*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: Tue, 15 Dec 2009 09:09:36 -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/Dec/15/t1260893424q3z1qqacufshu1r.htm/, Retrieved Tue, 15 Dec 2009 17:10:36 +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/Dec/15/t1260893424q3z1qqacufshu1r.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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594 0 595 0 591 0 589 0 584 0 573 0 567 0 569 0 621 0 629 0 628 0 612 0 595 0 597 0 593 0 590 0 580 0 574 0 573 0 573 0 620 0 626 0 620 0 588 0 566 0 557 0 561 0 549 0 532 0 526 0 511 0 499 0 555 0 565 0 542 0 527 0 510 0 514 0 517 0 508 0 493 0 490 0 469 0 478 0 528 0 534 0 518 1 506 1 502 1 516 1 528 1 533 1 536 1 537 1 524 1 536 1 587 1 597 1 581 1 564 1 558 1

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 WklBe[t] = + 626.710819672131 + 53.2918032786885X[t] -13.9899271402552M1[t] -17.5604007285975M2[t] -12.8985245901639M3[t] -14.6366484517304M4[t] -20.9747723132969M5[t] -23.5128961748634M6[t] -32.2510200364299M7[t] -27.5891438979964M8[t] + 26.0727322404372M9[t] + 36.5346083788707M10[t] + 15.9381238615665M11[t] -2.46187613843351t + 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) 626.710819672131 12.419659 50.4612 0 0 X 53.2918032786885 10.643633 5.0069 8e-06 4e-06 M1 -13.9899271402552 14.125909 -0.9904 0.327063 0.163532 M2 -17.5604007285975 14.825143 -1.1845 0.24217 0.121085 M3 -12.8985245901639 14.809874 -0.8709 0.388213 0.194107 M4 -14.6366484517304 14.799149 -0.989 0.327718 0.163859 M5 -20.9747723132969 14.792977 -1.4179 0.16282 0.08141 M6 -23.5128961748634 14.791364 -1.5896 0.118622 0.059311 M7 -32.2510200364299 14.794311 -2.18 0.034303 0.017152 M8 -27.5891438979964 14.801817 -1.8639 0.068589 0.034294 M9 26.0727322404372 14.813873 1.76 0.084911 0.042456 M10 36.5346083788707 14.830469 2.4635 0.017476 0.008738 M11 15.9381238615665 14.726423 1.0823 0.284649 0.142325 t -2.46187613843351 0.259735 -9.4784 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.86345984599006 R-squared 0.745562905637178 Adjusted R-squared 0.675186688047461 F-TEST (value) 10.5939610165426 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 5.63361357563963e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 23.2808972491104 Sum Squared Residuals 25474.0083060110 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 594 610.259016393444 -16.2590163934435 2 595 604.226666666667 -9.22666666666663 3 591 606.426666666667 -15.4266666666666 4 589 602.226666666667 -13.2266666666665 5 584 593.426666666667 -9.42666666666662 6 573 588.426666666667 -15.4266666666666 7 567 577.226666666667 -10.2266666666666 8 569 579.426666666667 -10.4266666666666 9 621 630.626666666667 -9.62666666666663 10 629 638.626666666667 -9.62666666666664 11 628 615.568306010929 12.4316939890711 12 612 597.168306010929 14.8316939890711 13 595 580.71650273224 14.2834972677598 14 597 574.684153005464 22.3158469945356 15 593 576.884153005464 16.1158469945355 16 590 572.684153005465 17.3158469945355 17 580 563.884153005464 16.1158469945355 18 574 558.884153005464 15.1158469945355 19 573 547.684153005465 25.3158469945355 20 573 549.884153005464 23.1158469945355 21 620 601.084153005464 18.9158469945355 22 626 609.084153005464 16.9158469945355 23 620 586.025792349727 33.9742076502732 24 588 567.625792349727 20.3742076502732 25 566 551.173989071038 14.8260109289619 26 557 545.141639344262 11.8583606557377 27 561 547.341639344262 13.6583606557377 