*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, 10 Dec 2010 11:49:39 +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/10/t1291981716mwn5ae58294aetd.htm/, Retrieved Fri, 10 Dec 2010 12:50:24 +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/2010/Dec/10/t1291981716mwn5ae58294aetd.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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8587 0 9743 9084 9081 9700 9731 0 8587 9743 9084 9081 9563 0 9731 8587 9743 9084 9998 0 9563 9731 8587 9743 9437 0 9998 9563 9731 8587 10038 0 9437 9998 9563 9731 9918 0 10038 9437 9998 9563 9252 0 9918 10038 9437 9998 9737 0 9252 9918 10038 9437 9035 0 9737 9252 9918 10038 9133 0 9035 9737 9252 9918 9487 0 9133 9035 9737 9252 8700 0 9487 9133 9035 9737 9627 0 8700 9487 9133 9035 8947 0 9627 8700 9487 9133 9283 0 8947 9627 8700 9487 8829 0 9283 8947 9627 8700 9947 0 8829 9283 8947 9627 9628 0 9947 8829 9283 8947 9318 0 9628 9947 8829 9283 9605 0 9318 9628 9947 8829 8640 0 9605 9318 9628 9947 9214 0 8640 9605 9318 9628 9567 0 9214 8640 9605 9318 8547 0 9567 9214 8640 9605 9185 0 8547 9567 9214 8640 9470 0 9185 8547 9567 9214 9123 0 9470 9185 8547 9567 9278 0 9123 9470 9185 8547 10170 0 9278 9123 9470 9185 9434 0 10170 9278 9123 9470 9655 0 9434 10170 9278 9123 9429 0 9655 9434 10170 9278 8739 0 9429 9655 9434 10170 9552 0 8739 9429 9655 9434 9687 0 9552 8739 9429 9655 9019 1 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 8 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Birth[t] = + 4305.74116897413 + 111.797019312520x[t] + 0.114363609997994`(t-1)(t-2)`[t] + 0.149789773200904`(t-3)`[t] + 0.231900872212371`(t-4)`[t] + 0.0557474763800532V6[t] -796.750672653784M1[t] + 162.958810260578M2[t] -278.392163261884M3[t] -15.4247378444073M4[t] -203.396827883287M5[t] + 379.290332668419M6[t] + 174.162983151376M7[t] -124.869742601276M8[t] -138.797056360805M9[t] -872.609202075023M10[t] -325.272262586112M11[t] + 3.6537386110181t + 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) 4305.74116897413 1689.958097 2.5478 0.013775 0.006888 x 111.797019312520 141.183035 0.7919 0.431973 0.215986 `(t-1)(t-2)` 0.114363609997994 0.144642 0.7907 0.432663 0.216331 `(t-3)` 0.149789773200904 0.138151 1.0842 0.283164 0.141582 `(t-4)` 0.231900872212371 0.138267 1.6772 0.099392 0.049696 V6 0.0557474763800532 0.146133 0.3815 0.704369 0.352184 M1 -796.750672653784 211.414945 -3.7687 0.000414 0.000207 M2 162.958810260578 238.077963 0.6845 0.496657 0.248328 M3 -278.392163261884 175.876762 -1.5829 0.119398 0.059699 M4 -15.4247378444073 241.109457 -0.064 0.949232 0.474616 M5 -203.396827883287 220.110027 -0.9241 0.359639 0.179819 M6 379.290332668419 191.053241 1.9853 0.052301 0.02615 M7 174.162983151376 208.618374 0.8348 0.407555 0.203778 M8 -124.869742601276 236.09641 -0.5289 0.599088 0.299544 M9 -138.797056360805 216.343578 -0.6416 0.523925 0.261962 M10 -872.609202075023 192.812809 -4.5257 3.4e-05 1.7e-05 M11 -325.272262586112 220.447326 -1.4755 0.145992 0.072996 t 3.6537386110181 3.515962 1.0392 0.303436 0.151718 Multiple Linear Regression - Regression Statistics Multiple R 0.885237692500969 R-squared 0.78364577222444 Adjusted R-squared 0.714249133126619 F-TEST (value) 11.2922726865752 F-TEST (DF numerator) 17 F-TEST (DF denominator) 53 p-value 4.21707113673619e-12 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 272.147760915185 Sum Squared Residuals 3925413.39987088 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8587 8634.22152834587 -47.2215283458666 2 9731 9530.27989199037 200.720108009635 3 9563 9203.24756631348 359.752433686525 4 9998 9390.6753230611 607.324676938906 5 9437 9431.79097520022 5.20902479978107 6 10038 10043.9482069436 -5.94820694356449 7 9918 9918.68636626116 -0.686366261159808 8 9252 9593.7611625277 -341.761162527693 9 9737 9597.44474028683 139.555259713166 10 9035 8828.66882371979 206.331176280213 11 9133 9210.88860954452 -77.8886095445173 12 9487 9521.2139274883 -34.2139274883007 13 8700 8647.52422290975 52.4757770902457 14 9627 9557.50042013785 69.4995798621518 15 8947 9195.48986163386 -248.489861633859 16 9283 9360.42751082836 -77.4275108283573 17 8829 9283.77713121297 -454.777131212974 18 9947 9762.51163273201 