$ontext Section 9.3.1 from Greene. Nonlinear Least Squares Regression example Erwin Kalvelagen, nov 2007 ============================================================== Model: C = alpha + beta * Y ** gamma Starting values: gamma = 1 alpha, beta from linear regression C = alpha + beta * Y Reference: Greene, Econometric Analysis, 5th Edition $offtext *----------------------------------------------------------------------------- * data *----------------------------------------------------------------------------- set i /i1*i204/; table data(i,*) 'Greene appendix table 5.1 (selected columns)' Year qtr realgdp realcons realinvs realgovt realdpi i1 1950 1 1610.5 1058.9 198.1 361.0 1186.1 i2 1950 2 1658.8 1075.9 220.4 366.4 1178.1 i3 1950 3 1723.0 1131.0 239.7 359.6 1196.5 i4 1950 4 1753.9 1097.6 271.8 382.5 1210.0 i5 1951 1 1773.5 1122.8 242.9 421.9 1207.9 i6 1951 2 1803.7 1091.4 249.2 480.1 1225.8 i7 1951 3 1839.8 1103.9 230.1 534.2 1235.8 i8 1951 4 1843.3 1110.5 210.6 563.7 1238.5 i9 1952 1 1864.7 1113.6 215.6 584.8 1238.5 i10 1952 2 1866.2 1135.1 197.7 604.4 1252.0 i11 1952 3 1878.0 1140.4 207.8 610.5 1276.1 i12 1952 4 1940.2 1180.5 223.3 620.8 1300.5 i13 1953 1 1976.0 1194.9 227.5 641.2 1317.5 i14 1953 2 1992.2 1202.5 228.5 655.9 1336.3 i15 1953 3 1979.5 1199.8 222.8 647.6 1330.2 i16 1953 4 1947.8 1191.8 205.0 645.4 1325.9 i17 1954 1 1938.1 1196.2 203.4 627.1 1330.3 i18 1954 2 1941.0 1211.3 203.0 606.1 1327.9 i19 1954 3 1962.0 1227.3 213.3 591.2 1344.2 i20 1954 4 2000.9 1252.6 223.3 587.4 1373.6 i21 1955 1 2058.1 1280.1 247.2 586.4 1392.7 i22 1955 2 2091.0 1304.3 262.8 579.9 1423.3 i23 1955 3 2118.9 1320.3 266.4 584.0 1451.1 i24 1955 4 2130.1 1336.7 272.0 571.3 1468.1 i25 1956 1 2121.0 1339.2 262.9 570.9 1480.9 i26 1956 2 2137.7 1343.7 260.0 582.6 1497.8 i27 1956 3 2135.3 1346.8 257.1 577.3 1504.1 i28 1956 4 2170.4 1365.3 254.4 592.5 1526.5 i29 1957 1 2182.7 1374.2 250.0 604.0 1527.5 i30 1957 2 2177.7 1376.5 249.9 600.6 1538.6 i31 1957 3 2198.9 1387.7 255.6 605.5 1548.7 i32 1957 4 2176.0 1388.8 234.1 616.6 1543.1 i33 1958 1 2117.4 1370.1 216.7 609.6 1524.7 i34 1958 2 2129.7 1380.9 211.3 625.0 1534.1 i35 1958 3 2177.5 1402.3 228.4 628.4 1568.1 i36 1958 4 2226.5 1418.8 249.6 641.5 1588.0 i37 1959 1 2273.0 1445.2 263.0 651.5 1599.5 i38 1959 2 2332.4 1468.2 286.2 663.9 1629.6 i39 1959 3 2331.4 1483.8 266.6 668.1 1627.0 i40 1959 4 2339.1 1485.6 275.6 662.2 1639.2 i41 1960 1 2391.0 1499.2 305.3 648.8 1657.7 i42 1960 2 2379.2 1518.1 274.0 657.4 1666.5 i43 1960 3 2383.6 1512.1 272.4 665.9 1667.7 i44 1960 4 2352.9 1513.5 239.5 673.1 1667.2 i45 1961 1 2366.5 1512.8 245.0 680.4 1680.6 i46 1961 2 2410.8 1535.2 263.3 687.2 1705.4 i47 1961 3 2450.4 1542.9 285.5 694.0 1729.4 i48 1961 4 2500.4 1574.2 