Model 5 - WZM & WZM<25j

*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 15:17:43 -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/t1258669116c1w0h1kf6vxdgi5.htm/, Retrieved Thu, 19 Nov 2009 23:18:48 +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/t1258669116c1w0h1kf6vxdgi5.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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7.5 20.3 8 8.2 6.8 15.8 7.5 8 6.5 15.8 6.8 7.5 6.6 15.8 6.5 6.8 7.6 23.2 6.6 6.5 8 23.2 7.6 6.6 8.1 23.2 8 7.6 7.7 20.9 8.1 8 7.5 20.9 7.7 8.1 7.6 20.9 7.5 7.7 7.8 19.8 7.6 7.5 7.8 19.8 7.8 7.6 7.8 19.8 7.8 7.8 7.5 20.6 7.8 7.8 7.5 20.6 7.5 7.8 7.1 20.6 7.5 7.5 7.5 21.1 7.1 7.5 7.5 21.1 7.5 7.1 7.6 21.1 7.5 7.5 7.7 22.4 7.6 7.5 7.7 22.4 7.7 7.6 7.9 22.4 7.7 7.7 8.1 20.5 7.9 7.7 8.2 20.5 8.1 7.9 8.2 20.5 8.2 8.1 8.2 18.4 8.2 8.2 7.9 18.4 8.2 8.2 7.3 18.4 7.9 8.2 6.9 17.6 7.3 7.9 6.6 17.6 6.9 7.3 6.7 17.6 6.6 6.9 6.9 18.5 6.7 6.6 7 18.5 6.9 6.7 7.1 18.5 7 6.9 7.2 17.3 7.1 7 7.1 17.3 7.2 7.1 6.9 17.3 7.1 7.2 7 16.2 6.9 7.1 6.8 16.2 7 6.9 6.4 16.2 6.8 7 6.7 18.5 6.4 6.8 6.6 18.5 6.7 6.4 6.4 18.5 6.6 6.7 6.3 16.3 6.4 6.6 6.2 16.3 6.3 6.4 6.5 16.3 6.2 6.3 6.8 16.8 6.5 6.2 6.8 16.8 6.8 6.5 6.4 16.8 6.8 6.8 6.1 14.8 6.4 6.8 5.8 14.8 6.1 6.4 6.1 14.8 5.8 6.1 7.2 21.4 6.1 5.8 7.3 21.4 7.2 6.1 6.9 21.4 7.3 7.2 6.1 16.1 6.9 7.3 5.8 16.1 6.1 6.9 6.2 16.1 5.8 6.1 7. etc...

