link 3

*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 10:25:03 -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/t1258738068qdxb5im8ew4mysd.htm/, Retrieved Fri, 20 Nov 2009 18:28:00 +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/t1258738068qdxb5im8ew4mysd.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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3956.2 3977.7 3142.7 3983.4 3884.3 4152.9 3892.2 4286.1 3613 4348.1 3730.5 3949.3 3481.3 4166.7 3649.5 4217.9 4215.2 4528.2 4066.6 4232.2 4196.8 4470.9 4536.6 5121.2 4441.6 4170.8 3548.3 4398.6 4735.9 4491.4 4130.6 4251.8 4356.2 4901.9 4159.6 4745.2 3988 4666.9 4167.8 4210.4 4902.2 5273.6 3909.4 4095.3 4697.6 4610.1 4308.9 4718.1 4420.4 4185.5 3544.2 4314.7 4433 4422.6 4479.7 5059.2 4533.2 5043.6 4237.5 4436.6 4207.4 4922.6 4394 4454.8 5148.4 5058.7 4202.2 4768.9 4682.5 5171.8 4884.3 4989.3 5288.9 5202.1 4505.2 4838.4 4611.5 4876.5 5104 5875.5 4586.6 5717.9 4529.3 4778.8 4504.1 6195.9 4604.9 4625.4 4795.4 5549.8 5391.1 6397.6 5213.9 5856.7 5415 6343.8 5990.3 6615.5 4241.8 5904.6 5677.6 6861 5164.2 6553.5 3962.3 5481 4011 5435.3 3310.3 5278 3837.3 4671.8 4145.3 4891.5 3796.7 4241.6 3849.6 4152.1 4285 4484.4 4189.6 4124.7

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] = + 1524.33606731476 + 0.642981455652727X[t] + 278.588595755845M1[t] -642.67177115473M2[t] + 57.6791311614409M3[t] -209.92069689072M4[t] -481.355455121435M5[t] -278.074378354788M6[t] -751.999266684727M7[t] -123.500037172763M8[t] -10.4870991147353M9[t] -173.353327124048M10[t] + 17.7112845633109M11[t] -3.82626082202624t + 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) 1524.33606731476 324.720802 4.6943 2.3e-05 1.2e-05 X 0.642981455652727 0.067045 9.5903 0 0 M1 278.588595755845 178.564072 1.5602 0.125431 0.062715 M2 -642.67177115473 186.741669 -3.4415 0.001224 0.000612 M3 57.6791311614409 186.011345 0.3101 0.757868 0.378934 M4 -209.92069689072 186.613208 -1.1249 0.266345 0.133172 M5 -481.355455121435 185.801943 -2.5907 0.012717 0.006359 M6 -278.074378354788 186.663784 -1.4897 0.142982 0.071491 M7 -751.999266684727 185.299926 -4.0583 0.000185 9.3e-05 M8 -123.500037172763 189.500814 -0.6517 0.517761 0.258881 M9 -10.4870991147353 185.085404 -0.0567 0.955056 0.477528 M10 -173.353327124048 186.395937 -0.93 0.357109 0.178555 M11 17.7112845633109 185.775273 0.0953 0.924452 0.462226 t -3.82626082202624 2.67948 -1.428 0.159907 0.079954 Multiple Linear Regression - Regression Statistics Multiple R 0.894325528268926 R-squared 0.799818150513493 Adjusted R-squared 0.744448702783183 F-TEST (value) 14.4451169968173 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 2.89546164822241e-12 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 292.466566643235 Sum Squared Residuals 4020224.55239184 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 3956.2 4356.68573839844 -400.485738398441 2 3142.7 3435.26410496306 -292.564104963056 3 3884.3 4240.77410319034 -356.474103190338 4 3892.2 4054.99314420909 -162.793144209095 5 3613 3819.59697540682 -206.596975406823 6 3730.5 3762.63078683714 -32.1307868371358 7 3481.3 3424.66380614407 56.636193855927 8 3649.5 4082.25742536343 -432.75742536343 9 4215.2 4390.96124828847 -175.761248288474 10 4066.6 4033.94624858393 32.6537514160729 11 4196.8 4374.66427291357 -177.864272913566 12 4536.6 4771.2575681392 -234.657568139197 13 4441.6 4434.93032762066 6.66967237933591 14 3548.3 3656.31487548575 -108.014875485755 15 4735.9 4412.50819606447 323.391803935527 16 4130.6 3987.02375041589 143.576249584109 17 4356.2 4129.76497568299 226.435024317011 18 4159.6 4228.46459752683 -68.864597526826 19 3988 3700.36800039725 287.631999602747 20 4167.8 4031.51993458172 