*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, 20 Nov 2009 01:56:31 -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/20/t1258707523ipmecoxbp69194f.htm/, Retrieved Fri, 20 Nov 2009 09:58:55 +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/20/t1258707523ipmecoxbp69194f.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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User-defined keywords:

Dataseries X:

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8.2 267722 8 266003 7.9 262971 7.6 265521 7.6 264676 8.3 270223 8.4 269508 8.4 268457 8.4 265814 8.4 266680 8.6 263018 8.9 269285 8.8 269829 8.3 270911 7.5 266844 7.2 271244 7.4 269907 8.8 271296 9.3 270157 9.3 271322 8.7 267179 8.2 264101 8.3 265518 8.5 269419 8.6 268714 8.5 272482 8.2 268351 8.1 268175 7.9 270674 8.6 272764 8.7 272599 8.7 270333 8.5 270846 8.4 270491 8.5 269160 8.7 274027 8.7 273784 8.6 276663 8.5 274525 8.3 271344 8 271115 8.2 270798 8.1 273911 8.1 273985 8 271917 7.9 273338 7.9 270601 8 273547 8 275363 7.9 281229 8 277793 7.7 279913 7.2 282500 7.5 280041 7.3 282166 7 290304 7 283519 7 287816 7.2 285226 7.3 287595

