*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: Tue, 15 Dec 2009 08:01:33 -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/15/t1260889424aomli66pkaqop53.htm/, Retrieved Tue, 15 Dec 2009 16:03:56 +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/15/t1260889424aomli66pkaqop53.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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8.7 0 8.2 0 8.3 0 8.5 0 8.6 0 8.5 0 8.2 0 8.1 0 7.9 0 8.6 0 8.7 0 8.7 0 8.5 0 8.4 0 8.5 0 8.7 0 8.7 0 8.6 0 8.5 0 8.3 0 8 0 8.2 0 8.1 0 8.1 0 8 0 7.9 0 7.9 0 8 0 8 0 7.9 0 8 0 7.7 0 7.2 0 7.5 0 7.3 0 7 0 7 0 7 0 7.2 0 7.3 0 7.1 0 6.8 0 6.4 0 6.1 0 6.5 0 7.7 0 7.9 0 7.5 0 6.9 1 6.6 1 6.9 1 7.7 1 8 1 8 1 7.7 1 7.3 1 7.4 1 8.1 1 8.3 1 8.2 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] = + 9.16083333333332 + 0.908333333333333X[t] -0.520763888888894M1[t] -0.680694444444444M2[t] -0.500624999999998M3[t] -0.180555555555555M4[t] -0.100486111111111M5[t] -0.180416666666666M6[t] -0.340347222222221M7[t] -0.560277777777777M8[t] -0.620208333333332M9[t] + 0.0398611111111113M10[t] + 0.119930555555556M11[t] -0.0400694444444444t + 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) 9.16083333333332 0.23375 39.1907 0 0 X 0.908333333333333 0.192142 4.7274 2.2e-05 1.1e-05 M1 -0.520763888888894 0.27086 -1.9226 0.060731 0.030365 M2 -0.680694444444444 0.270064 -2.5205 0.015251 0.007626 M3 -0.500624999999998 0.269341 -1.8587 0.069474 0.034737 M4 -0.180555555555555 0.268693 -0.672 0.50496 0.25248 M5 -0.100486111111111 0.26812 -0.3748 0.709546 0.354773 M6 -0.180416666666666 0.267622 -0.6741 0.503592 0.251796 M7 -0.340347222222221 0.2672 -1.2738 0.20915 0.104575 M8 -0.560277777777777 0.266855 -2.0996 0.041283 0.020641 M9 -0.620208333333332 0.266586 -2.3265 0.024452 0.012226 M10 0.0398611111111113 0.266393 0.1496 0.881709 0.440854 M11 0.119930555555556 0.266278 0.4504 0.65454 0.32727 t -0.0400694444444444 0.004529 -8.8477 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.826158445159561 R-squared 0.682537776508464 Adjusted R-squared 0.592820191608682 F-TEST (value) 7.60762538660493 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 1.03769617343374e-07 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.420961201695992 Sum Squared Residuals 8.15158333333333 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.7 8.60000000000002 0.0999999999999748 2 8.2 8.4 -0.200000000000002 3 8.3 8.54 -0.239999999999998 4 8.5 8.82 -0.319999999999998 5 8.6 8.86 -0.259999999999999 6 8.5 8.74 -0.239999999999998 7 8.2 8.54 -0.339999999999999 8 8.1 8.28 -0.179999999999999 9 7.9 8.18 -0.279999999999998 10 8.6 8.8 -0.199999999999998 11 8.7 8.84 -0.139999999999999 12 8.7 8.68 0.0200000000000014 13 8.5 8.11916666666666 0.380833333333340 14 8.4 7.91916666666666 0.480833333333336 15 8.5 8.05916666666667 0.440833333333334 16 8.7 8.33916666666667 0.360833333333333 17 8.7 8.37916666666666 0.320833333333334 18 8.6 8.25916666666667 0.340833333333334 19 8.5 8.05916666666667 0.440833333333334 20 8.3 7.79916666666667 0.500833333333335 21 8 7.69916666666667 0.300833333333334 22 8.2 8.31916666666667 -0.119166666666666 23 8.1 8.35916666666667 -0.259166666666666 24 8.1 8.19916666666666 -0.099166666666666 25 8 7.63833333333333 0.361666666666673 26 7.9 7.43833333333333 0.461666666666667 27 7.9 7.57833333333333 0.321666666666666 28 8 7.85833333333333 0.141666666666666 29 8 7.89833333333333 0.101666666666667 30 7.9 7.77833333333333 0.121666666666667 31 8 7.57833333333333 0.421666666666666 32 7.7 7.31833333333333 0.381666666666666 33 7.2 7.21833333333333 -0.0183333333333336 34 7.5 7.83833333333333 -0.338333333333334 35 7.3 7.87833333333333 -0.578333333333334 36 7 7.71833333333333 -0.718333333333333 37 7 7.1575 -0.157499999999995 38 7 6.9575 0.0424999999999987 39 7.2 7.0975 0.102499999999998 40 7.3 7.3775 -0.0775000000000014 41 7.1 7.4175 -0.317500000000001 42 6.8 7.2975 -0.497500000000001 43 6.4 7.0975 -0.6975 44 6.1 6.8375 -0.737500000000002 45 6.5 6.7375 -0.237500000000001 46 7.7 7.3575 0.342499999999999 47 7.9 7.3975 0.502499999999999 48 7.5 7.2375 0.262499999999999 49 6.9 7.585 -0.684999999999994 50 6.6 7.385 -0.785 51 6.9 7.525 -0.625000000000001 52 7.7 7.805 -0.105000000000000 53 8 7.845 0.155 54 8 7.725 0.274999999999999 55 7.7 7.525 0.174999999999999 56 7.3 7.265 0.0349999999999991 57 7.4 7.165 0.235000000000000 58 8.1 7.785 0.314999999999999 59 8.3 7.825 0.475 60 8.2 7.665 0.534999999999999 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0367275995485921 0.0734551990971841 0.963272400451408 18 0.00887396740833418 0.0177479348166684 0.991126032591666 19 0.00355962762817975 0.00711925525635949 0.99644037237182 20 0.00093520308811711 0.00187040617623422 0.999064796911883 21 0.000205639544342628 0.000411279088685257 0.999794360455657 22 0.00143105103014031 0.00286210206028062 0.99856894896986 23 0.00693245237184964 0.0138649047436993 0.99306754762815 24 0.0111216173762111 0.0222432347524222 0.988878382623789 25 0.0159672051098605 0.0319344102197211 0.98403279489014 26 0.0127719264216920 0.0255438528433840 0.987228073578308 27 0.0101942480201403 0.0203884960402806 0.98980575197986 28 0.00765725585317985 0.0153145117063597 0.99234274414682 29 0.00574866352191956 0.0114973270438391 0.99425133647808 30 0.00420403571412294 0.00840807142824589 0.995795964285877 31 0.00492693495367684 0.00985386990735367 0.995073065046323 32 0.0175612698862942 0.0351225397725885 0.982438730113706 33 0.0338015801894037 0.0676031603788074 0.966198419810596 34 0.039205657616669 0.078411315233338 0.960794342383331 35 0.0497419796941624 0.0994839593883249 0.950258020305838 36 0.0882935888525051 0.176587177705010 0.911706411147495 37 0.132833437938102 0.265666875876205 0.867166562061898 38 0.282646892686004 0.565293785372008 0.717353107313996 39 0.629499645838958 0.741000708322084 0.370500354161042 40 0.628064600833266 0.743870798333467 0.371935399166734 41 0.519012561318738 0.961974877362524 0.480987438681262 42 0.494551511501602 0.989103023003204 0.505448488498398 43 0.598149511284816 0.803700977430369 0.401850488715184 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 6 0.222222222222222 NOK 5% type I error level 15 0.555555555555556 NOK 10% type I error level 19 0.703703703703704 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') }

