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: Wed, 16 Dec 2009 09:36:16 -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/Dec/16/t12609816533c06kewqatcbkri.htm/, Retrieved Wed, 16 Dec 2009 17:41:04 +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/Dec/16/t12609816533c06kewqatcbkri.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:

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No (this computation is public)

User-defined keywords:

Dataseries X:

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9.3 145.0 8.7 137.7 8.2 148.3 8.3 152.2 8.5 169.4 8.6 168.6 8.5 161.1 8.2 174.1 8.1 179.0 7.9 190.6 8.6 190.0 8.7 181.6 8.7 174.8 8.5 180.5 8.4 196.8 8.5 193.8 8.7 197.0 8.7 216.3 8.6 221.4 8.5 217.9 8.3 229.7 8 227.4 8.2 204.2 8.1 196.6 8.1 198.8 8 207.5 7.9 190.7 7.9 201.6 8 210.5 8 223.5 7.9 223.8 8 231.2 7.7 244.0 7.2 234.7 7.5 250.2 7.3 265.7 7 287.6 7 283.3 7 295.4 7.2 312.3 7.3 333.8 7.1 347.7 6.8 383.2 6.4 407.1 6.1 413.6 6.5 362.7 7.7 321.9 7.9 239.4 7.5 191.0 6.9 159.7 6.6 163.4 6.9 157.6 7.7 166.2 8 176.7 8 198.3 7.7 226.2 7.3 216.2 7.4 235.9 8.1 226.9 8.3 242.3 8.2 253.1

