*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 06:03:53 -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/t1258722307bvacu4pavt5l593.htm/, Retrieved Fri, 20 Nov 2009 14:05:19 +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/t1258722307bvacu4pavt5l593.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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1.4816 133.91 1.4562 133.14 1.4268 135.31 1.4088 133.09 1.4016 135.39 1.3650 131.85 1.3190 130.25 1.3050 127.65 1.2785 118.30 1.3239 119.73 1.3449 122.51 1.2732 123.28 1.3322 133.52 1.4369 153.20 1.4975 163.63 1.5770 168.45 1.5553 166.26 1.5557 162.31 1.5750 161.56 1.5527 156.59 1.4748 157.97 1.4718 158.68 1.4570 163.55 1.4684 162.89 1.4227 164.95 1.3896 159.82 1.3622 159.05 1.3716 166.76 1.3419 164.55 1.3511 163.22 1.3516 160.68 1.3242 155.24 1.3074 157.60 1.2999 156.56 1.3213 154.82 1.2881 151.11 1.2611 149.65 1.2727 148.99 1.2811 148.53 1.2684 146.70 1.2650 145.11 1.2770 142.70 1.2271 143.59 1.2020 140.96 1.1938 140.77 1.2103 139.81 1.1856 140.58 1.1786 139.59 1.2015 138.05 1.2256 136.06 1.2292 135.98 1.2037 134.75 1.2165 132.22 1.2694 135.37 1.2938 138.84 1.3201 138.83 1.3014 136.55 1.3119 135.63 1.3408 139.14 1.2991 136.09

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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation dollar/euro[t] = + 0.87081207549798 + 0.00398330914477538`japanseyen/euro`[t] -0.00929114954903965M1[t] + 0.00203645132440579M2[t] + 1.65863052193402e-05M3[t] + 0.00459523507501139M4[t] + 0.00352491868082835M5[t] + 0.0213563932885019M6[t] + 0.0152530710875644M7[t] + 0.0190352757404278M8[t] -0.000333249651898802M9[t] + 0.0164825936044026M10[t] + 0.0185390565970668M11[t] -0.00381444702971641t + 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) 0.87081207549798 0.105136 8.2827 0 0 `japanseyen/euro` 0.00398330914477538 0.000691 5.7634 1e-06 0 M1 -0.00929114954903965 0.044826 -0.2073 0.836713 0.418357 M2 0.00203645132440579 0.04482 0.0454 0.963956 0.481978 M3 1.65863052193402e-05 0.044875 4e-04 0.999707 0.499853 M4 0.00459523507501139 0.044924 0.1023 0.918971 0.459486 M5 0.00352491868082835 0.044786 0.0787 0.937608 0.468804 M6 0.0213563932885019 0.044652 0.4783 0.634716 0.317358 M7 0.0152530710875644 0.044612 0.3419 0.73398 0.36699 M8 0.0190352757404278 0.044488 0.4279 0.670738 0.335369 M9 -0.000333249651898802 0.044457 -0.0075 0.994052 0.497026 M10 0.0164825936044026 0.044442 0.3709 0.71243 0.356215 M11 0.0185390565970668 0.044443 0.4171 0.678518 0.339259 t -0.00381444702971641 0.000534 -7.138 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.811712323002453 R-squared 0.658876895314039 Adjusted R-squared 0.562472539641919 F-TEST (value) 6.83451375947102 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 4.52281989304915e-07 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0702463354068596 Sum Squared Residuals 0.226989191352279 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.4816 1.39111140649609 0.09048859350391 2 1.4562 1.39555741229835 0.0606425877016533 3 1.4268 1.39836688109361 0.0284331189063933 4 1.4088 1.39028813653228 0.0185118634677189 5 1.4016 1.39456498414136 0.00703501585863508 6 1.365 1.39448109734682 -0.0294810973468171 7 1.319 1.37819003348452 -0.0591900334845227 8 1.305 1.36780118733125 -0.0628011873312537 9 1.2785 1.30737427440556 -0.0288742744055608 10 1.3239 1.32607180270917 -0.00217180270917455 11 1.3449 1.33538741809460 0.00951258190540206 12 1.2732 1.31610106250929 -0.0429010625092917 13 1.3322 1.34378455157304 -0.0115845515730356 14 1.4369 1.42968922938594 0.00721077061405595 15 1.4975 1.46540083171705 0.0320991682829516 16 1.577 1.48536458353494 0.0916354164650585 17 1.5553 1.47175637308398 0.083543626916016 18 1.5557 1.47003932954008 0.0856606704599218 19 1.575 1.45713407845084 0.117865921549157 20 