Copyright

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Software written by Ed van Stee & Patrick Wessa

Disclaimer

Information provided on this web site is provided "AS IS" without warranty of any kind, either express or implied, including, without limitation, warranties of merchantability, fitness for a particular purpose, and noninfringement. We use reasonable efforts to include accurate and timely information and periodically update the information, and software without notice. However, we make no warranties or representations as to the accuracy or completeness of such information (or software), and we assume no liability or responsibility for errors or omissions in the content of this web site, or any software bugs in online applications. Your use of this web site is AT YOUR OWN RISK. Under no circumstances and under no legal theory shall we be liable to you or any other person for any direct, indirect, special, incidental, exemplary, or consequential damages arising from your access to, or use of, this web site.

Privacy Policy

We may request personal information to be submitted to our servers in order to be able to:

• personalize online software applications according to your needs
• enforce strict security rules with respect to the data that you upload (e.g. statistical data)
• manage user sessions of online applications
• alert you about important changes or upgrades in resources or applications

We NEVER allow other companies to directly offer registered users information about their products and services. Banner references and hyperlinks of third parties NEVER contain any personal data of the visitor.

We do NOT sell, nor transmit by any means, personal information, nor statistical data series uploaded by you to third parties.

We carefully protect your data from loss, misuse, alteration, and destruction. However, at any time, and under any circumstance you are solely responsible for managing your passwords, and keeping them secret.

We store a unique ANONYMOUS USER ID in the form of a small 'Cookie' on your computer. This allows us to track your progress when using this website which is necessary to create state-dependent features. The cookie is used for NO OTHER PURPOSE. At any time you may opt to disallow cookies from this website - this will not affect other features of this website.

We examine cookies that are used by third-parties (banner and online ads) very closely: abuse from third-parties automatically results in termination of the advertising contract without refund. We have very good reason to believe that the cookies that are produced by third parties (banner ads) do NOT cause any privacy or security risk.

FreeStatistics.org is safe. There is no need to download any software to use the applications and services contained in this website. Hence, your system's security is not compromised by their use, and your personal data - other than data you submit in the account application form, and the user-agent information that is transmitted by your browser - is never transmitted to our servers.

As a general rule, we do not log on-line behavior of individuals (other than normal logging of webserver 'hits'). However, in cases of abuse, hacking, unauthorized access, Denial of Service attacks, illegal copying, hotlinking, non-compliance with international webstandards (such as robots.txt), or any other harmful behavior, our system engineers are empowered to log, track, identify, publish, and ban misbehaving individuals - even if this leads to ban entire blocks of IP addresses, or disclosing user's identity.

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