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*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 04:59:00 -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/t1258718399kqcww4n4nbdcclp.htm/, Retrieved Fri, 20 Nov 2009 13:00:11 +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/t1258718399kqcww4n4nbdcclp.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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1,6 0,55 1,6 1,6 1,59 1,58 1,6 0,56 1,6 1,6 1,6 1,59 1,61 0,56 1,6 1,6 1,6 1,6 1,61 0,56 1,61 1,6 1,6 1,6 1,62 0,56 1,61 1,61 1,6 1,6 1,63 0,56 1,62 1,61 1,61 1,6 1,63 0,55 1,63 1,62 1,61 1,61 1,63 0,56 1,63 1,63 1,62 1,61 1,63 0,55 1,63 1,63 1,63 1,62 1,63 0,55 1,63 1,63 1,63 1,63 1,64 0,56 1,63 1,63 1,63 1,63 1,64 0,55 1,64 1,63 1,63 1,63 1,64 0,55 1,64 1,64 1,63 1,63 1,65 0,55 1,64 1,64 1,64 1,63 1,65 0,55 1,65 1,64 1,64 1,64 1,65 0,53 1,65 1,65 1,64 1,64 1,65 0,53 1,65 1,65 1,65 1,64 1,65 0,53 1,65 1,65 1,65 1,65 1,66 0,53 1,65 1,65 1,65 1,65 1,67 0,54 1,66 1,65 1,65 1,65 1,68 0,54 1,67 1,66 1,65 1,65 1,68 0,54 1,68 1,67 1,66 1,65 1,68 0,55 1,68 1,68 1,67 1,66 1,68 0,55 1,68 1,68 1,68 1,67 1,69 0,54 1,68 1,68 1,68 1,68 1,7 0,55 1,69 1,68 1,68 1,68 1,7 0,56 1,7 1,69 1,68 1,68 1,71 0,58 1,7 1,7 1,69 1,68 1,73 0,59 1,71 1,7 1,7 1,69 1,73 0,6 1,73 1,71 1,7 1,7 1,73 0,6 1,73 1,73 1,71 1,7 1,74 0,6 1,73 1,73 1,73 1,71 1,74 0,59 1,74 1,73 1,73 1,73 1,74 0,6 1,74 1,74 1,73 1,73 1 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] = + 0.373041030730798 + 0.0731322482947126X[t] + 1.42861292764240Y1[t] -0.79401619507144Y2[t] + 0.441334049246045Y3[t] -0.334606997433555Y4[t] + 0.00358222217610623M1[t] -0.00127785491116625M2[t] -0.00305874236631328M3[t] -0.00477107569119665M4[t] + 0.00229643512668181M5[t] -0.00643096429226704M6[t] + 0.000599238113361633M7[t] -0.00360162979024544M8[t] -0.00326629003045313M9[t] -0.00469609721726716M10[t] + 0.000140090461648979M11[t] + 0.00106672568748966t + 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) 0.373041030730798 0.248106 1.5036 0.140752 0.070376 X 0.0731322482947126 0.070876 1.0318 0.308506 0.154253 Y1 1.42861292764240 0.156205 9.1458 0 0 Y2 -0.79401619507144 0.268437 -2.9579 0.005239 0.00262 Y3 0.441334049246045 0.264374 1.6694 0.103056 0.051528 Y4 -0.334606997433555 0.16104 -2.0778 0.044359 0.022179 M1 0.00358222217610623 0.005045 0.7101 0.481882 0.240941 M2 -0.00127785491116625 0.005022 -0.2544 0.800498 0.400249 M3 -0.00305874236631328 0.005116 -0.5979 0.553392 0.276696 M4 -0.00477107569119665 0.005114 -0.9329 0.356609 0.178305 M5 0.00229643512668181 0.005026 0.4569 0.650264 0.325132 M6 -0.00643096429226704 0.00496 -1.2967 0.202364 0.101182 M7 0.000599238113361633 0.005181 0.1157 0.908523 0.454261 M8 -0.00360162979024544 0.004982 -0.7229 0.47405 0.237025 M9 -0.00326629003045313 0.005012 -0.6517 0.518425 0.259212 M10 -0.00469609721726716 0.005314 -0.8837 0.382281 0.191141 M11 0.000140090461648979 0.005278 0.0265 0.978962 0.489481 t 0.00106672568748966 0.000645 1.6537 0.106219 0.053109 Multiple Linear Regression - Regression Statistics Multiple R 0.998175276796487 R-squared 0.996353883207744 Adjusted R-squared 0.994764550247017 F-TEST (value) 626.900660735122 F-TEST (DF numerator) 17 F-TEST (DF denominator) 39 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.00734352971707821 Sum Squared Residuals 0.00210316971951882 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.6 1.60630956962621 -0.00630956962621468 2 1.6 1.6043148112275 -0.00431481122750106 3 1.61 1.60025457948551 0.0097454205144918 4 1.61 1.61389510112454 -0.00389510112453837 5 1.62 1.61408917567919 