Model 2

*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 13:56:07 -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/t1258664331ei2dlj8r328wa7d.htm/, Retrieved Thu, 19 Nov 2009 21:59:03 +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/t1258664331ei2dlj8r328wa7d.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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109.8 8.4 111.7 8.4 98.6 8.4 96.9 8.6 95.1 8.9 97 8.8 112.7 8.3 102.9 7.5 97.4 7.2 111.4 7.4 87.4 8.8 96.8 9.3 114.1 9.3 110.3 8.7 103.9 8.2 101.6 8.3 94.6 8.5 95.9 8.6 104.7 8.5 102.8 8.2 98.1 8.1 113.9 7.9 80.9 8.6 95.7 8.7 113.2 8.7 105.9 8.5 108.8 8.4 102.3 8.5 99 8.7 100.7 8.7 115.5 8.6 100.7 8.5 109.9 8.3 114.6 8 85.4 8.2 100.5 8.1 114.8 8.1 116.5 8 112.9 7.9 102 7.9 106 8 105.3 8 118.8 7.9 106.1 8 109.3 7.7 117.2 7.2 92.5 7.5 104.2 7.3 112.5 7 122.4 7 113.3 7 100 7.2 110.7 7.3 112.8 7.1 109.8 6.8 117.3 6.4 109.1 6.1 115.9 6.5 96 7.7 99.8 7.9 116.8 7.5 115.7 6.9

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 X[t] = + 15.9233811697884 -0.0770963900381123Y[t] + 0.996295645871979M1[t] + 0.762999863713578M2[t] + 0.344480759308709M3[t] -0.0705681875557902M4[t] + 0.149521935264028M5[t] + 0.206663386712050M6[t] + 0.754543431491648M7[t] -0.0342476813499836M8[t] -0.366763349395718M9[t] + 0.311865128579307M10[t] -0.944976434817711M11[t] + 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) 15.9233811697884 1.594683 9.9853 0 0 Y -0.0770963900381123 0.015817 -4.8741 1.2e-05 6e-06 M1 0.996295645871979 0.424485 2.3471 0.023007 0.011504 M2 0.762999863713578 0.4263 1.7898 0.079663 0.039832 M3 0.344480759308709 0.398075 0.8654 0.391055 0.195528 M4 -0.0705681875557902 0.37734 -0.187 0.852421 0.426211 M5 0.149521935264028 0.377829 0.3957 0.694015 0.347008 M6 0.206663386712050 0.379752 0.5442 0.588767 0.294384 M7 0.754543431491648 0.428583 1.7606 0.084554 0.042277 M8 -0.0342476813499836 0.390916 -0.0876 0.930545 0.465272 M9 -0.366763349395718 0.386312 -0.9494 0.347078 0.173539 M10 0.311865128579307 0.44705 0.6976 0.48872 0.24436 M11 -0.944976434817711 0.414852 -2.2779 0.027134 0.013567 Multiple Linear Regression - Regression Statistics Multiple R 0.659051412074979 R-squared 0.434348763758023 Adjusted R-squared 0.295821930392641 F-TEST (value) 3.13548467979755 F-TEST (DF numerator) 12 F-TEST (DF denominator) 49 p-value 0.00228889097970797 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.595921221440596 Sum Squared Residuals 17.4009830059994 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.4 8.45449318947566 -0.0544931894756573 2 8.4 8.0747142662448 0.325285733755204 3 8.4 8.6661578713392 -0.266157871339199 4 8.6 8.38217278753949 0.217827212460509 5 8.9 8.74103641242791 0.158963587572089 6 8.8 8.65169472280352 0.148305277196481 7 8.3 7.98916144398475 0.310838556015246 8 7.5 7.95591495351662 -0.455914953516623 9 7.2 8.0474294306805 -0.847429430680506 10 7.4 7.64670844812196 -0.246708448121959 11 8.8 8.24018024563964 0.559819754360364 12 9.3 8.4604506140991 0.83954938590091 13 9.3 8.12297871231173 1.17702128768827 14 8.7 8.18264921229816 0.517350787701845 15 8.2 8.2575470041372 -0.0575470041372048 16 8.3 8.01981975436036 0.280180245639637 17 8.5 8.77958460744697 -0.279584607446968 18 8.6 8.73650075184544 -0.136500751845443 19 8.5 8.60593256428965 -0.105932564289653 20 8.2 7.96362459252043 0.236375407479564 21 8.1 7.99346195765383 0.106538042346171 22 7.9 7.45396747302668 0.446032526973322 23 8.6 8.74130678088737 -0.141306780887367 24 8.7 8.54525664314102 0.154743356858984 25 8.7 8.19236546334603 0.50763453665397 26 8.5 8.52187332846585 -0.0218733284658483 27 8.4 7.87977469295045 0.520225307049546 28 