*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, 16 Dec 2008 01:21:50 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/16/t1229415989y0n49mesw9ntw6n.htm/, Retrieved Tue, 16 Dec 2008 09:26:41 +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/2008/Dec/16/t1229415989y0n49mesw9ntw6n.htm/}, year = {2008}, } @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 = {2008}, 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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111,8 142 129,5 100,9 109 103,7 102,1 120,7 114,2 118 159,6 123,5 110,6 142,4 111,3 115,1 145,1 126,5 114,8 148,9 119,6 110,1 136,9 116,5 110,8 119,9 116,7 95,6 133,9 119,4 108,1 131 124 116 133,2 130,6 111,2 135 120,1 98,2 99,1 113,2 97,6 110,8 111,1 113,3 152,3 126 107 131,9 115,8 107,9 127,9 111 117,5 142 128,7 105,4 118,7 112,6 104,2 116,3 114,7 98 125,7 118,5 106,7 122,7 124,8 113,4 125,3 128,6 111,7 123,2 127 94,2 88,8 111,8 92,5 94,9 100,6 109,8 136,8 122,9 105,1 128,7 117,8 104,4 110,8 108,1 111,1 132,8 129,6 98,7 112 111,4 100,5 104,5 110 93,7 112 115,2 103,2 110,6 118,8 104,1 107,2 116,2 106,9 116,2 126,3 89,2 85,7 106,7 88,7 94,2 96,5 110,7 127,2 119,1 98,8 108,9 109,6 102,5 111,9 110,3 101,8 126,3 118,8 96 105,9 104,5 98,3 101,3 107,7 94 105,5 127,7 105,1 106,3 118,5 114 117,3 120,1 115,5 110,9 127,4 94,3 85,4 107,8 100,8 81,9 106,5 111,2 121,5 124,6 103,4 106,3 101,9 106,7 111,8 106,5 112,2 122,8 119,4 100,7 101,8 103,3 99 92,2 99,6 91,5 106,3 120,9 102,7 103 111,7 111,4 97,7 123,9

