Multiple Linear 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: Wed, 01 Dec 2010 14:26:43 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/01/t1291213548lh54hx16tqilrje.htm/, Retrieved Wed, 01 Dec 2010 15:25:59 +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/2010/Dec/01/t1291213548lh54hx16tqilrje.htm/}, year = {2010}, } @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 = {2010}, 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:

Linear Trend

Dataseries X:

» Textbox « » Textfile « » CSV «

94.6 116.1 95.9 107.5 104.7 116.7 102.8 112.5 98.1 113 113.9 126.4 80.9 114.1 95.7 112.5 113.2 112.4 105.9 113.1 108.8 116.3 102.3 111.7 99 118.8 100.7 116.5 115.5 125.1 100.7 113.1 109.9 119.6 114.6 114.4 85.4 114 100.5 117.8 114.8 117 116.5 120.9 112.9 115 102 117.3 106 119.4 105.3 114.9 118.8 125.8 106.1 117.6 109.3 117.6 117.2 114.9 92.5 121.9 104.2 117 112.5 106.4 122.4 110.5 113.3 113.6 100 114.2 110.7 125.4 112.8 124.6 109.8 120.2 117.3 120.8 109.1 111.4 115.9 124.1 96 120.2 99.8 125.5 116.8 116 115.7 117 99.4 105.7 94.3 102 91 106.4 93.2 96.9 103.1 107.6 94.1 98.8 91.8 101.1 102.7 105.7 82.6 104.6 89.1 103.2 104.5 101.6 105.1 106.7 95.1 99.5 88.7 101

