*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: Wed, 18 Nov 2009 08:17:01 -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/18/t1258557595z5tmdmggtyg1etc.htm/, Retrieved Wed, 18 Nov 2009 16:20:07 +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/18/t1258557595z5tmdmggtyg1etc.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:

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No (this computation is public)

User-defined keywords:

Dataseries X:

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819.3 31.3 785.8 849.4 30 819.3 880.4 31.3 849.4 900.1 33 880.4 937.2 31.3 900.1 948.9 29 937.2 952.6 28.7 948.9 947.3 28 952.6 974.2 29.7 947.3 1000.8 30.7 974.2 1032.8 24 1000.8 1050.7 29 1032.8 1057.3 33 1050.7 1075.4 28 1057.3 1118.4 28.7 1075.4 1179.8 31.7 1118.4 1227 34 1179.8 1257.8 35.3 1227 1251.5 27 1257.8 1236.3 31.3 1251.5 1170.6 38.7 1236.3 1213.1 37.3 1170.6 1265.5 37.3 1213.1 1300.8 37.7 1265.5 1348.4 34.7 1300.8 1371.9 34.7 1348.4 1403.3 33.7 1371.9 1451.8 38.3 1403.3 1474.2 38 1451.8 1438.2 38.3 1474.2 1513.6 42.7 1438.2 1562.2 41.7 1513.6 1546.2 39.7 1562.2 1527.5 39.3 1546.2 1418.7 39.3 1527.5 1448.5 37.7 1418.7 1492.1 38.3 1448.5 1395.4 37.7 1492.1 1403.7 37 1395.4 1316.6 34.3 1403.7 1274.5 29.7 1316.6 1264.4 34.7 1274.5 1323.9 32 1264.4 1332.1 30.3 1323.9 1250.2 28.3 1332.1 1096.7 31.3 1250.2 1080.8 17.7 1096.7 1039.2 15.7 1080.8 792 14.3 1039.2 746.6 13.3 792 688.8 11 746.6 715.8 2.7 688.8 672.9 3.3 715.8 629.5 3.7 672.9 681.2 1.4 629 etc...

