*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 09:55:45 -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/t1258649862dcfnrhhq832mq4d.htm/, Retrieved Thu, 19 Nov 2009 17:57:54 +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/t1258649862dcfnrhhq832mq4d.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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50.9 0 52.7 54.8 56 56.6 50.6 0 50.9 52.7 54.8 56 52.1 0 50.6 50.9 52.7 54.8 53.3 0 52.1 50.6 50.9 52.7 53.9 0 53.3 52.1 50.6 50.9 54.3 0 53.9 53.3 52.1 50.6 54.2 0 54.3 53.9 53.3 52.1 54.2 0 54.2 54.3 53.9 53.3 53.5 0 54.2 54.2 54.3 53.9 51.4 0 53.5 54.2 54.2 54.3 50.5 0 51.4 53.5 54.2 54.2 50.3 0 50.5 51.4 53.5 54.2 49.8 0 50.3 50.5 51.4 53.5 50.7 0 49.8 50.3 50.5 51.4 52.8 0 50.7 49.8 50.3 50.5 55.3 0 52.8 50.7 49.8 50.3 57.3 0 55.3 52.8 50.7 49.8 57.5 0 57.3 55.3 52.8 50.7 56.8 0 57.5 57.3 55.3 52.8 56.4 0 56.8 57.5 57.3 55.3 56.3 0 56.4 56.8 57.5 57.3 56.4 0 56.3 56.4 56.8 57.5 57 0 56.4 56.3 56.4 56.8 57.9 0 57 56.4 56.3 56.4 58.9 0 57.9 57 56.4 56.3 58.8 0 58.9 57.9 57 56.4 56.5 1 58.8 58.9 57.9 57 51.9 1 56.5 58.8 58.9 57.9 47.4 1 51.9 56.5 58.8 58.9 44.9 1 47.4 51.9 56.5 58.8 43.9 1 44.9 47.4 51.9 56.5 43.4 1 43.9 44.9 47.4 51.9 42.9 1 43.4 43.9 44.9 47.4 42.6 1 42.9 43.4 43.9 44.9 42.2 1 42.6 42.9 43.4 43.9 41.2 1 42.2 42.6 42.9 43.4 40.2 1 41.2 42.2 42.6 42.9 39 etc...

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] = + 2.16415698290064 -1.27984653729293X[t] + 2.05394600247998Y1[t] -1.72849331883367Y2[t] + 0.670044338186338Y3[t] -0.0341379716604865Y4[t] + 0.00238579303693468t + 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) 2.16415698290064 0.954132 2.2682 0.027669 0.013835 X -1.27984653729293 0.360586 -3.5493 0.000851 0.000426 Y1 2.05394600247998 0.136137 15.0874 0 0 Y2 -1.72849331883367 0.302339 -5.7171 1e-06 0 Y3 0.670044338186338 0.294646 2.2741 0.027288 0.013644 Y4 -0.0341379716604865 0.126848 -0.2691 0.78894 0.39447 t 0.00238579303693468 0.012017 0.1985 0.843435 0.421718 Multiple Linear Regression - Regression Statistics Multiple R 0.99880042950019 R-squared 0.997602297969763 Adjusted R-squared 0.997314573726135 F-TEST (value) 3467.216684243 F-TEST (DF numerator) 6 F-TEST (DF denominator) 50 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.52899962131653 Sum Squared Residuals 13.9920299676516 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 50.9 51.278336976999 -0.378336976999010 2 50.6 50.4298855122953 0.170114487704693 3 52.1 51.5612479342902 0.538752065709812 4 53.3 54.0287106584488 -0.728710658448789 5 53.9 53.7635267237442 0.136473276255827 6 54.3 53.9393960344464 0.360603965553641 7 54.2 54.479110485508 -0.279110485507947 8 54.2 53.9457653876827 0.254234612317348 9 53.5 54.3685354648812 -0.868535464881188 10 51.4 52.8524994336993 -1.45249943369931 11 50.5 49.7549577418779 0.745042258122101 12 50.3 51.0695970655031 -0.76959706550312 13 49.8 50.8336411149654 -1.03364111496538 14 50.7 49.6234024066484 1.07659759335162 15 52.8 52.2353015681913 0.564698431808687 16 55.3 54.6671354047248 0.632864595275185 17 57.3 56.794659124609 0.505340875391039 18 57.5 57.9600725612186 -0.460072561218556 19 56.8 56.519682022063 0.280317977937017 20 56.4 55.9933506968186 0.406649303181353 21 56.3 56.4498363363635 -0.149836336363460 22 56.4 56.4623662256233 -0.0623662256233217 23 57 56.5988747956794 0.401125204320569 24 57.9 57.6074296131665 0.292570386833451 25 58.9 58.4916890481199 0.408310951880054 26 58.8 59.3909896624323 -0.590989662432319 27 56.5 56.7621981204661 -0.262198120466067 28 51.9 52.8526776033743 -0.952677603374322 29 47.4 47.2813040128417 0.118695987158349 30 44.9 44.454313880691 0.445686119308982 31 43.9 44.0963679814415 -0.196367981441466 32 43.4 43.5078762168823 -0.107876216882307 33 42.9 42.6902923545193 0.209707645480736 34 42.6 41.9452523966979 0.654747603302083 35 42.2 41.894816850975 0.305183149024988 36 41.2 41.2762190554071 -0.076219055407127 37 40.2 39.7321118578719 0.467888142128111 38 39.3 39.1512686234861 0.148731376513872 39 38.5 38.3772071836026 0.122792816397411 40 38.3 37.65617379508 0.643826204919994 41 37.9 38.0616631099806 -0.161663109980644 42 37.6 37.0828578697377 0.517142130262305 43 37.3 37.0537586992552 0.246241300744770 44 36 36.6973185462558 -0.697318546255817 45 34.5 34.3607644189272 0.139235581072815 46 33.5 33.3385006127702 0.161499387229848 47 32.9 33.0188641334335 -0.118864133433515 48 32.9 32.5566884996953 0.343311500304746 49 32.8 32.9773329033368 -0.177332903336784 50 31.9 32.4064354648744 -0.506435464874398 51 30.5 30.753601970559 -0.253601970559012 52 29.2 29.3691029132556 -0.169102913255641 53 28.7 28.5216234422341 0.178376557765923 54 28.4 28.8367396495484 -0.436739649548355 55 28 28.2639238219406 -0.263923821940568 56 27.4 27.6726364037011 -0.272636403701079 57 26.9 26.9501076071578 -0.0501076071578286 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 10 0.901524698811363 0.196950602377274 0.0984753011886369 11 0.846112551154415 0.307774897691171 0.153887448845586 12 0.929202531630804 0.141594936738392 0.0707974683691961 13 0.98880877692813 0.0223824461437410 0.0111912230718705 14 0.985506083442567 0.0289878331148663 0.0144939165574331 15 0.980119150203583 0.0397616995928338 0.0198808497964169 16 0.997933721884726 0.00413255623054755 0.00206627811527377 17 0.998807487761217 0.00238502447756627 0.00119251223878314 18 0.999391387210264 0.00121722557947249 0.000608612789736243 19 0.999599216558343 0.00080156688331478 0.00040078344165739 20 0.999762252523554 0.000475494952891421 0.000237747476445710 21 0.99988997488039 0.000220050239222344 0.000110025119611172 22 0.999925485517358 0.000149028965282945 7.45144826414727e-05 23 0.999884211915945 0.00023157616811034 0.00011578808405517 24 0.999764934806881 0.00047013038623754 0.00023506519311877 25 0.999831366825912 0.000337266348176278 0.000168633174088139 26 0.999715880851508 0.000568238296984148 0.000284119148492074 27 0.99998106787976 3.78642404808122e-05 1.89321202404061e-05 28 0.99996980223743 6.03955251384981e-05 3.01977625692491e-05 29 0.999931962309783 0.000136075380433637 6.80376902168184e-05 30 0.999961384564023 7.72308719550925e-05 3.86154359775462e-05 31 0.999971529128322 5.69417433561704e-05 2.84708716780852e-05 32 0.99996035455622 7.92908875581408e-05 3.96454437790704e-05 33 0.99990222558633 0.000195548827339156 9.77744136695782e-05 34 0.999740695366481 0.00051860926703744 0.00025930463351872 35 0.999407282337377 0.00118543532524525 0.000592717662622624 36 0.999407194163458 0.00118561167308359 0.000592805836541794 37 0.998595856140662 0.00280828771867669 0.00140414385933835 38 0.997868095257654 0.00426380948469167 0.00213190474234584 39 0.998101787063814 0.00379642587237155 0.00189821293618577 40 0.995439925897525 0.00912014820495052 0.00456007410247526 41 0.998928439991366 0.00214312001726708 0.00107156000863354 42 0.996841320264963 0.0063173594700732 0.0031586797350366 43 0.995846258380053 0.00830748323989334 0.00415374161994667 44 0.990941646905787 0.0181167061884256 0.0090583530942128 45 0.981732795388951 0.0365344092220974 0.0182672046110487 46 0.948732000743258 0.102535998513485 0.0512679992567423 47 0.872858860382225 0.25428227923555 0.127141139617775 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 28 0.736842105263158 NOK 5% type I error level 33 0.868421052631579 NOK 10% type I error level 33 0.868421052631579 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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