Multiple regression

*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, 17 Nov 2009 12:34:37 -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/17/t1258486537s8dbs9nd1eov8be.htm/, Retrieved Tue, 17 Nov 2009 20:35:49 +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/17/t1258486537s8dbs9nd1eov8be.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:

JSSHWWS7P3

Dataseries X:

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8 11.1 8.1 10.9 7.7 10 7.5 9.2 7.6 9.2 7.8 9.5 7.8 9.6 7.8 9.5 7.5 9.1 7.5 8.9 7.1 9 7.5 10.1 7.5 10.3 7.6 10.2 7.7 9.6 7.7 9.2 7.9 9.3 8.1 9.4 8.2 9.4 8.2 9.2 8.2 9 7.9 9 7.3 9 6.9 9.8 6.6 10 6.7 9.8 6.9 9.3 7 9 7.1 9 7.2 9.1 7.1 9.1 6.9 9.1 7 9.2 6.8 8.8 6.4 8.3 6.7 8.4 6.6 8.1 6.4 7.7 6.3 7.9 6.2 7.9 6.5 8 6.8 7.9 6.8 7.6 6.4 7.1 6.1 6.8 5.8 6.5 6.1 6.9 7.2 8.2 7.3 8.7 6.9 8.3 6.1 7.9 5.8 7.5 6.2 7.8 7.1 8.3 7.7 8.4 7.9 8.2 7.7 7.7 7.4 7.2 7.5 7.3 8 8.1

