| Multiple Linear Regression - Estimated Regression Equation |
| S&P500[t] = + 1019.31468282777 + 0.422059103429782Gold[t] + e[t] |
| Multiple Linear Regression - Ordinary Least Squares | |||||
| Variable | Parameter | S.D. | T-STAT H0: parameter = 0 | 2-tail p-value | 1-tail p-value |
| (Intercept) | 1019.31468282777 | 57.591832 | 17.6989 | 0 | 0 |
| Gold | 0.422059103429782 | 0.091705 | 4.6023 | 2.3e-05 | 1.2e-05 |
| Multiple Linear Regression - Regression Statistics | |
| Multiple R | 0.517209606602194 |
| R-squared | 0.267505777161597 |
| Adjusted R-squared | 0.254876566423003 |
| F-TEST (value) | 21.1815118694738 |
| F-TEST (DF numerator) | 1 |
| F-TEST (DF denominator) | 58 |
| p-value | 2.32277443732443e-05 |
| Multiple Linear Regression - Residual Statistics | |
| Residual Standard Deviation | 123.692686659152 |
| Sum Squared Residuals | 887393.082511627 |
| Multiple Linear Regression - Actuals, Interpolation, and Residuals | |||
| Time or Index | Actuals | Interpolation Forecast | Residuals Prediction Error |
| 1 | 1111.92 | 1194.99678463041 | -83.0767846304139 |
| 2 | 1131.13 | 1187.44192667902 | -56.3119266790215 |
| 3 | 1144.94 | 1188.13832419968 | -43.1983241996808 |
| 4 | 1113.89 | 1199.63943476814 | -85.7494347681424 |
| 5 | 1107.3 | 1184.44530704467 | -77.1453070446703 |
| 6 | 1120.68 | 1186.95655871008 | -66.2765587100774 |
| 7 | 1140.84 | 1185.94361686185 | -45.1036168618461 |
| 8 | 1101.72 | 1184.55082182053 | -82.8308218205277 |
| 9 | 1104.24 | 1191.36707634092 | -87.1270763409187 |
| 10 | 1114.58 | 1195.77759397176 | -81.19759397176 |
| 11 | 1130.2 | 1200.42024410949 | -70.2202441094875 |
| 12 | 1173.78 | 1210.44414781594 | -36.6641478159449 |
| 13 | 1211.92 | 1199.85046431986 | 12.0695356801427 |
| 14 | 1181.27 | 1196.95935946136 | -15.6893594613633 |
| 15 | 1203.6 | 1202.25620120941 | 1.34379879059282 |
| 16 | 1180.59 | 1199.5972288578 | -19.0072288577995 |
| 17 | 1156.85 | 1199.91377318537 | -43.0637731853719 |
| 18 | 1191.5 | 1194.61693143733 | -3.11693143732804 |
| 19 | 1191.33 | 1201.89745097149 | -10.5674509714919 |
| 20 | 1234.18 | 1201.49649482323 | 32.6835051767666 |
| 21 | 1220.33 | 1204.8518646955 | 15.4781353044997 |
| 22 | 1228.81 | 1216.03643093639 | 12.7735690636104 |
| 23 | 1207.01 | 1213.25084085375 | -6.24084085375293 |
| 24 | 1249.48 | 1230.23871976680 | 19.2412802331983 |
| 25 | 1248.29 | 1243.00600764555 | 5.28399235444738 |
| 26 | 1280.08 | 1259.14976835174 | 20.9302316482582 |
| 27 | 1280.66 | 1257.46153193802 | 23.1984680619775 |
| 28 | 1302.88 | 1267.06337654105 | 35.8166234589499 |
| 29 | 1310.61 | 1298.29575019485 | 12.3142498051458 |
| 30 | 1270.05 | 1283.10162247138 | -13.0516224713820 |
| 31 | 1270.06 | 1282.23640130935 | -12.1764013093509 |
| 32 | 1278.53 | 1288.27184648840 | -9.74184648839677 |
