*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 07:14:17 -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/t1258726571gpca9r73cqpscp3.htm/, Retrieved Fri, 20 Nov 2009 15:16:24 +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/t1258726571gpca9r73cqpscp3.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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94,6 0 95,9 1 104,7 1 102,8 1 98,1 1 113,9 1 80,9 1 95,7 1 113,2 1 105,9 1 108,8 0 102,3 0 99 1 100,7 1 115,5 0 100,7 1 109,9 1 114,6 0 85,4 1 100,5 1 114,8 0 116,5 0 112,9 1 102 1 106 0 105,3 1 118,8 1 106,1 1 109,3 0 117,2 0 92,5 1 104,2 0 112,5 1 122,4 1 113,3 1 100 1 110,7 1 112,8 1 109,8 1 117,3 1 109,1 1 115,9 1 96 1 99,8 0 116,8 1 115,7 0 99,4 0 94,3 0 91 1 93,2 1 103,1 0 94,1 1 91,8 0 102,7 0 82,6 0 89,1 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 Y[t] = + 104.905263157895 -0.880938833570407X[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) 104.905263157895 2.277194 46.0678 0 0 X -0.880938833570407 2.801516 -0.3145 0.75439 0.377195 Multiple Linear Regression - Regression Statistics Multiple R 0.0427522005347858 R-squared 0.00182775065056654 Adjusted R-squared -0.0166569206336824 F-TEST (value) 0.0988792617656127 F-TEST (DF numerator) 1 F-TEST (DF denominator) 54 p-value 0.754390470610119 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 9.9260577212226 Sum Squared Residuals 5320.43758179232 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 94.6 104.905263157895 -10.3052631578949 2 95.9 104.024324324324 -8.12432432432433 3 104.7 104.024324324324 0.675675675675677 4 102.8 104.024324324324 -1.22432432432433 5 98.1 104.024324324324 -5.92432432432433 6 113.9 104.024324324324 9.87567567567568 7 80.9 104.024324324324 -23.1243243243243 8 95.7 104.024324324324 -8.32432432432432 9 113.2 104.024324324324 9.17567567567568 10 105.9 104.024324324324 1.87567567567568 11 108.8 104.905263157895 3.89473684210527 12 102.3 104.905263157895 -2.60526315789473 13 99 104.024324324324 -5.02432432432433 14 100.7 104.024324324324 -3.32432432432432 15 115.5 104.905263157895 10.5947368421053 16 100.7 104.024324324324 -3.32432432432432 17 109.9 104.024324324324 5.87567567567568 18 114.6 104.905263157895 9.69473684210526 19 85.4 104.024324324324 -18.6243243243243 20 100.5 104.024324324324 -3.52432432432433 21 114.8 104.905263157895 9.89473684210527 22 116.5 104.905263157895 11.5947368421053 23 112.9 104.024324324324 8.87567567567568 24 102 104.024324324324 -2.02432432432433 25 106 104.905263157895 1.09473684210527 26 105.3 104.024324324324 1.27567567567567 27 118.8 104.024324324324 14.7756756756757 28 106.1 104.024324324324 2.07567567567567 29 109.3 104.905263157895 4.39473684210527 30 117.2 104.905263157895 12.2947368421053 31 92.5 104.024324324324 -11.5243243243243 32 104.2 104.905263157895 -0.705263157894728 33 112.5 104.024324324324 8.47567567567567 34 122.4 104.024324324324 18.3756756756757 35 113.3 104.024324324324 9.27567567567567 36 100 104.024324324324 -4.02432432432433 37 110.7 104.024324324324 6.67567567567568 38 112.8 104.024324324324 8.77567567567567 39 109.8 104.024324324324 5.77567567567567 40 117.3 104.024324324324 13.2756756756757 41 109.1 104.024324324324 5.07567567567567 42 115.9 104.024324324324 11.8756756756757 43 96 104.024324324324 -8.02432432432433 44 99.8 104.905263157895 -5.10526315789473 45 116.8 104.024324324324 12.7756756756757 46 115.7 104.905263157895 10.7947368421053 47 99.4 104.905263157895 -5.50526315789472 48 94.3 104.905263157895 -10.6052631578947 49 91 104.024324324324 -13.0243243243243 50 93.2 104.024324324324 -10.8243243243243 51 103.1 104.905263157895 -1.80526315789474 52 94.1 104.024324324324 -9.92432432432433 53 91.8 104.905263157895 -13.1052631578947 54 102.7 104.905263157895 -2.20526315789473 55 82.6 104.905263157895 -22.3052631578947 56 89.1 104.024324324324 -14.9243243243243 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0769445609977064 0.153889121995413 0.923055439002294 6 0.242584774842380 0.485169549684759 0.75741522515762 7 0.71194176921252 0.576116461574961 0.288058230787481 8 0.615191684642902 0.769616630714196 0.384808315357098 9 0.67444073793005 0.6511185241399 0.32555926206995 10 0.58754701219993 0.82490597560014 0.41245298780007 11 0.567597664334845 0.86480467133031 0.432402335665155 12 0.465074326952916 0.930148653905831 0.534925673047084 13 0.375819259642542 0.751638519285083 0.624180740357458 14 0.289387510168005 0.578775020336009 0.710612489831995 15 0.323399985958005 0.64679997191601 0.676600014041995 16 0.247837963355304 0.495675926710608 0.752162036644696 17 0.224347393605276 0.448694787210552 0.775652606394724 18 0.212467008051156 0.424934016102313 0.787532991948844 19 0.363415338599363 0.726830677198727 0.636584661400637 20 0.2941976378515 0.5883952757030 0.7058023621485 21 0.277157501979427 0.554315003958854 0.722842498020573 22 0.282284241787079 0.564568483574158 0.717715758212921 23 0.296467008641802 0.592934017283605 0.703532991358198 24 0.234432659015944 0.468865318031888 0.765567340984056 25 0.184346437396219 0.368692874792437 0.815653562603781 26 0.140345602173351 0.280691204346703 0.859654397826649 27 0.220141584000948 0.440283168001896 0.779858415999052 28 0.169578728253236 0.339157456506471 0.830421271746764 29 0.135004121219399 0.270008242438799 0.8649958787806 30 0.166615811381745 0.333231622763491 0.833384188618255 31 0.192928567993643 0.385857135987286 0.807071432006357 32 0.157626244459011 0.315252488918021 0.84237375554099 33 0.146796196746138 0.293592393492276 0.853203803253862 34 0.289602131787374 0.579204263574748 0.710397868212626 35 0.279865126540874 0.559730253081748 0.720134873459126 36 0.226039466701503 0.452078933403005 0.773960533298497 37 0.192312621891145 0.38462524378229 0.807687378108855 38 0.181353128916851 0.362706257833702 0.81864687108315 39 0.149887674707983 0.299775349415967 0.850112325292017 40 0.216475449324352 0.432950898648705 0.783524550675648 41 0.192022245302091 0.384044490604182 0.80797775469791 42 0.315265681679281 0.630531363358562 0.684734318320719 43 0.254610075860285 0.50922015172057 0.745389924139715 44 0.198559934029665 0.397119868059329 0.801440065970335 45 0.534048722569906 0.931902554860188 0.465951277430094 46 0.854343265629178 0.291313468741643 0.145656734370822 47 0.804757688849777 0.390484622300446 0.195242311150223 48 0.7256806939706 0.5486386120588 0.2743193060294 49 0.625354213589077 0.749291572821846 0.374645786410923 50 0.494142039836367 0.988284079672733 0.505857960163633 51 0.482584036656949 0.965168073313897 0.517415963343051 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):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; 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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