Paper - multiple regression analysis (5)

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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Fri, 04 Dec 2009 04:24:48 -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/04/t1259925967okuus7xp45atu3o.htm/, Retrieved Fri, 04 Dec 2009 12:26:20 +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/04/t1259925967okuus7xp45atu3o.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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209465 555332 213587 216234 204045 543599 209465 213587 200237 536662 204045 209465 203666 542722 200237 204045 241476 593530 203666 200237 260307 610763 241476 203666 243324 612613 260307 241476 244460 611324 243324 260307 233575 594167 244460 243324 237217 595454 233575 244460 235243 590865 237217 233575 230354 589379 235243 237217 227184 584428 230354 235243 221678 573100 227184 230354 217142 567456 221678 227184 219452 569028 217142 221678 256446 620735 219452 217142 265845 628884 256446 219452 248624 628232 265845 256446 241114 612117 248624 265845 229245 595404 241114 248624 231805 597141 229245 241114 219277 593408 231805 229245 219313 590072 219277 231805 212610 579799 219313 219277 214771 574205 212610 219313 211142 572775 214771 212610 211457 572942 211142 214771 240048 619567 211457 211142 240636 625809 240048 211457 230580 619916 240636 240048 208795 587625 230580 240636 197922 565742 208795 230580 194596 557274 197922 208795 194581 560576 194596 197922 185686 54 etc...

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 Werkl[t] = -37913.5795481944 + 0.167921847385156x[t] + 0.983692389278356`y-1`[t] -0.262137203471688`y-2`[t] + 856.237856090604M1[t] + 1422.34146979028M2[t] + 645.643330794933M3[t] + 7996.00756864989M4[t] + 26773.9655269154M5[t] -109.931030073537M6[t] -10226.4147700189M7[t] -153.278421988093M8[t] -1876.05346206772M9[t] + 7808.93801645004M10[t] -1.11551941151744M11[t] -54.7353627623493t + 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) -37913.5795481944 24602.591742 -1.541 0.129489 0.064745 x 0.167921847385156 0.093463 1.7967 0.078311 0.039156 `y-1` 0.983692389278356 0.155181 6.339 0 0 `y-2` -0.262137203471688 0.14998 -1.7478 0.086514 0.043257 M1 856.237856090604 3091.676439 0.2769 0.782938 0.391469 M2 1422.34146979028 3123.560735 0.4554 0.650784 0.325392 M3 645.643330794933 3281.633294 0.1967 0.844809 0.422405 M4 7996.00756864989 3115.899663 2.5662 0.013262 0.006631 M5 26773.9655269154 5380.796392 4.9758 8e-06 4e-06 M6 -109.931030073537 5995.742542 -0.0183 0.985443 0.492722 M7 -10226.4147700189 3436.071346 -2.9762 0.004454 0.002227 M8 -153.278421988093 3999.797006 -0.0383 0.969581 0.48479 M9 -1876.05346206772 3544.949911 -0.5292 0.59895 0.299475 M10 7808.93801645004 3302.655746 2.3644 0.021902 0.010951 M11 -1.11551941151744 3287.674264 -3e-04 0.999731 0.499865 t -54.7353627623493 55.020009 -0.9948 0.324517 0.162259 Multiple Linear Regression - Regression Statistics Multiple R 0.988435732988545 R-squared 0.977005198248602 Adjusted R-squared 0.970242021262897 F-TEST (value) 144.45950480279 F-TEST (DF numerator) 15 F-TEST (DF denominator) 51 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 4996.85437277711 Sum Squared Residuals 1273396234.75983 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 209465 209561.228590528 -96.2285905275284 2 204045 204741.466955078 -696.4669550782 3 200237 198494.076400831 1742.92359916866 4 203666 204482.194695522 -816.194695522485 5 241476 236108.490186626 5367.50981337371 6 260307 248358.196230774 11948.8037692263 7 243324 247110.136264965 -3786.13626496471 8 244460 235269.732463264 9190.26753673597 9 233575 236180.537605615 -2605.53760561483 10 237217 235021.629618516 2195.37038148382 11 235243 232822.218503783 2420.78149621709 12 230354 229622.554323738 731.445676261644 13 227184 225300.862499134 1883.13750086607 14 221678 222073.295976653 -395.295976652897 15 217142 215708.876207892 1433.12379210799 16 219452 220249.776991623 -797.77699162257 17 256446 251117.118324051 5328.88167594948 18 265845 261332.060847585 4512.93915241516 19 248624 250599.577761978 -1975.57776197764 20 241114 238507.923965441 2606.07603455865 