Multiple Regression

*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 08:57:39 -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/t12586463451g737s4y8huzjz8.htm/, Retrieved Thu, 19 Nov 2009 16:59:18 +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/19/t12586463451g737s4y8huzjz8.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}, }

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Dataseries X:

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1.4 1.9 1.5 -0.7 -0.7 -2.9 -0.8 1 1 1.6 3 1.5 -0.7 -0.7 -2.9 -0.8 -0.8 0 3.2 3 1.5 -0.7 -0.7 -2.9 -2.9 -1.3 3.1 3.2 3 1.5 -0.7 -0.7 -0.7 -0.4 3.9 3.1 3.2 3 1.5 -0.7 -0.7 -0.3 1 3.9 3.1 3.2 3 1.5 1.5 1.4 1.3 1 3.9 3.1 3.2 3 3 2.6 0.8 1.3 1 3.9 3.1 3.2 3.2 2.8 1.2 0.8 1.3 1 3.9 3.1 3.1 2.6 2.9 1.2 0.8 1.3 1 3.9 3.9 3.4 3.9 2.9 1.2 0.8 1.3 1 1 1.7 4.5 3.9 2.9 1.2 0.8 1.3 1.3 1.2 4.5 4.5 3.9 2.9 1.2 0.8 0.8 0 3.3 4.5 4.5 3.9 2.9 1.2 1.2 0 2 3.3 4.5 4.5 3.9 2.9 2.9 1.6 1.5 2 3.3 4.5 4.5 3.9 3.9 2.5 1 1.5 2 3.3 4.5 4.5 4.5 3.2 2.1 1 1.5 2 3.3 4.5 4.5 3.4 3 2.1 1 1.5 2 3.3 3.3 2.3 4 3 2.1 1 1.5 2 2 1.9 5.1 4 3 2.1 1 1.5 1.5 1.7 4.5 5.1 4 3 2.1 1 1 1.9 4.2 4.5 5.1 4 3 2.1 2.1 3.3 3.3 4.2 4.5 5.1 4 3 3 3.8 2.7 3.3 4.2 4.5 5.1 4 4 4.4 1.8 2.7 3.3 4.2 4.5 5.1 5.1 4.5 1.4 1.8 2.7 3.3 4.2 4.5 4.5 3.5 0.5 1.4 1.8 2.7 3.3 4.2 4.2 3 -0.4 0.5 1.4 1.8 2.7 3.3 3.3 2.8 0.8 -0.4 0.5 1.4 1.8 2.7 2.7 2.9 0.7 0.8 -0.4 0.5 1.4 1.8 1.8 2.6 1.9 0.7 0.8 -0.4 0.5 1.4 1.4 2.1 2 1.9 0.7 0.8 -0.4 0.5 0.5 1.5 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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation bbp[t] = -0.760450316872803 + 0.774258904965288dnst[t] + 0.0362963516334644y1[t] + 0.219062946263493y2[t] -0.323223334732678y3[t] -0.0953145983971215y4[t] + 0.157222725472913y5[t] + 0.427847039627963y6[t] + 0.0931224218556833M1[t] + 0.164204507217995M2[t] + 0.621519667509817M3[t] + 0.538780505278127M4[t] + 0.811416463269424M5[t] + 0.545048444379198M6[t] + 0.510790444622955M7[t] + 0.465193494862052M8[t] + 0.481376617769967M9[t] + 0.660971252550343M10[t] + 0.500162621073792M11[t] -0.00411827080944167t + 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) -0.760450316872803 0.474901 -1.6013 0.117596 0.058798 dnst 0.774258904965288 0.132288 5.8528 1e-06 0 y1 0.0362963516334644 0.118406 0.3065 0.760866 0.380433 y2 0.219062946263493 0.17854 1.227 0.227382 0.113691 y3 -0.323223334732678 0.164078 -1.9699 0.056166 0.028083 y4 -0.0953145983971215 0.15149 -0.6292 0.532996 0.266498 y5 0.157222725472913 0.152997 1.0276 0.310626 0.155313 y6 0.427847039627963 0.143527 2.9809 0.004991 0.002496 M1 0.0931224218556833 0.437042 0.2131 0.832408 0.416204 M2 0.164204507217995 0.432991 0.3792 0.706625 0.353313 M3 0.621519667509817 0.440206 1.4119 0.166121 0.08306 M4 0.538780505278127 0.445565 1.2092 0.234053 0.117026 M5 0.811416463269424 0.432863 1.8745 0.068559 0.03428 M6 0.545048444379198 0.4463 1.2213 0.22951 0.114755 M7 0.510790444622955 0.443234 1.1524 0.256348 0.128174 M8 0.465193494862052 0.447887 1.0386 0.305536 0.152768 M9 0.481376617769967 0.448655 1.0729 0.290068 0.145034 M10 0.660971252550343 0.442265 1.4945 0.1433 0.07165 M11 0.500162621073792 0.450604 1.11 0.273981 0.136991 t -0.00411827080944167 0.005713 -0.7209 0.475411 0.237706 Multiple Linear Regression - Regression Statistics Multiple R 0.942883982470664 R-squared 0.889030204399739 Adjusted R-squared 0.833545306599608 F-TEST (value) 16.0229222662035 F-TEST (DF numerator) 19 F-TEST (DF denominator) 38 p-value 1.10933484620546e-12 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.631796656250199 