| 33 | 1303.8 | 1281.43448901283 | 22.3655109871657 |
| 34 | 1335.83 | 1272.80338034770 | 63.0266196523047 |
| 35 | 1377.76 | 1278.50117824400 | 99.2588217560027 |
| 36 | 1400.63 | 1293.12552617784 | 107.504473822161 |
| 37 | 1418.03 | 1289.32699424697 | 128.703005753029 |
| 38 | 1437.9 | 1297.95810291211 | 139.94189708789 |
| 39 | 1406.8 | 1302.26310576709 | 104.536894232906 |
| 40 | 1420.83 | 1297.13508766042 | 123.694912339578 |
| 41 | 1482.37 | 1303.61369489807 | 178.756305101931 |
| 42 | 1530.63 | 1300.61707526372 | 230.012924736282 |
| 43 | 1504.66 | 1295.65788079842 | 209.002119201582 |
| 44 | 1455.18 | 1300.30053093615 | 154.879469063855 |
| 45 | 1473.96 | 1302.93840033258 | 171.021599667418 |
| 46 | 1527.29 | 1332.69356712438 | 194.596432875619 |
| 47 | 1545.79 | 1352.84688931315 | 192.943110686846 |
| 48 | 1479.63 | 1350.31453469257 | 129.315465307425 |
| 49 | 1467.97 | 1376.69322865694 | 91.2767713430639 |
| 50 | 1378.6 | 1405.39324769016 | -26.7932476901615 |
| 51 | 1330.45 | 1436.52010656811 | -106.070106568108 |
| 52 | 1326.41 | 1393.99765189756 | -67.5876518975572 |
| 53 | 1385.97 | 1379.33109805337 | 6.63890194662774 |
| 54 | 1399.62 | 1394.20868144927 | 5.41131855072777 |
| 55 | 1276.69 | 1414.99509229319 | -138.305092293189 |
| 56 | 1269.42 | 1404.44361470744 | -135.023614707444 |
| 57 | 1287.83 | 1366.35278062291 | -78.5227806229066 |
| 58 | 1164.17 | 1390.72669384598 | -226.556693845976 |
| 59 | 968.67 | 1327.20679877979 | -358.536798779794 |
| 60 | 888.61 | 1347.67666529614 | -459.066665296139 |
| Goldfeld-Quandt test for Heteroskedasticity | |||
| p-values | Alternative Hypothesis | ||
| breakpoint index | greater | 2-sided | less |
| 5 | 0.0035477676060547 | 0.0070955352121094 | 0.996452232393945 |
| 6 | 0.000407009828533351 | 0.000814019657066702 | 0.999592990171467 |
| 7 | 6.82754319458242e-05 | 0.000136550863891648 | 0.999931724568054 |
| 8 | 2.51168489748499e-05 | 5.02336979496999e-05 | 0.999974883151025 |
| 9 | 4.78082437928866e-06 | 9.56164875857732e-06 | 0.999995219175621 |
| 10 | 5.94779920054944e-07 | 1.18955984010989e-06 | 0.99999940522008 |
| 11 | 1.05387500296421e-07 | 2.10775000592842e-07 | 0.9999998946125 |
| 12 | 1.44552344641224e-07 | 2.89104689282448e-07 | 0.999999855447655 |
| 13 | 2.29980676721762e-06 | 4.59961353443525e-06 | 0.999997700193233 |
| 14 | 1.23726479443126e-06 | 2.47452958886251e-06 | 0.999998762735206 |
| 15 | 7.83210511744976e-07 | 1.56642102348995e-06 | 0.999999216789488 |
| 16 | 2.44087304435337e-07 | 4.88174608870674e-07 | 0.999999755912696 |
| 17 | 5.85277579336617e-08 | 1.17055515867323e-07 | 0.999999941472242 |
| 18 | 4.25736282068503e-08 | 8.51472564137006e-08 | 0.999999957426372 |
| 19 | 1.34988251081253e-08 | 2.69976502162506e-08 | 0.999999986501175 |
| 20 | 2.00066395286088e-08 | 4.00132790572176e-08 | 0.99999997999336 |