136.28006541828 21 4902.2 4824.3244954677 77.8755045322973 22 3909.4 3900.00695744075 9.3930425592461 23 4697.6 4418.25216167611 279.347838323889 24 4308.9 4466.15661350127 -157.256613501269 25 4420.4 4398.46702515444 21.9329748455550 26 3544.2 3556.45360149218 -12.2536014921762 27 4433 4322.35594205125 110.644057948750 28 4479.7 4460.25184784559 19.4481521544112 29 4533.2 4174.96031808467 358.239681915334 30 4237.5 3984.12539044808 253.37460955192 31 4207.4 3818.86322874334 388.536771256659 32 4394 4142.74947247893 251.250527521068 33 5148.4 4640.23265078362 508.167349216384 34 4202.2 4287.20413610412 -85.0041361041163 35 4682.5 4733.49971545193 -50.9997154519336 36 4884.3 4594.61805440997 289.681945590027 37 5288.9 5006.20684310669 282.693156893307 38 4505.2 3847.26785995319 657.932140046805 39 4611.5 4568.29009490771 43.2099050922918 40 5104 4939.2024802306 164.797519769405 41 4586.6 4562.60758376698 23.9924162330158 42 4529.3 4158.23851470813 371.061485291871 43 4504.1 4591.65638636164 -87.556386361643 44 4604.9 4206.52697894897 398.373021051027 45 4795.4 4910.08571379036 -114.685713790357 46 5391.1 5288.5129030614 102.587096938601 47 5213.9 5127.96258456417 85.9374154358281 48 5415 5419.62130622728 -4.62130622727842 49 5990.3 5869.08170266194 121.218297338057 50 4241.8 4486.89955810582 -245.099558105818 51 5677.6 5798.37166378623 -120.771663786231 52 5164.2 5329.22877729883 -165.028777298830 53 3962.3 4364.37014705854 -402.070147058539 54 4011 4534.44071047983 -523.440710479829 55 3310.3 3955.54857835369 -645.24857835369 56 3837.3 4190.44618862694 -353.146188626944 57 4145.3 4440.89589166985 -295.595891669850 58 3796.7 3856.32975480980 -59.6297548098038 59 3849.6 3986.02126539422 -136.421265394218 60 4285 4178.14645772228 106.853542277719 61 4189.6 4221.62836305781 -32.0283630578138 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.170617936035025 0.34123587207005 0.829382063964975 18 0.189765042061470 0.379530084122939 0.81023495793853 19 0.0976045252144802 0.195209050428960 0.90239547478552 20 0.0493755011208002 0.0987510022416003 0.9506244988792 21 0.0239928513686637 0.0479857027373273 0.976007148631336 22 0.0715726636036019 0.143145327207204 0.928427336396398 23 0.0409263693241836 0.0818527386483672 0.959073630675816 24 0.0556433460573724 0.111286692114745 0.944356653942628 25 0.0603006268232277 0.120601253646455 0.939699373176772 26 0.0745142237021768 0.149028447404354 0.925485776297823 27 0.0662281881900997 0.132456376380199 0.9337718118099 28 0.100950381688143 0.201900763376286 0.899049618311857 29 0.0649632465460511 0.129926493092102 0.935036753453949 30 0.0394029706794745 0.078805941358949 0.960597029320525 31 0.0301686430029839 0.0603372860059678 0.969831356997016 32 0.0192694128562213 0.0385388257124426 0.980730587143779 33 0.0226623556853606 0.0453247113707211 0.97733764431464 34 0.0767834346546182 0.153566869309236 0.923216565345382 35 0.160819717029869 0.321639434059737 0.839180282970131 36 0.174012745880594 0.348025491761188 0.825987254119406 37 0.228799885801640 0.457599771603281 0.77120011419836 38 0.355616410273066 0.711232820546133 0.644383589726934 39 0.701642655697984 0.596714688604032 0.298357344302016 40 0.820624873762323 0.358750252475353 0.179375126237677 41 0.802025447951648 0.395949104096704 0.197974552048352 42 0.702611167877644 0.594777664244713 0.297388832122356 43 0.677061330984355 0.64587733803129 0.322938669015645 44 0.822635215561182 0.354729568877637 0.177364784438818 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 3 0.107142857142857 NOK 10% type I error level 7 0.25 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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