paper multiple regression met trend

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Title produced by software: Multiple Regression

Date of computation: Wed, 30 Dec 2009 09:26:00 -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/30/t1262190411bvrgxpgk4ykvdpv.htm/, Retrieved Wed, 30 Dec 2009 17:27:03 +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/Dec/30/t1262190411bvrgxpgk4ykvdpv.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}, }

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Dataseries X:

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4223.4 401 4627.3 394 5175.3 372 4550.7 334 4639.3 320 5498.7 334 5031.0 400 4033.3 427 4643.5 423 4873.2 395 4608.7 373 4733.5 377 3955.6 391 4590.9 398 5127.5 393 5257.3 375 5416.9 371 5813.3 364 5261.9 400 4669.2 406 5855.8 407 5274.6 397 5516.7 389 5819.5 394 5156.0 399 5377.3 401 6386.8 396 5144.0 392 6138.5 384 5567.8 370 5822.6 380 5145.5 376 5706.6 378 6078.5 376 6074.5 373 5577.6 374 5727.5 379 6067.0 376 7069.9 371 5490.0 375 5948.3 360 6177.5 338 6890.1 352 5756.2 344 6528.8 330 6792.0 334 6657.4 333 5753.7 343 5750.9 350 5968.4 341 5871.7 320 7004.9 302 6363.4 287 6694.7 304 7101.6 370 5364.0 385 6958.6 365 6503.3 333 5316.0 313 5312.7 330 4478.0 367