290.2 711.1 1764.4 i49 1962 1 2544.0 1590.6 307.3 723.4 1777.9 i50 1962 2 2571.5 1609.9 304.5 731.7 1799.3 i51 1962 3 2596.8 1622.9 310.0 740.8 1811.4 i52 1962 4 2603.3 1645.9 299.5 744.2 1825.5 i53 1963 1 2634.1 1657.1 315.4 740.0 1838.9 i54 1963 2 2668.4 1673.0 320.8 744.3 1857.2 i55 1963 3 2719.6 1695.7 331.5 765.9 1879.2 i56 1963 4 2739.4 1710.0 335.2 759.2 1910.5 i57 1964 1 2800.5 1743.8 348.9 763.1 1947.6 i58 1964 2 2833.8 1775.0 347.5 772.9 1999.4 i59 1964 3 2872.0 1807.8 355.7 766.4 2027.8 i60 1964 4 2879.5 1812.8 358.3 766.1 2052.6 i61 1965 1 2950.1 1852.5 394.9 765.5 2071.8 i62 1965 2 2989.9 1873.2 394.6 781.3 2096.4 i63 1965 3 3050.7 1905.3 408.4 800.3 2155.3 i64 1965 4 3123.6 1959.3 410.1 817.2 2200.4 i65 1966 1 3201.1 1988.6 444.1 832.5 2219.3 i66 1966 2 3213.2 1994.0 436.5 857.8 2224.6 i67 1966 3 3233.6 2016.6 432.7 870.1 2254.0 i68 1966 4 3261.8 2025.1 435.8 888.0 2280.5 i69 1967 1 3291.8 2037.3 424.9 925.6 2312.6 i70 1967 2 3289.7 2064.6 405.0 921.3 2329.9 i71 1967 3 3313.5 2075.2 415.2 926.8 2351.4 i72 1967 4 3338.3 2087.9 423.6 934.8 2367.9 i73 1968 1 3406.2 2136.2 433.8 951.4 2409.5 i74 1968 2 3464.8 2169.6 451.8 956.0 2451.2 i75 1968 3 3489.2 2210.7 437.3 958.3 2457.9 i76 1968 4 3504.1 2220.4 442.2 960.5 2474.3 i77 1969 1 3558.3 2244.8 470.8 956.9 2477.5 i78 1969 2 3567.6 2258.8 467.1 956.0 2501.5 i79 1969 3 3588.3 2269.0 477.2 954.1 2550.2 i80 1969 4 3571.4 2286.5 452.6 943.1 2568.1 i81 1970 1 3566.5 2300.8 438.0 936.2 2581.9 i82 1970 2 3573.9 2312.0 439.4 927.3 2626.0 i83 1970 3 3605.2 2332.2 446.5 930.9 2661.1 i84 1970 4 3566.5 2324.9 421.0 929.9 2650.9 i85 1971 1 3666.1 2369.8 475.9 918.6 2703.5 i86 1971 2 3686.2 2391.4 490.2 915.2 2742.6 i87 1971 3 3714.5 2409.8 496.5 911.9 2752.9 i88 1971 4 3723.8 2449.8 480.6 909.4 2782.1 i89 1972 1 3796.9 2482.2 513.6 920.8 2797.6 i90 1972 2 3883.8 2527.5 544.9 921.9 2822.9 i91 1972 3 3922.3 2565.9 554.1 907.6 2883.6 i92 1972 4 3990.5 2626.3 559.4 909.1 2993.0 i93 1973 1 4092.3 2674.2 595.2 914.5 3031.9 i94 1973 2 4133.3 2671.4 618.2 911.5 3059.6 i95 1973 3 4117.0 2682.5 597.5 898.5 3079.3 i96 1973 4 4151.1 2675.6 615.3 908.4 3118.3 i97 1974 1 4119.3 2652.4 579.2 920.0 3072.1 i98 1974 2 4130.4 2662.0 577.3 927.8 3045.5 i99 1974 3 4084.5 2672.2 543.4 924.2 3053.3 i100 1974 4 4062.0 2628.4 547.0 927.4 3036.7 i101 1975 1 4010.0 2648.8 450.8 940.8 3015.0 i102 1975 2 4045.2 2695.4 436.4 938.3 3156.6 i103 1975 3 4115.4 2734.7 474.9 941.8 3114.9 i104 1975 4 4167.2 2764.6 486.8 949.1 3147.6 i105 1976 1 4266.1 2824.7 535.1 952.5 3201.9 i106 1976 2 4301.5 2850.9 559.8 943.3 3229.0 i107 1976 3 4321.9 2880.3 561.1 938.9 3259.7 i108 1976 4 4357.4 2919.6 565.9 938.6 3283.5 i109 1977 1 4410.5 2954.7 595.5 945.3 3305.4 i110 1977 2 4489.8 