpc lpt lxp co1 si1 t ;
c: model w_chick = lpb lpp lpc lpt lxp co1 si1 t ;
srestrict
/*symmetry restrictions */
b.lpp = p.lpb ,
b.lpc = c.lpb ,
p.lpc = c.lpp ,
/* homogeneity restrictions*/
b.lpb + b.lpp + b.lpc + b.lpt = 0 ,
p.lpb + p.lpp + p.lpc + p.lpt = 0 ,
c.lpb + c.lpp + c.lpc + c.lpt = 0;
run;
/* Full Nonlinear AIDS Model */
proc model data=aids;
/* imposing homogeneity and symmetry and adding-up restrictions */
restrict gbb + gbp + gbc + gbt = 0 ,
gbp + gpp + gpc + gpt = 0 ,
gbc + gpc + gcc + gct = 0 ,
gbt + gpt + gct + gtt = 0 ,
ab + ap + ac + at = 1 ;
/*non-linear price index*/
a0 = 0; /* restrict the constant term in the nonlinear price index to be zero */
p = a0 + ab*lpb + ap*lpp + ac*lpc + at*lpt +
.5*(gbb*lpb*lpb + gbp*lpb*lpp + gbc*lpb*lpc + gbt*lpb*lpt +
gbp*lpp*lpb + gpp*lpp*lpp + gpc*lpp*lpc + gpt*lpp*lpt +
gbc*lpc*lpb + gpc*lpc*lpp + gcc*lpc*lpc + gct*lpc*lpt +
gbt*lpt*lpb + gpt*lpt*lpp + gct*lpt*lpc + gtt*lpt*lpt );
/*share equation*/
w_beef = ab + gbb*lpb + gbp*lpp + gbc*lpc + gbt*lpt + bb*(lx-p)
+ abco1*co1 + absi1*si1 + ab_t*t ;
w_pork = ap + gbp*lpb + gpp*lpp + gpc*lpc + gpt*lpt + bp*(lx-p)
+ apco1*co1 + apsi1*si1 + ap_t*t ;
w_chick = ac + gbc*lpb + gpc*lpp + gcc*lpc + gct*lpt + bc*(lx-p)
+ acco1*co1 + acsi1*si1 + ac_t*t ;
fit w_beef w_pork w_chick / itsur nestit outs=rest estdata=fin0 outest=fin2
out = resid2 converge = .00001 maxit = 1000 ;
parms ab bb gbb gbp gbc gbt abco1 absi1 ab_t
ap bp gpp gpc gpt apco1 apsi1 ap_t
ac bc gcc gct acco1 acsi1 ac_t
at gtt ;
run;
quit;
data resid2;
merge resid2 aids_;
run;
proc gplot data = resid2;
plot w_beef*date w_pork*date w_chick*date / overlay cframe=ligr
haxis=axis1 vaxis=axis2 vref=0;
title 'Residual Plots';
footnote1 c=blue ' * beef '
c=red ' . pork '
c=green ' o chicken ';
symbol1 c=blue i=join v=star;
symbol2 c=red i=join v=dot;
symbol3 c=green i = join v = circle ;
axis1 label=('Time') ;
axis2 label=(angle=90 'Residuals');
run;
quit;
