*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 07:12:05 -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/t1259331192s875vzx0fpwkqg4.htm/, Retrieved Fri, 27 Nov 2009 15:13:24 +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/t1259331192s875vzx0fpwkqg4.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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation S&P500[t] = + 966.129982902622 + 0.43035784722686Gold[t] + 79.2882926988925M1[t] + 60.6832961939366M2[t] + 44.0736579306548M3[t] + 45.6598105362746M4[t] + 63.6943256675065M5[t] + 78.8140598459922M6[t] + 48.70465990289M7[t] + 40.1346426021993M8[t] + 56.910689036992M9[t] + 40.5558293597744M10[t] + 19.6090020272143M11[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) 966.129982902622 87.792906 11.0046 0 0 Gold 0.43035784722686 0.100599 4.278 9.2e-05 4.6e-05 M1 79.2882926988925 85.72947 0.9249 0.35976 0.17988 M2 60.6832961939366 85.608012 0.7089 0.481919 0.240959 M3 44.0736579306548 85.539391 0.5152 0.608798 0.304399 M4 45.6598105362746 85.58575 0.5335 0.596203 0.298102 M5 63.6943256675065 85.570508 0.7443 0.460371 0.230185 M6 78.8140598459922 85.581723 0.9209 0.361794 0.180897 M7 48.70465990289 85.546944 0.5693 0.571842 0.285921 M8 40.1346426021993 85.549242 0.4691 0.641138 0.320569 M9 56.910689036992 85.607917 0.6648 0.509438 0.254719 M10 40.5558293597744 85.519799 0.4742 0.637534 0.318767 M11 19.6090020272143 85.559441 0.2292 0.819719 0.409859 Multiple Linear Regression - Regression Statistics Multiple R 0.539236875404102 R-squared 0.290776407795579 Adjusted R-squared 0.109698043828493 F-TEST (value) 1.60580425747845 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0.122579274521098 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 135.206812857309 Sum Squared Residuals 859201.465422471 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1111.92 1224.55472950969 -112.634729509693 2 1131.13 1198.24632753938 -67.1163275393772 3 1144.94 1182.34677972402 -37.4067797240199 4 1113.89 1195.66018366657 -81.7701836665715 5 1107.3 1198.20181629764 -90.9018162976366 6 1120.68 1215.88217966712 -95.2021796671222 7 1140.84 1184.73992089068 -43.8999208906755 8 1101.72 1174.74972269414 -73.0297226941361 9 1104.24 1198.47604836164 -94.2360483616427 10 1114.58 1186.61842818795 -72.0384281879458 11 1130.2 1170.40553717488 -40.2055371748809 12 1173.78 1161.01753401930 12.7624659806953 13 1211.92 1229.50384475280 -17.5838447528029 14 1181.27 1207.95089699434 -26.6808969943431 15 1203.6 1196.74224971376 6.85775028624147 16 1180.59 1195.61714788185 -15.027147881849 17 1156.85 1213.9744313985 -57.1244313985011 18 1191.5 1223.69317459429 -32.1931745942897 19 1191.33 1201.00744751585 -9.67744751585082 20 1234.18 1192.02859026029 42.1514097397056 21 1220.33 1212.22598158054 8.1040184194591 22 1228.81 1207.27560485483 21.5343951451650 23 1207.01 1183.48841573058 23.5215842694224 24 1249.48 1181.20131705424 68.2786829457556 25 1248.29 1273.50793463175 -25.2179346317495 26 1280.08 1271.36412578322 8.715874216779 27 1280.66 1253.03305613103 27.6269438689684 28 1302.88 1264.40984976106 38.4701502389376 29 1310.61 1314.29084558708 -3.68084558708223 30 1270.05 1313.9176972654 -43.8676972654009 31 1270.06 1282.92606373548 -12.8660637354836 32 1278.53 1280.51016365014 -1.98016365013698 33 1303.8 1290.31441295985 13.4855870401454 34 1335.83 1265.15873530685 