1.5527 1.43730478962446 0.115395210375544 21 1.4748 1.41961878382220 0.0551812161777969 22 1.4718 1.43544832954158 0.0363516704584212 23 1.457 1.45308906103958 0.00391093896041746 24 1.4684 1.42810657337725 0.0402934266227523 25 1.4227 1.42320659363673 -0.000506593636728716 26 1.3896 1.41028537156776 -0.0206853715677602 27 1.3622 1.40138391147738 -0.0391839114773802 28 1.3716 1.43285942672367 -0.0612594267236742 29 1.3419 1.41917155008982 -0.077271550089821 30 1.3511 1.42789077650523 -0.0767907765052269 31 1.3516 1.40785540204684 -0.0562554020468437 32 1.3242 1.38615395792241 -0.0619539579224124 33 1.3074 1.37237159508204 -0.0649715950820395 34 1.2999 1.38123034979806 -0.081330349798058 35 1.3213 1.37254140784910 -0.0512414078490966 36 1.2881 1.33540982729520 -0.0473098272951967 37 1.2611 1.31648859936507 -0.0553885993650685 38 1.2727 1.32137276917325 -0.0486727691732459 39 1.2811 1.31370613491775 -0.0326061349177464 40 1.2684 1.30718088092288 -0.038780880922883 41 1.265 1.29596265595879 -0.0309626559587909 42 1.277 1.30037990849784 -0.0233799084978392 43 1.2271 1.29400728440604 -0.0669072844060354 44 1.202 1.28349893897842 -0.081498938978423 45 1.1938 1.25955913781887 -0.0657591378188728 46 1.2103 1.26873655726647 -0.0584365572664735 47 1.1856 1.27004572127090 -0.0844457212708983 48 1.1786 1.24374874159079 -0.0651487415907873 49 1.2015 1.22450884892908 -0.0230088489290772 50 1.2256 1.22409521757470 0.00150478242529683 51 1.2292 1.21794224079422 0.0112577592057818 52 1.2037 1.21380697228622 -0.0101069722862203 53 1.2165 1.19884443672604 0.0176555632739608 54 1.2694 1.22540888811004 0.0439911118899614 55 1.2938 1.22931320161176 0.0644867983882446 56 1.3201 1.22924112614345 0.0908588738565454 57 1.3014 1.19697620887132 0.104423791128676 58 1.3119 1.20631296068472 0.105587039315285 59 1.3408 1.21853639174582 0.122263608254175 60 1.2991 1.18403379522748 0.115066204772523 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0462577247309380 0.0925154494618759 0.953742275269062 18 0.0589432929326276 0.117886585865255 0.941056707067372 19 0.158141076586022 0.316282153172044 0.841858923413978 20 0.311195381102368 0.622390762204735 0.688804618897632 21 0.366321090259379 0.732642180518759 0.63367890974062 22 0.561022133657758 0.877955732684484 0.438977866342242 23 0.779490597153117 0.441018805693765 0.220509402846883 24 0.9244278699661 0.151144260067801 0.0755721300339006 25 0.95339126682738 0.0932174663452398 0.0466087331726199 26 0.973353676376728 0.0532926472465446 0.0266463236232723 27 0.980536098848033 0.0389278023039349 0.0194639011519675 28 0.974435881061493 0.051128237877014 0.025564118938507 29 0.980817309842959 0.0383653803140823 0.0191826901570411 30 0.995725482996184 0.00854903400763244 0.00427451700381622 31 0.991130030705296 0.0177399385894082 0.00886996929470412 32 0.988610148079184 0.0227797038416310 0.0113898519208155 33 0.984117383938584 0.0317652321228326 0.0158826160614163 34 0.99588429589016 0.00823140821967907 0.00411570410983954 35 0.993983135274175 0.0120337294516490 0.00601686472582448 36 0.999239412377729 0.00152117524454288 0.000760587622271439 37 0.99891353887708 0.00217292224583884 0.00108646112291942 38 0.99858218117545 0.00283563764909772 0.00141781882454886 39 0.99764595975693 0.00470808048614025 0.00235404024307013 40 0.994570434124123 0.0108591317517543 0.00542956587587717 41 0.997168956905494 0.00566208618901259 0.00283104309450629 42 0.995897252939372 0.00820549412125556 0.00410274706062778 43 0.99372381767437 0.0125523646512592 0.00627618232562958 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 8 0.296296296296296 NOK 5% type I error level 16 0.592592592592593 NOK 10% type I error level 20 0.740740740740741 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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