Systems of Regression Equations

Cross-Sectional Time Series of Investment Data

Boot, J. and G. deWitt (1960). “Investment Demand: An Empirical Contribution to the Aggregation Problem,” International Economic Review, Vol. 1, pp. 3-30

Grunfeld’s Investment Data

• Cross-Section: n=10 Firms (GM, US Steel, GE, Chrysler, Atlantic Refining, IBM, Union Oil, Westinghouse, Goodyear, Diamond Match)

• Time Series: T=20 years per firm (1935-1954)

• Dependent Variable: Gross Investment (Y, in millions of 1947 $)

• Independent Variables: Value of Firm (X1, in millions of 1947 $)

Stock of Plant/Equipment (X2, in millions of 1947 $)

Regression Model

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Beta_OLS SE(B_OLS) t(B_OLS) P-value-43.2055 9.5727 -4.5134 0.00000.1140 0.0058 19.5709 0.00000.2346 0.0255 9.1837 0.0000

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Firm s2_olsGM 14609.92US Steel 34423.61GE 31275.77Chrysler 706.28Atl Ref 2929.09IBM 2992.58U Oil 263.84Westinghouse 617.57Goodyear 843.16D Match 1357.71

Beta_GLS1 SE(B_GLS1) t(B_GLS1) P-value-23.4460 4.1376 -5.6666 0.00000.1067 0.0049 21.7549 0.00000.1616 0.0131 12.3557 0.0000

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Iteration1 Iteration2 Iteration3 Iteration4 Iteration5 Iteration6 Iteration7 Iteration8 Iteration9 Iteration10 Iteration11 SE(Beta) t(Beta)Beta0 -23.4460 -8.3433 -2.8501 -1.8618 -1.2250 -1.0015 -0.9251 -0.8961 -0.8843 -0.8793 -0.8772 0.4107 -2.1360Beta1 0.1067 0.0915 0.0757 0.0612 0.0527 0.0501 0.0492 0.0490 0.0488 0.0488 0.0488 0.0029 17.0946Beta2 0.1616 0.1227 0.1119 0.1127 0.1119 0.1098 0.1082 0.1073 0.1068 0.1065 0.1064 0.0047 22.7862

SumD^2 228.094164 30.175123 0.97697413 0.40559749 0.04996573 0.00583529 0.00084574 0.00013979 2.4685E-05 4.5185E-06

Firm GM US Steel GE Chrysler Atl Ref IBM U Oil Westinghouse Goodyear D Matchs2_gls 154220.4 92140.0 2002.2 2375.0 234.8 759.9 140.5 92.8 100.9 2.3

V(B_GLS1)0.168655006 -0.00064054 8.57E-05-0.00064054 8.14316E-06 -6.3E-068.56758E-05 -6.28303E-06 2.18E-05

Cross-Sectional Correlation Over Time - I

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Cross-Sectional Correlation Over Time - II

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V(B(GLS3))0.750832 -0.000945 -0.001895-0.000945 0.000006 -0.000001-0.001895 -0.000001 0.000016

S_ij Firm1 Firm2 Firm3 Firm4 Firm5 Firm6 Firm7 Firm8 Firm9 Firm10Firm1 32292.9 20896.6 -11977.9 3444.6 -4696.7 -4172.5 226.2 -3174.2 -2656.9 1532.6Firm2 20896.6 42939.3 -24709.8 4255.7 -4358.0 1928.1 355.1 -3452.0 -3608.0 3174.2Firm3 -11977.9 -24709.8 21162.0 -2523.6 2370.9 -214.4 -614.1 2969.9 2717.9 -2352.8Firm4 3444.6 4255.7 -2523.6 713.7 -637.4 -213.9 24.5 -455.1 -458.5 313.0Firm5 -4696.7 -4358.0 2370.9 -637.4 1010.3 350.0 51.7 575.1 480.7 -319.5Firm6 -4172.5 1928.1 -214.4 -213.9 350.0 2003.0 -28.5 373.1 201.2 106.3Firm7 226.2 355.1 -614.1 24.5 51.7 -28.5 111.9 -33.4 -33.7 66.0Firm8 -3174.2 -3452.0 2969.9 -455.1 575.1 373.1 -33.4 610.6 477.2 -313.8Firm9 -2656.9 -3608.0 2717.9 -458.5 480.7 201.2 -33.7 477.2 460.0 -312.0Firm10 1532.6 3174.2 -2352.8 313.0 -319.5 106.3 66.0 -313.8 -312.0 288.9

