Model 3

*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 06:13:45 -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/t1258723192kkqvzswfcp2gfx1.htm/, Retrieved Fri, 20 Nov 2009 14:20:04 +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/t1258723192kkqvzswfcp2gfx1.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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108.5 98.71 112.3 98.54 116.6 98.2 115.5 96.92 120.1 99.06 132.9 99.65 128.1 99.82 129.3 99.99 132.5 100.33 131 99.31 124.9 101.1 120.8 101.1 122 100.93 122.1 100.85 127.4 100.93 135.2 99.6 137.3 101.88 135 101.81 136 102.38 138.4 102.74 134.7 102.82 138.4 101.72 133.9 103.47 133.6 102.98 141.2 102.68 151.8 102.9 155.4 103.03 156.6 101.29 161.6 103.69 160.7 103.68 156 104.2 159.5 104.08 168.7 104.16 169.9 103.05 169.9 104.66 185.9 104.46 190.8 104.95 195.8 105.85 211.9 106.23 227.1 104.86 251.3 107.44 256.7 108.23 251.9 108.45 251.2 109.39 270.3 110.15 267.2 109.13 243 110.28 229.9 110.17 187.2 109.99 178.2 109.26 175.2 109.11 192.4 107.06 187 109.53 184 108.92 194.1 109.24 212.7 109.12 217.5 109 200.5 107.23 205.9 109.49 196.5 109.04 206.3 109.02