Sum Squared Residuals 15.1683465642594 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.4 1.50548374758740 -0.105483747587396 2 1 0.566568388583449 0.433431611416551 3 -0.8 -1.14687540374691 0.346875403746912 4 -2.9 -1.95335009054473 -0.94664990945527 5 -0.7 -0.842595170715503 0.142595170715503 6 -0.7 0.224692356539836 -0.924692356539836 7 1.5 1.30233048233115 0.197669517668852 8 3 3.16023978316684 -0.160239783166836 9 3.2 3.49458229512739 -0.294582295127388 10 3.1 3.68388487006568 -0.583884870065677 11 3.9 3.27184682005959 0.628153179940412 12 1 1.15430378772647 -0.154303787726467 13 1.3 0.351323672415560 0.94867632758444 14 0.8 -0.405209971031664 1.20520997103166 15 1.2 0.565300059613197 0.634699940386803 16 2.9 2.32437554514818 0.575624454851822 17 3.9 3.95332267485612 -0.0533226748561192 18 4.5 4.25206550701232 0.247934492987681 19 4.5 4.13363995069621 0.366360049303791 20 3.3 2.52299205467446 0.777007945325538 21 2 1.79606033580044 0.203939664199558 22 1.5 1.58589135360422 -0.0858913536042175 23 1 1.59476148897646 -0.594761488976461 24 2.1 2.70743046743632 -0.607430467436317 25 3 3.71957740545422 -0.719577405454224 26 4 4.78278556784361 -0.782785567843612 27 5.1 5.07757525351036 0.0224247464896440 28 4.5 4.17440221601209 0.325597783987906 29 4.2 3.56164358410561 0.63835641589439 30 3.3 2.91352264765176 0.38647735234824 31 2.7 3.14055088189794 -0.440550881897936 32 1.8 2.26548318527320 -0.465483185273203 33 1.4 1.54830578237450 -0.148305782374504 34 0.5 0.673734774585324 -0.173734774585324 35 -0.4 0.345682711805831 -0.74568271180583 36 0.8 0.516396724863678 0.283603275136322 37 0.7 1.34117661486606 -0.641176614866062 38 1.9 2.07118838664694 -0.17118838664694 39 2 2.39066426506064 -0.390664265060643 40 1.1 1.54208550040772 -0.442085500407717 41 0.9 1.58802630408160 -0.688026304081604 42 0.4 0.791966786906765 -0.391966786906765 43 0.7 0.848777573926151 -0.148777573926151 44 2.1 2.31829417823242 -0.218294178232419 45 2.8 2.96269024082294 -0.162690240822945 46 3.9 3.36592379683282 0.53407620316718 47 3.5 2.78770897915812 0.71229102084188 48 2 1.52186901997354 0.478130980026464 49 2 1.48243855967676 0.51756144032324 50 1.5 2.18466762795766 -0.684667627957663 51 2.5 3.11333582556272 -0.613335825562715 52 3.1 2.61248682897674 0.487513171023258 53 2.7 2.73960260767217 -0.0396026076721690 54 2.8 2.11775270188932 0.682247298110681 55 2.5 2.47470111114856 0.0252988888514431 56 3 2.93299079865308 0.0670092013469196 57 3.2 2.79836134587472 0.401638654125278 58 2.8 2.49056520491196 0.309434795088039 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 23 0.495101178150941 0.990202356301883 0.504898821849059 24 0.39244355083016 0.78488710166032 0.60755644916984 25 0.918673940884308 0.162652118231385 0.0813260591156923 26 0.938600998932204 0.122798002135592 0.0613990010677959 27 0.90671861853416 0.186562762931679 0.0932813814658394 28 0.924967945675359 0.150064108649282 0.0750320543246409 29 0.960026306465154 0.079947387069693 0.0399736935348465 30 0.938328967647873 0.123342064704254 0.0616710323521272 31 0.962604037318086 0.0747919253638286 0.0373959626819143 32 0.933323436923023 0.133353126153953 0.0666765630769767 33 0.875877791348684 0.248244417302632 0.124122208651316 34 0.877543523175374 0.244912953649253 0.122456476824626 35 0.756631768557978 0.486736462884043 0.243368231442022 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 2 0.153846153846154 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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