model 1

*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, 17 Nov 2009 07:36:29 -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/17/t1258468648hnp3qef5ppr2hlt.htm/, Retrieved Tue, 17 Nov 2009 15:37:43 +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/17/t1258468648hnp3qef5ppr2hlt.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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5.7 97.33 91.4 6.1 97.89 91.1 6 98.69 104.4 5.9 99.01 97.6 5.8 99.18 93.7 5.7 98.45 104.5 5.6 98.13 95.4 5.4 98.29 86.5 5.4 99.1 102.9 5.5 99.26 101.9 5.6 98.85 103.7 5.7 98.05 100.7 5.9 98.53 94.2 6.1 99.34 93.6 6 100.14 104.7 5.8 100.3 101 5.8 100.22 97.6 5.7 99.9 105.8 5.5 99.58 93.7 5.3 99.9 91.2 5.2 100.78 106.3 5.2 100.78 103.4 5 100.46 107.4 5.1 100.06 101.2 5.1 100.28 96.9 5.2 100.78 96.3 4.9 101.58 109.8 4.8 102.06 97.9 4.5 102.02 105.1 4.5 101.68 107.9 4.4 101.32 95 4.4 101.81 95.2 4.2 102.3 105.8 4.1 102.12 110.1 3.9 102.1 112.2 3.8 101.75 102.5 3.9 101.5 103.7 4.2 102.16 102 4.1 103.47 112.3 3.8 104.05 103.3 3.6 104.09 106.9 3.7 103.55 104.6 3.5 102.77 100.7 3.4 102.89 99 3.1 103.6 106.5 3.1 103.76 114.9 3.1 103.92 114.1 3.2 103.35 102.2 3.3 103.32 107 3.5 104.2 107.4 3.6 105.44 107.4 3.5 105.81 110.1 3.3 106.25 105.6 3.2 105.94 110.9 3.1 105.82 101.9 3.2 105.96 93.2 3 106.49 110.5 3 106.32 113.1 3.1 105.88 101.7 3.4 105.07 96.7

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 manwerk[t] = + 42.9053953972478 -0.371174663740103infl[t] -0.00596100651594128indprod[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) 42.9053953972478 2.145333 19.9994 0 0 infl -0.371174663740103 0.024574 -15.1046 0 0 indprod -0.00596100651594128 0.009699 -0.6146 0.541276 0.270638 Multiple Linear Regression - Regression Statistics Multiple R 0.923950647602837 R-squared 0.853684799205702 Adjusted R-squared 0.848550932511165 F-TEST (value) 166.284956349597 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.41123506047334 Sum Squared Residuals 9.63951367286317 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 5.7 6.23412937986647 -0.53412937986647 2 6.1 6.02805987012679 0.0719401298732074 3 6 5.65183875247269 0.348161247527311 4 5.9 5.57359770438425 0.326402295615746 5 5.8 5.53374593696061 0.266254063039393 6 5.7 5.74032457111872 -0.0403245711187177 7 5.6 5.91334562281062 -0.313345622810620 8 5.4 5.90701063460408 -0.507010634604076 9 5.4 5.50859865011316 -0.108598650113159 10 5.5 5.45517171043068 0.0448282895693196 11 5.6 5.59662351083543 0.00337648916456707 12 5.7 5.91144626137534 -0.211446261375338 13 5.9 5.7720289651337 0.127971034866295 14 6.1 5.47495409141379 0.625045908586214 15 6 5.11184718809475 0.888152811905245 16 5.8 5.07451496600532 0.725485033994677 17 5.8 5.12447636125873 0.675523638741269 18 5.7 5.19437200022484 0.505627999775157 19 5.5 5.38527607146457 0.114723928535432 20 5.3 5.28140269535759 0.0185973046424141 21 5.2 4.86475779287558 0.335242207124417 22 5.2 4.88204471177181 0.317955288228188 23 5 4.97697657810488 0.0230234218951166 24 5.1 5.16240468399976 -0.0624046839997579 25 5.1 5.10637858599548 -0.00637858599548303 26 5.2 4.924367858035 0.275632141965004 27 4.9 4.54695453907771 0.353045460922294 28 4.8 4.43972667802216 0.360273321977843 29 4.5 4.41165441765699 0.088345582343014 30 4.5 4.52116298508398 -0.0211629850839815 31 4.4 4.73168284808607 -0.331682848086066 32 4.4 4.54861506155022 -0.148615061550224 33 4.2 4.3035528072486 -0.103552807248597 34 4.1 4.34473191870327 -0.244731918703266 35 3.9 4.33963729829459 -0.439637298294595 36 3.8 4.52737019380826 -0.72737019380826 37 3.9 4.61301065192416 -0.713010651924156 38 4.2 4.37816908493279 -0.178169084932789 39 4.1 3.83053190831906 0.269468091680942 40 3.8 3.66889966199327 0.131100338006730 41 3.6 3.63259305198627 -0.0325930519862744 42 3.7 3.8467376853926 -0.146737685392598 43 3.5 4.15950184852205 -0.65950184852205 44 3.4 4.12509459995034 -0.725094599950336 45 3.1 3.81685303982531 -0.716853039825305 46 3.1 3.70739263889298 -0.607392638892978 47 3.1 3.65277349790732 -0.552773497907315 48 3.2 3.93527903377888 -0.735279033778878 49 3.3 3.91780144241456 -0.617801442414564 50 3.5 3.58878333571689 -0.0887833357168928 51 3.6 3.12852675267917 0.471473247320834 52 3.5 2.97509740950228 0.524902590497715 53 3.3 2.83860508677838 0.461394913221624 54 3.2 2.92207589800332 0.27792410199668 55 3.1 3.02026591629561 0.0797340837043944 56 3.2 3.02016222006068 0.17983777993932 57 3 2.72031423555264 0.279685764447359 58 3 2.76791531144701 0.232084688552988 59 3.1 2.99918763777439 0.100812362225613 60 3.4 3.32964414798358 0.0703558520164221 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.133654654729225 0.267309309458449 0.866345345270775 7 0.0979029324828419 0.195805864965684 0.902097067517158 8 0.117675259472100 0.235350518944201 0.8823247405279 9 0.147082653542542 0.294165307085084 0.852917346457458 10 0.097912471093227 0.195824942186454 0.902087528906773 11 0.0561045026838678 0.112209005367736 0.943895497316132 12 0.030162622879887 0.060325245759774 0.969837377120113 13 0.0193878970320463 0.0387757940640925 0.980612102967954 14 0.0261584552537144 0.0523169105074287 0.973841544746286 15 0.0268962203271467 0.0537924406542933 0.973103779672853 16 0.0212995745305414 0.0425991490610827 0.978700425469459 17 0.0175592379635252 0.0351184759270504 0.982440762036475 18 0.0145143555366644 0.0290287110733288 0.985485644463336 19 0.0148091564933966 0.0296183129867932 0.985190843506603 20 0.0226163761169921 0.0452327522339841 0.977383623883008 21 0.0457191101840913 0.0914382203681827 0.954280889815909 22 0.0624831999165149 0.124966399833030 0.937516800083485 23 0.103569397511921 0.207138795023843 0.896430602488079 24 0.121614930147984 0.243229860295967 0.878385069852016 25 0.130545926812266 0.261091853624533 0.869454073187734 26 0.152427924500632 0.304855849001265 0.847572075499368 27 0.232261531137293 0.464523062274585 0.767738468862707 28 0.329100884488003 0.658201768976005 0.670899115511997 29 0.446635330251446 0.893270660502892 0.553364669748554 30 0.574327428930753 0.851345142138495 0.425672571069247 31 0.6600179970986 0.6799640058028 0.3399820029014 32 0.720244471385001 0.559511057229998 0.279755528614999 33 0.780693605916276 0.438612788167449 0.219306394083724 34 0.833383783387174 0.333232433225652 0.166616216612826 35 0.868438136568065 0.26312372686387 0.131561863431935 36 0.902818202314845 0.19436359537031 0.097181797685155 37 0.917267182206942 0.165465635586116 0.0827328177930578 38 0.947619543910288 0.104760912179423 0.0523804560897117 39 0.990342097425197 0.0193158051496058 0.0096579025748029 40 0.994954705716354 0.0100905885672920 0.00504529428364601 41 0.99488399715649 0.0102320056870192 0.0051160028435096 42 0.99714706196525 0.00570587606949953 0.00285293803474976 43 0.996654209346526 0.00669158130694823 0.00334579065347411 44 0.99500733042256 0.00998533915488111 0.00499266957744056 45 0.993820153176877 0.0123596936462463 0.00617984682312316 46 0.991195772399331 0.0176084552013381 0.00880422760066904 47 0.988802631930184 0.0223947361396326 0.0111973680698163 48 0.987571672618272 0.0248566547634566 0.0124283273817283 49 0.992389897525642 0.0152202049487156 0.0076101024743578 50 0.991995108609736 0.0160097827805277 0.00800489139026383 51 0.990281155930347 0.0194376881393068 0.0097188440696534 52 0.995791508123941 0.00841698375211731 0.00420849187605866 53 0.998832492925788 0.00233501414842308 0.00116750707421154 54 0.996691481770487 0.00661703645902615 0.00330851822951308 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 6 0.122448979591837 NOK 5% type I error level 22 0.448979591836735 NOK 10% type I error level 26 0.530612244897959 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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