0.00591082432080794 6 1.63 1.62512797171662 0.00487202828338250 7 1.63 1.63549347467816 -0.00549347467816241 8 1.63 1.62956383348674 0.000436166513261634 9 1.63 1.63130184696920 -0.00130184696919800 10 1.63 1.62759269549554 0.00240730450446184 11 1.64 1.63422693134489 0.00577306865510891 12 1.64 1.64870837336421 -0.00870837336420861 13 1.64 1.64541715927709 -0.0054171592770901 14 1.65 1.64603714836977 0.00396285163023228 15 1.65 1.6562630459042 -0.00626304590419878 16 1.65 1.64621463135020 0.00378536864980361 17 1.65 1.65876220834802 -0.00876220834802496 18 1.65 1.64775546464223 0.00224453535776977 19 1.66 1.65585239273535 0.00414760726465145 20 1.67 1.66773570227860 0.00226429772139774 21 1.68 1.67548373505159 0.00451626494840622 22 1.68 1.68587996137044 -0.00587996137043944 23 1.68 1.68564130578720 -0.00564130578720288 24 1.68 1.68763521153117 -0.00763521153116847 25 1.69 1.68820676693748 0.00179323306251834 26 1.7 1.69943086729707 0.000569132702930058 27 1.7 1.70579399533807 -0.00579399533806928 28 1.71 1.70308421120832 0.00691578879168415 29 1.73 1.72730316999118 0.00269683000882003 30 1.73 1.73765984537047 -0.0076598453704659 31 1.73 1.73428979005462 -0.00428979005461587 32 1.74 1.73663625884908 0.00336374115091619 33 1.74 1.74490099114117 -0.00490099114117154 34 1.74 1.73732907017408 0.00267092982592012 35 1.75 1.74764532403295 0.00235467596705387 36 1.78 1.76097466352677 0.0190253364732305 37 1.82 1.80273580471776 0.0172641952822358 38 1.83 1.83887379251284 -0.00887379251283594 39 1.84 1.83423567613653 0.00576432386347366 40 1.85 1.85120780028641 -0.00120780028641301 41 1.86 1.85671706471261 0.00328293528739105 42 1.86 1.85939491875677 0.000605081243227267 43 1.87 1.8606189554173 0.00938104458269844 44 1.87 1.87210689051279 -0.00210689051278590 45 1.87 1.86368536900091 0.00631463099908783 46 1.87 1.86919827295994 0.000801727040057486 47 1.87 1.87248643883496 -0.00248643883495991 48 1.87 1.87268175157785 -0.00268175157785346 49 1.87 1.87733069944145 -0.00733069944144935 50 1.88 1.87134338059283 0.00865661940717465 51 1.88 1.88345270313570 -0.00345270313569739 52 1.87 1.87559825603054 -0.00559825603053637 53 1.87 1.87312838126899 -0.00312838126899405 54 1.87 1.87006179951391 -6.17995139136242e-05 55 1.87 1.87374538711457 -0.00374538711457161 56 1.87 1.87395731487279 -0.00395731487278968 57 1.87 1.87462805783712 -0.00462805783712452 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.20574896448881 0.41149792897762 0.79425103551119 22 0.155010020193866 0.310020040387732 0.844989979806134 23 0.103805504613915 0.207611009227829 0.896194495386085 24 0.072149606260111 0.144299212520222 0.927850393739889 25 0.105339142127560 0.210678284255120 0.89466085787244 26 0.0565881976078421 0.113176395215684 0.943411802392158 27 0.0430153598453238 0.0860307196906475 0.956984640154676 28 0.0244276697020466 0.0488553394040933 0.975572330297953 29 0.0135233843555140 0.0270467687110279 0.986476615644486 30 0.0111422663646205 0.0222845327292410 0.98885773363538 31 0.00841971055135736 0.0168394211027147 0.991580289448643 32 0.00453812422004067 0.00907624844008134 0.99546187577996 33 0.0139135408381284 0.0278270816762568 0.986086459161872 34 0.0228205556149223 0.0456411112298447 0.977179444385078 35 0.0707688155057276 0.141537631011455 0.929231184494272 36 0.877724232731431 0.244551534537138 0.122275767268569 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.0625 NOK 5% type I error level 7 0.4375 NOK 10% type I error level 8 0.5 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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