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: Tue, 23 Dec 2008 13:22:18 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/23/t1230063848f4f2czh3o741r0h.htm/, Retrieved Tue, 23 Dec 2008 21:24:08 +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/2008/Dec/23/t1230063848f4f2czh3o741r0h.htm/}, year = {2008}, } @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 = {2008}, 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:

» Textbox « » Textfile « » CSV «

104.3 0 103.9 0 103.9 0 103.9 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 0 108.0 1 108.0 1 108.0 1 108.0 1 108.0 1 108.2 1 112.3 1 111.3 1 111.3 1 115.3 1 117.2 1 118.3 1 118.3 1 118.3 1 119.0 1 120.6 1 122.6 1 122.6 1 127.4 1 125.9 1 121.5 1 118.8 1 121.6 1 122.3 1 122.7 1 120.8 1 120.1 1 120.1 1 120.1 1 120.1 1 128.4 1 129.8 1 129.8 1 128.6 1 128.6 1 133.7 1 130.0 1 125.9 1 129.4 1 129.4 1 130.6 1 130.6 1 130.6 1 130.8 1 129.7 1 125.8 1 126.0 1 125.6 1 125.4 1 124.7 1 126.9 1 129.1 1

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 6 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation y[t] = + 100.760435897436 + 0.370153846153845x[t] + 0.925943223443226M1[t] + 1.01766300366300M2[t] + 1.52430952380952M3[t] + 1.16364835164835M4[t] + 4.31965384615385M5[t] + 3.77565934065935M6[t] + 2.46499816849817M7[t] + 1.43767032967033M8[t] + 1.86034249084249M9[t] + 2.54968131868132M10[t] + 1.51066117216117M11[t] + 0.393994505494505t + 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) 100.760435897436 1.650305 61.0556 0 0 x 0.370153846153845 1.475439 0.2509 0.802766 0.401383 M1 0.925943223443226 1.950324 0.4748 0.636679 0.318339 M2 1.01766300366300 1.950283 0.5218 0.603729 0.301864 M3 1.52430952380952 2.028288 0.7515 0.455275 0.227637 M4 1.16364835164835 2.027155 0.574 0.568093 0.284047 M5 4.31965384615385 2.026511 2.1316 0.037147 0.018574 M6 3.77565934065935 2.026358 1.8633 0.06732 0.03366 M7 2.46499816849817 2.026696 1.2163 0.228648 0.114324 M8 1.43767032967033 2.027524 0.7091 0.481023 0.240512 M9 1.86034249084249 2.028841 0.9169 0.362841 0.18142 M10 2.54968131868132 2.030648 1.2556 0.214129 0.107064 M11 1.51066117216117 2.02106 0.7475 0.457706 0.228853 t 0.393994505494505 0.03153 12.496 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.938648554720469 R-squared 0.881061109278825 Adjusted R-squared 0.855291016289237 F-TEST (value) 34.189287156814 F-TEST (DF numerator) 13 F-TEST (DF denominator) 60 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.50015320700787 Sum Squared Residuals 735.064348351648 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 104.3 102.080373626374 2.21962637362638 2 103.9 102.566087912088 1.33391208791209 3 103.9 103.466728937729 0.433271062271068 4 103.9 103.500062271062 0.399937728937733 5 108 107.050062271062 0.949937728937727 6 108 106.900062271062 1.09993772893772 7 108 105.983395604396 2.01660439560439 8 108 105.350062271062 2.64993772893773 9 108 106.166728937729 1.83327106227106 10 108 107.250062271062 0.749937728937732 11 108 106.605036630037 1.39496336996337 12 108 105.48836996337 2.51163003663003 13 108 106.808307692308 1.19169230769230 14 108 107.294021978022 0.705978021978017 15 108 108.194663003663 -0.194663003663004 16 108 108.227996336996 -0.227996336996338 17 108 111.777996336996 -3.77799633699634 18 108 111.627996336996 -3.62799633699634 19 108 110.711329670330 -2.71132967032967 20 108 110.077996336996 -2.07799633699634 21 108 110.894663003663 -2.894663003663 22 108 111.977996336996 -3.97799633699633 23 108 111.703124542125 -3.70312454212454 24 108 110.586457875458 -2.58645787545787 25 108 111.906395604396 -3.9063956043956 26 108 112.39210989011 -4.39210989010989 27 108 113.292750915751 -5.29275091575091 28 108.2 113.326084249084 -5.12608424908424 29 112.3 116.876084249084 -4.57608424908425 30 111.3 116.726084249084 -5.42608424908426 31 111.3 115.809417582418 -4.50941758241759 32 115.3 115.176084249084 0.123915750915752 33 117.2 115.992750915751 1.20724908424909 34 118.3 117.076084249084 1.22391575091575 35 118.3 116.431058608059 1.86894139194139 36 118.3 115.314391941392 2.98560805860806 37 119 116.634329670330 2.36567032967033 38 120.6 117.120043956044 3.47995604395604 39 122.6 118.020684981685 4.57931501831502 40 122.6 