28 549 543.141639344262 5.85836065573769 29 532 534.341639344262 -2.34163934426231 30 526 529.341639344262 -3.34163934426232 31 511 518.141639344262 -7.14163934426232 32 499 520.341639344262 -21.3416393442623 33 555 571.541639344262 -16.5416393442623 34 565 579.541639344262 -14.5416393442623 35 542 556.483278688525 -14.4832786885246 36 527 538.083278688525 -11.0832786885246 37 510 521.631475409836 -11.6314754098359 38 514 515.59912568306 -1.59912568306017 39 517 517.79912568306 -0.799125683060167 40 508 513.59912568306 -5.59912568306017 41 493 504.79912568306 -11.7991256830602 42 490 499.79912568306 -9.79912568306015 43 469 488.59912568306 -19.5991256830601 44 478 490.79912568306 -12.7991256830602 45 528 541.99912568306 -13.9991256830602 46 534 549.99912568306 -15.9991256830601 47 518 580.232568306011 -62.232568306011 48 506 561.832568306011 -55.8325683060109 49 502 545.380765027322 -43.3807650273222 50 516 539.348415300546 -23.3484153005465 51 528 541.548415300546 -13.5484153005465 52 533 537.348415300546 -4.34841530054648 53 536 528.548415300546 7.45158469945352 54 537 523.548415300546 13.4515846994535 55 524 512.348415300546 11.6515846994535 56 536 514.548415300546 21.4515846994535 57 587 565.748415300546 21.2515846994535 58 597 573.748415300546 23.2515846994535 59 581 550.690054644809 30.3099453551912 60 564 532.290054644809 31.7099453551912 61 558 515.83825136612 42.1617486338799 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.000650868310468905 0.00130173662093781 0.999349131689531 18 4.09820766975986e-05 8.19641533951972e-05 0.999959017923302 19 1.35579630392022e-05 2.71159260784044e-05 0.99998644203696 20 1.45730083118351e-06 2.91460166236703e-06 0.999998542699169 21 1.52917493011367e-07 3.05834986022734e-07 0.999999847082507 22 2.85187681892524e-08 5.70375363785047e-08 0.999999971481232 23 6.44737124097064e-08 1.28947424819413e-07 0.999999935526288 24 1.49916225224408e-05 2.99832450448815e-05 0.999985008377478 25 0.000161927051770797 0.000323854103541594 0.999838072948229 26 0.00137762425920953 0.00275524851841905 0.99862237574079 27 0.00181968822565924 0.00363937645131848 0.99818031177434 28 0.00377536787069353 0.00755073574138706 0.996224632129306 29 0.00970590044899297 0.0194118008979859 0.990294099551007 30 0.0138491561116238 0.0276983122232476 0.986150843888376 31 0.0353027476544418 0.0706054953088836 0.964697252345558 32 0.0789299074710405 0.157859814942081 0.92107009252896 33 0.113997796675585 0.227995593351169 0.886002203324415 34 0.183588281536904 0.367176563073807 0.816411718463096 35 0.357577807191265 0.71515561438253 0.642422192808735 36 0.597102705870928 0.805794588258144 0.402897294129072 37 0.819527817668375 0.360944364663249 0.180472182331625 38 0.92968132080419 0.140637358391618 0.0703186791958092 39 0.988341236982567 0.0233175260348660 0.0116587630174330 40 0.9992755149195 0.00144897016100054 0.000724485080500272 41 0.999540816261806 0.000918367476388102 0.000459183738194051 42 0.999826745097998 0.00034650980400484 0.00017325490200242 43 0.999420570048014 0.00115885990397243 0.000579429951986215 44 0.996222324566964 0.00755535086607102 0.00377767543303551 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 17 0.607142857142857 NOK 5% type I error level 20 0.714285714285714 NOK 10% type I error level 21 0.75 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 = 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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