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] = + 1.00534172096945 + 0.104125740041533X[t] + 0.953792248854353`y(t-1)`[t] -0.353608708255487`y(t-2)`[t] -0.120038817733203M1[t] -0.0200034714829910M2[t] -0.083701842016721M3[t] -0.093534271571743M4[t] + 0.120752434670518M5[t] -0.3598303808755M6[t] -0.287254502541916M7[t] -0.287172802964734M8[t] -0.217506131383949M9[t] + 0.0272760961556319M10[t] + 0.126370704489030M11[t] -0.000124744305243137t + 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) 1.00534172096945 0.380607 2.6414 0.010933 0.005466 X 0.104125740041533 0.024644 4.2251 9.9e-05 4.9e-05 `y(t-1)` 0.953792248854353 0.154434 6.1761 0 0 `y(t-2)` -0.353608708255487 0.12094 -2.9238 0.005146 0.002573 M1 -0.120038817733203 0.125125 -0.9594 0.341907 0.170953 M2 -0.0200034714829910 0.129414 -0.1546 0.877771 0.438886 M3 -0.083701842016721 0.132667 -0.6309 0.530911 0.265456 M4 -0.093534271571743 0.134679 -0.6945 0.490522 0.245261 M5 0.120752434670518 0.169183 0.7137 0.478642 0.239321 M6 -0.3598303808755 0.127765 -2.8164 0.00689 0.003445 M7 -0.287254502541916 0.143099 -2.0074 0.050022 0.025011 M8 -0.287172802964734 0.137875 -2.0828 0.042298 0.021149 M9 -0.217506131383949 0.149302 -1.4568 0.151296 0.075648 M10 0.0272760961556319 0.146293 0.1864 0.852833 0.426416 M11 0.126370704489030 0.129964 0.9723 0.335466 0.167733 t -0.000124744305243137 0.001499 -0.0832 0.934009 0.467004 Multiple Linear Regression - Regression Statistics Multiple R 0.963304547754958 R-squared 0.927955651725384 Adjusted R-squared 0.906766137526968 F-TEST (value) 43.7931536813967 F-TEST (DF numerator) 15 F-TEST (DF denominator) 51 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.199820452761184 Sum Squared Residuals 2.03633888042591 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7.5 7.72967726491395 -0.22967726491395 2 6.8 6.95484765389596 -0.154847653895955 3 6.5 6.40017431898668 0.0998256810133238 4 6.6 6.35160556624895 0.248394433751053 5 7.6 7.53775984185539 0.0622401581446084 6 8 7.97548366003293 0.0245163399670651 7 8.1 8.07584298534753 0.0241570146524692 8 7.7 7.79024648010718 -0.0902464801071824 9 7.5 7.44291063701543 0.0570893629845652 10 7.6 7.6382531537811 -0.0382531537810969 11 7.8 7.7887856703001 0.0112143296999025 12 7.8 7.81768780045115 -0.0176878004511470 13 7.8 7.6268024967616 0.173197503238397 14 7.5 7.8100136907398 -0.310013690739799 15 7.5 7.46005290124452 0.0399470987554803 16 7.1 7.5561783398609 -0.456178339860901 17 7.5 7.44088627227694 0.0591137277230566 18 7.5 7.48313909526962 0.0168609047303809 19 7.6 7.41414674599577 0.185853254004235 20 7.7 7.64484638820713 0.0551536117928682 21 7.7 7.77440666954256 -0.0744066695425606 22 7.9 7.98370328195135 -0.0837032819513493 23 8.1 8.07559268967146 0.0244073103285375 24 8.2 8.06913394899696 0.130866051003037 25 8.2 7.97362787019285 0.226372129807146 26 8.2 7.81951354722505 0.380486452774946 27 7.9 7.75569043238608 0.14430956761392 28 7.3 7.45959558386951 -0.159595583869511 29 6.9 7.12426421693733 -0.224264216937335 30 6.6 6.47420498249763 0.125795017502374 31 6.7 6.40196192517185 0.298038074828145 32 6.9 6.69709388384326 0.202906116156745 33 7 6.92203339006412 0.0779666099358808 34 7.1 7.1913483565328 -0.0913483565327949 35 7.2 7.225385686571 -0.0253856865709956 36 7.1 7.15890859183661 -0.0589085918366104 37 6.9 6.90800493408718 -0.00800493408717858 38 7 6.73797964304114 0.262020356958860 39 6.8 6.8402574947387 -0.0402574947386995 40 6.4 6.60418100028201 -0.204181000282014 41 6.7 6.74703700642392 -0.0470370064239158 42 6.6 6.69391060453116 -0.0939106045311558 43 6.4 6.56489990119741 -0.164899901197414 44 6.3 6.18038264943266 0.119617350567341 45 6.2 6.22526709347386 -0.0252670934738619 46 6.5 6.40990622264831 0.0900937773516862 47 6.8 6.88243750217909 -0.0824375021790897 48 6.8 6.93599711556448 -0.135997115564476 49 6.4 6.70975094104938 -0.309750941049384 50 6.1 6.21989316336955 -0.119893163369546 51 5.8 6.01137585717646 -0.211375857176461 52 6.1 5.82136362113654 0.278636378863464 53 7.2 7.11497575448062 0.085024245519375 54 7.3 7.5773570558925 -0.277357055892507 55 6.9 7.35621783572525 -0.456217835725247 56 6.1 6.38743059840977 -0.287430598409771 57 5.8 5.83538220990402 -0.0353822099040236 58 6.2 6.07678898508644 0.123211014913555 59 7.1 7.02779845127835 0.0722015487216452 60 7.7 7.6182725431508 0.0817274568491964 61 7.9 7.75213649299503 0.147863507004969 62 7.7 7.7577523017285 -0.0577523017285054 63 7.4 7.43244899546756 -0.0324489954675638 64 7.5 7.20707588860209 0.292924111397909 65 8 7.93507690802579 0.0649230919742107 66 8.1 7.89590460177616 0.204095398223843 67 8 7.88693060656219 0.113069393437811 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.332229881297581 0.664459762595162 0.667770118702419 20 0.188239028032669 0.376478056065338 0.811760971967331 21 0.281226203205016 0.562452406410031 0.718773796794984 22 0.198039274471929 0.396078548943859 0.80196072552807 23 0.120925082598873 0.241850165197746 0.879074917401127 24 0.0985220455025425 0.197044091005085 0.901477954497458 25 0.0706106224314515 0.141221244862903 0.929389377568548 26 0.169664683165319 0.339329366330637 0.830335316834681 27 0.181585675886515 0.363171351773031 0.818414324113485 28 0.145696791788923 0.291393583577846 0.854303208211077 29 0.359500901624064 0.719001803248128 0.640499098375936 30 0.278762929323195 0.557525858646389 0.721237070676805 31 0.442465027247184 0.884930054494368 0.557534972752816 32 0.427682299674425 0.85536459934885 0.572317700325575 33 0.420640315736847 0.841280631473694 0.579359684263153 34 0.419922416055795 0.83984483211159 0.580077583944205 35 0.357794674660713 0.715589349321426 0.642205325339287 36 0.311997520699315 0.623995041398631 0.688002479300684 37 0.259969487158367 0.519938974316734 0.740030512841633 38 0.359580776380143 0.719161552760287 0.640419223619857 39 0.369297732500309 0.738595465000619 0.63070226749969 40 0.403752068603968 0.807504137207936 0.596247931396032 41 0.309537602621365 0.61907520524273 0.690462397378635 42 0.257727750703199 0.515455501406399 0.7422722492968 43 0.278482008842884 0.556964017685768 0.721517991157116 44 0.574906676795454 0.850186646409093 0.425093323204546 45 0.498458312705182 0.996916625410364 0.501541687294818 46 0.566118724190153 0.867762551619694 0.433881275809847 47 0.552672427027186 0.894655145945629 0.447327572972814 48 0.919504750405178 0.160990499189644 0.0804952495948221 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):

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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