multiple regression without seasonal dummies

*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 11:12:11 -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/t1258741320lsijg7sxti2ec1q.htm/, Retrieved Fri, 20 Nov 2009 19:22:12 +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/t1258741320lsijg7sxti2ec1q.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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462 1919 455 1911 461 1870 461 2263 463 1802 462 1863 456 1989 455 2197 456 2409 472 2502 472 2593 471 2598 465 2053 459 2213 465 2238 468 2359 467 2151 463 2474 460 3079 462 2312 461 2565 476 1972 476 2484 471 2202 453 2151 443 1976 442 2012 444 2114 438 1772 427 1957 424 2070 416 1990 406 2182 431 2008 434 1916 418 2397 412 2114 404 1778 409 1641 412 2186 406 1773 398 1785 397 2217 385 2153 390 1895 413 2475 413 1793 401 2308 397 2051 397 1898 409 2142 419 1874 424 1560 428 1808 430 1575 424 1525 433 1997 456 1753 459 1623 446 2251 441 1890

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 wkl[t] = + 356.359058985737 + 0.0361691651132457bvg[t] + 8.56292555607903M1[t] + 4.52298938484529M2[t] + 9.20429259096884M3[t] + 6.34447970174317M4[t] + 17.7168814951074M5[t] + 7.72003391933123M6[t] -2.02485392329182M7[t] -1.57777765723701M8[t] -7.0784462199644M9[t] + 15.7665893416910M10[t] + 19.1439730815084M11[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) 356.359058985737 32.20814 11.0643 0 0 bvg 0.0361691651132457 0.01276 2.8346 0.006695 0.003348 M1 8.56292555607903 16.386103 0.5226 0.603676 0.301838 M2 4.52298938484529 17.322916 0.2611 0.795133 0.397567 M3 9.20429259096884 17.231166 0.5342 0.595693 0.297846 M4 6.34447970174317 16.749714 0.3788 0.706521 0.35326 M5 17.7168814951074 17.943204 0.9874 0.328404 0.164202 M6 7.72003391933123 17.242416 0.4477 0.656359 0.328179 M7 -2.02485392329182 16.703121 -0.1212 0.904018 0.452009 M8 -1.57777765723701 17.052526 -0.0925 0.926666 0.463333 M9 -7.0784462199644 16.667794 -0.4247 0.672969 0.336484 M10 15.7665893416910 16.78322 0.9394 0.352216 0.176108 M11 19.1439730815084 16.922399 1.1313 0.263558 0.131779 Multiple Linear Regression - Regression Statistics Multiple R 0.448758728566603 R-squared 0.201384396464714 Adjusted R-squared 0.00173049558089289 F-TEST (value) 1.00866747693500 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 0.455970418781994 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 26.1987945445670 Sum Squared Residuals 32946.0881082447 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 462 434.330612394134 27.6693876058664 2 455 430.001322901995 24.9986770980054 3 461 433.199690338475 27.8003096615250 4 461 444.554359338755 16.4456406612451 5 463 439.252776014913 23.7472239850872 6 462 431.462247511045 30.5377524889553 7 456 426.274674472691 29.7253255273094 8 455 434.2449370823 20.7550629176995 9 456 436.412131523581 19.5878684764188 10 472 462.620899440768 9.37910055923155 11 472 469.289677205891 2.71032279410877 12 471 450.326549949949 20.6734500500510 13 465 439.177280519309 25.8227194806908 14 459 440.924410766195 18.0755892338053 15 465 446.509943100149 18.4900568998505 16 468 448.026599189626 19.9734008103735 17 467 451.875814639436 15.1241853605644 18 463 453.561607395238 9.43839260476218 19 460 465.699064446128 -5.69906444612843 20 462 438.404391070324 23.5956089296762 21 461 442.054521281248 18.9454787187525 22 476 443.451241930748 32.5487580692518 23 476 465.347238208547 10.6527617914526 24 471 436.003560565104 34.9964394348962 25 453 442.721858700407 10.2781412995928 26 443 432.352318634356 10.6476813656445 27 442 