WS 7 Multiple Regression - Linear Trend

*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 06:24:16 -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/t1258723518c2p3fpmuxqii2wa.htm/, Retrieved Fri, 20 Nov 2009 14:25:31 +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/t1258723518c2p3fpmuxqii2wa.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:

SHW WS 7 Multiple Regression - Linear Trend

Dataseries X:

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14.2 -0.8 13.5 -0.2 11.9 0.2 14.6 1 15.6 0 14.1 -0.2 14.9 1 14.2 0.4 14.6 1 17.2 1.7 15.4 3.1 14.3 3.3 17.5 3.1 14.5 3.5 14.4 6 16.6 5.7 16.7 4.7 16.6 4.2 16.9 3.6 15.7 4.4 16.4 2.5 18.4 -0.6 16.9 -1.9 16.5 -1.9 18.3 0.7 15.1 -0.9 15.7 -1.7 18.1 -3.1 16.8 -2.1 18.9 0.2 19 1.2 18.1 3.8 17.8 4 21.5 6.6 17.1 5.3 18.7 7.6 19 4.7 16.4 6.6 16.9 4.4 18.6 4.6 19.3 6 19.4 4.8 17.6 4 18.6 2.7 18.1 3 20.4 4.1 18.1 4 19.6 2.7 19.9 2.6 19.2 3.1 17.8 4.4 19.2 3 22 2 21.1 1.3 19.5 1.5 22.2 1.3 20.9 3.2 22.2 1.8 23.5 3.3 21.5 1 24.3 2.4 22.8 0.4 20.3 -0.1 23.7 1.3 23.3 -1.1 19.6 -4.4 18 -7.5 17.3 -12.2 16.8 -14.5 18.2 -16 16.5 -16.7 16 -16.3 18.4 -16.9