21 229245 231050.670664757 -1805.67066475674 22 231805 231265.812459148 539.187540852268 23 219277 228403.730288793 -9126.73028879309 24 219313 214795.153668799 4517.84633120140 25 212610 217191.062835046 -4581.06283504648 26 214771 210159.949247053 4611.05075294656 27 211142 212971.252431636 -1829.25243163623 28 211457 216158.626077849 -4701.62607784869 29 240048 243972.363821706 -3924.36382170613 30 240636 246124.075956097 -5488.07595609693 31 230580 228046.939747185 2533.06025281487 32 208795 222596.829016315 -13801.8290163150 33 197922 198350.997844826 -428.997844825983 34 194596 201574.263385931 -6978.26338593104 35 194581 193842.409353981 738.590646019228 36 185686 192677.522568489 -6991.5225684888 37 178106 181847.948057314 -3741.94805731378 38 172608 176268.416112548 -3660.41611254765 39 167302 169516.796845914 -2214.79684591444 40 168053 170955.983464945 -2902.98346494474 41 202300 201330.027793155 969.972206844983 42 202388 209432.79481873 -7044.79481873001 43 182516 186519.608191053 -4003.60819105272 44 173476 174517.529954869 -1041.52995486892 45 166444 166164.344960978 279.655039021949 46 171297 171987.867705375 -690.86770537542 47 169701 171187.630588166 -1486.63058816635 48 164182 166719.969429707 -2537.96942970671 49 161914 160002.260125262 1911.73987473831 50 159612 159329.863188347 282.136811653336 51 151001 153321.281279458 -2320.28127945761 52 158114 154130.596183914 3983.40381608574 53 186530 190638.348128768 -4108.34812876756 54 187069 190662.77796118 -3593.77796118002 55 174330 170945.740980101 3384.25901989892 56 169362 166314.984600111 3047.01539988933 57 166827 162266.448923824 4560.55107617561 58 178037 173102.426831030 4934.57316897038 59 186412 178958.011265277 7453.98873472312 60 189226 184945.800009268 4280.19999073248 61 191563 186938.637892717 4624.36210728341 62 188906 189047.008520321 -141.008520321151 63 186005 182816.716834268 3188.28316573162 64 195309 190073.822586147 5235.17741385274 65 223532 227165.651745694 -3633.65174569448 66 226899 227234.094185635 -335.094185634512 67 214126 210277.997054719 3848.00294528128 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.0183002867240648 0.0366005734481295 0.981699713275935 20 0.0180782783603871 0.0361565567207741 0.981921721639613 21 0.00626333974865208 0.0125266794973042 0.993736660251348 22 0.00660126009128533 0.0132025201825707 0.993398739908715 23 0.173771737021593 0.347543474043186 0.826228262978407 24 0.181470550125243 0.362941100250486 0.818529449874757 25 0.161119876550444 0.322239753100889 0.838880123449556 26 0.513402138448295 0.97319572310341 0.486597861551705 27 0.449791180164339 0.899582360328679 0.55020881983566 28 0.357673836074425 0.71534767214885 0.642326163925575 29 0.596115230800752 0.807769538398495 0.403884769199248 30 0.961247379020376 0.0775052419592484 0.0387526209796242 31 0.982731193232279 0.0345376135354421 0.0172688067677211 32 0.97109954541039 0.05780090917922 0.02890045458961 33 0.987560697773815 0.02487860445237 0.012439302226185 34 0.988949127687506 0.0221017446249891 0.0110508723124945 35 0.99550683100598 0.00898633798803914 0.00449316899401957 36 0.992149268260826 0.0157014634783478 0.0078507317391739 37 0.991357160432364 0.0172856791352727 0.00864283956763636 38 0.983531396289012 0.0329372074219753 0.0164686037109877 39 0.991575753574717 0.0168484928505664 0.00842424642528322 40 0.99679437857408 0.00641124285184015 0.00320562142592007 41 0.999640693447595 0.000718613104810173 0.000359306552405086 42 0.998945980162276 0.00210803967544833 0.00105401983772417 43 0.996820939149883 0.0063581217002347 0.00317906085011735 44 0.99155936472354 0.0168812705529203 0.00844063527646015 45 0.989353983475544 0.0212920330489121 0.0106460165244560 46 0.994636036783788 0.0107279264324235 0.00536396321621177 47 0.981950607500409 0.0360987849991821 0.0180493924995911 48 0.99857916414625 0.00284167170749937 0.00142083585374969 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 6 0.2 NOK 5% type I error level 21 0.7 NOK 10% type I error level 23 0.766666666666667 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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