Workshop 7: Multiple lineair regression software

*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: Thu, 19 Nov 2009 09:34:41 -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/19/t1258648559iw8c6vqv2bstyva.htm/, Retrieved Thu, 19 Nov 2009 17:36:11 +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/19/t1258648559iw8c6vqv2bstyva.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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0,7461 0,5270 0,7775 0,4720 0,7790 0,0000 0,7744 0,0520 0,7905 0,3130 0,7719 0,3640 0,7811 0,3630 0,7557 -0,1550 0,7637 0,0520 0,7595 0,5680 0,7471 0,6680 0,7615 1,3780 0,7487 0,2520 0,7389 -0,4020 0,7337 -0,0500 0,7510 0,5550 0,7382 0,0500 0,7159 0,1500 0,7542 0,4500 0,7636 0,2990 0,7433 0,1990 0,7658 0,4960 0,7627 0,4440 0,7480 -0,3930 0,7692 -0,4440 0,7850 0,1980 0,7913 0,4940 0,7720 0,1330 0,7880 0,3880 0,8070 0,4840 0,8268 0,2780 0,8244 0,3690 0,8487 0,1650 0,8572 0,1550 0,8214 0,0870 0,8827 0,4140 0,9216 0,3600 0,8865 0,9750 0,8816 0,2700 0,8884 0,3590 0,9466 0,1690 0,9180 0,3810 0,9337 0,1540 0,9559 0,4860 0,9626 0,9250 0,9434 0,7280 0,8639 -0,0140 0,7996 0,0460 0,6680 -0,8190 0,6572 -1,6740 0,6928 -0,7880 0,6438 0,2790 0,6454 0,3960 0,6873 -0,1410 0,7265 -0,0190 0,7912 0,0990 0,8114 0,7420 0,8281 0,0050 0,8393 0,4480

