*The author of this computation has been verified*

R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Fri, 27 Nov 2009 06:52:51 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Nov/27/t1259330006mqq9ugf283jfw6u.htm/, Retrieved Fri, 27 Nov 2009 14:53:41 +0100

BibTeX entries for LaTeX users:

@Manual{KEY, author = {{YOUR NAME}}, publisher = {Office for Research Development and Education}, title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2009/Nov/27/t1259330006mqq9ugf283jfw6u.htm/}, year = {2009}, } @Manual{R, title = {R: A Language and Environment for Statistical Computing}, author = {{R Development Core Team}}, organization = {R Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2009}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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416.25 1111.92 398.35 1131.13 400.00 1144.94 427.25 1113.89 391.25 1107.30 397.20 1120.68 394.80 1140.84 391.50 1101.72 407.65 1104.24 418.10 1114.58 429.10 1130.20 452.85 1173.78 427.75 1211.92 420.90 1181.27 433.45 1203.60 427.15 1180.59 427.90 1156.85 415.35 1191.50 432.60 1191.33 431.65 1234.18 439.60 1220.33 466.10 1228.81 459.50 1207.01 499.75 1249.48 530.00 1248.29 568.25 1280.08 564.25 1280.66 587.00 1302.88 661.00 1310.61 625.00 1270.05 622.95 1270.06 637.25 1278.53 621.05 1303.80 600.60 1335.83 614.10 1377.76 648.75 1400.63 639.75 1418.03 660.20 1437.90 670.40 1406.80 658.25 1420.83 673.60 1482.37 666.50 1530.63 654.75 1504.66 665.75 1455.18 672.00 1473.96 742.50 1527.29 790.25 1545.79 784.25 1479.63 846.75 1467.97 914.75 1378.60 988.50 1330.45 887.75 1326.41 853.00 1385.97 888.25 1399.62 937.50 1276.69 912.50 1269.42 822.25 1287.83 880.00 1164.17 729.50 968.67 778.00 888.61

Output produced by software:

Enter (or paste) a matrix (table) containing all data (time) series. Every column represents a different variable and must be delimited by a space or Tab. Every row represents a period in time (or category) and must be delimited by hard returns. The easiest way to enter data is to copy and paste a block of spreadsheet cells. Please, do not use commas or spaces to seperate groups of digits! Summary of computational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 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

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

R code (references can be found in the software module):