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 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = -1581.71366541556 + 17.1393927894853X[t] -1.45486092444313M1[t] -0.824024568713355M2[t] + 5.5934443397917M3[t] + 41.7883174989466M4[t] + 8.69965578100341M5[t] + 10.2346763403491M6[t] + 4.92475170042933M7[t] + 7.20871783851072M8[t] + 11.3211930468029M9[t] + 30.1172787296379M10[t] -3.60510496166605M11[t] -1.50025676429478t + 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) -1581.71366541556 282.09094 -5.6071 1e-06 1e-06 X 17.1393927894853 2.872522 5.9667 0 0 M1 -1.45486092444313 12.93169 -0.1125 0.910903 0.455452 M2 -0.824024568713355 13.566718 -0.0607 0.951825 0.475912 M3 5.5934443397917 13.56087 0.4125 0.681871 0.340935 M4 41.7883174989466 14.652208 2.852 0.006437 0.003218 M5 8.69965578100341 13.534018 0.6428 0.523478 0.261739 M6 10.2346763403491 13.514681 0.7573 0.45265 0.226325 M7 4.92475170042933 13.525463 0.3641 0.717409 0.358705 M8 7.20871783851072 13.524981 0.533 0.59655 0.298275 M9 11.3211930468029 13.523237 0.8372 0.406736 0.203368 M10 30.1172787296379 13.812041 2.1805 0.03426 0.01713 M11 -3.60510496166605 13.539742 -0.2663 0.791203 0.395601 t -1.50025676429478 0.612783 -2.4483 0.018142 0.009071 Multiple Linear Regression - Regression Statistics Multiple R 0.905968603940365 R-squared 0.820779111325654 Adjusted R-squared 0.771207376160409 F-TEST (value) 16.5574012809846 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 2.46247466861860e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 21.3062643774529 Sum Squared Residuals 21335.9743809301 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 108.5 107.160679145797 1.33932085420329 2 112.3 103.377561963018 8.9224380369823 3 116.6 102.467380558803 14.1326194411971 4 115.5 115.223574183122 0.276425816878101 5 120.1 117.312956270382 2.78704372961763 6 132.9 127.459961811230 5.44003818877033 7 128.1 123.563477181227 4.53652281877262 8 129.3 127.260883329227 2.03911667077351 9 132.5 135.700495321649 -3.20049532164891 10 131 135.514143594914 -4.51414359491423 11 124.9 130.971016232494 -6.071016232494 12 120.8 133.075864429865 -12.2758644298653 13 122 127.207049966915 -5.20704996691508 14 122.1 124.966478135191 -2.86647813519102 15 127.4 131.254841702560 -3.85484170256033 16 135.2 143.154065687405 -7.95406568740485 17 137.3 147.642962765193 -10.3429627651933 18 135 146.477969064980 -11.4779690649804 19 136 149.437241550772 -13.4372415507723 20 138.4 156.391132328774 -17.9911323287736 21 134.7 160.374502195930 -25.6745021959297 22 138.4 158.816999046036 -20.4169990460363 23 133.9 153.588295972037 -19.6882959720368 24 133.6 147.294841702560 -13.6948417025603 25 141.2 139.197906176977 2.00209382302308 26 151.8 142.099152182099 9.7008478179014 27 155.4 149.244485388942 6.1555146110581 28 156.6 154.116558330098 2.48344166990224 29 161.6 160.662182542624 0.937817457375732 30 160.7 160.525552409781 0.174447590219479 31 156 162.627855256098 -6.62785525609819 32 159.5 161.354837495146 -1.85483749514650 33 168.7 165.338207362303 3.36179263769735 34 169.9 163.609310284514 6.29068971548574 35 169.9 155.981092219987 13.9189077800132 36 185.9 154.658061859461 31.2419381405390 37 190.8 160.101246637571 30.698753362429 38 195.8 174.657279739543 21.1427202604575 39 211.9 186.087461143757 25.8125388562426 40 227.1 197.301109417023 29.7988905829774 41 251.3 206.931824331657 44.3681756683434 42 256.7 220.506708430401 36.193291569599 43 251.9 217.467193439873 34.4328065601268 44 251.2 234.361932035776 16.8380679642241 45 270.3 250.000088999782 20.2999110002179 46 267.2 249.813737273047 17.3862627269528 47 243 234.301398525357 8.69860147464335 48 229.9 234.520913515885 -4.62091351588456 49 187.2 228.480705125039 -41.2807051250392 50 178.2 215.09952798015 -36.8995279801501 51 175.2 217.445831205937 -42.2458312059375 52 192.4 217.004692382353 -24.6046923823529 53 187 224.750074090143 -37.7500740901435 54 184 214.329808283608 -30.3298082836084 55 194.1 213.004232572029 -18.904232572029 56 212.7 211.731214811078 0.968785188922443 57 217.5 212.286706120337 5.21329387966342 58 200.5 199.245809801488 1.25419019851200 59 205.9 202.758197050126 3.14180294987420 60 196.5 197.150318492229 -0.65031849222889 61 206.3 193.852412947701 12.4475870522989 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.00212462027808544 0.00424924055617088 0.997875379721915 18 0.000692759857787616 0.00138551971557523 0.999307240142212 19 0.000135820792718360 0.000271641585436719 0.999864179207282 20 3.01176146051441e-05 6.02352292102883e-05 0.999969882385395 21 1.5122937970789e-05 3.0245875941578e-05 0.99998487706203 22 2.66196150550665e-06 5.32392301101329e-06 0.999997338038495 23 4.89844127617984e-07 9.79688255235967e-07 0.999999510155872 24 2.75890669522639e-07 5.51781339045279e-07 0.99999972410933 25 3.17759141901912e-07 6.35518283803824e-07 0.999999682240858 26 9.41698248552693e-07 1.88339649710539e-06 0.999999058301752 27 8.36624539081892e-07 1.67324907816378e-06 0.99999916337546 28 2.4593381047624e-07 4.9186762095248e-07 0.99999975406619 29 9.530040488934e-08 1.9060080977868e-07 0.999999904699595 30 2.24664589048073e-08 4.49329178096146e-08 0.999999977533541 31 7.9373040972106e-09 1.58746081944212e-08 0.999999992062696 32 3.40417084429783e-09 6.80834168859566e-09 0.99999999659583 33 2.60978357669981e-09 5.21956715339962e-09 0.999999997390216 34 2.88085556322412e-09 5.76171112644824e-09 0.999999997119144 35 9.18010066600592e-09 1.83602013320118e-08 0.9999999908199 36 9.74112769617927e-07 1.94822553923585e-06 0.99999902588723 37 0.000374097117799097 0.000748194235598194 0.9996259028822 38 0.00583504778395467 0.0116700955679093 0.994164952216045 39 0.0201564796083976 0.0403129592167952 0.979843520391602 40 0.0490322668703238 0.0980645337406477 0.950967733129676 41 0.0700843453675452 0.140168690735090 0.929915654632455 42 0.0724986856499544 0.144997371299909 0.927501314350046 43 0.0927558348738917 0.185511669747783 0.907244165126108 44 0.0991995667485483 0.198399133497097 0.900800433251452 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 21 0.75 NOK 5% type I error level 23 0.821428571428571 NOK 10% type I error level 24 0.857142857142857 NOK

Charts produced by software:

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Parameters (Session):

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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