*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, 24 Nov 2009 03:34:40 -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/24/t125905895976xgn6sjj63oxi0.htm/, Retrieved Tue, 24 Nov 2009 11:36:11 +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/24/t125905895976xgn6sjj63oxi0.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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103,52 0 103,5 0 103,52 0 103,53 0 103,53 0 103,53 0 103,52 0 103,54 0 103,59 0 103,59 0 103,59 0 103,59 0 103,63 0 103,74 0 103,7 0 103,72 0 103,81 0 103,8 0 104,22 0 106,91 1 107,06 1 107,17 1 107,25 1 107,28 1 107,24 1 107,23 1 107,34 1 107,34 1 107,3 1 107,24 1 107,3 1 107,32 1 107,28 1 107,33 1 107,33 1 107,33 1 107,28 1 107,28 1 107,29 1 107,29 1 107,23 1 107,24 1 107,24 1 107,2 1 107,23 1 107,2 1 107,21 1 107,24 1 107,21 1 113,89 1 114,05 1 114,05 1 114,05 1 114,05 1 115,12 1 115,68 1 116,05 1 116,18 1 116,35 1 116,44 1 117 1 117,61 1 118,17 1 118,33 1 118,33 1 118,42 1 118,5 1 118,67 1 119,09 1 119,14 1 119,23 1 119,33 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 100.335200181349 -2.32705377694079X[t] + 0.149120980679454M1[t] + 1.09636674339122M2[t] + 0.95194583943642M3[t] + 0.702524935481617M4[t] + 0.419770698193481M5[t] + 0.143683127572011M6[t] + 0.132595556950542M7[t] + 0.809350282485877M8[t] + 0.691596045197743M9[t] + 0.462175141242941M10[t] + 0.239420903954802M11[t] + 0.281087570621469t + 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) 100.335200181349 1.030615 97.3547 0 0 X -2.32705377694079 0.902121 -2.5795 0.01245 0.006225 M1 0.149120980679454 1.249786 0.1193 0.905436 0.452718 M2 1.09636674339122 1.248478 0.8782 0.383479 0.191739 M3 0.95194583943642 1.247459 0.7631 0.448492 0.224246 M4 0.702524935481617 1.24673 0.5635 0.575271 0.287635 M5 0.419770698193481 1.246293 0.3368 0.737472 0.368736 M6 0.143683127572011 1.246147 0.1153 0.908604 0.454302 M7 0.132595556950542 1.246293 0.1064 0.915639 0.457819 M8 0.809350282485877 1.244656 0.6503 0.518092 0.259046 M9 0.691596045197743 1.243634 0.5561 0.580276 0.290138 M10 0.462175141242941 1.242904 0.3719 0.711359 0.355679 M11 0.239420903954802 1.242465 0.1927 0.847868 0.423934 t 0.281087570621469 0.019061 14.7469 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.93492067185643 R-squared 0.874076662664479 Adjusted R-squared 0.845852466365138 F-TEST (value) 30.9690541191741 F-TEST (DF numerator) 13 F-TEST (DF denominator) 58 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 2.15175963867601 Sum Squared Residuals 268.544033472836 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 103.52 100.765408732649 2.7545912673507 2 103.5 101.993742065983 1.50625793401684 3 103.52 102.130408732650 1.38959126735019 4 103.53 102.162075399316 1.36792460068352 5 103.53 102.160408732650 1.36959126735019 6 103.53 102.165408732650 1.36459126735020 7 103.52 102.435408732650 1.08459126735019 8 103.54 103.393251028807 0.146748971193397 9 103.59 103.55658436214 0.0334156378600615 10 103.59 103.608251028807 -0.0182510288066067 11 103.59 103.66658436214 -0.0765843621399367 12 103.59 103.708251028807 -0.118251028806602 13 103.63 104.138459580108 -0.508459580107533 14 103.74 105.366792913441 -1.62679291344077 15 103.7 105.503459580107 -1.80345958010743 16 103.72 105.535126246774 -1.8151262467741 17 103.81 105.533459580107 -1.72345958010743 18 103.8 105.538459580107 -1.73845958010743 19 104.22 105.808459580107 -1.58845958010743 20 106.91 104.439248099323 2.47075190067656 21 107.06 104.602581432657 2.45741856734324 22 107.17 104.654248099323 2.51575190067657 23 107.25 104.712581432657 2.53741856734324 24 107.28 104.754248099323 2.52575190067657 25 107.24 105.184456650624 2.05554334937564 26 107.23 106.412789983958 0.817210016042422 27 107.34 106.549456650624 0.790543349375747 28 107.34 106.581123317291 0.758876682709082 29 107.3 106.579456650624 0.720543349375744 30 107.24 106.584456650624 0.655543349375742 31 107.3 106.854456650624 0.445543349375744 32 107.32 107.812298946781 -0.492298946781062 33 107.28 107.975632280114 -0.695632280114389 34 107.33 108.027298946781 -0.697298946781059 35 