438.335711784556 3.66428821544409 28 444 439.165153736881 4.8348462631187 29 438 438.167701061515 -0.167701061515477 30 427 434.86214903169 -7.86214903168978 31 424 429.204376846864 -5.2043768468635 32 416 426.757919903859 -10.7579199038586 33 406 428.201731042874 -22.2017310428744 34 431 444.753331874825 -13.7533318748251 35 434 444.803152424224 -10.8031524242238 36 418 443.056547762187 -25.0565477621867 37 412 441.383599591217 -29.3835995912172 38 404 425.190823941933 -21.1908239419329 39 409 424.916951527542 -15.9169515275417 40 412 441.769333625035 -29.769333625035 41 406 438.203870226629 -32.2038702266287 42 398 428.641052632212 -30.6410526322115 43 397 434.521244118511 -37.5212441185106 44 385 432.653493817318 -47.6534938173177 45 390 417.821180655373 -27.8211806553729 46 413 461.644331982711 -48.6443319827108 47 413 440.354345115295 -27.3543451152946 48 401 439.837492067108 -38.8374920671078 49 397 439.104942189083 -42.1049421890827 50 397 429.531123755522 -32.5311237555223 51 409 443.037703249278 -34.0377032492779 52 419 430.484554109702 -11.4845541097023 53 424 430.499838057507 -6.49983805750738 54 428 429.472943429816 -1.47294342981616 55 430 411.300640115807 18.6993598841931 56 424 409.939258126199 14.0607418738006 57 433 421.510435496924 11.4895645030760 58 456 435.530194770947 20.4698052290526 59 459 434.205587046043 24.7944129539571 60 446 437.775849655653 8.22415034434721 61 441 433.28170660585 7.71829339414989 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.00111383901519994 0.00222767803039988 0.9988861609848 17 9.2964761969175e-05 0.00018592952393835 0.999907035238031 18 2.85027988114868e-05 5.70055976229736e-05 0.999971497201188 19 3.55272712958011e-06 7.10545425916021e-06 0.99999644727287 20 2.04131820005764e-06 4.08263640011528e-06 0.9999979586818 21 6.05032366324585e-07 1.21006473264917e-06 0.999999394967634 22 3.57327540352414e-07 7.14655080704828e-07 0.99999964267246 23 1.77416735414836e-07 3.54833470829673e-07 0.999999822583265 24 5.33112449775572e-08 1.06622489955114e-07 0.999999946688755 25 5.02709000663388e-07 1.00541800132678e-06 0.999999497291 26 7.04174380946219e-06 1.40834876189244e-05 0.99999295825619 27 0.000231628379451259 0.000463256758902518 0.999768371620549 28 0.00123781890701027 0.00247563781402053 0.99876218109299 29 0.00815255594983233 0.0163051118996647 0.991847444050168 30 0.0467573760150095 0.093514752030019 0.95324262398499 31 0.0804843016363668 0.160968603272734 0.919515698363633 32 0.183049811287510 0.366099622575019 0.81695018871249 33 0.322872119172126 0.645744238344253 0.677127880827874 34 0.367262172366226 0.734524344732453 0.632737827633774 35 0.314648911409605 0.629297822819209 0.685351088590395 36 0.426973330048694 0.853946660097387 0.573026669951306 37 0.494309231534686 0.988618463069372 0.505690768465314 38 0.470609296367544 0.941218592735088 0.529390703632456 39 0.47889719546208 0.95779439092416 0.52110280453792 40 0.458652907353971 0.917305814707941 0.541347092646029 41 0.412219952982272 0.824439905964543 0.587780047017728 42 0.419750490326861 0.839500980653722 0.580249509673139 43 0.344311163653681 0.688622327307363 0.655688836346319 44 0.289228983478521 0.578457966957041 0.71077101652148 45 0.383858990299359 0.767717980598718 0.616141009700641 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 13 0.433333333333333 NOK 5% type I error level 14 0.466666666666667 NOK 10% type I error level 15 0.5 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; 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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