WS7(4)

*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: Sat, 21 Nov 2009 03:55:20 -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/21/t1258800966zkrb8i1kzkp5zkb.htm/, Retrieved Sat, 21 Nov 2009 11:56:18 +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/21/t1258800966zkrb8i1kzkp5zkb.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)

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Dataseries X:

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9,5 7,8 9,2 9,2 10 10,9 9,6 7,8 9,5 9,2 9,2 10 9,5 7,8 9,6 9,5 9,2 9,2 9,1 7,5 9,5 9,6 9,5 9,2 8,9 7,5 9,1 9,5 9,6 9,5 9 7,1 8,9 9,1 9,5 9,6 10,1 7,5 9 8,9 9,1 9,5 10,3 7,5 10,1 9 8,9 9,1 10,2 7,6 10,3 10,1 9 8,9 9,6 7,7 10,2 10,3 10,1 9 9,2 7,7 9,6 10,2 10,3 10,1 9,3 7,9 9,2 9,6 10,2 10,3 9,4 8,1 9,3 9,2 9,6 10,2 9,4 8,2 9,4 9,3 9,2 9,6 9,2 8,2 9,4 9,4 9,3 9,2 9 8,2 9,2 9,4 9,4 9,3 9 7,9 9 9,2 9,4 9,4 9 7,3 9 9 9,2 9,4 9,8 6,9 9 9 9 9,2 10 6,6 9,8 9 9 9 9,8 6,7 10 9,8 9 9 9,3 6,9 9,8 10 9,8 9 9 7 9,3 9,8 10 9,8 9 7,1 9 9,3 9,8 10 9,1 7,2 9 9 9,3 9,8 9,1 7,1 9,1 9 9 9,3 9,1 6,9 9,1 9,1 9 9 9,2 7 9,1 9,1 9,1 9 8,8 6,8 9,2 9,1 9,1 9,1 8,3 6,4 8,8 9,2 9,1 9,1 8,4 6,7 8,3 8,8 9,2 9,1 8,1 6,6 8,4 8,3 8,8 9,2 7,7 6,4 8,1 8,4 8,3 8,8 7,9 6,3 7,7 8,1 8,4 8,3 7,9 6,2 7,9 7,7 8,1 8,4 8 6,5 7,9 7,9 7,7 8,1 7,9 6,8 8 7,9 7,9 7,7 7,6 6,8 7,9 8 7,9 7,9 7,1 6,4 7,6 7,9 8 7,9 6,8 6,1 7,1 7,6 7,9 8 6,5 5,8 6,8 7,1 7,6 7,9 6,9 6,1 6,5 6,8 7,1 7,6 8,2 7,2 6,9 6,5 6,8 7,1 8,7 7,3 8,2 6, 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t][t] = + 0.896550866003112 + 0.0673089460667498`X[t]`[t] + 1.40499954314054Y1[t] -0.52400110570065Y2[t] -0.374098836785457Y3[t] + 0.366078798454609Y4[t] -0.223823163888711M1[t] -0.429401490852397M2[t] -0.320995317768826M3[t] -0.266240827611457M4[t] -0.340227919439894M5[t] -0.113023296599354M6[t] + 0.486610505980998M7[t] -0.457802485721289M8[t] -0.434078057541609M9[t] + 0.0287063053531581M10[t] -0.170539719028066M11[t] -0.00408144041800594t + 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) 0.896550866003112 0.696565 1.2871 0.205843 0.102922 `X[t]` 0.0673089460667498 0.051976 1.295 0.203136 0.101568 Y1 1.40499954314054 0.156182 8.9959 0 0 Y2 -0.52400110570065 0.275272 -1.9036 0.064559 0.03228 Y3 -0.374098836785457 0.272264 -1.374 0.177483 0.088742 Y4 0.366078798454609 0.145139 2.5223 0.015969 0.007985 M1 -0.223823163888711 0.136939 -1.6345 0.110417 0.055209 M2 -0.429401490852397 0.141424 -3.0363 0.004311 0.002155 M3 -0.320995317768826 0.140445 -2.2856 0.027948 0.013974 M4 -0.266240827611457 0.139002 -1.9154 0.062993 0.031497 M5 -0.340227919439894 0.131747 -2.5824 0.013788 0.006894 M6 -0.113023296599354 0.129688 -0.8715 0.388952 0.194476 M7 0.486610505980998 0.135915 3.5803 0.000959 0.00048 M8 -0.457802485721289 0.179057 -2.5567 0.014683 0.007342 M9 -0.434078057541609 0.191835 -2.2628 0.029448 0.014724 M10 0.0287063053531581 0.17802 0.1613 0.872748 0.436374 M11 -0.170539719028066 0.142509 -1.1967 0.238838 0.119419 t -0.00408144041800594 0.003494 -1.1682 0.250008 0.125004 Multiple Linear Regression - Regression Statistics Multiple R 0.985302965025289 R-squared 0.970821932887626 Adjusted R-squared 0.957768587074195 F-TEST (value) 74.3734171118595 F-TEST (DF numerator) 17 F-TEST (DF denominator) 38 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.187914623193336 Sum Squared Residuals 1.34185241317596 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 9.5 9.54811220076473 -0.0481122007647298 2 9.6 9.72976044714444 -0.129760447144436 3 9.5 9.52452176365017 -0.0245217636501714 4 9.1 9.24987241364975 -0.149872413649753 5 8.9 8.734617930575 0.165382069425006 6 9 8.93343583174699 0.0665641682530137 7 10.1 9.91424360265894 0.185756397341062 8 10.3 10.3872368053984 -0.0872368053984253 9 10.2 10.0075837367547 0.192416263245295 10 9.6 9.85281653776541 -0.252816537765412 11 9.2 9.1867563685949 0.0132436314050998 12 9.3 9.22970292595195 0.0702970740480514 13 9.4 9.55321192967871 -0.153211929678711 14 9.4 9.3683751562891 0.0316248437109006 15 9.2 9.23645837532421 -0.0364583753242111 16 9 9.00532951260238 -0.00532951260237865 17 9 8.7674764888934 0.232523511106605 18 9 9.1298342921731 -0.1298342921731 19 9.8 9.70006708357491 0.0999329164250862 20 10 9.78216384245611 0.217836157543888 21 9.8 9.67033674889205 0.129663251107952 22 9.3 9.45742226138556 -0.157422261385556 23 9 8.88116941216946 0.118830587830545 24 9 9.04289480234236 -0.0428948023423637 25 9.1 9.09275508305432 0.00724491694567558 26 9.1 8.94605462718834 0.153945372811657 27 9.1 8.87469382053411 0.225306179465889 28 9.2 8.8946878812016 0.305312118798396 29 8.8 8.98026539390133 -0.180265393901325 30 8.3 8.56206507007088 -0.262065070070878 31 8.4 8.64750090308469 -0.24750090308469 32 8.1 8.28102349808174 -0.181023498081745 33 7.7 7.85392262212873 -0.153922622128725 34 7.9 7.68064588154694 0.219354118453061 35 7.9 8.1100254039305 -0.210025403930501 36 8 8.23169204039826 -0.231692040398255 37 7.9 7.94322878748668 -0.0432287874866822 38 7.6 7.6138847149118 -0.0138847149117937 39 7.1 7.28477623310001 -0.184776233100014 40 6.8 6.84397492268328 -0.0439749226832812 41 6.5 6.66183616771515 -0.161836167715152 42 6.9 6.71807828158209 0.181921718417913 43 8.2 8.0360608851926 0.163939114807396 44 8.7 8.71360232298069 -0.0136023229806874 45 8.3 8.46815689222452 -0.168156892224523 46 7.9 7.7091153193021 0.190884680697906 47 7.5 7.42204881530514 0.0779511846948565 48 7.8 7.59571023130743 0.204289768692568 49 8.3 8.06269199901555 0.237308000984447 50 8.4 8.44192505446633 -0.0419250544663281 51 8.2 8.1795498073915 0.0204501926085077 52 7.7 7.80613526986298 -0.106135269862983 53 7.2 7.25580401891513 -0.0558040189151345 54 7.3 7.15658652442695 0.143413475573052 55 8.1 8.30212752548885 -0.202127525488854 56 8.5 8.43597353108303 0.0640264689169696 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.128972209923009 0.257944419846019 0.87102779007699 22 0.0677485936852214 0.135497187370443 0.932251406314779 23 0.0366932996967069 0.0733865993934138 0.963306700303293 24 0.0208611640694361 0.0417223281388721 0.979138835930564 25 0.00765372561380247 0.0153074512276049 0.992346274386197 26 0.00295105054371247 0.00590210108742495 0.997048949456288 27 0.00621527802386244 0.0124305560477249 0.993784721976138 28 0.222176822473349 0.444353644946699 0.77782317752665 29 0.244266937499125 0.488533874998249 0.755733062500875 30 0.189297328449588 0.378594656899176 0.810702671550412 31 0.726418359696651 0.547163280606698 0.273581640303349 32 0.768478472075714 0.463043055848572 0.231521527924286 33 0.863494842631606 0.273010314736788 0.136505157368394 34 0.858323829590447 0.283352340819107 0.141676170409553 35 0.796954891427427 0.406090217145146 0.203045108572573 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.0666666666666667 NOK 5% type I error level 4 0.266666666666667 NOK 10% type I error level 5 0.333333333333333 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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