JJ Workshop 7, Multiple Regression

*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 12:02:44 -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/t1258743980v4vobiw3w7nnp4h.htm/, Retrieved Fri, 20 Nov 2009 20:06:32 +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/t1258743980v4vobiw3w7nnp4h.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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95,1 93,8 97 93,8 112,7 107,6 102,9 101 97,4 95,4 111,4 96,5 87,4 89,2 96,8 87,1 114,1 110,5 110,3 110,8 103,9 104,2 101,6 88,9 94,6 89,8 95,9 90 104,7 93,9 102,8 91,3 98,1 87,8 113,9 99,7 80,9 73,5 95,7 79,2 113,2 96,9 105,9 95,2 108,8 95,6 102,3 89,7 99 92,8 100,7 88 115,5 101,1 100,7 92,7 109,9 95,8 114,6 103,8 85,4 81,8 100,5 87,1 114,8 105,9 116,5 108,1 112,9 102,6 102 93,7 106 103,5 105,3 100,6 118,8 113,3 106,1 102,4 109,3 102,1 117,2 106,9 92,5 87,3 104,2 93,1 112,5 109,1 122,4 120,3 113,3 104,9 100 92,6 110,7 109,8 112,8 111,4 109,8 117,9 117,3 121,6 109,1 117,8 115,9 124,2 96 106,8 99,8 102,7 116,8 116,8 115,7 113,6 99,4 96,1 94,3 85

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 TIA[t] = + 39.3947409484312 + 0.663042805057097IAidM[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) 39.3947409484312 6.225231 6.3282 0 0 IAidM 0.663042805057097 0.062142 10.6698 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.813932329770039 R-squared 0.662485837444884 Adjusted R-squared 0.656666627745657 F-TEST (value) 113.844640713495 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 2.66453525910038e-15 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.27333687938982 Sum Squared Residuals 1612.86874692490 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 95.1 101.588156062786 -6.48815606278646 2 97 101.588156062787 -4.58815606278681 3 112.7 110.738146772575 1.96185322742522 4 102.9 106.362064259198 -3.46206425919794 5 97.4 102.649024550878 -5.2490245508782 6 111.4 103.378371636441 8.021628363559 7 87.4 98.5381591595242 -11.1381591595242 8 96.8 97.1457692689043 -0.345769268904290 9 114.1 112.660970907240 1.43902909275963 10 110.3 112.859883748757 -2.55988374875750 11 103.9 108.483801235381 -4.58380123538065 12 101.6 98.339246318007 3.26075368199292 13 94.6 98.9359848425585 -4.33598484255846 14 95.9 99.0685934035699 -3.16859340356987 15 104.7 101.654460343293 3.04553965670745 16 102.8 99.930549050144 2.8694509498559 17 98.1 97.6098992324443 0.490100767555738 18 113.9 105.500108612624 8.39989138737629 19 80.9 88.1283871201278 -7.22838712012777 20 95.7 91.9077311089532 3.79226889104678 21 113.2 103.643588758464 9.55641124153616 22 105.9 102.516415989867 3.38358401013322 23 108.8 102.781633111890 6.01836688811038 24 102.3 98.8696805620527 3.43031943794725 25 99 100.925113257730 -1.92511325772974 26 100.7 97.7425077934557 2.95749220654432 27 115.5 106.428368539704 9.07163146029635 28 100.7 100.858808977224 -0.158808977224035 29 109.9 102.914241672901 6.98575832709897 30 114.6 108.218584113358 6.38141588664218 31 85.4 93.6316424021017 -8.23164240210167 32 100.5 97.1457692689043 3.35423073109571 33 114.8 109.610974003978 5.18902599602227 34 116.5 111.069668175103 5.43033182489667 35 112.9 107.422932747289 5.47706725271071 36 102 101.521851782281 0.478148217718865 37 106 108.019671271841 -2.01967127184069 38 105.3 106.096847137175 -0.796847137175103 39 118.8 114.517490761400 4.28250923859976 40 106.1 107.290324186278 -1.19032418627789 41 109.3 107.091411344761 2.20858865523925 42 117.2 110.274016809035 6.92598319096518 43 92.5 97.2783778299157 -4.77837782991571 44 104.2 101.124026099247 3.07597390075313 45 112.5 111.732710980160 0.767289019839573 46 122.4 119.1587903968 3.24120960320009 47 113.3 108.947931198921 4.35206880107937 48 