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: Thu, 19 Nov 2009 12:50:16 -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/19/t1258660276kmouwf09ytwx8er.htm/, Retrieved Thu, 19 Nov 2009 20:51:28 +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/19/t1258660276kmouwf09ytwx8er.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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Dataseries X:

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19 595 19 18 19 22 591 19 19 18 23 589 22 19 19 20 584 23 22 19 14 573 20 23 22 14 567 14 20 23 14 569 14 14 20 15 621 14 14 14 11 629 15 14 14 17 628 11 15 14 16 612 17 11 15 20 595 16 17 11 24 597 20 16 17 23 593 24 20 16 20 590 23 24 20 21 580 20 23 24 19 574 21 20 23 23 573 19 21 20 23 573 23 19 21 23 620 23 23 19 23 626 23 23 23 27 620 23 23 23 26 588 27 23 23 17 566 26 27 23 24 557 17 26 27 26 561 24 17 26 24 549 26 24 17 27 532 24 26 24 27 526 27 24 26 26 511 27 27 24 24 499 26 27 27 23 555 24 26 27 23 565 23 24 26 24 542 23 23 24 17 527 24 23 23 21 510 17 24 23 19 514 21 17 24 22 517 19 21 17 22 508 22 19 21 18 493 22 22 19 16 490 18 22 22 14 469 16 18 22 12 478 14 16 18 14 528 12 14 16 16 534 14 12 14 8 518 16 14 12 3 506 8 16 14 0 502 3 8 16 5 516 0 3 8 1 528 5 0 3 1 533 1 5 0 3 536 1 1 5 6 537 3 1 1 7 524 6 3 1 8 536 7 6 3 14 587 8 7 6 14 597 14 8 7 13 581 14 14 8