118.054018315018 4.54598168498168 41 127.4 121.604018315018 5.79598168498169 42 125.9 121.454018315018 4.44598168498169 43 121.5 120.537351648352 0.962648351648353 44 118.8 119.904018315018 -1.10401831501832 45 121.6 120.720684981685 0.879315018315014 46 122.3 121.804018315018 0.495981684981686 47 122.7 121.158992673993 1.54100732600733 48 120.8 120.042326007326 0.757673992673989 49 120.1 121.362263736264 -1.26226373626374 50 120.1 121.847978021978 -1.74797802197803 51 120.1 122.748619047619 -2.64861904761905 52 120.1 122.781952380952 -2.68195238095239 53 128.4 126.331952380952 2.06804761904762 54 129.8 126.181952380952 3.61804761904763 55 129.8 125.265285714286 4.5347142857143 56 128.6 124.631952380952 3.96804761904762 57 128.6 125.448619047619 3.15138095238095 58 133.7 126.531952380952 7.16804761904761 59 130 125.886926739927 4.11307326007326 60 125.9 124.77026007326 1.12973992673993 61 129.4 126.090197802198 3.30980219780220 62 129.4 126.575912087912 2.82408791208792 63 130.6 127.476553113553 3.12344688644688 64 130.6 127.509886446886 3.09011355311355 65 130.6 131.059886446886 -0.459886446886451 66 130.8 130.909886446886 -0.109886446886437 67 129.7 129.993219780220 -0.293219780219789 68 125.8 129.359886446886 -3.55988644688645 69 126 130.176553113553 -4.17655311355311 70 125.6 131.259886446886 -5.65988644688645 71 125.4 130.614860805861 -5.2148608058608 72 124.7 129.498194139194 -4.79819413919413 73 126.9 130.818131868132 -3.91813186813187 74 129.1 131.303846153846 -2.20384615384616 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.069579804762308 0.139159609524616 0.930420195237692 18 0.0526203854940858 0.105240770988172 0.947379614505914 19 0.0317032594793010 0.0634065189586021 0.968296740520699 20 0.0171255815257111 0.0342511630514222 0.982874418474289 21 0.00845869479870889 0.0169173895974178 0.991541305201291 22 0.00386648223211917 0.00773296446423834 0.99613351776788 23 0.00141754552915488 0.00283509105830977 0.998582454470845 24 0.000472689076199105 0.00094537815239821 0.9995273109238 25 0.000179006627938791 0.000358013255877582 0.999820993372061 26 7.10306331255767e-05 0.000142061266251153 0.999928969366874 27 3.41332265617906e-05 6.82664531235813e-05 0.999965866773438 28 1.85056790732549e-05 3.70113581465098e-05 0.999981494320927 29 3.54273957076497e-05 7.08547914152994e-05 0.999964572604292 30 5.3986836641093e-05 0.000107973673282186 0.999946013163359 31 7.96051582885303e-05 0.000159210316577061 0.999920394841711 32 0.000709569181621591 0.00141913836324318 0.999290430818378 33 0.00647751400285187 0.0129550280057037 0.993522485997148 34 0.0292025823445879 0.0584051646891759 0.970797417655412 35 0.067311958689985 0.13462391737997 0.932688041310015 36 0.0962779579903187 0.192555915980637 0.903722042009681 37 0.123196434950648 0.246392869901297 0.876803565049352 38 0.167256947624052 0.334513895248104 0.832743052375948 39 0.243865746266528 0.487731492533055 0.756134253733472 40 0.28214886270465 0.5642977254093 0.71785113729535 41 0.356750146514000 0.713500293028001 0.643249853486 42 0.352076534738209 0.704153069476418 0.647923465261791 43 0.305895875974853 0.611791751949707 0.694104124025147 44 0.276893180355737 0.553786360711474 0.723106819644263 45 0.213171146581652 0.426342293163305 0.786828853418348 46 0.175146695548696 0.350293391097393 0.824853304451304 47 0.126531516378551 0.253063032757102 0.873468483621449 48 0.0904770337772993 0.180954067554599 0.9095229662227 49 0.0956868506218758 0.191373701243752 0.904313149378124 50 0.155961525845126 0.311923051690251 0.844038474154874 51 0.361695734205674 0.723391468411347 0.638304265794326 52 0.873316759284334 0.253366481431332 0.126683240715666 53 0.888387935183533 0.223224129632933 0.111612064816467 54 0.887177347683372 0.225645304633257 0.112822652316629 55 0.863934453756804 0.272131092486392 0.136065546243196 56 0.757150882764308 0.485698234471384 0.242849117235692 57 0.598036028399444 0.803927943201112 0.401963971600556 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 11 0.268292682926829 NOK 5% type I error level 14 0.341463414634146 NOK 10% type I error level 16 0.390243902439024 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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