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] = + 7.02041292526928 + 0.439446655138649X[t] + 0.351957019937916M1[t] + 0.154179253734453M2[t] -0.162075980749598M3[t] -0.462275880747515M4[t] -0.423098710389799M5[t] -0.356655138648721M6[t] -0.393533302777141M7[t] -0.52252213587783M8[t] -0.685144169670202M9[t] -0.83261047105148M10[t] -0.688865705535531M11[t] -0.0358554344882185t + 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) 7.02041292526928 0.951326 7.3796 0 0 X 0.439446655138649 0.115365 3.8092 0.000412 0.000206 M1 0.351957019937916 0.296141 1.1885 0.240744 0.120372 M2 0.154179253734453 0.296242 0.5205 0.605246 0.302623 M3 -0.162075980749598 0.29914 -0.5418 0.590567 0.295284 M4 -0.462275880747515 0.300707 -1.5373 0.131072 0.065536 M5 -0.423098710389799 0.295482 -1.4319 0.158935 0.079467 M6 -0.356655138648721 0.292618 -1.2188 0.229118 0.114559 M7 -0.393533302777141 0.292964 -1.3433 0.18577 0.092885 M8 -0.52252213587783 0.292417 -1.7869 0.080543 0.040272 M9 -0.685144169670202 0.292048 -2.346 0.023342 0.011671 M10 -0.83261047105148 0.293032 -2.8414 0.006673 0.003337 M11 -0.688865705535531 0.295493 -2.3312 0.024177 0.012089 t -0.0358554344882185 0.004192 -8.5533 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.913204880251137 R-squared 0.833943153314494 Adjusted R-squared 0.78701404446859 F-TEST (value) 17.7702746509171 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 9.85878045867139e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.461454472004603 Sum Squared Residuals 9.79525056772017 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 11.1 10.8520877518282 0.24791224817184 2 10.9 10.6623992166503 0.237600783349653 3 10 10.1345098856226 -0.134509885622617 4 9.2 9.71056522010875 -0.510565220108753 5 9.2 9.75783162149211 -0.557831621492115 6 9.5 9.8763090897727 -0.376309089772704 7 9.6 9.80357549115607 -0.203575491156066 8 9.5 9.63873122356716 -0.138731223567157 9 9.1 9.30841975874497 -0.208419758744974 10 8.9 9.12509802287548 -0.225098022875477 11 9 9.05720869184775 -0.0572086918477461 12 10.1 9.88599762495052 0.21400237504948 13 10.3 10.2020992104002 0.0979007895997833 14 10.2 10.0124106752224 0.187589324777599 15 9.6 9.704244671764 -0.104244671763996 16 9.2 9.36818933727786 -0.168189337277861 17 9.3 9.45940040417509 -0.159400404175086 18 9.4 9.57787787245568 -0.177877872455676 19 9.4 9.5490889393529 -0.149088939352902 20 9.2 9.384244671764 -0.184244671763995 21 9 9.1857672034834 -0.185767203483405 22 9 8.87061147107231 0.129388528927686 23 9 8.71483280901685 0.285167190983145 24 9.8 9.19206441800871 0.607935581991293 25 10 9.37633200691681 0.623667993083189 26 9.8 9.186643471739 0.613356528261006 27 9.3 8.92242213379446 0.377577866205546 28 9 8.63031146482218 0.369688535177816 29 9 8.67757786620555 0.322422133794454 30 9.1 8.75211066897227 0.347889331027729 31 9.1 8.63543240484177 0.464567595158232 32 9.1 8.38269880622513 0.71730119377487 33 9.2 8.2281660034584 0.971833996541593 34 8.8 7.95695493656118 0.843045063438822 35 8.3 7.88906560553345 0.410934394466552 36 8.4 8.67390987312236 -0.273909873122356 37 8.1 8.94606679305819 -0.84606679305819 38 7.7 8.62454426133878 -0.924544261338778 39 7.9 8.22848892685264 -0.328488926852643 40 7.9 7.84848892685264 0.0515110731473577 41 8 7.98364465926373 0.0163553407362655 42 7.9 8.14606679305819 -0.246066793058188 43 7.6 8.07333319444155 -0.473333194441551 44 7.1 7.73271026479718 -0.632710264797183 45 6.8 7.402398799975 -0.602398799974999 46 6.5 7.0872430675639 -0.587243067563908 47 6.9 7.32696639513323 -0.426966395133232 48 8.2 8.46336798683306 -0.263367986833060 49 8.7 8.82341423779662 -0.123414237796622 50 8.3 8.41400237504948 -0.11400237504948 51 7.9 7.71033438196629 0.189665618033709 52 7.5 7.24244505093856 0.257554949061439 53 7.8 7.42154544886352 0.378454551136482 54 8.3 7.84763557574116 0.452364424258839 55 8.4 8.03856997020771 0.361430029792288 56 8.2 7.96161503364653 0.238384966353466 57 7.7 7.67524823433821 0.0247517656617854 58 7.2 7.36009250192712 -0.160092501927124 59 7.3 7.51192649846872 -0.211926498468719 60 8.1 8.38466009708536 -0.284660097085356 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.00397750473921186 0.00795500947842373 0.996022495260788 18 0.00110549530872612 0.00221099061745224 0.998894504691274 19 0.00122324959130601 0.00244649918261201 0.998776750408694 20 0.00114966847633331 0.00229933695266662 0.998850331523667 21 0.00075082889122571 0.00150165778245142 0.999249171108774 22 0.000437732532951463 0.000875465065902925 0.999562267467049 23 0.000223363230442993 0.000446726460885987 0.999776636769557 24 0.00029891075028015 0.0005978215005603 0.99970108924972 25 0.000252917129962766 0.000505834259925532 0.999747082870037 26 0.000137201164717379 0.000274402329434758 0.999862798835283 27 5.60228511141323e-05 0.000112045702228265 0.999943977148886 28 9.68041331186885e-05 0.000193608266237377 0.999903195866881 29 0.000108017920919255 0.00021603584183851 0.99989198207908 30 5.59848605319028e-05 0.000111969721063806 0.999944015139468 31 2.24091247070007e-05 4.48182494140014e-05 0.999977590875293 32 3.57314642886258e-05 7.14629285772517e-05 0.999964268535711 33 0.000772223662349088 0.00154444732469818 0.999227776337651 34 0.00604848829477893 0.0120969765895579 0.993951511705221 35 0.131662862634265 0.263325725268531 0.868337137365735 36 0.833163241391016 0.333673517217968 0.166836758608984 37 0.975486591954285 0.0490268160914304 0.0245134080457152 38 0.991464517569267 0.0170709648614660 0.00853548243073302 39 0.982957973477203 0.0340840530455943 0.0170420265227971 40 0.962474862837816 0.0750502743243675 0.0375251371621838 41 0.917391958795115 0.165216082409771 0.0826080412048853 42 0.866961830632167 0.266076338735666 0.133038169367833 43 0.929996635185024 0.140006729629952 0.070003364814976 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 17 0.62962962962963 NOK 5% type I error level 21 0.777777777777778 NOK 10% type I error level 22 0.814814814814815 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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