*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, 18 Nov 2009 10:34:19 -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/18/t1258565801p4289xn3wz12pg6.htm/, Retrieved Wed, 18 Nov 2009 18:36:53 +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/18/t1258565801p4289xn3wz12pg6.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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Dataseries X:

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24 24 22 23 25 24 24 24 29 27 26 28 26 25 21 19 23 19 22 19 21 20 16 16 19 22 16 21 25 25 27 29 23 28 22 25 23 26 20 24 24 28 23 28 20 28 21 28 22 32 17 31 21 22 19 29 23 31 22 29 15 32 23 32 21 31 18 29 18 28 18 28 18 29 10 22 13 26 10 24 9 27 9 27 6 23 11 21 9 19 10 17 9 19 16 21 10 13 7 8 7 5 14 10 11 6 10 6 6 8 8 11 13 12 12 13 15 19 16 19 16 18

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 s[t] = + 21.3485818218273 + 0.212254119197142consv[t] -0.569446095385408M1[t] -5.12036481419157M2[t] -0.951260284566528M3[t] -0.703819760211766M4[t] -0.389420173623291M5[t] -1.17786482015882M6[t] -3.49366193821264M7[t] -1.55475411322988M8[t] + 0.00209629719801545M9[t] -0.628799173176945M10[t] -1.12665370578561M11[t] -0.241752058106757t + 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) 21.3485818218273 2.909627 7.3372 0 0 consv 0.212254119197142 0.075131 2.8251 0.006915 0.003458 M1 -0.569446095385408 2.133534 -0.2669 0.790712 0.395356 M2 -5.12036481419157 2.254835 -2.2708 0.027781 0.01389 M3 -0.951260284566528 2.255604 -0.4217 0.675143 0.337572 M4 -0.703819760211766 2.234698 -0.315 0.754193 0.377097 M5 -0.389420173623291 2.23139 -0.1745 0.862207 0.431104 M6 -1.17786482015882 2.230011 -0.5282 0.599854 0.299927 M7 -3.49366193821264 2.228325 -1.5678 0.123627 0.061813 M8 -1.55475411322988 2.230761 -0.697 0.48926 0.24463 M9 0.00209629719801545 2.227292 9e-04 0.999253 0.499627 M10 -0.628799173176945 2.227939 -0.2822 0.779004 0.389502 M11 -1.12665370578561 2.224166 -0.5066 0.614838 0.307419 t -0.241752058106757 0.030543 -7.915 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.864621245324081 R-squared 0.747569897865765 Adjusted R-squared 0.677748805786083 F-TEST (value) 10.7069350478307 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 4.74262407124115e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.51640624782329 Sum Squared Residuals 581.16030628734 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 24 25.6314825290665 -1.63148252906652 2 22 20.6265576329564 1.37344236704355 3 25 24.7661642236719 0.233835776328138 4 24 24.7718526899199 -0.771852689919875 5 29 25.481262575993 3.518737424007 6 26 24.6633199905479 1.33668000945214 7 26 21.4690084567959 4.53099154320413 8 21 21.892639508489 -0.892639508489017 9 23 23.2077378608102 -0.207737860810164 10 22 22.3350903323284 -0.335090332328447 11 21 21.8077378608102 -0.80773786081016 12 16 21.8436230317005 -5.84362303170045 13 19 22.3059495933911 -3.30594959339114 14 16 17.3010246972811 -1.30102469728107 15 25 22.0773936455879 2.92260635441207 16 27 22.9320985886245 4.0679014113755 17 23 22.7924919979091 0.207508002090925 18 22 21.1255329356754 0.874467064324638 19 23 18.7802378787119 4.21976212128807 20 20 20.0528854071936 -0.0528854071936464 21 24 22.2170002363034 1.78299976369664 22 23 21.3443527078216 1.65564729217836 23 20 20.6047461171062 -0.604746117106214 24 21 21.4896477647851 -0.489647764785071 25 22 21.5274660880815 0.472533911918524 26 