8.5 7.96585228133369 0.534147718666315 29 8.7 8.44036049127927 0.259639508720726 30 8.7 8.3664380796625 0.333561920337495 31 8.6 7.77329155187804 0.82670844812196 32 8.5 8.12552701160047 0.374472988399530 33 8.3 7.0837245552041 1.21627544479590 34 8 7.4 0.6 35 8.2 8.39437302571586 -0.194373025715862 36 8.1 8.17519397095808 -0.0751939709580769 37 8.1 8.06901123928505 0.0309887607149493 38 8 7.70465159406186 0.295348405938142 39 7.9 7.56367949379419 0.336320506205807 40 7.9 7.98898119834512 -0.088981198345118 41 8 7.90068576101249 0.0993142389875127 42 8 8.01179468548719 -0.0117946854871877 43 7.9 7.51887346475227 0.381126535247730 44 8 7.70920650539466 0.290793494605335 45 7.7 7.12998238922697 0.57001761077303 46 7.2 7.19954938590091 0.000450614099091947 47 7.5 7.84698865644526 -0.346988656445265 48 7.3 7.88993732781706 -0.589937327817061 49 7 8.24633293637271 -1.24633293637271 50 7 7.249782892837 -0.249782892836995 51 7 7.53284093777895 -0.532840937778949 52 7.2 8.14317397842134 -0.943173978421343 53 7.3 7.53833272783336 -0.238332727833359 54 7.1 7.43357176020135 -0.333571760201346 55 6.8 8.21274097509528 -1.41274097509528 56 6.4 6.8457269369678 -0.445726936967806 57 6.1 7.14540166723459 -1.04540166723459 58 6.5 7.29977469295045 -0.799774692950454 59 7.7 7.57715129131187 0.122848708688129 60 7.9 8.22916144398476 -0.329161443984755 61 7.5 7.91481845920883 -0.414818459208826 62 6.9 7.76632870609235 -0.866328706092347 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.288656256314636 0.577312512629273 0.711343743685364 17 0.185537990109705 0.37107598021941 0.814462009890295 18 0.0975842354058738 0.195168470811748 0.902415764594126 19 0.0512089367009808 0.102417873401962 0.94879106329902 20 0.0540480842533272 0.108096168506654 0.945951915746673 21 0.0764995145054434 0.152999029010887 0.923500485494557 22 0.0550627286983965 0.110125457396793 0.944937271301604 23 0.0291282623335279 0.0582565246670557 0.970871737666472 24 0.0229918614122505 0.0459837228245009 0.97700813858775 25 0.0160172664236255 0.0320345328472509 0.983982733576374 26 0.00781855059471062 0.0156371011894212 0.99218144940529 27 0.00401307438741618 0.00802614877483237 0.995986925612584 28 0.00260537272923674 0.00521074545847348 0.997394627270763 29 0.00117749152481466 0.00235498304962932 0.998822508475185 30 0.000555115648320069 0.00111023129664014 0.99944488435168 31 0.000535562917236048 0.00107112583447210 0.999464437082764 32 0.000831477721426997 0.00166295544285399 0.999168522278573 33 0.00177050821657676 0.00354101643315353 0.998229491783423 34 0.00199631325396319 0.00399262650792637 0.998003686746037 35 0.00160814230362017 0.00321628460724034 0.99839185769638 36 0.00412641200389955 0.0082528240077991 0.9958735879961 37 0.00733390153005977 0.0146678030601195 0.99266609846994 38 0.0109233368972966 0.0218466737945931 0.989076663102703 39 0.0109605826596985 0.0219211653193970 0.989039417340301 40 0.0114011619941477 0.0228023239882954 0.988598838005852 41 0.0095906890805261 0.0191813781610522 0.990409310919474 42 0.00866992213408834 0.0173398442681767 0.991330077865912 43 0.0270958802743233 0.0541917605486465 0.972904119725677 44 0.0527791089135693 0.105558217827139 0.947220891086431 45 0.433671202761077 0.867342405522153 0.566328797238923 46 0.526056830913735 0.94788633817253 0.473943169086265 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 10 0.32258064516129 NOK 5% type I error level 19 0.612903225806452 NOK 10% type I error level 21 0.67741935483871 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = No 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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