Cross-Sectional Correlation- Iterated EGLS – (ML)

Beta(GLS4) SE(B(GLS4)) t(B(GLS4))-0.0300 0.1481 -0.20280.0143 0.0013 11.25000.0895 0.0021 41.7738

S_ij Firm1 Firm2 Firm3 Firm4 Firm5 Firm6 Firm7 Firm8 Firm9 Firm10Firm1 297969.9 188790.0 23270.6 38280.7 6565.4 23509.2 10084.9 14627.1 5679.7 988.4Firm2 188790.0 139571.2 16102.0 25764.2 5484.2 15804.8 6471.6 10301.1 3985.1 680.1Firm3 23270.6 16102.0 2437.7 2992.3 526.3 1911.5 754.3 1311.9 508.8 88.3Firm4 38280.7 25764.2 2992.3 5284.7 944.6 3190.9 1290.8 1921.5 742.5 136.8Firm5 6565.4 5484.2 526.3 944.6 379.1 509.0 265.0 392.0 145.3 20.6Firm6 23509.2 15804.8 1911.5 3190.9 509.0 2019.1 766.7 1196.2 442.6 90.2Firm7 10084.9 6471.6 754.3 1290.8 265.0 766.7 398.7 506.5 221.4 31.7Firm8 14627.1 10301.1 1311.9 1921.5 392.0 1196.2 506.5 821.0 330.6 51.8Firm9 5679.7 3985.1 508.8 742.5 145.3 442.6 221.4 330.6 198.5 17.6Firm10 988.4 680.1 88.3 136.8 20.6 90.2 31.7 51.8 17.6 4.7

V(B(GLS4))0.021930 -0.000062 0.000123-0.000062 0.000002 -0.0000010.000123 -0.000001 0.000005

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GM 1 5.111 0.515US Steel 2 159.750 0.530GE 3 -169.686 0.447Chrysler 4 21.869 0.173Atl Ref 5 -35.285 0.799IBM 6 14.894 -0.007U Oil 7 -0.168 0.668Westinghouse 8 -10.467 0.456Goodyear 9 -22.831 0.572D Match 10 36.812 0.538

Beta(GLS5,0)-33.33960.09740.2898

s2(u0)7316.8512899.5811461.25367.38721.40

2125.17238.08388.34641.61185.86

Beta(GLS5) SE(BGLS5) t(BGLS5)-14.2464 4.4042 -3.23480.0952 0.0062 15.40900.1872 0.0195 9.6049

V(B(GLS5))19.3966963 -0.0133321 -0.0302569-0.0133321 0.0000382 -0.0000393-0.0302569 -0.0000393 0.0003797

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Cross-Sectional and Autocorrelation - I

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14681.18 5227.22 -5261.80 1418.18 -1405.15 -2474.47 -128.53 -1526.80 -1315.03 316.685227.22 14142.76 -6530.77 1181.59 -634.52 1439.22 -414.02 -890.21 -1230.05 432.06-5261.80 -6530.77 7998.74 -892.83 937.44 521.47 358.52 1496.99 1430.92 -371.151418.18 1181.59 -892.83 382.67 -158.90 -204.60 -40.86 -197.28 -232.13 53.90-1405.15 -634.52 937.44 -158.90 349.37 171.80 75.05 215.78 169.96 -53.59-2474.47 1439.22 521.47 -204.60 171.80 1650.97 -71.77 343.66 208.04 9.55-128.53 -414.02 358.52 -40.86 75.05 -71.77 115.51 89.38 96.14 -17.22-1526.80 -890.21 1496.99 -197.28 215.78 343.66 89.38 388.74 309.03 -60.49-1315.03 -1230.05 1430.92 -232.13 169.96 208.04 96.14 309.03 342.57 -72.28316.68 432.06 -371.15 53.90 -53.59 9.55 -17.22 -60.49 -72.28 23.20