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 7 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation I.P.C.N.[t] = + 35.3607050481266 + 0.80254709385946T.I.P.[t] + 4.40769055462786M1[t] -1.67120091994249M2[t] -1.61314465658158M3[t] -3.05293292720597M4[t] -2.50303586532052M5[t] -5.22204938138058M6[t] + 13.1270665500966M7[t] + 5.15709840515961M8[t] -10.8793637664784M9[t] -8.40882886848744M10[t] -6.11396816149799M11[t] -0.120470689324152t + 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) 35.3607050481266 8.439546 4.1899 0.000125 6.3e-05 T.I.P. 0.80254709385946 0.0814 9.8593 0 0 M1 4.40769055462786 2.479298 1.7778 0.082048 0.041024 M2 -1.67120091994249 2.48515 -0.6725 0.504646 0.252323 M3 -1.61314465658158 2.657938 -0.6069 0.546891 0.273445 M4 -3.05293292720597 2.513266 -1.2147 0.230669 0.115334 M5 -2.50303586532052 2.503426 -0.9998 0.322617 0.161308 M6 -5.22204938138058 2.74159 -1.9048 0.063076 0.031538 M7 13.1270665500966 2.600069 5.0487 7e-06 4e-06 M8 5.15709840515961 2.45626 2.0996 0.041281 0.020641 M9 -10.8793637664784 2.72884 -3.9868 0.000237 0.000119 M10 -8.40882886848744 2.758882 -3.0479 0.003812 0.001906 M11 -6.11396816149799 2.546047 -2.4014 0.020433 0.010216 t -0.120470689324152 0.030364 -3.9675 0.000252 0.000126 Multiple Linear Regression - Regression Statistics Multiple R 0.889237640042933 R-squared 0.790743580469125 Adjusted R-squared 0.731605896688661 F-TEST (value) 13.3712301517351 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 1.52531320907201e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.87919616466738 Sum Squared Residuals 692.215492662624 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 116.1 115.568879992535 0.531120007464972 2 107.5 110.412829050658 -2.91282905065792 3 116.7 117.412829050658 -0.71282905065791 4 112.5 114.327730612376 -1.82773061237640 5 113 110.985185643798 2.01481435620177 6 126.4 120.825945521393 5.57405447860652 7 114.1 112.570536666184 1.52946333381559 8 112.5 116.357794821043 -3.85779482104322 9 112.4 114.245436102622 -1.84543610262157 10 113.1 110.736906526114 2.36309347388565 11 116.3 115.238683115972 1.06131688402794 12 111.7 116.015624478059 -4.31562447805942 13 118.8 117.654438933627 1.14556106637307 14 116.5 112.819406829293 3.6805931707065 15 125.1 124.634689392450 0.465310607549739 16 113.1 111.196733443382 1.90326655661828 17 119.6 119.00959307945 0.590406920549954 18 114.4 119.942080215205 -5.54208021520527 19 114 114.736350316662 -0.73635031666215 20 117.8 118.764372599679 -0.964372599678796 21 117 114.083863180907 2.91613681909310 22 120.9 117.798257449135 3.10174255086521 23 115 117.083477928906 -2.08347792890604 24 117.3 114.329212078012 2.97078792198823 25 119.4 121.826620318753 -2.42662031875331 26 114.9 115.065475189157 -0.165475189157182 27 125.8 125.837446530297 -0.0374465302966489 28 117.6 114.084839478333 3.51516052166703 29 117.6 117.082416551245 0.517583448755456 30 114.9 120.58305438735 -5.68305438735005 31 121.9 118.988786411174 2.91121358882552 32 117 120.288148575069 -3.28814857506897 33 106.4 110.792356593140 -4.39235659314032 34 110.5 121.087637031016 -10.5876370310158 35 113.6 115.95884849456 -2.35884849456000 36 114.2 111.278469618403 2.92153038159698 37 125.4 124.152943388003 1.24705661199705 38 124.6 119.638930121213 4.96106987878668 39 120.2 117.168874413672 3.03112558632831 40 120.8 121.627718657669 -0.827718657669097 41 111.4 115.476258860583 -4.07625886058282 42 124.1 118.094094893443 6.00590510655705 43 120.2 120.352052967793 -0.152052967792771 44 125.5 115.311293090198 10.1887069098025 45 116 112.797660824846 3.20233917515382 46 117 114.264923230268 2.73507676973241 47 105.7 103.357795618024 2.34220438197630 48 102 105.258302911514 -3.25830291151428 49 106.4 106.897117367082 -0.497117367081779 50 96.9 102.463358809678 -5.56335880967808 51 107.6 110.346160612923 -2.74616061292350 52 98.8 101.562977808240 -2.76297780823982 53 101.1 100.146545864924 0.95345413507564 54 105.7 106.054824982608 -0.354824982608252 55 104.6 108.152273638186 -3.55227363818620 56 103.2 105.278390914011 -2.07839091401149 57 101.6 101.480683298485 0.119316701514982 58 106.7 104.312275763467 2.38772423653252 59 99.5 98.4611948425382 1.03880515746181 60 101 99.3183909140115 1.68160908598851 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.117915118832620 0.235830237665240 0.88208488116738 18 0.669119897642092 0.661760204715816 0.330880102357908 19 0.534201641346125 0.93159671730775 0.465798358653875 20 0.426092123603887 0.852184247207775 0.573907876396113 21 0.368988038765255 0.737976077530509 0.631011961234745 22 0.286366660834793 0.572733321669585 0.713633339165207 23 0.219543857156316 0.439087714312632 0.780456142843684 24 0.227455230415668 0.454910460831337 0.772544769584332 25 0.181112730615597 0.362225461231194 0.818887269384403 26 0.119980692282968 0.239961384565935 0.880019307717032 27 0.0737994389073882 0.147598877814776 0.926200561092612 28 0.0693029613401554 0.138605922680311 0.930697038659845 29 0.0467686664687126 0.0935373329374253 0.953231333531287 30 0.0657919213349363 0.131583842669873 0.934208078665064 31 0.0725837543393367 0.145167508678673 0.927416245660663 32 0.0543034651069253 0.108606930213851 0.945696534893075 33 0.0429249078545572 0.0858498157091145 0.957075092145443 34 0.452841518433914 0.905683036867828 0.547158481566086 35 0.520089485042161 0.959821029915678 0.479910514957839 36 0.432761168990491 0.865522337980982 0.567238831009509 37 0.353614586807036 0.707229173614072 0.646385413192964 38 0.431911901468352 0.863823802936704 0.568088098531648 39 0.355621736083964 0.711243472167929 0.644378263916036 40 0.247886374143954 0.495772748287908 0.752113625856046 41 0.498751580339638 0.997503160679276 0.501248419660362 42 0.404151730578281 0.808303461156563 0.595848269421719 43 0.276446229730004 0.552892459460008 0.723553770269996 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 0.0740740740740741 OK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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