70.6712646931523 35 1377.76 1250.02173891185 127.73826108815 36 1400.63 1245.32463629105 155.305363708954 37 1418.03 1320.73970836490 97.2902916351026 38 1437.9 1310.93552983573 126.964470164269 39 1406.8 1298.71554161416 108.084458385837 40 1420.83 1295.07284637598 125.757153624024 41 1482.37 1319.71335446214 162.656645537859 42 1530.63 1331.77754792532 198.852452074685 43 1504.66 1296.61144327730 208.048556722702 44 1455.18 1292.77536229610 162.404637703898 45 1473.96 1312.24114527606 161.718854723937 46 1527.29 1326.22651382834 201.063486171661 47 1545.79 1325.82927370086 219.960726299138 48 1479.63 1303.63812459029 175.991875409714 49 1467.97 1409.82378274086 58.1462172591427 50 1378.6 1420.48311984733 -41.8831198473281 51 1330.45 1435.61237281703 -105.162372817027 52 1326.41 1393.83997231454 -67.4299723145406 53 1385.97 1396.91955225464 -10.9495522546392 54 1399.62 1427.20940054787 -27.5894005478719 55 1276.69 1418.29512458069 -141.605124580692 56 1269.42 1398.96616109933 -129.54616109933 57 1287.83 1376.9024118219 -89.0724118218989 58 1164.17 1385.40071782203 -221.230717822032 59 968.67 1299.68503448183 -331.01503448183 60 888.61 1300.94838804512 -412.338388045118 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0261477322182245 0.052295464436449 0.973852267781776 17 0.00694502732620366 0.0138900546524073 0.993054972673796 18 0.00189743461728283 0.00379486923456566 0.998102565382717 19 0.000451033835614018 0.000902067671228036 0.999548966164386 20 0.000149900895380581 0.000299801790761163 0.99985009910462 21 4.44691646432391e-05 8.89383292864782e-05 0.999955530835357 22 8.66445843210221e-06 1.73289168642044e-05 0.999991335541568 23 1.57261827453181e-06 3.14523654906362e-06 0.999998427381726 24 3.68782019236857e-07 7.37564038473714e-07 0.99999963121798 25 2.70900057744506e-06 5.41800115489012e-06 0.999997290999423 26 2.20614485988824e-06 4.41228971977648e-06 0.99999779385514 27 8.47269116020372e-07 1.69453823204074e-06 0.999999152730884 28 2.01188439286955e-07 4.0237687857391e-07 0.99999979881156 29 8.94396251969942e-08 1.78879250393988e-07 0.999999910560375 30 8.03497798793192e-08 1.60699559758638e-07 0.99999991965022 31 4.45784081335017e-08 8.91568162670033e-08 0.999999955421592 32 1.9931616406731e-08 3.9863232813462e-08 0.999999980068384 33 5.21678241684605e-09 1.04335648336921e-08 0.999999994783218 34 2.10739001335331e-09 4.21478002670661e-09 0.99999999789261 35 1.55453700447082e-09 3.10907400894164e-09 0.999999998445463 36 7.45988643266476e-10 1.49197728653295e-09 0.999999999254011 37 6.6279610160492e-10 1.32559220320984e-09 0.999999999337204 38 3.61574508232931e-10 7.23149016465861e-10 0.999999999638425 39 8.63274522516247e-11 1.72654904503249e-10 0.999999999913673 40 3.29534367840143e-11 6.59068735680287e-11 0.999999999967047 41 6.50325314630043e-11 1.30065062926009e-10 0.999999999934967 42 2.23947639891672e-10 4.47895279783344e-10 0.999999999776052 43 2.05135208348553e-10 4.10270416697106e-10 0.999999999794865 44 4.29151088698244e-11 8.58302177396488e-11 0.999999999957085 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 27 0.93103448275862 NOK 5% type I error level 28 0.96551724137931 NOK 10% type I error level 29 1 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 = Include Monthly 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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