Beta(GLS6) SE(B(GLS6)) t(B(GLS6))-13.1656 0.7570 -17.39280.0902 0.0033 27.36640.1829 0.0099 18.4119

V(B(GLS6))0.572984 -0.000303 -0.004268-0.000303 0.000011 -0.000011-0.004268 -0.000011 0.000099

S_ij Firm1 Firm2 Firm3 Firm4 Firm5 Firm6 Firm7 Firm8 Firm9 Firm10Firm1 16611.3 6791.0 -5856.7 1778.2 -1470.5 -2100.3 -137.5 -1547.6 -1434.5 347.7Firm2 6791.0 15145.5 -6374.6 1498.0 -618.6 1581.1 -387.5 -795.8 -1219.5 424.9Firm3 -5856.7 -6374.6 6935.8 -1028.0 824.4 293.5 298.0 1260.4 1254.6 -317.7Firm4 1778.2 1498.0 -1028.0 447.7 -175.3 -154.0 -45.2 -212.3 -258.6 60.9Firm5 -1470.5 -618.6 824.4 -175.3 332.7 131.3 67.1 187.5 150.4 -47.2Firm6 -2100.3 1581.1 293.5 -154.0 131.3 1434.7 -79.5 277.9 154.9 18.1Firm7 -137.5 -387.5 298.0 -45.2 67.1 -79.5 111.5 74.8 85.6 -14.2Firm8 -1547.6 -795.8 1260.4 -212.3 187.5 277.9 74.8 329.7 266.8 -49.2Firm9 -1434.5 -1219.5 1254.6 -258.6 150.4 154.9 85.6 266.8 311.7 -63.2Firm10 347.7 424.9 -317.7 60.9 -47.2 18.1 -14.2 -49.2 -63.2 20.1

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Firm Results - IFirm Coefficient beta(OLS) SE(B(OLS)) t(B(OLS)) p-value Firm Coefficient beta(OLS) SE(B(OLS)) t(B(OLS)) p-value

1 b0_1 -149.7825 105.8421 -1.4151 0.1586 6 b0_6 5.4770 4.4526 1.2301 0.22011 b1_1 0.1193 0.0258 4.6172 0.0000 6 b1_6 0.0109 0.0057 1.9237 0.05581 b2_1 0.3714 0.0371 10.0193 0.0000 6 b2_6 0.4243 0.0501 8.4616 0.00002 b0_2 -49.1983 148.0754 -0.3323 0.7400 7 b0_7 -4.4995 11.2894 -0.3986 0.69062 b1_2 0.1749 0.0742 2.3566 0.0194 7 b1_7 0.0875 0.0656 1.3337 0.18382 b2_2 0.3896 0.1424 2.7369 0.0068 7 b2_7 0.1238 0.0171 7.2536 0.00003 b0_3 -9.9191 31.3186 -0.3167 0.7518 8 b0_8 -0.5094 8.0153 -0.0636 0.94943 b1_3 0.0265 0.0155 1.7065 0.0895 8 b1_8 0.0529 0.0157 3.3677 0.00093 b2_3 0.1517 0.0257 5.9130 0.0000 8 b2_8 0.0924 0.0561 1.6472 0.10114 b0_4 -6.1900 13.5065 -0.4583 0.6472 9 b0_9 -7.7228 9.3593 -0.8251 0.41034 b1_4 0.0779 0.0200 3.9026 0.0001 9 b1_9 0.0754 0.0340 2.2204 0.02754 b2_4 0.3157 0.0288 10.9574 0.0000 9 b2_9 0.0821 0.0280 2.9331 0.00375 b0_5 23.2578 7.2151 3.2235 0.0015 10 b0_10 0.1615 2.0656 0.0782 0.93785 b1_5 0.1521 0.0599 2.5397 0.0119 10 b1_10 0.0046 0.0272 0.1684 0.86655 b2_5 0.0075 0.0231 0.3244 0.7460 10 b2_10 0.4374 0.0796 5.4954 0.0000