library(lattice) library(lmtest) n25 <- 25 #minimum number of obs. for Goldfeld-Quandt test par1 <- as.numeric(par1) x <- t(y) k <- length(x[1,]) n <- length(x[,1]) x1 <- cbind(x[,par1], x[,1:k!=par1]) mycolnames <- c(colnames(x)[par1], colnames(x)[1:k!=par1]) colnames(x1) <- mycolnames #colnames(x)[par1] x <- x1 if (par3 == 'First Differences'){ x2 <- array(0, dim=c(n-1,k), dimnames=list(1:(n-1), paste('(1-B)',colnames(x),sep=''))) for (i in 1:n-1) { for (j in 1:k) { x2[i,j] <- x[i+1,j] - x[i,j] } } x <- x2 } if (par2 == 'Include Monthly Dummies'){ x2 <- array(0, dim=c(n,11), dimnames=list(1:n, paste('M', seq(1:11), sep =''))) for (i in 1:11){ x2[seq(i,n,12),i] <- 1 } x <- cbind(x, x2) } if (par2 == 'Include Quarterly Dummies'){ x2 <- array(0, dim=c(n,3), dimnames=list(1:n, paste('Q', seq(1:3), sep =''))) for (i in 1:3){ x2[seq(i,n,4),i] <- 1 } x <- cbind(x, x2) } k <- length(x[1,]) if (par3 == 'Linear Trend'){ x <- cbind(x, c(1:n)) colnames(x)[k+1] <- 't' } x k <- length(x[1,]) df <- as.data.frame(x) (mylm <- lm(df)) (mysum <- summary(mylm)) if (n > n25) { kp3 <- k + 3 nmkm3 <- n - k - 3 gqarr <- array(NA, dim=c(nmkm3-kp3+1,3)) numgqtests <- 0 numsignificant1 <- 0 numsignificant5 <- 0 numsignificant10 <- 0 for (mypoint in kp3:nmkm3) { j <- 0 numgqtests <- numgqtests + 1 for (myalt in c('greater', 'two.sided', 'less')) { j <- j + 1 gqarr[mypoint-kp3+1,j] <- gqtest(mylm, point=mypoint, alternative=myalt)$p.value } if (gqarr[mypoint-kp3+1,2] < 0.01) numsignificant1 <- numsignificant1 + 1 if (gqarr[mypoint-kp3+1,2] < 0.05) numsignificant5 <- numsignificant5 + 1 if (gqarr[mypoint-kp3+1,2] < 0.10) numsignificant10 <- numsignificant10 + 1 } gqarr } bitmap(file='test0.png') plot(x[,1], type='l', main='Actuals and Interpolation', ylab='value of Actuals and Interpolation (dots)', xlab='time or index') points(x[,1]-mysum$resid) grid() dev.off() bitmap(file='test1.png') plot(mysum$resid, type='b', pch=19, main='Residuals', ylab='value of Residuals', xlab='time or index') grid() dev.off() bitmap(file='test2.png') hist(mysum$resid, main='Residual Histogram', xlab='values of Residuals') grid() dev.off() bitmap(file='test3.png') densityplot(~mysum$resid,col='black',main='Residual Density Plot', xlab='values of Residuals') dev.off() bitmap(file='test4.png') qqnorm(mysum$resid, main='Residual Normal Q-Q Plot') qqline(mysum$resid) grid() dev.off() (myerror <- as.ts(mysum$resid)) bitmap(file='test5.png') dum <- cbind(lag(myerror,k=1),myerror) dum dum1 <- dum[2:length(myerror),] dum1 z <- as.data.frame(dum1) z plot(z,main=paste('Residual Lag plot, lowess, and regression line'), ylab='values of Residuals', xlab='lagged values of Residuals') lines(lowess(z)) abline(lm(z)) grid() dev.off() bitmap(file='test6.png') acf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Autocorrelation Function') grid() dev.off() bitmap(file='test7.png') pacf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Partial Autocorrelation Function') grid() dev.off() bitmap(file='test8.png') opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0)) plot(mylm, las = 1, sub='Residual Diagnostics') par(opar) dev.off() if (n > n25) { bitmap(file='test9.png') plot(kp3:nmkm3,gqarr[,2], main='Goldfeld-Quandt test',ylab='2-sided p-value',xlab='breakpoint') grid() dev.off() } load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Estimated Regression Equation', 1, TRUE) a<-table.row.end(a) myeq <- colnames(x)[1] myeq <- paste(myeq, '[t] = ', sep='') for (i in 1:k){ if (mysum$coefficients[i,1] > 0) myeq <- paste(myeq, '+', '') myeq <- paste(myeq, mysum$coefficients[i,1], sep=' ') if (rownames(mysum$coefficients)[i] != '(Intercept)') { myeq <- paste(myeq, rownames(mysum$coefficients)[i], sep='') if (rownames(mysum$coefficients)[i] != 't') myeq <- paste(myeq, '[t]', sep='') } } myeq <- paste(myeq, ' + e[t]') a<-table.row.start(a) a<-table.element(a, myeq) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable1.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/ols1.htm','Multiple Linear Regression - Ordinary Least Squares',''), 6, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Variable',header=TRUE) a<-table.element(a,'Parameter',header=TRUE) a<-table.element(a,'S.D.',header=TRUE) a<-table.element(a,'T-STAT<br />H0: parameter = 0',header=TRUE) a<-table.element(a,'2-tail p-value',header=TRUE) a<-table.element(a,'1-tail p-value',header=TRUE) a<-table.row.end(a) for (i in 1:k){ a<-table.row.start(a) a<-table.element(a,rownames(mysum$coefficients)[i],header=TRUE) a<-table.element(a,mysum$coefficients[i,1]) a<-table.element(a, round(mysum$coefficients[i,2],6)) a<-table.element(a, round(mysum$coefficients[i,3],4)) a<-table.element(a, round(mysum$coefficients[i,4],6)) a<-table.element(a, round(mysum$coefficients[i,4]/2,6)) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable2.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Regression Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple R',1,TRUE) a<-table.element(a, sqrt(mysum$r.squared)) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'R-squared',1,TRUE) a<-table.element(a, mysum$r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Adjusted R-squared',1,TRUE) a<-table.element(a, mysum$adj.r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (value)',1,TRUE) a<-table.element(a, mysum$fstatistic[1]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF numerator)',1,TRUE) a<-table.element(a, mysum$fstatistic[2]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF denominator)',1,TRUE) a<-table.element(a, mysum$fstatistic[3]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'p-value',1,TRUE) a<-table.element(a, 1-pf(mysum$fstatistic[1],mysum$fstatistic[2],mysum$fstatistic[3])) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Residual Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Residual Standard Deviation',1,TRUE) a<-table.element(a, mysum$sigma) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Sum Squared Residuals',1,TRUE) a<-table.element(a, sum(myerror*myerror)) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable3.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Actuals, Interpolation, and Residuals', 4, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Time or Index', 1, TRUE) a<-table.element(a, 'Actuals', 1, TRUE) a<-table.element(a, 'Interpolation<br />Forecast', 1, TRUE) a<-table.element(a, 'Residuals<br />Prediction Error', 1, TRUE) a<-table.row.end(a) for (i in 1:n) { a<-table.row.start(a) a<-table.element(a,i, 1, TRUE) a<-table.element(a,x[i]) a<-table.element(a,x[i]-mysum$resid[i]) a<-table.element(a,mysum$resid[i]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable4.tab') if (n > n25) { a<-table.start() a<-table.row.start(a) a<-table.element(a,'Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'p-values',header=TRUE) a<-table.element(a,'Alternative Hypothesis',3,header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'breakpoint index',header=TRUE) a<-table.element(a,'greater',header=TRUE) a<-table.element(a,'2-sided',header=TRUE) a<-table.element(a,'less',header=TRUE) a<-table.row.end(a) for (mypoint in kp3:nmkm3) { a<-table.row.start(a) a<-table.element(a,mypoint,header=TRUE) a<-table.element(a,gqarr[mypoint-kp3+1,1]) a<-table.element(a,gqarr[mypoint-kp3+1,2]) a<-table.element(a,gqarr[mypoint-kp3+1,3]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable5.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Description',header=TRUE) a<-table.element(a,'# significant tests',header=TRUE) a<-table.element(a,'% significant tests',header=TRUE) a<-table.element(a,'OK/NOK',header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'1% type I error level',header=TRUE) a<-table.element(a,numsignificant1) a<-table.element(a,numsignificant1/numgqtests) if (numsignificant1/numgqtests < 0.01) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'5% type I error level',header=TRUE) a<-table.element(a,numsignificant5) a<-table.element(a,numsignificant5/numgqtests) if (numsignificant5/numgqtests < 0.05) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'10% type I error level',header=TRUE) a<-table.element(a,numsignificant10) a<-table.element(a,numsignificant10/numgqtests) if (numsignificant10/numgqtests < 0.1) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable6.tab') }

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