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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation GP[t] = + 778.153450228708 -68.6145719886735TW[t] -11.7049313874972M1[t] -47.8474972772816M2[t] -56.3904116750164M3[t] -42.204371596602M4[t] -11.1122914397735M5[t] + 2.81229143977347M6[t] + 5.57854280113265M7[t] + 5.59562840339795M8[t] -7.04416031365714M9[t] -20.1456175125245M10[t] + 10.7754171204530M11[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) 778.153450228708 80.142163 9.7097 0 0 TW -68.6145719886735 9.588368 -7.156 0 0 M1 -11.7049313874972 28.738457 -0.4073 0.685605 0.342802 M2 -47.8474972772816 30.095462 -1.5899 0.118433 0.059216 M3 -56.3904116750164 30.302479 -1.8609 0.068887 0.034443 M4 -42.204371596602 30.14491 -1.4 0.167929 0.083965 M5 -11.1122914397735 30.007967 -0.3703 0.71278 0.35639 M6 2.81229143977347 30.007967 0.0937 0.925723 0.462862 M7 5.57854280113265 30.022669 0.1858 0.853376 0.426688 M8 5.59562840339795 30.14491 0.1856 0.853522 0.426761 M9 -7.04416031365714 30.483974 -0.2311 0.818237 0.409118 M10 -20.1456175125245 30.667392 -0.6569 0.514378 0.257189 M11 10.7754171204530 30.009805 0.3591 0.721122 0.360561 Multiple Linear Regression - Regression Statistics Multiple R 0.752809599250182 R-squared 0.56672229272322 Adjusted R-squared 0.458402865904025 F-TEST (value) 5.23195431664518 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 1.55958954396462e-05 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 47.4457925164591 Sum Squared Residuals 108052.954920714 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 145 128.332999346547 16.6670006534528 2 137.7 133.359176649967 4.34082335003272 3 148.3 159.123548246569 -10.8235482465694 4 152.2 166.448131126116 -14.2481311261163 5 169.4 183.817296885210 -14.4172968852102 6 168.6 190.880422565890 -22.2804225658898 7 161.1 200.508131126116 -39.4081311261163 8 174.1 221.109588324984 -47.0095883249838 9 179 215.331256806796 -36.3312568067959 10 190.6 215.952714005663 -25.3527140056633 11 190 198.843548246569 -8.8435482465694 12 181.6 181.206673927249 0.393326072750949 13 174.8 169.501742539752 5.29825746024819 14 180.5 147.082091047702 33.4179089522979 15 196.8 145.400633848835 51.3993661511654 16 193.8 152.725216728382 41.0747832716184 17 197 170.094382487476 26.9056175125244 18 216.3 184.018965367022 32.2810346329775 19 221.4 193.646673927249 27.753326072751 20 217.9 200.525216728382 17.3747832716184 21 229.7 201.608342409061 28.0916575909388 22 227.4 209.091256806796 18.3087431932041 23 204.2 226.289377042039 -22.0893770420388 24 196.6 222.375417120453 -25.7754171204531 25 198.8 210.670485732956 -11.8704857329559 26 207.5 181.389377042039 26.1106229579612 27 190.7 179.707919843171 10.9920801568286 28 201.6 193.893959921586 7.70604007841429 29 210.5 218.124582879547 -7.62458287954692 30 223.5 232.049165759094 -8.54916575909386 31 223.8 241.676874319320 -17.8768743193204 32 231.2 234.832502722718 -3.63250272271837 33 244 242.777085602265 1.22291439773470 34 234.7 263.982914397735 -29.2829143977347 35 250.2 274.31957743411 -24.1195774341102 36 265.7 277.267074711392 -11.5670747113919 37 287.6 286.146514920497 1.45348507950335 38 283.3 250.003949030712 33.2960509692877 39 295.4 241.461034632978 53.9389653670224 40 312.3 241.924160313657 70.3758396863429 41 333.8 266.154783271618 67.6452167283816 42 347.7 293.8022805489 53.8977194511 43 383.2 317.152903506861 66.0470964931388 44 407.1 344.615817904596 62.4841820954041 45 413.6 352.560400784143 61.0395992158571 46 362.7 312.013114789806 50.6868852101939 47 321.9 260.596663036375 61.3033369636245 48 239.4 236.098331518188 3.30166848181227 49 191 251.83922892616 -60.83922892616 50 159.7 256.865406229580 -97.1654062295796 51 163.4 268.906863428447 -105.506863428447 52 157.6 262.508531910259 -104.908531910259 53 166.2 238.708954476149 -72.508954476149 54 176.7 232.049165759094 -55.3491657590939 55 198.3 234.815417120453 -36.5154171204531 56 226.2 255.416874319320 -29.2168743193204 57 216.2 270.222914397735 -54.0229143977347 58 235.9 250.26 -14.3600000000000 59 226.9 233.150834240906 -6.25083424090613 60 242.3 208.652502722718 33.6474972772817 61 253.1 203.809028534089 49.2909714659114 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.235075301653969 0.470150603307939 0.764924698346031 17 0.137590070705310 0.275180141410619 0.86240992929469 18 0.107540038444535 0.215080076889069 0.892459961555465 19 0.102798285272010 0.205596570544021 0.89720171472799 20 0.0663566929569409 0.132713385913882 0.933643307043059 21 0.0475708920195838 0.0951417840391675 0.952429107980416 22 0.0288882681629786 0.0577765363259573 0.971111731837021 23 0.0169141740231301 0.0338283480462602 0.98308582597687 24 0.0096392870227353 0.0192785740454706 0.990360712977265 25 0.00662817088008041 0.0132563417601608 0.99337182911992 26 0.00541548591656098 0.0108309718331220 0.99458451408344 27 0.00288774281017935 0.0057754856203587 0.99711225718982 28 0.00148082253685615 0.00296164507371230 0.998519177463144 29 0.000625184277404634 0.00125036855480927 0.999374815722595 30 0.000251737674574878 0.000503475349149755 0.999748262325425 31 9.77832193518592e-05 0.000195566438703718 0.999902216780648 32 4.21273513613147e-05 8.42547027226293e-05 0.999957872648639 33 1.80629971241985e-05 3.61259942483969e-05 0.999981937002876 34 6.44591366356302e-06 1.28918273271260e-05 0.999993554086337 35 3.15089279319709e-06 6.30178558639417e-06 0.999996849107207 36 1.88317633961417e-06 3.76635267922833e-06 0.99999811682366 37 9.1892515605494e-07 1.83785031210988e-06 0.999999081074844 38 1.55460653861463e-06 3.10921307722927e-06 0.999998445393461 39 2.44612350469799e-05 4.89224700939597e-05 0.999975538764953 40 0.00280859934134784 0.00561719868269568 0.997191400658652 41 0.0168750423999310 0.0337500847998621 0.983124957600069 42 0.0165739340611925 0.0331478681223849 0.983426065938808 43 0.0149343445744819 0.0298686891489637 0.985065655425518 44 0.00937150312751316 0.0187430062550263 0.990628496872487 45 0.0145072038604476 0.0290144077208951 0.985492796139552 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 14 0.466666666666667 NOK 5% type I error level 23 0.766666666666667 NOK 10% type I error level 25 0.833333333333333 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 2 ; par2 = Include Monthly 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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