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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.778012058274589 + 0.0734527913500218X[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) 0.778012058274589 0.010219 76.1341 0 0 X 0.0734527913500218 0.02055 3.5744 0.000723 0.000362 Multiple Linear Regression - Regression Statistics Multiple R 0.42790305973798 R-squared 0.183101028533125 Adjusted R-squared 0.168769467630197 F-TEST (value) 12.7760702252412 F-TEST (DF numerator) 1 F-TEST (DF denominator) 57 p-value 0.000723181931649775 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0707069736074677 Sum Squared Residuals 0.284970138653447 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 0.7461 0.816721679316048 -0.0706216793160478 2 0.7775 0.812681775791799 -0.0351817757917988 3 0.779 0.778012058274589 0.000987941725411454 4 0.7744 0.78183160342479 -0.00743160342478972 5 0.7905 0.801002781967145 -0.0105027819671454 6 0.7719 0.804748874325997 -0.0328488743259965 7 0.7811 0.804675421534646 -0.0235754215346464 8 0.7557 0.766626875615335 -0.0109268756153352 9 0.7637 0.78183160342479 -0.0181316034247897 10 0.7595 0.8197332437614 -0.060233243761401 11 0.7471 0.827078522896403 -0.079978522896403 12 0.7615 0.879230004754919 -0.117730004754919 13 0.7487 0.796522161694794 -0.047822161694794 14 0.7389 0.74848403615188 -0.00958403615187983 15 0.7337 0.774339418707087 -0.0406394187070875 16 0.751 0.81877835747385 -0.0677783574738506 17 0.7382 0.78168469784209 -0.0434846978420897 18 0.7159 0.789029976977092 -0.0731299769770919 19 0.7542 0.811065814382098 -0.0568658143820984 20 0.7636 0.799974442888245 -0.0363744428882451 21 0.7433 0.792629163753243 -0.049329163753243 22 0.7658 0.8144446427842 -0.0486446427841993 23 0.7627 0.810625097633998 -0.0479250976339982 24 0.748 0.74914511127403 -0.00114511127403003 25 0.7692 0.745399018915179 0.0238009810848211 26 0.785 0.792555710961893 -0.00755571096189284 27 0.7913 0.8142977372015 -0.0229977372014993 28 0.772 0.787781279524141 -0.0157812795241414 29 0.788 0.806511741318397 -0.0185117413183970 30 0.807 0.813563209287999 -0.00656320928799903 31 0.8268 0.798431934269895 0.0283680657301054 32 0.8244 0.805116138282747 0.0192838617172534 33 0.8487 0.790131768847342 0.0585682311526579 34 0.8572 0.789397240933842 0.067802759066158 35 0.8214 0.78440245112204 0.0369975488779596 36 0.8827 0.808421513893498 0.0742784861065025 37 0.9216 0.804455063160596 0.117144936839404 38 0.8865 0.84962852984086 0.0368714701591402 39 0.8816 0.797844311939094 0.0837556880609056 40 0.8884 0.804381610369246 0.0840183896307536 41 0.9466 0.790425580012742 0.156174419987258 42 0.918 0.805997571778947 0.112002428221053 43 0.9337 0.789323788142492 0.144376211857508 44 0.9559 0.813710114870699 0.142189885129301 45 0.9626 0.845955890273359 0.116644109726641 46 0.9434 0.831485690377404 0.111914309622596 47 0.8639 0.776983719195688 0.0869162808043117 48 0.7996 0.78139088667669 0.0182091133233104 49 0.668 0.717854222158921 -0.0498542221589208 50 0.6572 0.655052085554652 0.00214791444534780 51 0.6928 0.720131258690771 -0.0273312586907715 52 0.6438 0.798505387061245 -0.154705387061245 53 0.6454 0.807099363649197 -0.161699363649197 54 0.6873 0.767655214694235 -0.0803552146942355 55 0.7265 0.776616455238938 -0.0501164552389381 56 0.7912 0.78528388461824 0.00591611538175929 57 0.8114 0.832514029456305 -0.0211140294563047 58 0.8281 0.778379322231339 0.0497206777686613 59 0.8393 0.810918908799398 0.0283810912006017 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0165688956336528 0.0331377912673057 0.983431104366347 6 0.00309874464390805 0.0061974892878161 0.996901255356092 7 0.00065635086985981 0.00131270173971962 0.99934364913014 8 0.000356162630945574 0.000712325261891147 0.999643837369054 9 7.34169727859286e-05 0.000146833945571857 0.999926583027214 10 1.83338498926448e-05 3.66676997852896e-05 0.999981666150107 11 7.95925270696285e-06 1.59185054139257e-05 0.999992040747293 12 2.28330208265459e-06 4.56660416530917e-06 0.999997716697917 13 1.05026197637992e-06 2.10052395275984e-06 0.999998949738024 14 9.88925907002844e-07 1.97785181400569e-06 0.999999011074093 15 8.11687246403366e-07 1.62337449280673e-06 0.999999188312754 16 2.80108237438389e-07 5.60216474876778e-07 0.999999719891763 17 1.34242029875230e-07 2.68484059750461e-07 0.99999986575797 18 4.29209975251621e-07 8.58419950503242e-07 0.999999570790025 19 1.40600440065859e-07 2.81200880131718e-07 0.99999985939956 20 4.13046127298554e-08 8.26092254597109e-08 0.999999958695387 21 1.539094945175e-08 3.07818989035e-08 0.99999998460905 22 5.26633064813193e-09 1.05326612962639e-08 0.99999999473367 23 1.75817056812536e-09 3.51634113625073e-09 0.99999999824183 24 3.97810634667618e-10 7.95621269335237e-10 0.99999999960219 25 1.46526596062830e-10 2.93053192125661e-10 0.999999999853473 26 1.05172266509937e-10 2.10344533019874e-10 0.999999999894828 27 1.08782194500936e-10 2.17564389001871e-10 0.999999999891218 28 3.51792781369143e-11 7.03585562738287e-11 0.99999999996482 29 2.50627801030645e-11 5.0125560206129e-11 0.999999999974937 30 6.49205728836221e-11 1.29841145767244e-10 0.99999999993508 31 7.2603958370207e-10 1.45207916740414e-09 0.99999999927396 32 2.47479208671765e-09 4.94958417343531e-09 0.999999997525208 33 3.69445973507212e-08 7.38891947014423e-08 0.999999963055403 34 3.38940500137604e-07 6.77881000275208e-07 0.9999996610595 35 2.92558262821573e-07 5.85116525643146e-07 0.999999707441737 36 3.08296909186209e-06 6.16593818372419e-06 0.999996917030908 37 9.16470963533985e-05 0.000183294192706797 0.999908352903647 38 0.000136095482420294 0.000272190964840589 0.99986390451758 39 0.000237591574200279 0.000475183148400558 0.9997624084258 40 0.000372531738470647 0.000745063476941294 0.99962746826153 41 0.00404919306103503 0.00809838612207007 0.995950806938965 42 0.0080113831225582 0.0160227662451164 0.991988616877442 43 0.0291697467072898 0.0583394934145795 0.97083025329271 44 0.0868632361288316 0.173726472257663 0.913136763871168 45 0.156968160175232 0.313936320350464 0.843031839824768 46 0.304727794252157 0.609455588504315 0.695272205747843 47 0.436708429297389 0.873416858594777 0.563291570702611 48 0.39679566544004 0.79359133088008 0.60320433455996 49 0.323012699969495 0.646025399938991 0.676987300030504 50 0.231056235585689 0.462112471171379 0.76894376441431 51 0.154476397092660 0.308952794185321 0.84552360290734 52 0.287830419624042 0.575660839248084 0.712169580375958 53 0.723377556881684 0.553244886236632 0.276622443118316 54 0.764141590218419 0.471716819563162 0.235858409781581 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 36 0.72 NOK 5% type I error level 38 0.76 NOK 10% type I error level 39 0.78 NOK

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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