2970.5 635.0 955.1 3326.8 i111 1977 3 4570.6 2999.1 670.7 956.0 3376.5 i112 1977 4 4576.1 3044.0 656.4 954.5 3433.8 i113 1978 1 4588.9 3060.8 667.2 956.7 3466.3 i114 1978 2 4765.7 3127.0 709.7 982.1 3513.0 i115 1978 3 4811.7 3143.1 728.8 990.3 3548.1 i116 1978 4 4876.0 3167.8 746.3 999.6 3582.6 i117 1979 1 4888.3 3188.6 746.0 990.6 3620.7 i118 1979 2 4891.4 3184.3 745.7 1000.5 3607.1 i119 1979 3 4926.2 3213.9 732.1 1002.4 3628.8 i120 1979 4 4942.6 3225.7 717.8 1010.8 3657.8 i121 1980 1 4958.9 3222.4 711.7 1025.6 3678.5 i122 1980 2 4857.8 3149.2 647.4 1028.7 3612.2 i123 1980 3 4850.3 3181.2 599.8 1015.4 3637.6 i124 1980 4 4936.6 3219.4 662.2 1013.9 3703.8 i125 1981 1 5032.5 3233.1 726.3 1027.5 3713.5 i126 1981 2 4997.3 3235.5 693.4 1030.1 3696.6 i127 1981 3 5056.8 3250.5 733.9 1027.8 3777.0 i128 1981 4 4997.1 3225.0 708.8 1034.8 3777.2 i129 1982 1 4914.3 3244.3 634.8 1033.6 3769.4 i130 1982 2 4935.5 3253.4 631.6 1039.5 3791.4 i131 1982 3 4912.1 3274.6 623.5 1046.8 3799.4 i132 1982 4 4915.6 3329.6 571.1 1064.0 3806.4 i133 1983 1 4972.4 3360.1 590.7 1069.8 3831.2 i134 1983 2 5089.8 3430.1 650.7 1078.2 3857.8 i135 1983 3 5180.4 3484.7 691.4 1097.0 3928.6 i136 1983 4 5286.8 3542.2 762.2 1078.8 4010.2 i137 1984 1 5402.3 3579.7 845.0 1091.0 4103.0 i138 1984 2 5493.8 3628.3 873.2 1115.2 4182.4 i139 1984 3 5541.3 3653.5 890.7 1123.1 4258.8 i140 1984 4 5583.1 3700.9 876.9 1144.2 4286.1 i141 1985 1 5629.7 3756.8 848.9 1157.6 4287.6 i142 1985 2 5673.8 3791.5 862.8 1180.5 4368.7 i143 1985 3 5758.6 3860.9 854.1 1209.2 4346.6 i144 1985 4 5806.0 3874.2 887.8 1214.7 4388.3 i145 1986 1 5858.9 3907.9 886.2 1224.0 4444.5 i146 1986 2 5883.3 3950.4 868.3 1248.0 4489.3 i147 1986 3 5937.9 4019.7 838.0 1277.4 4507.9 i148 1986 4 5969.5 4046.8 838.2 1271.5 4504.5 i149 1987 1 6013.3 4049.7 863.4 1278.4 4556.9 i150 1987 2 6077.2 4101.5 863.9 1289.1 4512.7 i151 1987 3 6128.1 4147.0 860.5 1292.4 4600.7 i152 1987 4 6234.4 4155.3 929.3 1310.0 4659.6 i153 1988 1 6275.9 4228.0 884.6 1300.1 4724.1 i154 1988 2 6349.8 4256.8 902.5 1302.4 4758.9 i155 1988 3 6382.3 4291.6 907.5 1300.3 4801.9 i156 1988 4 6465.2 4341.4 916.7 1327.2 4851.4 i157 1989 1 6543.8 4357.1 952.7 1319.3 4903.5 i158 1989 2 6579.4 4374.8 941.1 1340.6 4891.0 i159 1989 3 6610.6 4413.4 929.3 1353.5 4902.7 i160 1989 4 6633.5 4429.4 922.9 1360.4 4928.8 i161 1990 1 6716.3 4466.0 934.0 1381.2 5001.6 i162 1990 2 6731.7 4478.8 933.0 1384.7 5026.6 i163 1990 3 6719.4 4495.6 912.6 1384.8 5032.7 i164 1990 4 6664.2 4457.7 849.6 1398.6 4995.8 i165 1991 1 6631.4 4437.5 815.1 1404.7 4999.5 i166 1991 2 6668.5 4469.9 808.8 1408.9 5033.3 i167 1991 3 6684.9 4484.3 829.8 1403.0 5045.4 i168 1991 4 6720.9 4474.8 864.2 1397.0 5053.8 i169 1992 1 6783.3 4544.8 843.8 1407.6 5138.8 