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: Sat, 21 Nov 2009 08:19:01 -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/21/t125881709153eu5lsk795vxfg.htm/, Retrieved Sat, 21 Nov 2009 16:25:03 +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/21/t125881709153eu5lsk795vxfg.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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562000 4814 561000 3908 555000 5250 544000 3937 537000 4004 543000 5560 594000 3922 611000 3759 613000 4138 611000 4634 594000 3996 595000 4308 591000 4143 589000 4429 584000 5219 573000 4929 567000 5755 569000 5592 621000 4163 629000 4962 628000 5208 612000 4755 595000 4491 597000 5732 593000 5731 590000 5040 580000 6102 574000 4904 573000 5369 573000 5578 620000 4619 626000 4731 620000 5011 588000 5299 566000 4146 557000 4625 561000 4736 549000 4219 532000 5116 526000 4205 511000 4121 499000 5103 555000 4300 565000 4578 542000 3809 527000 5526 510000 4247 514000 3830 517000 4394 508000 4826 493000 4409 490000 4569 469000 4106 478000 4794 528000 3914 534000 3793 518000 4405 506000 4022 502000 4100 516000 4788

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 werkloos[t] = + 566709.674302673 + 8.7012473517102bouw[t] -1553.21484229700t + 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) 566709.674302673 34135.292382 16.6019 0 0 bouw 8.7012473517102 6.904233 1.2603 0.212704 0.106352 t -1553.21484229700 237.173027 -6.5489 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.675859075262134 R-squared 0.456785489614188 Adjusted R-squared 0.437725331355036 F-TEST (value) 23.9654615351828 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 2.79694436500222e-08 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 31471.1759586322 Sum Squared Residuals 56454790224.4939 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 562000 607044.264211509 -45044.2642115087 2 561000 597607.719268562 -36607.7192685623 3 555000 607731.57837226 -52731.5783722604 4 544000 594753.625757168 -50753.6257571679 5 537000 593783.394487435 -56783.3944874355 6 543000 605769.3205244 -62769.3205243995 7 594000 589963.462520001 4036.53747999876 8 611000 586991.944359376 24008.0556406245 9 613000 588736.502263377 24263.4977366233 10 611000 591499.106107528 19500.8938924721 11 594000 584394.49545484 9605.5045451602 12 595000 585556.069786276 9443.93021372361 13 591000 582567.149130947 8432.85086905279 14 589000 583502.491031239 5497.50896876067 15 584000 588823.261596793 -4823.26159679338 16 573000 584746.6850225 -11746.6850225004 17 567000 590380.700492716 -23380.7004927160 18 569000 587409.18233209 -18409.1823320903 19 621000 573421.885024199 47578.1149758006 20 629000 578820.966815919 50179.0331840811 21 628000 579408.258822143 48591.7411778574 22 612000 573913.378929521 38086.6210704791 23 595000 570063.034786372 24936.9652136276 24 597000 579308.067907548 17691.9320924523 25 593000 577746.151817899 15253.8481821010 26 590000 570180.37505557 19819.6249444297 27 580000 577867.88490079 2132.11509921049 28 574000 565890.575731144 8109.4242688563 29 573000 568383.440907392 4616.55909260805 30 573000 568648.786761602 4351.21323839762 31 620000 558751.075709015 61248.9242909847 32 626000 558172.40057011 67827.5994298901 33 620000 559055.534986292 60944.4650137083 34 588000 560008.279381287 27991.7206187128 35 566000 548422.526342468 17577.4736575316 36 557000 551037.208981641 5962.79101835942 37 561000 550449.832595383 10550.1674046166 38 549000 544398.072872252 4601.92712774775 39 532000 550649.876904439 -18649.8769044393 40 526000 541169.825724734 -15169.8257247343 41 511000 538885.706104894 -27885.7061048937 42 499000 545877.116161976 -46877.1161619761 43 555000 537336.799696256 17663.2003037442 44 565000 538202.531617734 26797.4683822658 45 542000 529958.057561972 