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 92.4076230512834 + 5.41714696378699X[t] + 0.790454329335046Y1[t] -61.837030919492M1[t] -49.7448230196926M2[t] -25.9779829341109M3[t] -19.7830480253596M4[t] -23.1312901068847M5[t] -41.7635956991291M6[t] + 11.7246097530180M7[t] -3.12435921568095M8[t] -54.2119348467759M9[t] -60.7348926951578M10[t] -17.8696049973595M11[t] + 0.709615656970414t + 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) 92.4076230512834 43.538863 2.1224 0.03934 0.01967 X 5.41714696378699 2.355752 2.2995 0.026168 0.013084 Y1 0.790454329335046 0.080168 9.8599 0 0 M1 -61.837030919492 35.591971 -1.7374 0.08916 0.04458 M2 -49.7448230196926 35.500603 -1.4012 0.168002 0.084001 M3 -25.9779829341109 35.499696 -0.7318 0.468097 0.234048 M4 -19.7830480253596 35.409876 -0.5587 0.579145 0.289572 M5 -23.1312901068847 35.363891 -0.6541 0.516381 0.25819 M6 -41.7635956991291 35.348805 -1.1815 0.243622 0.121811 M7 11.7246097530180 35.310167 0.332 0.741396 0.370698 M8 -3.12435921568095 35.259175 -0.0886 0.929784 0.464892 M9 -54.2119348467759 35.300485 -1.5357 0.131607 0.065804 M10 -60.7348926951578 36.051694 -1.6847 0.098978 0.049489 M11 -17.8696049973595 35.221736 -0.5073 0.61439 0.307195 t 0.709615656970414 0.883399 0.8033 0.426036 0.213018 Multiple Linear Regression - Regression Statistics Multiple R 0.983388137329576 R-squared 0.967052228640533 Adjusted R-squared 0.956801810884254 F-TEST (value) 94.3427137931215 F-TEST (DF numerator) 14 F-TEST (DF denominator) 45 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 55.682735703824 Sum Squared Residuals 139525.517495786 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 819.3 821.975919746773 -2.67591974677275 2 849.4 854.215672283344 -4.81567228334418 3 880.4 909.527094391804 -29.1270943918044 4 900.1 950.14487900535 -50.0448790053506 5 937.2 953.869053030259 -16.6690530302585 6 948.9 952.812780696605 -3.91278069660477 7 952.6 1014.63377336981 -62.0337733698061 8 947.3 999.627098201966 -52.3270982019664 9 974.2 954.268880120804 19.9311198791961 10 1000.8 975.135906352292 25.6640936477077 11 1032.8 1003.44201021 29.3579897899996 12 1050.7 1074.40150422199 -23.7015042219866 13 1057.3 1049.09180930971 8.2081906902895 14 1075.4 1040.02489662116 35.3751033788436 15 1118.4 1082.60057859932 35.7994214006762 16 1179.8 1139.74610621781 40.0538937821864 17 1227 1198.10081363114 28.8991863688593 18 1257.8 1224.52985909340 33.2701409065958 19 1251.5 1258.11135374661 -6.61135374660902 20 1236.3 1262.28587010435 -25.9858701043538 21 1170.6 1239.97989185636 -69.3798918563602 22 1213.1 1174.64969447833 38.4503055216656 23 1265.5 1251.81890682984 13.6810931701575 24 1300.8 1313.98479312684 -13.1847931268438 25 1348.4 1264.50897479849 83.8910252015118 26 1371.9 1314.93642443161 56.9635755683938 27 1403.3 1352.57140994974 50.7285900502549 28 1451.8 1409.21510249001 42.5848975099927 29 1474.2 1443.28836694907 30.9116330509339 30 1438.2 1444.69699808003 -6.49699808003346 31 1513.6 1494.27390997375 19.3260900262478 32 1562.2 1534.31766613010 27.8823338699012 33 1546.2 1511.52149263408 34.6785073659163 34 1527.5 1490.89402238780 36.6059776122033 35 1418.7 1519.687429784 -100.987429784 36 1448.5 1443.59778426462 4.90221573538222 37 1492.1 1409.27619619455 82.8238038054471 38 1395.4 1453.29154033206 -57.8915403320582 39 1403.7 1397.53905955326 6.16094044673937 40 1316.6 1396.37808425024 -79.7780842502385 41 1274.5 1299.97200970718 -25.4720097071809 42 1264.4 1275.85692732584 -11.4569273258365 43 1323.9 1307.44486290645 16.4551370935548 44 1332.1 1331.12839235171 0.97160764828582 45 1250.2 1276.39786395056 -26.1978639505628 46 1096.7 1222.09775307797 -125.397753077972 47 1080.8 1070.66471817231 10.1352818276917 48 1039.2 1065.84142106264 -26.6414210626368 49 792 964.247099950476 -172.247099950476 50 746.6 776.231466331835 -29.6314663318349 51 688.8 752.361857505866 -63.561857505866 52 715.8 668.61582803659 47.18417196341 53 672.9 690.569756682354 -17.6697566823538 54 629.5 640.903434804121 -11.4034348041211 55 681.2 648.336100003388 32.8638999966125 56 755.4 705.940973211867 49.4590267881332 57 760.6 719.63187143819 40.9681285618105 58 765.9 741.222623703604 24.6773762963956 59 836.8 788.986935003849 47.8130649961511 60 904.9 846.274497323915 58.6255026760848 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.0387625954978061 0.0775251909956121 0.961237404502194 19 0.0120227114788719 0.0240454229577438 0.987977288521128 20 0.00325605085823501 0.00651210171647002 0.996743949141765 21 0.0171715220758201 0.0343430441516401 0.98282847792418 22 0.00752115622085109 0.0150423124417022 0.992478843779149 23 0.00394294142774668 0.00788588285549336 0.996057058572253 24 0.00308468908071918 0.00616937816143836 0.99691531091928 25 0.00137300328602345 0.00274600657204690 0.998626996713976 26 0.000503559068111095 0.00100711813622219 0.99949644093189 27 0.000190634219282138 0.000381268438564275 0.999809365780718 28 6.38767551873373e-05 0.000127753510374675 0.999936123244813 29 2.10937169765883e-05 4.21874339531765e-05 0.999978906283023 30 4.19865491984932e-05 8.39730983969863e-05 0.999958013450801 31 6.94907217422868e-05 0.000138981443484574 0.999930509278258 32 0.000116343410435834 0.000232686820871669 0.999883656589564 33 5.60350947121674e-05 0.000112070189424335 0.999943964905288 34 0.000154146245093731 0.000308292490187463 0.999845853754906 35 0.00800885475735836 0.0160177095147167 0.991991145242642 36 0.00840752205448907 0.0168150441089781 0.99159247794551 37 0.148092165358697 0.296184330717394 0.851907834641303 38 0.238677624092011 0.477355248184022 0.761322375907989 39 0.441288135225498 0.882576270450997 0.558711864774502 40 0.613297750434857 0.773404499130286 0.386702249565143 41 0.499135650257685 0.99827130051537 0.500864349742315 42 0.425111162736333 0.850222325472665 0.574888837263667 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 13 0.52 NOK 5% type I error level 18 0.72 NOK 10% type I error level 19 0.76 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = 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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