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 3 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation Cons.[t] = + 47.420460898742 + 0.475118615783917Interm.[t] + 0.162045348343677Invest.[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) 47.420460898742 12.109421 3.916 0.000243 0.000122 Interm. 0.475118615783917 0.1587 2.9938 0.00407 0.002035 Invest. 0.162045348343677 0.06852 2.3649 0.021461 0.01073 Multiple Linear Regression - Regression Statistics Multiple R 0.693925573186993 R-squared 0.481532701122897 Adjusted R-squared 0.463340866074578 F-TEST (value) 26.4697156633125 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 7.40557926093288e-09 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 6.31100112621111 Sum Squared Residuals 2270.23790725716 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 129.5 123.549161608186 5.9508383918136 2 103.7 113.0228722008 -9.32287220080007 3 114.2 115.488945115362 -1.28894511536182 4 123.5 129.346895156895 -5.84689515689515 5 111.3 123.043837408583 -11.7438374085829 6 126.5 125.619393620138 0.880606379861531 7 119.6 126.092630359109 -6.49263035910928 8 116.5 121.915028684801 -5.41502868480073 9 116.7 119.492840794007 -2.79284079400695 10 119.4 114.539672710903 4.8603272890971 11 124 120.008723898005 3.9912761019948 12 130.6 124.118660729054 6.48133927094576 13 120.1 122.12977300031 -2.02977300031006 14 113.2 110.135802989581 3.06419701041890 15 111.1 111.746662395732 -0.646662395731787 16 126 125.930906619802 0.0690933801980991 17 115.8 119.631934234152 -3.83193423415220 18 111 119.411359594983 -8.41135959498302 19 128.7 126.257337718154 2.44266228184552 20 112.6 116.732745850761 -4.1327458507614 21 114.7 115.773694675796 -1.07369467579586 22 118.5 114.351185532366 4.14881446763385 23 124.8 117.998581444655 6.8014185553448 24 128.6 121.603194076101 6.996805923899 25 127 120.455197197747 6.54480280225338 26 111.8 106.566261438506 5.23373856149444 27 100.6 106.747036416569 -6.14703641656934 28 122.9 121.756288565231 1.14371143476882 29 117.8 118.210663749463 -0.410663749462989 30 108.1 114.977468983062 -6.87746898306243 31 129.6 121.725761372376 7.87423862762442 32 111.4 112.463747291107 -1.06374729110650 33 110 112.10362068694 -2.10362068693998 34 115.2 110.088154212187 5.11184578781308 35 118.8 114.374917574453 4.42508242554701 36 116.2 114.25157014429 1.94842985571 37 126.3 117.040310403578 9.25968959642192 38 106.7 103.688327779721 3.01167222027943 39 96.5 104.82815393275 -8.32815393274988 40 119.1 120.628259975337 -1.52825997533742 41 109.6 112.008918572819 -2.40891857281950 42 110.3 114.252993496251 -3.95299349625103 43 118.8 116.253863481351 2.54613651864876 44 104.5 110.192450403593 -5.6924504035935 45 107.7 110.539814617516 -2.83981461751559 46 127.7 109.177395032688 18.5226049673118 47 118.5 114.580847946565 3.91915205343539 48 120.1 120.591902458822 -0.491902458821932 49 127.4 120.267490153098 7.13250984690174 50 107.8 106.062819115715 1.73718088428455 51 106.5 108.583931399108 -2.08393139910804 52 124.6 119.942160797670 4.65783920232959 53 101.9 113.773146299732 -11.8731462997319 54 106.5 116.232287147709 -9.7322871477091 55 119.4 120.627938366301 -1.22793836630110 56 103.3 111.761121969569 -8.46112196956883 57 99.6 109.397784978637 -9.79778497863687 58 120.9 108.119234771903 12.7807652280967 59 111.7 112.905813619149 -1.20581361914908 60 123.9 116.180505230248 7.71949476975234 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.83740730375589 0.325185392488222 0.162592696244111 7 0.766938386228055 0.466123227543891 0.233061613771946 8 0.664873109574175 0.670253780851649 0.335126890425825 9 0.545874389398454 0.908251221203093 0.454125610601546 10 0.539635280363172 0.920729439273657 0.460364719636828 11 0.549944837725796 0.900110324548409 0.450055162274204 12 0.667148076532748 0.665703846934503 0.332851923467252 13 0.574638039578486 0.850723920843028 0.425361960421514 14 0.505470494026119 0.989059011947762 0.494529505973881 15 0.41031495355646 0.82062990711292 0.58968504644354 16 0.331995144139003 0.663990288278006 0.668004855860997 17 0.276795284136625 0.553590568273249 0.723204715863375 18 0.325314710211256 0.650629420422511 0.674685289788744 19 0.283695458643088 0.567390917286176 0.716304541356912 20 0.239170873721784 0.478341747443567 0.760829126278216 21 0.180954157830413 0.361908315660825 0.819045842169587 22 0.154483524850757 0.308967049701514 0.845516475149243 23 0.176079193571072 0.352158387142144 0.823920806428928 24 0.195674774480766 0.391349548961532 0.804325225519234 25 0.191743923822445 0.383487847644891 0.808256076177555 26 0.167406639456136 0.334813278912273 0.832593360543864 27 0.172775755580564 0.345551511161128 0.827224244419436 28 0.130944601425062 0.261889202850124 0.869055398574938 29 0.096968707441923 0.193937414883846 0.903031292558077 30 0.110251428189372 0.220502856378744 0.889748571810628 31 0.118838195292139 0.237676390584278 0.881161804707861 32 0.0862095016933643 0.172419003386729 0.913790498306636 33 0.0622865355616643 0.124573071123329 0.937713464438336 34 0.0536463362651251 0.107292672530250 0.946353663734875 35 0.0417277012964187 0.0834554025928375 0.958272298703581 36 0.0276470723102178 0.0552941446204357 0.972352927689782 37 0.0387838346878433 0.0775676693756866 0.961216165312157 38 0.0277732545744876 0.0555465091489752 0.972226745425512 39 0.035960468478744 0.071920936957488 0.964039531521256 40 0.0240009129924857 0.0480018259849713 0.975999087007514 41 0.0161176793862904 0.0322353587725809 0.98388232061371 42 0.0122892753364875 0.0245785506729750 0.987710724663513 43 0.00761801451660584 0.0152360290332117 0.992381985483394 44 0.0078788186560309 0.0157576373120618 0.99212118134397 45 0.00526879856158545 0.0105375971231709 0.994731201438415 46 0.0823541722418639 0.164708344483728 0.917645827758136 47 0.0610047643241979 0.122009528648396 0.938995235675802 48 0.0372711979209331 0.0745423958418661 0.962728802079067 49 0.0412505331089624 0.0825010662179248 0.958749466891038 50 0.0259134759486009 0.0518269518972019 0.97408652405140 51 0.0156239213920284 0.0312478427840569 0.984376078607972 52 0.0109815274291376 0.0219630548582753 0.989018472570862 53 0.0204754616807633 0.0409509233615266 0.979524538319237 54 0.0273060162096448 0.0546120324192897 0.972693983790355 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 9 0.183673469387755 NOK 10% type I error level 18 0.36734693877551 NOK

Charts produced by software:

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Parameters (Session):

par1 = 3 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 3 ; 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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