Multiple regression

*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: Mon, 27 Dec 2010 10:06:13 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/27/t1293444272dyug7zoyn80o460.htm/, Retrieved Mon, 27 Dec 2010 11:04:32 +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/2010/Dec/27/t1293444272dyug7zoyn80o460.htm/}, year = {2010}, } @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 = {2010}, 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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14450609152 9119000 15369578496 9166000 16440317952 9218000 17674829824 9283000 19782035456 9367000 21531359232 9448000 23118405632 9508000 24779487232 9557000 26495297536 9590000 29388689408 9613000 32676600000 9638000 35799700000 9673000 40077000000 9709000 45536800000 9738000 53409700000 9768000 59108700000 9795000 67180700000 9811000 72656600000 9822000 78026600000 9830000 83450800000 9837000 90756100000 9847000 95153800000 9852000 102966000000 9856000 109085000000 9856000 117854000000 9853000 125345000000 9858000 131200000000 9862000 136486000000 9870000 146033000000 9902000 158348000000 9938000 167909000000 9967400 175906000000 10004500 184714000000 10045000 190243000000 10084500 200495000000 10115600 207782000000 10136800 211399000000 10157000 221184000000 10181000 229572000000 10203000 238248000000 10226000 251741000000 10252000 258883000000 10287000 267652000000 10333000 274726000000 10376080,14 290825000000 10421120,61 302845000000 10478650 318193000000 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 5 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 R Framework error message The field 'Names of X columns' contains a hard return which cannot be interpreted. Please, resubmit your request without hard returns in the 'Names of X columns'. Multiple Linear Regression - Estimated Regression Equation Population [t] = + 9269874.88784229 + 1.37551473772754e-07GDP[t] + 24836.8461074138t + 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) 9269874.88784229 38791.749789 238.9651 0 0 GDP 1.37551473772754e-07 1e-06 0.1923 0.848332 0.424166 t 24836.8461074138 5090.182232 4.8794 1.3e-05 6e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.970485263345497 R-squared 0.941841646370779 Adjusted R-squared 0.939366822812088 F-TEST (value) 380.569209899239 F-TEST (DF numerator) 2 F-TEST (DF denominator) 47 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 95424.510321447 Sum Squared Residuals 427974346994.134 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 9119000 9296699.43653547 -177699.436535474 2 9166000 9321662.6882305 -155662.688230507 3 9218000 9346646.81612812 -128646.816128121 4 9283000 9371653.47116292 -88653.471162918 5 9367000 9396780.16651056 -29780.1665105557 6 9448000 9421857.63468146 26142.3653185359 7 9508000 9446912.78136014 61087.2186398563 8 9557000 9471978.1116897 85021.8883103057 9 9590000 9497050.97003314 92949.0299668622 10 9613000 9522285.80645675 90714.1935432527 11 9638000 9547574.90951172 90425.0904882762 12 9673000 9572841.34262688 100158.657373123 13 9709000 9598266.53765306 110733.462346941 14 9738000 9623854.38729698 114145.612703022 15 9768000 9649774.16240226 118225.837597743 16 9795000 9675394.9143587 119605.085641298 17 9811000 9701342.0759624 109657.924037591 18 9822000 9726932.14018505 95067.8598149447 19 9830000 9752507.63770663 77492.3622933712 20 9837000 9778090.59051808 58909.4094819192 21 9847000 9803932.29140685 43067.7085931533 22 9852000 9829374.04763047 22625.9523695291 23 9856000 9855285.4733613 714.526638707701 24 9856000 9880963.99693672 -24963.9969367216 25 9853000 9907007.03191765 -54007.0319176487 26 9858000 9932874.2761151 -74874.2761150943 27 9862000 9958516.48610145 -96516.4861014475 28 9870000 9984080.42929922 -114080.429299224 29 9902000 10010230.4793267 -108230.479326746 30 9938000 10036761.2718337 -98761.2718336718 31 9967400 10062913.2475818 -95513.247581827 32 10004500 10088850.092825 -84350.0928250015 33 10045000 10114898.4923134 -69898.4923134057 34 10084500 10140495.8605193 -55995.8605193091 35 10115600 10166742.8843358 -51142.8843358412 36 10136800 10192582.0680326 -55782.0680326371 37 10157000 10217916.4378207 -60916.437820687 38 10181000 10244099.2250990 -63099.2250989672 39 10203000 10270089.8529684 -67089.8529683869 40 10226000 10296120.0956623 -70120.0956622531 41 10252000 10322812.9238053 -70812.9238052827 42 10287000 10348632.1625384 -61632.1625383816 43 10333000 10374675.1975193 -41675.1975193087 44 10376080.14 10400485.0827522 -24404.9427521903 45 10421120.61 10427536.3700359 -6415.7600358729 46 10478650 10454026.5848580 24623.4151419654 47 10547958 10480974.5709849 66983.4290150873 48 10625700 10508116.0920354 117583.907964611 49 10708433 10534291.0388797 174141.961120336 50 10788760 10558111.1044929 230648.89550705 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.000144988924353674 0.000289977848707347 0.999855011075646 7 6.86218830065369e-05 0.000137243766013074 0.999931378116994 8 0.000706354526053009 0.00141270905210602 0.999293645473947 9 0.00982478650988757 0.0196495730197751 0.990175213490112 10 0.0452864120109638 0.0905728240219275 0.954713587989036 11 0.0411102451402387 0.0822204902804774 0.958889754859761 12 0.0265089916866627 0.0530179833733254 0.973491008313337 13 0.0215677157547557 0.0431354315095114 0.978432284245244 14 0.0241323003266545 0.0482646006533089 0.975867699673346 15 0.0433268542903101 0.0866537085806201 0.95667314570969 16 0.0342072923326275 0.0684145846652549 0.965792707667373 17 0.0252505611889102 0.0505011223778204 0.97474943881109 18 0.0142621151507254 0.0285242303014508 0.985737884849275 19 0.0089616142217367 0.0179232284434734 0.991038385778263 20 0.00761436192255568 0.0152287238451114 0.992385638077444 21 0.00582208000014989 0.0116441600002998 0.99417791999985 22 0.00988872152805141 0.0197774430561028 0.990111278471949 23 0.0122742439322062 0.0245484878644124 0.987725756067794 24 0.0193204020045003 0.0386408040090006 0.9806795979955 25 0.0167819664814713 0.0335639329629426 0.983218033518529 26 0.0117910249863205 0.0235820499726409 0.98820897501368 27 0.0083124061160059 0.0166248122320118 0.991687593883994 28 0.00656769398304549 0.0131353879660910 0.993432306016955 29 0.0053041256533737 0.0106082513067474 0.994695874346626 30 0.0398853283813295 0.079770656762659 0.96011467161867 31 0.147442820250706 0.294885640501412 0.852557179749294 32 0.286349701015347 0.572699402030695 0.713650298984653 33 0.437601639351946 0.875203278703892 0.562398360648054 34 0.549141488260151 0.901717023479698 0.450858511739849 35 0.658814783762889 0.682370432474222 0.341185216237111 36 0.731689400125684 0.536621199748632 0.268310599874316 37 0.806005029329621 0.387989941340757 0.193994970670379 38 0.882170600744 0.235658798511998 0.117829399255999 39 0.937912377199328 0.124175245601345 0.0620876228006724 40 0.967463623826817 0.0650727523463658 0.0325363761731829 41 0.982697893223315 0.0346042135533700 0.0173021067766850 42 0.988791153103486 0.0224176937930278 0.0112088468965139 43 0.995224945328499 0.00955010934300243 0.00477505467150122 44 0.996353948903291 0.00729210219341713 0.00364605109670856 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 5 0.128205128205128 NOK 5% type I error level 22 0.564102564102564 NOK 10% type I error level 30 0.76923076923077 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; par2 = Do not include Seasonal 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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