107.33 108.085632280114 -0.755632280114389 36 107.33 108.127298946781 -0.797298946781055 37 107.28 108.557507498082 -1.27750749808197 38 107.28 109.785840831415 -2.50584083141520 39 107.29 109.922507498082 -2.63250749808187 40 107.29 109.954174164749 -2.66417416474854 41 107.23 109.952507498082 -2.72250749808187 42 107.24 109.957507498082 -2.71750749808188 43 107.24 110.227507498082 -2.98750749808188 44 107.2 111.185349794239 -3.98534979423868 45 107.23 111.348683127572 -4.11868312757201 46 107.2 111.400349794239 -4.20034979423868 47 107.21 111.458683127572 -4.24868312757202 48 107.24 111.500349794239 -4.26034979423868 49 107.21 111.930558345540 -4.72055834553961 50 113.89 113.158891678873 0.731108321127171 51 114.05 113.295558345540 0.754441654460494 52 114.05 113.327225012206 0.722774987793827 53 114.05 113.325558345540 0.724441654460494 54 114.05 113.330558345540 0.719441654460497 55 115.12 113.600558345539 1.51944165446050 56 115.68 114.558400641696 1.12159935830370 57 116.05 114.721733975030 1.32826602497036 58 116.18 114.773400641696 1.40659935830370 59 116.35 114.831733975030 1.51826602497036 60 116.44 114.873400641696 1.56659935830370 61 117 115.303609192997 1.69639080700277 62 117.61 116.531942526330 1.07805747366955 63 118.17 116.668609192997 1.50139080700287 64 118.33 116.700275859664 1.62972414033621 65 118.33 116.698609192997 1.63139080700287 66 118.42 116.703609192997 1.71639080700288 67 118.5 116.973609192997 1.52639080700288 68 118.67 117.931451489154 0.738548510846074 69 119.09 118.094784822487 0.995215177512742 70 119.14 118.146451489154 0.993548510846073 71 119.23 118.204784822487 1.02521517751275 72 119.33 118.246451489154 1.08354851084607 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 1.54955075409298e-05 3.09910150818596e-05 0.999984504492459 18 4.27673861299270e-07 8.55347722598539e-07 0.999999572326139 19 2.94340827849628e-06 5.88681655699257e-06 0.999997056591722 20 1.78954359541076e-07 3.57908719082152e-07 0.99999982104564 21 1.15789556670435e-08 2.31579113340871e-08 0.999999988421044 22 9.59869217606954e-10 1.91973843521391e-09 0.99999999904013 23 1.04205943435395e-10 2.08411886870791e-10 0.999999999895794 24 1.28638055784226e-11 2.57276111568452e-11 0.999999999987136 25 1.31242701928871e-12 2.62485403857742e-12 0.999999999998687 26 9.7874818907804e-14 1.95749637815608e-13 0.999999999999902 27 7.03301506835766e-15 1.40660301367153e-14 0.999999999999993 28 5.00242106693344e-16 1.00048421338669e-15 1 29 4.05395777031992e-17 8.10791554063984e-17 1 30 4.52501780906316e-18 9.05003561812632e-18 1 31 1.90904042905979e-18 3.81808085811959e-18 1 32 2.19960918995568e-19 4.39921837991137e-19 1 33 2.76567829207708e-20 5.53135658415417e-20 1 34 4.57596443666624e-21 9.15192887333249e-21 1 35 1.33891726382325e-21 2.6778345276465e-21 1 36 8.23991845632549e-22 1.64798369126510e-21 1 37 9.59646329539283e-22 1.91929265907857e-21 1 38 1.65864520190152e-22 3.31729040380303e-22 1 39 2.70303962986345e-23 5.4060792597269e-23 1 40 3.90707311348483e-24 7.81414622696966e-24 1 41 1.06546983322795e-24 2.13093966645590e-24 1 42 1.53972633725262e-25 3.07945267450524e-25 1 43 1.43311665075113e-25 2.86623330150225e-25 1 44 2.06171215882732e-26 4.12342431765465e-26 1 45 5.37940717330637e-27 1.07588143466127e-26 1 46 4.79289350038597e-27 9.58578700077195e-27 1 47 1.93496730183953e-26 3.86993460367906e-26 1 48 3.33130939550566e-24 6.66261879101131e-24 1 49 1.99361679206807e-09 3.98723358413615e-09 0.999999998006383 50 0.9686141508827 0.0627716982346015 0.0313858491173008 51 0.991508920567788 0.0169821588644248 0.00849107943221239 52 0.994999691612553 0.0100006167748937 0.00500030838744685 53 0.996505082957263 0.00698983408547404 0.00349491704273702 54 0.999883840136367 0.000232319727265039 0.000116159863632519 55 0.99996043910997 7.91217800613651e-05 3.95608900306826e-05 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 36 0.923076923076923 NOK 5% type I error level 38 0.974358974358974 NOK 10% type I error level 39 1 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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