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 13.0121448258197 + 0.256102930340177X[t] + 1.61764090464894M1[t] -0.368594387114941M2[t] -1.26533457658441M3[t] + 0.947682584358821M4[t] + 1.44220586859912M5[t] + 0.81233944587344M6[t] + 0.151780623896049M7[t] + 0.213377436825695M8[t] -0.102263491369389M9[t] + 2.06583577579154M10[t] + 0.236982839056767M11[t] + 0.11686151426312t + 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) 13.0121448258197 0.587842 22.1354 0 0 X 0.256102930340177 0.029271 8.7494 0 0 M1 1.61764090464894 0.6812 2.3747 0.020834 0.010417 M2 -0.368594387114941 0.710692 -0.5186 0.605951 0.302975 M3 -1.26533457658441 0.710527 -1.7808 0.080087 0.040043 M4 0.947682584358821 0.709946 1.3349 0.187049 0.093524 M5 1.44220586859912 0.708683 2.0351 0.046346 0.023173 M6 0.81233944587344 0.70755 1.1481 0.255561 0.12778 M7 0.151780623896049 0.706947 0.2147 0.830743 0.415371 M8 0.213377436825695 0.706391 0.3021 0.763663 0.381832 M9 -0.102263491369389 0.706137 -0.1448 0.885346 0.442673 M10 2.06583577579154 0.705959 2.9263 0.004864 0.002432 M11 0.236982839056767 0.705858 0.3357 0.73826 0.36913 t 0.11686151426312 0.007561 15.456 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.910055945215638 R-squared 0.828201823422328 Adjusted R-squared 0.790347987905214 F-TEST (value) 21.8789407231372 F-TEST (DF numerator) 13 F-TEST (DF denominator) 59 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.22252474545640 Sum Squared Residuals 88.1794384419416 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 14.2 14.5417649004596 -0.341764900459630 2 13.5 12.826052881163 0.673947118836999 3 11.9 12.1486153780927 -0.248615378092716 4 14.6 14.6833763975712 -0.0833763975712067 5 15.6 15.0386582657345 0.561341734265546 6 14.1 14.4744327712039 -0.374432771203856 7 14.9 14.2380589798978 0.661941020102203 8 14.2 14.2628555488865 -0.062855548886458 9 14.6 14.2177378931586 0.382262106841401 10 17.2 16.6819707258208 0.518029274179232 11 15.4 15.3285234058254 0.0714765941746366 12 14.3 15.2596226670998 -0.959622667099753 13 17.5 16.9429044999438 0.557095500056226 14 14.5 15.1759718945791 -0.675971894579086 15 14.4 15.0363505452232 -0.636350545223174 16 16.6 17.2893983413275 -0.689398341327476 17 16.7 17.6446802094907 -0.944680209490723 18 16.6 17.0036238358581 -0.403623835858072 19 16.9 16.3062647699397 0.593735230060303 20 15.7 16.6896054414046 -0.989605441404605 21 16.4 16.0042304598263 0.395769540173696 22 18.4 17.4952721571958 0.904727842804195 23 16.9 15.4503469252819 1.44965307471808 24 16.5 15.3302256004883 1.16977439951172 25 18.3 17.7305956382848 0.56940436171521 26 15.1 15.4514571722398 -0.351457172239753 27 15.7 14.4666961527613 1.23330384723874 28 18.1 16.4380307254914 1.66196927450864 29 16.8 17.3055184543350 -0.505518454334962 30 18.9 17.3815502856548 1.51844971434519 31 19 17.0939559082807 1.90604409171929 32 18.1 17.9382818543579 0.161718145642063 33 17.8 17.790723026494 0.0092769735059921 34 21.5 20.7415514268025 0.758448573197485 35 17.1 18.6966261948886 -1.59662619488863 36 18.7 19.1655416098774 -0.465541609877394 37 19 20.1573455308029 -1.15734553080294 38 16.4 18.7745673209485 -2.37456732094852 39 16.9 17.4312621989938 -0.531262198993776 40 18.6 19.8123614602682 -1.21236146026816 41 19.3 20.7822903612478 -1.48229036124783 42 19.4 19.9619619363771 -0.561961936377062 43 17.6 19.2133822843906 -1.61338228439065 44 18.6 19.0589068021412 -0.458906802141183 45 18.1 18.9369582673113 -0.836958267311272 46 20.4 21.5036322721095 -1.10363227210951 47 18.1 19.7660305566038 -1.66603055660384 48 19.6 19.3129754223680 0.287024577632032 49 19.9 21.021867548246 -1.12186754824601 50 19.2 19.2805452359153 -0.0805452359153402 51 17.8 18.8336003701512 -1.03360037015121 52 19.2 20.8049349428813 -1.60493494288133 53 22 21.1602168110446 0.839783188955432 54 21.1 20.4679398513439 0.632060148656116 55 19.5 19.9754631296976 -0.475463129697647 56 22.2 20.1027008708224 2.09729912917762 57 20.9 20.3905170245367 0.50948297546325 58 22.2 22.3169337034845 -0.116933703484549 59 23.5 20.9890966765232 2.51090332347684 60 21.5 20.2799386119471 1.22006138805289 61 24.3 22.3729851333354 1.92701486666459 62 22.8 19.9914054951543 2.80859450484570 63 20.3 19.0834753547779 1.21652464522214 64 23.7 21.7718981324605 1.92810186753954 65 23.3 21.7686358981475 1.53136410185254 66 19.6 20.4104913195623 -0.810491319562317 67 18 19.0728749277935 -1.0728749277935 68 17.3 18.0476494823874 -0.747649482387438 69 16.8 17.2598333286731 -0.459833328673067 70 18.2 19.1606397145868 -0.960639714586848 71 16.5 17.2693762408771 -0.769376240877074 72 16 17.2516960882195 -1.25169608821950 73 18.4 18.8325367489274 -0.43253674892745 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.183982716470532 0.367965432941063 0.816017283529468 18 0.089408229599902 0.178816459199804 0.910591770400098 19 0.0369825528045665 0.073965105609133 0.963017447195433 20 0.0168593194481958 0.0337186388963915 0.983140680551804 21 0.00606274128471611 0.0121254825694322 0.993937258715284 22 0.00213873367570922 0.00427746735141844 0.99786126632429 23 0.000877952850211543 0.00175590570042309 0.999122047149788 24 0.000500127227713009 0.00100025445542602 0.999499872772287 25 0.000175949347069217 0.000351898694138435 0.99982405065293 26 0.000327766353261325 0.000655532706522649 0.999672233646739 27 0.000176170282482571 0.000352340564965143 0.999823829717517 28 0.000139280419034825 0.00027856083806965 0.999860719580965 29 0.000387296825746611 0.000774593651493222 0.999612703174253 30 0.00121702507689980 0.00243405015379961 0.9987829749231 31 0.00966153107247792 0.0193230621449558 0.990338468927522 32 0.00791705079302794 0.0158341015860559 0.992082949206972 33 0.00881278572329983 0.0176255714465997 0.9911872142767 34 0.0249667376777880 0.0499334753555761 0.975033262322212 35 0.0631751007789455 0.126350201557891 0.936824899221054 36 0.0460755160593644 0.0921510321187288 0.953924483940636 37 0.0506429218469786 0.101285843693957 0.949357078153021 38 0.100352052426032 0.200704104852063 0.899647947573968 39 0.116760443859033 0.233520887718066 0.883239556140967 40 0.131330755850001 0.262661511700003 0.868669244149999 41 0.101606439795886 0.203212879591773 0.898393560204114 42 0.0967916323784512 0.193583264756902 0.903208367621549 43 0.208065877742286 0.416131755484572 0.791934122257714 44 0.155928621388159 0.311857242776318 0.844071378611841 45 0.135619788802747 0.271239577605494 0.864380211197253 46 0.126782148691516 0.253564297383032 0.873217851308484 47 0.171892418163188 0.343784836326376 0.828107581836812 48 0.176900752829718 0.353801505659436 0.823099247170282 49 0.141657152873211 0.283314305746421 0.85834284712679 50 0.148293896358292 0.296587792716583 0.851706103641708 51 0.144784379555341 0.289568759110682 0.85521562044466 52 0.518158062653828 0.963683874692344 0.481841937346172 53 0.574300613255084 0.851398773489832 0.425699386744916 54 0.47290921779597 0.94581843559194 0.52709078220403 55 0.405269165199608 0.810538330399216 0.594730834800392 56 0.627485981897932 0.745028036204136 0.372514018102068 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 9 0.225 NOK 5% type I error level 15 0.375 NOK 10% type I error level 17 0.425 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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