*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 01:47:11 -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/t12586206143pctc3md3gmv49t.htm/, Retrieved Thu, 19 Nov 2009 09:50:26 +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/t12586206143pctc3md3gmv49t.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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3353 1 3186 1 3902 1 4164 1 3499 1 4145 1 3796 1 3711 1 3949 1 3740 1 3243 1 4407 1 4814 1 3908 1 5250 1 3937 1 4004 1 5560 1 3922 1 3759 1 4138 1 4634 1 3996 1 4308 1 4143 0 4429 0 5219 0 4929 0 5755 0 5592 0 4163 0 4962 0 5208 0 4755 0 4491 0 5732 0 5731 0 5040 0 6102 0 4904 0 5369 0 5578 0 4619 0 4731 0 5011 0 5299 0 4146 0 4625 0 4736 0 4219 0 5116 0 4205 0 4121 0 5103 1 4300 1 4578 1 3809 1 5526 1 4247 1 3830 1 4394 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 5051.60408163265 -785.340136054422X[t] -130.434013605440M1[t] -581.068027210885M2[t] + 380.331972789115M3[t] -309.668027210883M4[t] -187.868027210884M5[t] + 615.199999999999M6[t] -420.4M7[t] -232.2M8[t] -157.400000000000M9[t] + 210.4M10[t] -555.8M11[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) 5051.60408163265 243.685798 20.73 0 0 X -785.340136054422 134.144617 -5.8544 0 0 M1 -130.434013605440 311.72392 -0.4184 0.6775 0.33875 M2 -581.068027210885 326.387939 -1.7803 0.081357 0.040679 M3 380.331972789115 326.387939 1.1653 0.249666 0.124833 M4 -309.668027210883 326.387939 -0.9488 0.347489 0.173744 M5 -187.868027210884 326.387939 -0.5756 0.567576 0.283788 M6 615.199999999999 325.283408 1.8913 0.064631 0.032315 M7 -420.4 325.283408 -1.2924 0.202402 0.101201 M8 -232.2 325.283408 -0.7138 0.478784 0.239392 M9 -157.400000000000 325.283408 -0.4839 0.630667 0.315334 M10 210.4 325.283408 0.6468 0.520829 0.260415 M11 -555.8 325.283408 -1.7087 0.093973 0.046986 Multiple Linear Regression - Regression Statistics Multiple R 0.749715982507451 R-squared 0.562074054427113 Adjusted R-squared 0.452592568033891 F-TEST (value) 5.1339644075376 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 1.93964197101604e-05 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 514.318227918436 Sum Squared Residuals 12697115.4993197 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 3353 4135.82993197278 -782.82993197278 2 3186 3685.19591836735 -499.195918367347 3 3902 4646.59591836735 -744.595918367348 4 4164 3956.59591836735 207.404081632653 5 3499 4078.39591836735 -579.395918367346 6 4145 4881.46394557823 -736.463945578232 7 3796 3845.86394557823 -49.8639455782308 8 3711 4034.06394557823 -323.063945578231 9 3949 4108.86394557823 -159.863945578231 10 3740 4476.66394557823 -736.663945578231 11 3243 3710.46394557823 -467.463945578232 12 4407 4266.26394557823 140.736054421769 13 4814 4135.82993197279 678.170068027209 14 3908 3685.19591836735 222.804081632653 15 5250 4646.59591836735 603.404081632653 16 3937 3956.59591836735 -19.5959183673476 17 4004 4078.39591836735 -74.3959183673475 18 5560 4881.46394557823 678.536054421769 19 3922 3845.86394557823 76.1360544217683 20 3759 4034.06394557823 -275.063945578232 21 4138 4108.86394557823 29.1360544217685 22 4634 4476.66394557823 157.336054421768 23 3996 3710.46394557823 285.536054421769 24 4308 4266.26394557823 41.7360544217684 25 4143 4921.17006802721 -778.170068027212 26 4429 4470.53605442177 -41.5360544217689 27 5219 5431.93605442177 -212.936054421769 28 4929 4741.93605442177 