*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 07:15:41 -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/t1258726829chlubag35fvnqe4.htm/, Retrieved Fri, 20 Nov 2009 15:20:41 +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/t1258726829chlubag35fvnqe4.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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613 0 611 0 594 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

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 WklBe[t] = + 668.809090909091 + 48.5649350649351X[t] -25.2756132756135M1[t] -45.9339105339106M2[t] -58.8805194805196M3[t] -54.0271284271285M4[t] -49.3737373737373M5[t] -51.1203463203463M6[t] -57.4669552669553M7[t] -60.0135642135643M8[t] -68.7601731601732M9[t] -64.1067821067821M10[t] -10.4533910533911M11[t] -2.45339105339105t + 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) 668.809090909091 12.174422 54.9356 0 0 X 48.5649350649351 10.017937 4.8478 1.4e-05 7e-06 M1 -25.2756132756135 13.53622 -1.8673 0.068109 0.034055 M2 -45.9339105339106 14.124017 -3.2522 0.002123 0.001061 M3 -58.8805194805196 14.086711 -4.1799 0.000126 6.3e-05 M4 -54.0271284271285 14.053249 -3.8445 0.000362 0.000181 M5 -49.3737373737373 14.023656 -3.5207 0.000968 0.000484 M6 -51.1203463203463 13.997959 -3.652 0.000653 0.000326 M7 -57.4669552669553 13.976178 -4.1118 0.000156 7.8e-05 M8 -60.0135642135643 13.958332 -4.2995 8.6e-05 4.3e-05 M9 -68.7601731601732 13.944437 -4.931 1.1e-05 5e-06 M10 -64.1067821067821 13.934502 -4.6006 3.2e-05 1.6e-05 M11 -10.4533910533911 13.928539 -0.7505 0.456694 0.228347 t -2.45339105339105 0.235352 -10.4243 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.886111711290853 R-squared 0.785193964886805 Adjusted R-squared 0.72577952964273 F-TEST (value) 13.2155420086250 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 1.37907463226838e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 22.0198090459688 Sum Squared Residuals 22788.9835497836 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 613 641.080086580087 -28.0800865800875 2 611 617.968398268398 -6.96839826839825 3 594 602.568398268398 -8.56839826839822 4 595 604.968398268398 -9.96839826839813 5 591 607.168398268398 -16.1683982683981 6 589 602.968398268398 -13.9683982683981 7 584 594.168398268398 -10.1683982683983 8 573 589.168398268398 -16.1683982683982 9 567 577.968398268398 -10.9683982683983 10 569 580.168398268398 -11.1683982683982 11 621 631.368398268398 -10.3683982683982 12 629 639.368398268398 -10.3683982683982 13 628 611.639393939394 16.3606060606063 14 612 588.527705627706 23.4722943722944 15 595 573.127705627706 21.8722943722944 16 597 575.527705627706 21.4722943722944 17 593 577.727705627706 15.2722943722944 18 590 573.527705627706 16.4722943722944 19 580 564.727705627706 15.2722943722944 20 574 559.727705627706 14.2722943722944 21 573 548.527705627706 24.4722943722944 22 573 550.727705627706 22.2722943722944 23 620 601.927705627706 18.0722943722944 24 626 609.927705627706 16.0722943722944 25 620 582.198701298701 37.8012987012989 26 588 559.087012987013 28.912987012987 27 566 543.687012987013 22.3129870129870 28 557 546.087012987013 10.9129870129870 29 561 548.287012987013 12.7129870129870 30 549 544.087012987013 4.91298701298698 31 532 535.287012987013 -3.28701298701298 32 526 530.287012987013 -4.28701298701298 33 511 519.087012987013 -8.08701298701298 34 499 521.287012987013 -22.287012987013 35 555 572.487012987013 -17.487012987013 36 565 580.487012987013 -15.4870129870130 37 542 552.758008658009 -10.7580086580085 38 527 529.64632034632 -2.64632034632037 39 510 514.24632034632 -4.24632034632038 40 514 516.64632034632 -2.64632034632038 41 517 518.84632034632 -1.84632034632040 42 508 514.64632034632 -6.64632034632039 43 493 505.84632034632 -12.8463203463204 44 490 500.84632034632 -10.8463203463204 45 469 489.64632034632 -20.6463203463204 46 478 491.84632034632 -13.8463203463204 47 528 543.04632034632 -15.0463203463204 48 534 551.04632034632 -17.0463203463204 49 518 571.882251082251 -53.8822510822509 50 506 548.770562770563 -42.7705627705628 51 502 533.370562770563 -31.3705627705628 52 516 535.770562770563 -19.7705627705628 53 528 537.970562770563 -9.97056277056282 54 533 533.770562770563 -0.770562770562831 55 536 524.970562770563 11.0294372294372 56 537 519.970562770563 17.0294372294372 57 524 508.770562770563 15.2294372294372 58 536 510.970562770563 25.0294372294372 59 587 562.170562770563 24.8294372294372 60 597 570.170562770563 26.8294372294372 61 581 542.441558441558 38.5584415584417 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0137069167720528 0.0274138335441056 0.986293083227947 18 0.00261418502913884 0.00522837005827768 0.997385814970861 19 0.000810416712025571 0.00162083342405114 0.999189583287974 20 0.000136107809728030 0.000272215619456059 0.999863892190272 21 2.52444453492562e-05 5.04888906985124e-05 0.99997475555465 22 3.7787626663723e-06 7.5575253327446e-06 0.999996221237334 23 6.65012551220894e-07 1.33002510244179e-06 0.999999334987449 24 1.49629717717385e-07 2.99259435434771e-07 0.999999850370282 25 4.33471352725046e-08 8.66942705450092e-08 0.999999956652865 26 2.30758647195019e-05 4.61517294390037e-05 0.99997692413528 27 0.000345163769839914 0.000690327539679827 0.99965483623016 28 0.00328170832464049 0.00656341664928098 0.99671829167536 29 0.00507032033222853 0.0101406406644571 0.994929679667771 30 0.0107371342954134 0.0214742685908267 0.989262865704587 31 0.0247357870158941 0.0494715740317882 0.975264212984106 32 0.0324432626105482 0.0648865252210964 0.967556737389452 33 0.0635857058247206 0.127171411649441 0.93641429417528 34 0.113816586300741 0.227633172601482 0.886183413699259 35 0.134418845234776 0.268837690469553 0.865581154765223 36 0.180556880102299 0.361113760204599 0.8194431198977 37 0.320985646536842 0.641971293073685 0.679014353463158 38 0.537250724376461 0.925498551247077 0.462749275623539 39 0.719449073784487 0.561101852431027 0.280550926215513 40 0.859676155455267 0.280647689089466 0.140323844544733 41 0.960994273147536 0.0780114537049287 0.0390057268524644 42 0.994378350869577 0.0112432982608458 0.0056216491304229 43 0.994780563880792 0.010438872238416 0.005219436119208 44 0.997165062396721 0.00566987520655767 0.00283493760327884 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 12 0.428571428571429 NOK 5% type I error level 18 0.642857142857143 NOK 10% type I error level 20 0.714285714285714 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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