| 21 | 7.9960738916306e-09 | 1.59921477832612e-08 | 0.999999992003926 |
| 22 | 2.02890238353062e-09 | 4.05780476706123e-09 | 0.999999997971098 |
| 23 | 6.02166283058325e-10 | 1.20433256611665e-09 | 0.999999999397834 |
| 24 | 1.92785090031776e-10 | 3.85570180063552e-10 | 0.999999999807215 |
| 25 | 1.32841688049061e-10 | 2.65683376098122e-10 | 0.999999999867158 |
| 26 | 5.61731041387487e-11 | 1.12346208277497e-10 | 0.999999999943827 |
| 27 | 1.638373488418e-11 | 3.276746976836e-11 | 0.999999999983616 |
| 28 | 4.11973029829683e-12 | 8.23946059659366e-12 | 0.99999999999588 |
| 29 | 3.95034061705103e-12 | 7.90068123410207e-12 | 0.99999999999605 |
| 30 | 3.10630349055194e-12 | 6.21260698110387e-12 | 0.999999999996894 |
| 31 | 1.70497996750304e-12 | 3.40995993500608e-12 | 0.999999999998295 |
| 32 | 8.04299539326316e-13 | 1.60859907865263e-12 | 0.999999999999196 |
| 33 | 2.29145858131885e-13 | 4.5829171626377e-13 | 0.999999999999771 |
| 34 | 1.59276533510214e-13 | 3.18553067020428e-13 | 0.99999999999984 |
| 35 | 3.04850598793166e-13 | 6.09701197586332e-13 | 0.999999999999695 |
| 36 | 2.70385212266161e-13 | 5.40770424532321e-13 | 0.99999999999973 |
| 37 | 4.44142043450258e-13 | 8.88284086900516e-13 | 0.999999999999556 |
| 38 | 4.84925954158558e-13 | 9.69851908317116e-13 | 0.999999999999515 |
| 39 | 1.37136531850835e-13 | 2.74273063701670e-13 | 0.999999999999863 |
| 40 | 5.89440304497423e-14 | 1.17888060899485e-13 | 0.999999999999941 |
| 41 | 1.20811905864285e-13 | 2.4162381172857e-13 | 0.99999999999988 |
| 42 | 2.28190556485355e-12 | 4.5638111297071e-12 | 0.999999999997718 |
| 43 | 8.22368348652693e-12 | 1.64473669730539e-11 | 0.999999999991776 |
| 44 | 4.24329990252475e-12 | 8.4865998050495e-12 | 0.999999999995757 |
| 45 | 4.85867190302916e-12 | 9.71734380605832e-12 | 0.999999999995141 |
| 46 | 2.55891999010117e-11 | 5.11783998020234e-11 | 0.99999999997441 |
| 47 | 7.40424024997684e-10 | 1.48084804999537e-09 | 0.999999999259576 |
| 48 | 1.2716511688935e-07 | 2.543302337787e-07 | 0.999999872834883 |
| 49 | 2.28955006653279e-05 | 4.57910013306557e-05 | 0.999977104499335 |
| 50 | 0.00060210078870353 | 0.00120420157740706 | 0.999397899211296 |
| 51 | 0.0112230004426738 | 0.0224460008853475 | 0.988776999557326 |
| 52 | 0.012484068106448 | 0.024968136212896 | 0.987515931893552 |
| 53 | 0.0387582704982267 | 0.0775165409964533 | 0.961241729501773 |
| 54 | 0.088177137863861 | 0.176354275727722 | 0.911822862136139 |
| 55 | 0.0735672224288728 | 0.147134444857746 | 0.926432777571127 |
| Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity | |||
| Description | # significant tests | % significant tests | OK/NOK |
| 1% type I error level | 46 | 0.901960784313726 | NOK |
| 5% type I error level | 48 | 0.941176470588235 | NOK |
| 10% type I error level | 49 | 0.96078431372549 | NOK |