187.063945578231 29 5755 4863.73605442177 891.263945578231 30 5592 5666.80408163265 -74.8040816326526 31 4163 4631.20408163265 -468.204081632653 32 4962 4819.40408163265 142.595918367347 33 5208 4894.20408163265 313.795918367347 34 4755 5262.00408163265 -507.004081632653 35 4491 4495.80408163265 -4.80408163265294 36 5732 5051.60408163265 680.395918367347 37 5731 4921.17006802721 809.829931972788 38 5040 4470.53605442177 569.463945578231 39 6102 5431.93605442177 670.063945578231 40 4904 4741.93605442177 162.063945578231 41 5369 4863.73605442177 505.263945578231 42 5578 5666.80408163265 -88.8040816326526 43 4619 4631.20408163265 -12.2040816326532 44 4731 4819.40408163265 -88.404081632653 45 5011 4894.20408163265 116.795918367347 46 5299 5262.00408163265 36.9959183673471 47 4146 4495.80408163265 -349.804081632653 48 4625 5051.60408163265 -426.604081632653 49 4736 4921.17006802721 -185.170068027213 50 4219 4470.53605442177 -251.536054421769 51 5116 5431.93605442177 -315.936054421769 52 4205 4741.93605442177 -536.936054421769 53 4121 4863.73605442177 -742.73605442177 54 5103 4881.46394557823 221.536054421769 55 4300 3845.86394557823 454.136054421768 56 4578 4034.06394557823 543.936054421769 57 3809 4108.86394557823 -299.863945578231 58 5526 4476.66394557823 1049.33605442177 59 4247 3710.46394557823 536.536054421769 60 3830 4266.26394557823 -436.263945578231 61 4394 4135.82993197279 258.170068027209 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.961850591808232 0.0762988163835368 0.0381494081917684 17 0.9367134211605 0.126573157678999 0.0632865788394996 18 0.96596612213837 0.0680677557232602 0.0340338778616301 19 0.935324904119 0.129350191762 0.064675095881 20 0.903484191656526 0.193031616686947 0.0965158083434737 21 0.849313679554635 0.301372640890730 0.150686320445365 22 0.836166917088347 0.327666165823306 0.163833082911653 23 0.801925311410642 0.396149377178716 0.198074688589358 24 0.722675186903864 0.554649626192272 0.277324813096136 25 0.70785900541675 0.584281989166502 0.292140994583251 26 0.658954644121423 0.682090711757154 0.341045355878577 27 0.578833042515344 0.842333914969311 0.421166957484656 28 0.504341770075477 0.991316459849045 0.495658229924523 29 0.672199020651598 0.655601958696804 0.327800979348402 30 0.58158609193225 0.836827816135501 0.418413908067750 31 0.543566989127349 0.912866021745303 0.456433010872651 32 0.462388452269735 0.924776904539469 0.537611547730265 33 0.406901493255821 0.813802986511642 0.593098506744179 34 0.415719549571325 0.83143909914265 0.584280450428675 35 0.321602085420696 0.643204170841392 0.678397914579304 36 0.46089066273325 0.9217813254665 0.53910933726675 37 0.608540808498613 0.782918383002773 0.391459191501387 38 0.624752665877781 0.750494668244437 0.375247334122219 39 0.702229520213994 0.595540959572011 0.297770479786006 40 0.68414882939501 0.631702341209981 0.315851170604991 41 0.888564076574528 0.222871846850944 0.111435923425472 42 0.81184976203163 0.37630047593674 0.18815023796837 43 0.696443289347153 0.607113421305694 0.303556710652847 44 0.563047290967419 0.873905418065162 0.436952709032581 45 0.709567711052726 0.580864577894548 0.290432288947274 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 0.0666666666666667 OK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; 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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