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 wkh[t] = + 21.4194329301315 -4.87828410082541e-05los[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) 21.4194329301315 2.667658 8.0293 0 0 los -4.87828410082541e-05 1e-05 -4.983 6e-06 3e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.547512698653501 R-squared 0.29977015518684 Adjusted R-squared 0.287697226827992 F-TEST (value) 24.8299456665917 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 5.97563241044874e-06 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.468312733722697 Sum Squared Residuals 12.7203753608759 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.2 8.35919316971971 -0.159193169719713 2 8 8.44305087341287 -0.443050873412869 3 7.9 8.5909604473499 -0.690960447349894 4 7.6 8.46656420277885 -0.866564202778847 5 7.6 8.50778570343082 -0.907785703430822 6 8.3 8.23718728435804 0.062812715641965 7 8.4 8.27206701567894 0.127932984321063 8 8.4 8.32333778157861 0.0766622184213878 9 8.4 8.45227083036343 -0.0522708303634279 10 8.4 8.41002489005028 -0.0100248900502799 11 8.6 8.5886676538225 0.0113323461774927 12 8.9 8.28294558922378 0.617054410776222 13 8.8 8.25640772371529 0.543592276284713 14 8.3 8.20362468974436 0.0963753102556438 15 7.5 8.40202450412493 -0.902024504124927 16 7.2 8.1873800036886 -0.987380003688608 17 7.4 8.25260266211664 -0.852602662116644 18 8.8 8.18484329595618 0.615156704043822 19 9.3 8.24040695186458 1.05959304813542 20 9.3 8.18357494208997 1.11642505791004 21 8.7 8.38568225238716 0.314317747612838 22 8.2 8.53583583701057 -0.335835837010568 23 8.3 8.46671055130187 -0.166710551301871 24 8.5 8.27640868852867 0.223591311471328 25 8.6 8.31080059143949 0.289199408560508 26 8.5 8.12698684652039 0.37301315347961 27 8.2 8.32850876272549 -0.128508762725488 28 8.1 8.33709454274294 -0.237094542742941 29 7.9 8.21518622306331 -0.315186223063313 30 8.6 8.11323008535606 0.486769914643938 31 8.7 8.12127925412242 0.578720745877575 32 8.7 8.23182117184713 0.468178828152871 33 8.5 8.2067955744099 0.293204425590107 34 8.4 8.22411348296782 0.175886517032177 35 8.5 8.28904344434981 0.21095655565019 36 8.7 8.05161735716264 0.648382642837362 37 8.7 8.06347158752764 0.636528412472357 38 8.6 7.92302578826488 0.67697421173512 39 8.5 8.02732350234053 0.472676497659474 40 8.3 8.18250171958778 0.117498280412218 41 8 8.19367299017867 -0.193672990178673 42 8.2 8.20913715077829 -0.00913715077829032 43 8.1 8.0572761667196 0.0427238332804052 44 8.1 8.05366623648498 0.046333763515016 45 8 8.15454915169005 -0.154549151690053 46 7.9 8.08522873461732 -0.185228734617324 47 7.9 8.21874737045692 -0.318747370456915 48 8 8.0750331208466 -0.075033120846599 49 8 7.98644348157561 0.0135565184243906 50 7.9 7.70028333622119 0.19971666377881 51 8 7.86790117792555 0.132098822074448 52 7.7 7.76448155498805 -0.0644815549880528 53 7.2 7.6382803452997 -0.438280345299699 54 7.5 7.758237351339 -0.258237351338997 55 7.3 7.65457381419646 -0.354573814196457 56 7 7.25757905407128 -0.257579054071284 57 7 7.58857063031229 -0.588570630312289 58 7 7.37895076249982 -0.378950762499821 59 7.2 7.5052983207112 -0.305298320711199 60 7.3 7.38973177036264 -0.0897317703626448 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.214752998101 0.429505996202 0.785247001899 6 0.100078044619913 0.200156089239825 0.899921955380087 7 0.0521128142796056 0.104225628559211 0.947887185720394 8 0.0321186750932832 0.0642373501865663 0.967881324906717 9 0.0581476098140676 0.116295219628135 0.941852390185932 10 0.0496380146656687 0.0992760293313374 0.950361985334331 11 0.171427335533106 0.342854671066211 0.828572664466894 12 0.228384788310244 0.456769576620488 0.771615211689756 13 0.202155572055187 0.404311144110373 0.797844427944813 14 0.160694453853460 0.321388907706919 0.83930554614654 15 0.353402983288506 0.706805966577012 0.646597016711494 16 0.829598283125576 0.340803433748848 0.170401716874424 17 0.938175395808247 0.123649208383507 0.0618246041917533 18 0.949299658267228 0.101400683465545 0.0507003417327724 19 0.991335037963229 0.0173299240735422 0.0086649620367711 20 0.999066584758756 0.00186683048248802 0.000933415241244011 21 0.998759259736064 0.00248148052787249 0.00124074026393625 22 0.998898943953453 0.00220211209309409 0.00110105604654704 23 0.998676898843033 0.00264620231393418 0.00132310115696709 24 0.997590239227534 0.00481952154493191 0.00240976077246596 25 0.996052622218407 0.00789475556318548 0.00394737778159274 26 0.994159457354516 0.0116810852909669 0.00584054264548343 27 0.992203002044186 0.0155939959116276 0.0077969979558138 28 0.99233348791385 0.0153330241722978 0.00766651208614888 29 0.99495953648325 0.0100809270334982 0.00504046351674908 30 0.993501945878986 0.0129961082420281 0.00649805412101406 31 0.993551780216625 0.0128964395667507 0.00644821978337533 32 0.991855281704508 0.0162894365909837 0.00814471829549187 33 0.987008896699603 0.0259822066007948 0.0129911033003974 34 0.978731653584947 0.0425366928301064 0.0212683464150532 35 0.966744128334715 0.0665117433305706 0.0332558716652853 36 0.979078818313538 0.0418423633729231 0.0209211816864615 37 0.99019466586564 0.0196106682687179 0.00980533413435894 38 0.999001855589569 0.00199628882086258 0.000998144410431288 39 0.999826967757307 0.000346064485386081 0.000173032242693040 40 0.999711305633452 0.000577388733096799 0.000288694366548400 41 0.999527268206338 0.000945463587324972 0.000472731793662486 42 0.998974317187662 0.00205136562467588 0.00102568281233794 43 0.998384066105031 0.00323186778993723 0.00161593389496862 44 0.997547084023913 0.00490583195217398 0.00245291597608699 45 0.995319043588899 0.00936191282220207 0.00468095641110104 46 0.992016945733395 0.0159661085332100 0.00798305426660498 47 0.9926615931083 0.0146768137833972 0.00733840689169862 48 0.98631943937794 0.027361121244118 0.013680560622059 49 0.97397949570932 0.0520410085813581 0.0260205042906790 50 0.98519830942063 0.0296033811587389 0.0148016905793694 51 0.991664711059211 0.0166705778815772 0.00833528894078861 52 0.994510505261063 0.0109789894778732 0.0054894947389366 53 0.9873093166327 0.0253813667346006 0.0126906833673003 54 0.977703029819198 0.0445939403616041 0.0222969701808020 55 0.946604712509954 0.106790574980092 0.0533952874900459 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 14 0.274509803921569 NOK 5% type I error level 34 0.666666666666667 NOK 10% type I error level 38 0.745098039215686 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 = Do not include Seasonal 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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