firm(i) s2(i)1 8423.882 9299.603 774.694 176.325 90.576 109.777 88.678 104.319 82.7910 1.18

Coefficient bbarb0 -19.8925b1 0.0782b2 0.2396

gamma2409.5477 -1.0070 -3.2525

-1.0070 0.0033 -0.0018-3.2525 -0.0018 0.0267

Note: Gamma estimate does notSubtract off the average of the V matrices (not positive definite)

Firm Results - IIV_i

Firm(i) Coefficient b0_i b1_i b2_i Firm(i) Coefficient b0_i b1_i b2_i1 b0_1 11202.5554 -2.6234 0.9069 6 b0_6 19.8256 0.0041 -0.15801 b1_1 -2.6234 0.0007 -0.0004 6 b1_6 0.0041 0.0000 -0.00021 b2_1 0.9069 -0.0004 0.0014 6 b2_6 -0.1580 -0.0002 0.00252 b0_2 21926.3137 -10.4230 -3.0831 7 b0_7 127.4504 -0.6368 -0.08772 b1_2 -10.4230 0.0055 -0.0015 7 b1_7 -0.6368 0.0043 0.00002 b2_2 -3.0831 -0.0015 0.0203 7 b2_7 -0.0877 0.0000 0.00033 b0_3 980.8548 -0.4499 -0.1719 8 b0_8 64.2449 -0.1095 0.16893 b1_3 -0.4499 0.0002 0.0000 8 b1_8 -0.1095 0.0002 -0.00073 b2_3 -0.1719 0.0000 0.0007 8 b2_8 0.1689 -0.0007 0.00314 b0_4 182.4250 -0.2547 0.0243 9 b0_9 87.5972 -0.2132 -0.04144 b1_4 -0.2547 0.0004 -0.0002 9 b1_9 -0.2132 0.0012 -0.00064 b2_4 0.0243 -0.0002 0.0008 9 b2_9 -0.0414 -0.0006 0.00085 b0_5 52.0571 -0.2455 0.0191 10 b0_10 4.2666 -0.0542 -0.06075 b1_5 -0.2455 0.0036 -0.0012 10 b1_10 -0.0542 0.0007 0.00035 b2_5 0.0191 -0.0012 0.0005 10 b2_10 -0.0607 0.0003 0.0063

betahat SE(BETA)-7.0015 17.31650.0711 0.02020.2317 0.0543

V(Beta-hat)299.8623 -0.1307 -0.3406

-0.1307 0.0004 -0.0002-0.3406 -0.0002 0.0029

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Firm Coefficient blupb SE(blupb) Firm Coefficient blupb SE(blupb)1 beta0_1 -49.4403 33.2132 6 beta0_6 6.4308 48.93381 beta1_1 0.0964 0.0567 6 beta1_6 0.0127 0.05741 beta2_1 0.3747 0.1605 6 beta2_6 0.4060 0.15782 beta0_2 -29.3664 26.3478 7 beta0_7 -4.3169 48.50662 beta1_2 0.1691 0.0526 7 beta1_7 0.0846 0.04352 beta2_2 0.3299 0.1312 7 beta2_7 0.1247 0.16273 beta0_3 -4.2017 43.4327 8 beta0_8 -0.0329 48.63593 beta1_3 0.0239 0.0564 8 beta1_8 0.0503 0.05613 beta2_3 0.1520 0.1619 8 beta2_8 0.1078 0.15684 beta0_4 -6.1463 47.8278 9 beta0_9 -8.3235 48.48624 beta1_4 0.0782 0.0552 9 beta1_9 0.0761 0.05154 beta2_4 0.3129 0.1615 9 beta2_9 0.0835 0.16185 beta0_5 25.5448 48.7507 10 beta0_10 -0.1623 49.05935 beta1_5 0.1080 0.0448 10 beta1_10 0.0121 0.05365 beta2_5 0.0236 0.1627 10 beta2_10 0.4022 0.1514