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 Werkloosheid[t] = + 434.393784762256 -0.0075438885580698Export[t] -0.788029037341926t + 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) 434.393784762256 27.049263 16.0594 0 0 Export -0.0075438885580698 0.005609 -1.3451 0.183839 0.09192 t -0.788029037341926 0.247405 -3.1852 0.002329 0.001164 Multiple Linear Regression - Regression Statistics Multiple R 0.59474503976666 R-squared 0.353721662327046 Adjusted R-squared 0.331436202407289 F-TEST (value) 15.8723070378932 F-TEST (DF numerator) 2 F-TEST (DF denominator) 58 p-value 3.17810599714807e-06 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 25.3767396765567 Sum Squared Residuals 37350.7771634801 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 401 401.744896788762 -0.744896788762101 2 394 397.909891162815 -3.90989116281551 3 372 392.987811195651 -20.9878111956514 4 334 396.91169495168 -62.9116949516798 5 320 395.455277388093 -75.455277388093 6 334 388.184030523946 -54.1840305239458 7 400 390.924278165213 9.07572183478686 8 427 397.662786742257 29.3372132577426 9 423 392.271476906781 30.7285230932187 10 395 389.750616667651 5.24938333234922 11 373 390.957946153918 -17.9579461539183 12 377 389.228439824529 -12.2284398245293 13 391 394.30880169651 -3.30880169650984 14 398 388.728140258226 9.27185974177382 15 393 383.892060620624 9.107939379376 16 375 382.124834848445 -7.12483484844461 17 371 380.132801197235 -9.13280119723475 18 364 376.354374735474 -12.3543747354740 19 400 379.726045849052 20.2739541509483 20 406 383.409279560078 22.5907204399222 21 407 373.66967235973 33.3303276402698 22 397 377.266151352338 19.7338486476616 23 389 374.651746895088 14.3482531049122 24 394 371.579428402362 22.4205715976376 25 399 375.7967694233 23.2032305767003 26 401 373.339277848057 27.660722151943 27 396 364.935693311344 31.0643066886564 28 392 373.523208973971 18.4767910260292 29 384 365.232782765628 18.7672172343715 30 370 368.750050928377 1.24994907162302 31 380 366.039839086439 13.9601609135611 32 376 370.359776991766 5.640223008234 33 378 365.338872084491 12.6611279155089 34 376 361.745270892403 14.2547291075970 35 373 360.987417409293 12.0125825907066 36 374 363.947946596456 10.0520534035437 37 379 362.02908866426 16.9709113357403 38 376 358.679909461453 17.3200905385469 39 371 350.326114589223 20.673885410777 40 375 361.456675084776 13.5433249152245 41 360 357.211281921270 2.78871807872977 42 338 354.694193626419 -16.6941936264187 43 352 348.530389602596 3.46961039740375 44 344 356.29637580125 -12.2963758012497 45 330 349.679938463943 -19.679938463943 46 334 346.906357958117 -12.9063579581171 47 333 347.133736320691 -14.1337363206914 48 343 353.163119373277 -10.1631193732771 49 350 352.396213223898 -2.39621322389781 50 341 349.967388425176 -8.9673884251757 51 320 349.908853411399 -29.9088534113991 52 302 340.572089860053 -38.5720898600525 53 287 344.623465332712 -57.6234653327123 54 304 341.336146016082 -37.3361460160819 55 370 337.478508724461 32.5214912755386 56 385 349.798740445622 35.2012595543785 57 365 336.981226713582 28.0187732864185 58 333 339.627930136729 -6.62793013672876 59 313 347.796759984383 -34.7967599843831 60 330 347.033625779283 -17.0336257792828 61 367 352.542480521362 14.4575194786383 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.128576740255717 0.257153480511434 0.871423259744283 7 0.992375863344742 0.0152482733105155 0.00762413665525777 8 0.996077462220734 0.00784507555853108 0.00392253777926554 9 0.995870990328615 0.00825801934276989 0.00412900967138494 10 0.991421617237413 0.0171567655251744 0.0085783827625872 11 0.993190649140393 0.0136187017192134 0.0068093508596067 12 0.991488704806038 0.0170225903879249 0.00851129519396243 13 0.990554780978801 0.0188904380423976 0.0094452190211988 14 0.984058835973443 0.0318823280531135 0.0159411640265567 15 0.975771054708717 0.0484578905825654 0.0242289452912827 16 0.969525058176965 0.0609498836460701 0.0304749418230350 17 0.965433010901005 0.0691339781979901 0.0345669890989951 18 0.965861959025217 0.0682760819495657 0.0341380409747829 19 0.95333000557903 0.093339988841942 0.046669994420971 20 0.93017930392075 0.139641392158502 0.0698206960792508 21 0.931785447538073 0.136429104923854 0.068214552461927 22 0.901613928020361 0.196772143959278 0.098386071979639 23 0.865716998713558 0.268566002572885 0.134283001286442 24 0.82028839792877 0.359423204142461 0.179711602071230 25 0.767773825527692 0.464452348944616 0.232226174472308 26 0.70894467150943 0.582110656981139 0.291055328490570 27 0.674650110820677 0.650699778358645 0.325349889179323 28 0.630747206164286 0.738505587671429 0.369252793835714 29 0.564441926251334 0.871116147497332 0.435558073748666 30 0.572175968135985 0.85564806372803 0.427824031864015 31 0.512404162500219 0.975191674999562 0.487595837499781 32 0.500502869715229 0.998994260569542 0.499497130284771 33 0.440490472877435 0.88098094575487 0.559509527122565 34 0.376768036033145 0.753536072066290 0.623231963966855 35 0.319224901976116 0.638449803952233 0.680775098023884 36 0.274522635502399 0.549045271004798 0.725477364497601 37 0.230518663702754 0.461037327405507 0.769481336297246 38 0.1977133797135 0.395426759427 0.8022866202865 39 0.193592820767430 0.387185641534861 0.80640717923257 40 0.183983051769039 0.367966103538078 0.816016948230961 41 0.173901411572426 0.347802823144852 0.826098588427574 42 0.180568200793371 0.361136401586741 0.81943179920663 43 0.163224192638866 0.326448385277733 0.836775807361134 44 0.153678095151314 0.307356190302628 0.846321904848686 45 0.139127322545848 0.278254645091696 0.860872677454152 46 0.111320987914333 0.222641975828667 0.888679012085667 47 0.0862353674524685 0.172470734904937 0.913764632547531 48 0.0673187175791484 0.134637435158297 0.932681282420852 49 0.059379432701863 0.118758865403726 0.940620567298137 50 0.0555690887294804 0.111138177458961 0.94443091127052 51 0.0464503350329135 0.0929006700658269 0.953549664967086 52 0.0381954039131798 0.0763908078263596 0.96180459608682 53 0.100931429772212 0.201862859544423 0.899068570227788 54 0.408274702313542 0.816549404627084 0.591725297686458 55 0.277808946594597 0.555617893189193 0.722191053405403 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 2 0.04 NOK 5% type I error level 9 0.18 NOK 10% type I error level 15 0.3 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; par2 = Do not include Seasonal 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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