12041.9424380279 46 527000 543344.884422562 -16344.8844225615 47 510000 530662.774217427 -20662.7742174272 48 514000 525481.139229467 -11481.139229467 49 517000 528835.427893535 -11835.4278935346 50 508000 531041.151907176 -23041.1519071764 51 493000 525859.516919216 -32859.5169192162 52 490000 525698.501653193 -35698.5016531928 53 469000 520116.609287054 -51116.609287054 54 478000 524549.852622734 -46549.8526227336 55 528000 515339.540110932 12660.4598890683 56 534000 512733.474339078 21266.5256609222 57 518000 516505.422876027 1494.57712397261 58 506000 511619.630298025 -5619.63029802538 59 502000 510745.112749162 -8745.1127491618 60 516000 515178.356084841 821.643915158606 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.00594499727243689 0.0118899945448738 0.994055002727563 7 0.457778805931046 0.915557611862092 0.542221194068954 8 0.537328090090084 0.925343819819832 0.462671909909916 9 0.478681672649291 0.957363345298582 0.521318327350709 10 0.374373442213613 0.748746884427226 0.625626557786387 11 0.329155367709836 0.658310735419673 0.670844632290164 12 0.27235207325347 0.54470414650694 0.72764792674653 13 0.25334464387983 0.50668928775966 0.74665535612017 14 0.231745554439286 0.463491108878572 0.768254445560714 15 0.196092996413406 0.392185992826813 0.803907003586594 16 0.239012927310714 0.478025854621428 0.760987072689286 17 0.24500182496577 0.49000364993154 0.75499817503423 18 0.263675052244097 0.527350104488195 0.736324947755903 19 0.203726416215138 0.407452832430277 0.796273583784862 20 0.217785096568249 0.435570193136497 0.782214903431751 21 0.214597567807323 0.429195135614645 0.785402432192677 22 0.163631254703716 0.327262509407432 0.836368745296284 23 0.184396541426512 0.368793082853024 0.815603458573488 24 0.136055438208542 0.272110876417084 0.863944561791458 25 0.101630972728842 0.203261945457684 0.898369027271158 26 0.100934616515958 0.201869233031916 0.899065383484042 27 0.0794340772528003 0.158868154505601 0.9205659227472 28 0.130675944783649 0.261351889567298 0.869324055216351 29 0.141341487881805 0.282682975763610 0.858658512118195 30 0.132986338984222 0.265972677968445 0.867013661015778 31 0.131113409529845 0.262226819059689 0.868886590470155 32 0.190416830872246 0.380833661744493 0.809583169127754 33 0.335794291455472 0.671588582910944 0.664205708544528 34 0.436006643826852 0.872013287653705 0.563993356173148 35 0.61189372112511 0.776212557749779 0.388106278874890 36 0.69703938116491 0.605921237670179 0.302960618835090 37 0.751233481282891 0.497533037434217 0.248766518717109 38 0.786882205727622 0.426235588544757 0.213117794272378 39 0.817189727234405 0.365620545531190 0.182810272765595 40 0.835841209809425 0.328317580381149 0.164158790190575 41 0.879424388839048 0.241151222321905 0.120575611160952 42 0.923707024032912 0.152585951934175 0.0762929759670875 43 0.908023772639457 0.183952454721085 0.0919762273605427 44 0.952654899606659 0.0946902007866824 0.0473451003933412 45 0.952380392342353 0.0952392153152936 0.0476196076576468 46 0.968956135623123 0.0620877287537536 0.0310438643768768 47 0.953782889758618 0.0924342204827637 0.0462171102413818 48 0.925673642562453 0.148652714875093 0.0743263574375466 49 0.917255161865474 0.165489676269052 0.0827448381345262 50 0.931284386184135 0.13743122763173 0.068715613815865 51 0.891909434280995 0.216181131438009 0.108090565719005 52 0.835805940209643 0.328388119580713 0.164194059790357 53 0.887074120015914 0.225851759968172 0.112925879984086 54 0.967278796204977 0.0654424075900452 0.0327212037950226 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 1 0.0204081632653061 OK 10% type I error level 6 0.122448979591837 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; 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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