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1 7160.29 -1967.05 605.37 -282.76 -237.32 -104.47 371.87 126.18 146.31 -21.652 -1967.05 7904.66 976.82 367.84 205.33 395.51 -211.84 511.50 208.24 62.443 605.37 976.82 658.49 -21.10 6.64 66.59 -4.02 176.31 89.05 14.424 -282.76 367.84 -21.10 149.87 8.69 40.47 -12.94 13.31 7.19 1.385 -237.32 205.33 6.64 8.69 76.99 3.30 9.73 2.24 -11.47 1.856 -104.47 395.51 66.59 40.47 3.30 93.30 -8.11 39.78 -8.35 2.767 371.87 -211.84 -4.02 -12.94 9.73 -8.11 75.37 11.26 13.89 -1.908 126.18 511.50 176.31 13.31 2.24 39.78 11.26 88.66 41.38 5.409 146.31 208.24 89.05 7.19 -11.47 -8.35 13.89 41.38 70.37 2.18

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Firm Example - II

firm (i) Coefficient beta(SUR) SE(B(SUR))1 b0_1 -162.1917 73.34891 b1_1 0.1197 0.01701 b2_1 0.3876 0.02972 b0_2 -15.3907 79.49272 b1_2 0.1625 0.03852 b2_2 0.3573 0.10013 b0_3 -25.0153 21.55133 b1_3 0.0384 0.00983 b2_3 0.1316 0.02054 b0_4 6.3077 11.05624 b1_4 0.0615 0.01614 b2_4 0.3064 0.02545 b0_5 26.2813 6.30785 b1_5 0.1381 0.04725 b2_5 0.0079 0.01856 b0_6 9.1207 3.87036 b1_6 0.0121 0.00396 b2_6 0.3838 0.03827 b0_7 -11.9221 8.79067 b1_7 0.1260 0.04607 b2_7 0.1291 0.01468 b0_8 1.0093 5.21688 b1_8 0.0567 0.00878 b2_8 0.0446 0.03519 b0_9 -7.9773 7.59519 b1_9 0.0975 0.02389 b2_9 0.0581 0.022810 b0_10 1.9083 1.185110 b1_10 -0.0168 0.015910 b2_10 0.3986 0.0570

Estimated GLS

firm (i) Coefficient beta(SUR,ML) SE(B(SUR))1 b0_1 -289.4632 39.27431 b1_1 0.1401 0.00661 b2_1 0.4480 0.02172 b0_2 -50.2824 37.83082 b1_2 0.2122 0.01562 b2_2 0.1435 0.04713 b0_3 -44.0237 16.09493 b1_3 0.0521 0.00593 b2_3 0.1127 0.01914 b0_4 24.0540 9.88074 b1_4 0.0398 0.01374 b2_4 0.2843 0.02515 b0_5 34.3090 5.46005 b1_5 0.1084 0.02405 b2_5 0.0055 0.01186 b0_6 14.9753 3.52406 b1_6 0.0147 0.00326 b2_6 0.3147 0.02927 b0_7 -17.4418 7.22507 b1_7 0.1125 0.03457 b2_7 0.1530 0.01398 b0_8 -0.0493 4.62478 b1_8 0.0654 0.00708 b2_8 -0.0113 0.02869 b0_9 -20.9977 7.10229 b1_9 0.1969 0.01489 b2_9 -0.0095 0.021010 b0_10 10.2370 0.649210 b1_10 -0.0976 0.006410 b2_10 -0.0388 0.0493

ML (Iterated GLS)