i170 1992 2 6846.8 4566.7 901.8 1405.7 5172.5 i171 1992 3 6899.7 4600.5 912.1 1413.1 5174.2 i172 1992 4 6990.6 4665.9 941.6 1413.7 5271.5 i173 1993 1 6988.7 4674.9 964.8 1396.4 5181.2 i174 1993 2 7031.2 4721.5 967.0 1398.0 5258.6 i175 1993 3 7062.0 4776.9 964.1 1398.4 5266.8 i176 1993 4 7168.7 4822.3 1015.6 1402.2 5338.5 i177 1994 1 7229.4 4866.6 1057.3 1388.0 5293.2 i178 1994 2 7330.2 4907.9 1118.5 1390.4 5381.2 i179 1994 3 7370.2 4944.5 1101.8 1417.5 5420.9 i180 1994 4 7461.1 4993.6 1150.5 1404.5 5493.4 i181 1995 1 7488.7 5011.6 1162.4 1407.3 5515.4 i182 1995 2 7503.3 5059.6 1128.5 1414.0 5509.0 i183 1995 3 7561.4 5099.2 1119.1 1410.8 5546.6 i184 1995 4 7621.9 5132.1 1152.4 1393.5 5585.3 i185 1996 1 7676.4 5174.3 1172.3 1404.8 5622.0 i186 1996 2 7802.9 5229.5 1233.4 1430.4 5649.4 i187 1996 3 7841.9 5254.3 1281.4 1422.0 5709.7 i188 1996 4 7931.3 5291.9 1283.7 1430.6 5729.9 i189 1997 1 8016.4 5350.7 1325.4 1434.6 5771.8 i190 1997 2 8131.9 5375.7 1400.6 1457.0 5821.2 i191 1997 3 8216.6 5462.1 1408.6 1464.8 5877.3 i192 1997 4 8272.9 5507.1 1438.5 1465.3 5947.5 i193 1998 1 8396.3 5576.3 1543.3 1456.1 6064.5 i194 1998 2 8442.9 5660.2 1516.8 1482.6 6153.6 i195 1998 3 8528.5 5713.7 1559.7 1489.9 6209.9 i196 1998 4 8667.9 5784.7 1612.1 1504.8 6246.6 i197 1999 1 8733.5 5854.0 1641.8 1512.3 6268.2 i198 1999 2 8771.2 5936.1 1617.4 1516.8 6300.0 i199 1999 3 8871.5 6000.0 1655.8 1533.2 6332.4 i200 1999 4 9049.9 6083.6 1725.4 1564.8 6379.2 i201 2000 1 9102.5 6171.7 1722.9 1560.4 6431.6 i202 2000 2 9229.4 6226.3 1801.6 1577.2 6523.7 i203 2000 3 9260.1 6292.1 1788.8 1570.0 6566.5 i204 2000 4 9303.9 6341.1 1778.3 1582.8 6634.9 ; *----------------------------------------------------------------------------- * statistical model *----------------------------------------------------------------------------- variables alpha 'parameter to estimate' beta 'parameter to estimate' gamma 'parameter to estimate' sse 'sum of squared errors' ; equations obj 'dummy obj' linfit(i) 'linear fit' nlfit(i) 'non-linear fit' ; obj.. sse =n= 0; linfit(i).. data(i,'realcons') =e= alpha + beta*data(i,'realdpi'); nlfit(i).. data(i,'realcons') =e= alpha + beta*data(i,'realdpi')**gamma; model linear /obj,linfit/; model nonlinear /obj,nlfit/; *----------------------------------------------------------------------------- * initial values for alpha and beta by OLS *----------------------------------------------------------------------------- option lp=ls; solve linear using lp minimizing sse; *----------------------------------------------------------------------------- * nonlinear fit *----------------------------------------------------------------------------- gamma.l = 1; option nlp=nls; solve nonlinear using nlp minimizing sse;