184.488367267991 19 9628 9660.90238989545 -32.9023898954528 20 9318 9409.95453368235 -91.95453368235 21 9605 9550.40112264026 54.5988773597412 22 8640 8794.97954227136 -154.979542271357 23 9214 9188.92628628081 25.0737137191913 24 9567 9488.22370112505 78.7762988749507 25 8547 8613.69163526503 -66.6916352650307 26 9185 9592.59455047553 -407.594550475526 27 9470 9188.935789411 281.064210589001 28 9123 9366.85642709664 -243.856427096638 29 9278 9276.63431892557 1.36568107443183 30 10170 9930.38316484827 239.616835151732 31 9434 9789.55773701722 -355.557737017217 32 9655 9560.2198715013 94.7801284986962 33 9429 9680.47181793883 -251.471817938827 34 8739 8836.61848183618 -97.6184818361834 35 9552 9285.06573043731 266.934269562693 36 9687 9563.52499821418 123.475001785821 37 9019 8846.83272495757 172.167275042432 38 9672 9904.09232479283 -232.092324792831 39 9206 9517.64427475753 -311.644274757528 40 9069 9681.4008451006 -612.400845100591 41 9788 9525.80460012419 262.195399875815 42 10312 10102.1890315722 209.810968427847 43 10105 10010.5920557503 94.4079442496724 44 9863 9929.128965353 -66.1289653530095 45 9656 10021.7714060889 -365.771406088937 46 9295 9212.89880367672 82.1011963232786 47 9946 9623.93799682872 322.062003171277 48 9701 9911.74623017709 -210.746230177087 49 9049 9092.88741155926 -43.8874115592631 50 10190 10075.8296937688 114.170306231216 51 9706 9650.43429916945 55.565700830553 52 9765 9867.75610678556 -102.756106785564 53 9893 9845.93849871287 47.0615012871326 54 9994 10407.1233849730 -413.123384973042 55 10433 10223.0739625391 209.926037460885 56 10073 10026.0017800295 46.9982199705025 57 10112 10070.872680787 41.1273192129988 58 9266 9398.68511413702 -132.685114137015 59 9820 9799.75480746787 20.2451925321346 60 10097 10054.2911429954 42.708857004617 61 9115 9181.84247696252 -66.8424769625176 62 10411 10155.7031188346 255.296881165353 63 9678 9814.2482087147 -136.248208714693 64 10408 9978.88378712776 429.116212872245 65 10153 10014.0544758242 138.945524175813 66 10368 10582.8445789310 -214.844578930964 67 10581 10496.1874885367 84.812511463271 68 10597 10238.9336869061 358.066313093854 69 10680 10298.0382322581 381.961767741859 70 9738 9641.14923435894 96.8507656410638 71 9556 10112.4265694408 -556.426569440779 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.822170479400922 0.355659041198156 0.177829520599078 22 0.726284324877526 0.547431350244949 0.273715675122474 23 0.734119060199927 0.531761879600146 0.265880939800073 24 0.695321580669837 0.609356838660326 0.304678419330163 25 0.616612185510457 0.766775628979085 0.383387814489543 26 0.564900672616462 0.870198654767076 0.435099327383538 27 0.770796157934591 0.458407684130818 0.229203842065409 28 0.729656332695517 0.540687334608966 0.270343667304483 29 0.667193483222985 0.665613033554029 0.332806516777015 30 0.835345797250065 0.329308405499870 0.164654202749935 31 0.802955190485801 0.394089619028397 0.197044809514199 32 0.790149238737184 0.419701522525633 0.209850761262816 33 0.717302877979736 0.565394244040528 0.282697122020264 34 0.716175228965786 0.567649542068428 0.283824771034214 35 0.701417381982728 0.597165236034545 0.298582618017272 36 0.677272261782925 0.64545547643415 0.322727738217075 37 0.625859979038627 0.748280041922745 0.374140020961373 38 0.557127243813863 0.885745512372273 0.442872756186136 39 0.490098592858451 0.980197185716903 0.509901407141549 40 0.70738397190228 0.585232056195442 0.292616028097721 41 0.770074873307943 0.459850253384113 0.229925126692057 42 0.736410679988357 0.527178640023286 0.263589320011643 43 0.703411309279631 0.593177381440737 0.296588690720369 44 0.64568282354887 0.708634352902261 0.354317176451130 45 0.570421962329166 0.859156075341667 0.429578037670834 46 0.474535533795958 0.949071067591917 0.525464466204042 47 0.732009170832274 0.535981658335451 0.267990829167726 48 0.609532596311674 0.780934807376653 0.390467403688326 49 0.834080592115458 0.331838815769083 0.165919407884542 50 0.707734470138124 0.584531059723752 0.292265529861876 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 0 0 OK

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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