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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation y[t] = -40.2944805912345 + 0.0638539628062868x[t] + 0.682999055030689y1[t] + 0.122458369917985y2[t] + 0.218214695228774y3[t] + 3.70568054851267M1[t] + 2.87391682631078M2[t] + 1.61334266086646M3[t] + 1.73017385141576M4[t] + 0.646074597671226M5[t] + 2.809494330732M6[t] + 1.90389388989760M7[t] + 0.793880895008001M8[t] -1.33056690921134M9[t] -0.0375894807118368M10[t] -2.25856294729839M11[t] + 0.107006757579222t + 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) -40.2944805912345 15.29372 -2.6347 0.01183 0.005915 x 0.0638539628062868 0.023147 2.7586 0.008634 0.004317 y1 0.682999055030689 0.149917 4.5558 4.6e-05 2.3e-05 y2 0.122458369917985 0.185172 0.6613 0.512105 0.256052 y3 0.218214695228774 0.159979 1.364 0.180003 0.090002 M1 3.70568054851267 2.219616 1.6695 0.102635 0.051317 M2 2.87391682631078 2.310706 1.2437 0.22066 0.11033 M3 1.61334266086646 2.245781 0.7184 0.476592 0.238296 M4 1.73017385141576 2.165772 0.7989 0.428968 0.214484 M5 0.646074597671226 2.189838 0.295 0.769457 0.384729 M6 2.809494330732 2.186799 1.2848 0.20609 0.103045 M7 1.90389388989760 2.236367 0.8513 0.399529 0.199765 M8 0.793880895008001 2.303007 0.3447 0.732071 0.366035 M9 -1.33056690921134 2.444624 -0.5443 0.589195 0.294597 M10 -0.0375894807118368 2.300216 -0.0163 0.987041 0.493521 M11 -2.25856294729839 2.326337 -0.9709 0.337308 0.168654 t 0.107006757579222 0.057718 1.854 0.070948 0.035474 Multiple Linear Regression - Regression Statistics Multiple R 0.931011076253719 R-squared 0.866781624107108 Adjusted R-squared 0.814793965222077 F-TEST (value) 16.6728343360097 F-TEST (DF numerator) 16 F-TEST (DF denominator) 41 p-value 4.49307258065801e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.16773018165331 Sum Squared Residuals 411.415094654049 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 19 20.8386264980515 -1.83862649805149 2 22 19.762697356893 2.23730264310701 3 23 20.7486338837362 2.25136611626385 4 20 21.7035761826179 -1.70357618261788 5 14 18.7521953860957 -4.75219538609566 6 14 16.3923433551886 -2.39234335518862 7 14 14.3320632923518 -0.332063292351784 8 15 15.3401749495957 -0.340174949595674 9 11 14.5165646604365 -3.51656466043655 10 17 13.2431570335042 3.75684296649580 11 16 13.9339024653372 2.06609753466275 12 20 14.3928471860701 5.60715281392989 13 24 22.252068439352 1.74793156064801 14 23 24.2755106280701 -1.27551062807010 15 20 23.6100745373425 -3.61007453734249 16 21 21.8967761033132 -0.896776103313184 17 19 20.6339690803581 -1.63396908035812 18 23 20.9423577823621 2.05764221763789 19 23 22.8490582746225 0.150941725377507 20 23 24.900592378422 -1.90059237842198 21 23 24.1391338895347 -1.13913388953469 22 27 25.1559942987757 1.84400570122431 23 26 23.7306970000899 2.26930299991007 24 17 24.4983139478705 -7.49831394787049 25 24 22.3397245044267 1.66027549557329 26 26 25.3310367517534 0.668963248246625 27 24 23.6704962326411 0.329503767358865 28 27 23.2152383094388 3.78476169056121 29 27 24.0955318521494 2.90446814785059 30 26 25.3390946199915 0.660905380008493 31 24 23.7458984137165 0.254101586283472 32 23 24.8302576135788 -1.83025761357884 33 23 22.3052257049062 0.694774295093838 34 24 21.6776809860648 2.32231901393525 35 17 19.0706891947650 -2.07068919476503 36 21 15.6922065166389 5.30779348336107 37 19 21.8533119998816 -2.85331199988161 38 22 18.9164494266869 3.08355057331306 39 22 19.8651355597365 2.13486444026355 40 18 19.0621097850671 -1.06210978506707 41 16 15.8161032660465 0.183896733953522 42 14 14.8897649480211 -0.88976494802113 43 12 12.1820832992101 -0.182083299210095 44 14 12.3244309618592 1.67556903814084 45 16 11.3747656718246 4.62523432817538 46 8 12.9275719124426 -4.92757191244255 47 3 5.26471133980779 -2.26471133980779 48 0 3.41663234942047 -3.41663234942047 49 5 3.71626855828819 1.28373144171181 50 1 5.71430583659659 -4.71430583659659 51 1 2.10565978654377 -1.10565978654377 52 3 3.12229961956307 -0.122299619563073 53 6 2.70220041535034 3.29779958464966 54 7 6.43643929443664 0.563560705563365 55 8 7.8908967200991 0.109103279900901 56 14 11.6045440965443 2.39545590345566 57 14 14.6643100732980 -0.664310073297984 58 13 15.9955957692128 -2.9955957692128 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 20 0.312621957274185 0.62524391454837 0.687378042725815 21 0.205630555503526 0.411261111007052 0.794369444496474 22 0.104679547652515 0.209359095305030 0.895320452347485 23 0.275712351356802 0.551424702713603 0.724287648643198 24 0.765483570208344 0.469032859583312 0.234516429791656 25 0.786068589218885 0.42786282156223 0.213931410781115 26 0.696843194816621 0.606313610366758 0.303156805183379 27 0.623260599781946 0.753478800436108 0.376739400218054 28 0.590577284915783 0.818845430168434 0.409422715084217 29 0.542539137114812 0.914921725770375 0.457460862885188 30 0.441512895664596 0.883025791329192 0.558487104335404 31 0.33231324725719 0.66462649451438 0.66768675274281 32 0.392164404561732 0.784328809123463 0.607835595438268 33 0.604516689657925 0.79096662068415 0.395483310342075 34 0.50063825906849 0.998723481863021 0.499361740931510 35 0.525856617710022 0.948286764579956 0.474143382289978 36 0.421154391020655 0.84230878204131 0.578845608979345 37 0.81422602099987 0.37154795800026 0.18577397900013 38 0.73557260757972 0.528854784840561 0.264427392420281 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):

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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