100 100.792504696718 -0.792504696718322 49 110.7 112.196840943700 -1.49684094370039 50 112.8 113.257709431792 -0.45770943179176 51 109.8 117.567487664663 -7.76748766466289 52 117.3 120.020746043374 -2.72074604337414 53 109.1 117.501183384157 -8.40118338415718 54 115.9 121.744657336523 -5.8446573365226 55 96 110.207712528529 -14.2077125285291 56 99.8 107.489237027795 -7.68923702779501 57 116.8 116.838140579100 -0.0381405791000808 58 115.7 114.716403602917 0.983596397082638 59 99.4 103.113154514418 -3.71315451441816 60 94.3 95.7533793782844 -1.45337937828439 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0151066619131761 0.0302133238263521 0.984893338086824 6 0.677475946997449 0.645048106005102 0.322524053002551 7 0.680802457542027 0.638395084915947 0.319197542457973 8 0.716162431321369 0.567675137357263 0.283837568678631 9 0.610167217385887 0.779665565228226 0.389832782614113 10 0.553825906778263 0.892348186443473 0.446174093221737 11 0.498573762428518 0.997147524857037 0.501426237571482 12 0.551630435117172 0.896739129765656 0.448369564882828 13 0.473849311248627 0.947698622497253 0.526150688751373 14 0.389750346783943 0.779500693567885 0.610249653216057 15 0.384346234214776 0.768692468429552 0.615653765785224 16 0.3666861715693 0.7333723431386 0.6333138284307 17 0.30134106917798 0.60268213835596 0.69865893082202 18 0.459053252697399 0.918106505394797 0.540946747302601 19 0.471007948755572 0.942015897511143 0.528992051244428 20 0.484924557928896 0.96984911585779 0.515075442071104 21 0.66104201889386 0.677915962212281 0.338957981106141 22 0.615046510550671 0.769906978898658 0.384953489449329 23 0.628412762507958 0.743174474984085 0.371587237492042 24 0.584726503218605 0.83054699356279 0.415273496781395 25 0.518497350767787 0.963005298464426 0.481502649232213 26 0.466095250167815 0.932190500335631 0.533904749832185 27 0.584359091762546 0.831281816474908 0.415640908237454 28 0.508472429042935 0.98305514191413 0.491527570957065 29 0.554855708303303 0.890288583393394 0.445144291696697 30 0.572816012429167 0.854367975141666 0.427183987570833 31 0.654935817634854 0.690128364730293 0.345064182365146 32 0.614714007738845 0.77057198452231 0.385285992261155 33 0.60374504371331 0.792509912573379 0.396254956286689 34 0.606105040382586 0.787789919234828 0.393894959617414 35 0.619591739893866 0.760816520212268 0.380408260106134 36 0.547558899286647 0.904882201426705 0.452441100713353 37 0.4951905907498 0.9903811814996 0.5048094092502 38 0.425362452510621 0.850724905021242 0.574637547489379 39 0.421985015371418 0.843970030742836 0.578014984628582 40 0.356523555334657 0.713047110669314 0.643476444665343 41 0.311320999313173 0.622641998626346 0.688679000686827 42 0.428790077916929 0.857580155833858 0.571209922083071 43 0.388105295208232 0.776210590416463 0.611894704791768 44 0.371122564362602 0.742245128725203 0.628877435637398 45 0.328959664313403 0.657919328626806 0.671040335686597 46 0.3800243902191 0.7600487804382 0.6199756097809 47 0.492195136632261 0.984390273264523 0.507804863367739 48 0.422926502111185 0.84585300422237 0.577073497888815 49 0.374328294462446 0.748656588924892 0.625671705537554 50 0.354654785019243 0.709309570038487 0.645345214980757 51 0.357237018421745 0.71447403684349 0.642762981578255 52 0.288343265665315 0.57668653133063 0.711656734334685 53 0.265304399267584 0.530608798535167 0.734695600732416 54 0.186256829392694 0.372513658785387 0.813743170607306 55 0.695817319781347 0.608365360437306 0.304182680218653 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 1 0.0196078431372549 OK 10% type I error level 1 0.0196078431372549 OK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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