17 16.5225411919714 0.477458808028586 27 21 18.5396065907154 2.46039340928457 28 19 20.0310738913434 -1.03107389134342 29 23 20.5282296582194 2.47177034178058 30 22 19.0735247151828 2.92647528481715 31 15 17.1527378966137 -2.1527378966137 32 23 18.8498936634897 4.1501063365103 33 21 19.9527378966137 1.0472621033863 34 18 18.6555821297377 -0.655582129737703 35 18 17.7037214198251 0.296278580174866 36 18 18.588623067504 -0.588623067503993 37 18 17.9896790332090 0.0103209667910265 38 10 11.7112294219161 -1.71122942191606 39 13 16.4875983702229 -3.48759837022292 40 10 16.0687785980766 -6.06877859807663 41 9 16.7781884841498 -7.77818848414978 42 9 15.7479917795075 -6.74799177950749 43 6 12.3414261265583 -6.34142612655835 44 11 13.6140736550401 -2.61407365504006 45 9 14.5046637689669 -5.50466376896693 46 10 13.2075080020909 -3.20750800209093 47 9 12.8924096497698 -3.89240964976978 48 16 14.2018195358429 1.79818046415708 49 10 11.6925884287736 -1.69258842877363 50 7 5.838647055875 1.16135294412500 51 7 9.12923716980186 -2.12923716980186 52 14 10.1961962320356 3.80380376796443 53 11 9.41982728372872 1.58017271627128 54 10 8.38963057908644 1.61036942091356 55 6 6.25658964132015 -0.256589641320147 56 8 8.59050776578757 -0.590507765787571 57 13 10.1178602373059 2.88213976269414 58 12 9.45746682802128 2.54253317197872 59 15 9.9913849524887 5.00861504751129 60 16 10.8762866001676 5.12371339983243 61 16 9.85283432747826 6.14716567252174 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.131321011662442 0.262642023324883 0.868678988337558 18 0.0775578710148501 0.155115742029700 0.92244212898515 19 0.0385097278373563 0.0770194556747126 0.961490272162644 20 0.0207005925653359 0.0414011851306718 0.979299407434664 21 0.0108430990361390 0.0216861980722780 0.989156900963861 22 0.00458058278498988 0.00916116556997975 0.99541941721501 23 0.00233301401978601 0.00466602803957202 0.997666985980214 24 0.00110186331896363 0.00220372663792727 0.998898136681036 25 0.000366087392167951 0.000732174784335901 0.999633912607832 26 0.000207432999181084 0.000414865998362168 0.99979256700082 27 0.000206698545444532 0.000413397090889064 0.999793301454555 28 0.000205068357120666 0.000410136714241332 0.99979493164288 29 0.000147236232778612 0.000294472465557224 0.999852763767221 30 0.000202691821046355 0.000405383642092710 0.999797308178954 31 0.00796411138970885 0.0159282227794177 0.992035888610291 32 0.0269567419588238 0.0539134839176477 0.973043258041176 33 0.0464365042976975 0.092873008595395 0.953563495702302 34 0.0612827373952374 0.122565474790475 0.938717262604763 35 0.104391044040296 0.208782088080592 0.895608955959704 36 0.102317975122531 0.204635950245062 0.897682024877469 37 0.254231426618622 0.508462853237245 0.745768573381378 38 0.237096250135469 0.474192500270938 0.762903749864531 39 0.53142964504802 0.937140709903960 0.468570354951980 40 0.556322236639797 0.887355526720406 0.443677763360203 41 0.647130350559114 0.705739298881773 0.352869649440886 42 0.698132084465303 0.603735831069395 0.301867915534698 43 0.634804289279024 0.730391421441951 0.365195710720976 44 0.61136623789619 0.777267524207621 0.388633762103811 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 9 0.321428571428571 NOK 5% type I error level 12 0.428571428571429 NOK 10% type I error level 15 0.535714285714286 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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