WS 7 Multiple Regression (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: Sat, 21 Nov 2009 00:09:30 -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/21/t1258787455l3834w12dqw09no.htm/, Retrieved Sat, 21 Nov 2009 08:11:07 +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/21/t1258787455l3834w12dqw09no.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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83.4 108.8 113.6 128.4 112.9 121.1 104 119.5 109.9 128.7 99 108.7 106.3 105.5 128.9 119.8 111.1 111.3 102.9 110.6 130 120.1 87 97.5 87.5 107.7 117.6 127.3 103.4 117.2 110.8 119.8 112.6 116.2 102.5 111 112.4 112.4 135.6 130.6 105.1 109.1 127.7 118.8 137 123.9 91 101.6 90.5 112.8 122.4 128 123.3 129.6 124.3 125.8 120 119.5 118.1 115.7 119 113.6 142.7 129.7 123.6 112 129.6 116.8 151.6 127 110.4 112.1 99.2 114.2 130.5 121.1 136.2 131.6 129.7 125 128 120.4 121.6 117.7 135.8 117.5 143.8 120.6 147.5 127.5 136.2 112.3 156.6 124.5 123.3 115.2 104.5 104.7 139.8 130.9 136.5 129.2 112.1 113.5 118.5 125.6 94.4 107.6 102.3 107 111.4 121.6 99.2 110.7 87.8 106.3 115.8 118.6 79.7 104.6

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 8 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation inv[t] = -68.9708525393686 + 1.58049859530267cons[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) -68.9708525393686 22.08574 -3.1229 0.002793 0.001397 cons 1.58049859530267 0.1876 8.4248 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.741829610929678 R-squared 0.550311171652077 Adjusted R-squared 0.542557915990906 F-TEST (value) 70.978076269054 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 1.19335652470909e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 12.0632206961269 Sum Squared Residuals 8440.23502668097 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 83.4 102.987394629562 -19.5873946295617 2 113.6 133.965167097494 -20.3651670974937 3 112.9 122.427527351784 -9.52752735178425 4 104 119.8987295993 -15.8987295993 5 109.9 134.439316676085 -24.5393166760845 6 99 102.829344770031 -3.82934477003122 7 106.3 97.7717492650627 8.5282507349373 8 128.9 120.372879177891 8.52712082210921 9 111.1 106.938641117818 4.16135888218185 10 102.9 105.832292101106 -2.93229210110627 11 130 120.847028756482 9.1529712435184 12 87 85.1277605026414 1.87223949735863 13 87.5 101.248846174729 -13.7488461747286 14 117.6 132.226618642661 -14.6266186426608 15 103.4 116.263582830104 -12.8635828301039 16 110.8 120.372879177891 -9.5728791778908 17 112.6 114.683084234801 -2.08308423480122 18 102.5 106.464491539227 -3.96449153922735 19 112.4 108.677189572651 3.72281042734892 20 135.6 137.442264007160 -1.84226400715958 21 105.1 103.461544208152 1.63845579184772 22 127.7 118.792380582588 8.90761941741187 23 137 126.852923418632 10.1470765813683 24 91 91.6078047433823 -0.607804743382288 25 90.5 109.309389010772 -18.8093890107721 26 122.4 133.332967659373 -10.9329676593727 27 123.3 135.861765411857 -12.5617654118569 28 124.3 129.855870749707 -5.55587074970679 29 120 119.8987295993 0.101270400699997 30 118.1 113.892834937150 4.20716506285011 31 119 110.573787887014 8.42621211298573 32 142.7 136.019815271387 6.68018472861282 33 123.6 108.04499013453 15.5550098654700 34 129.6 115.631383391983 13.9686166080172 35 151.6 131.75246906407 19.84753093593 36 110.4 108.203039994060 2.19696000593973 37 99.2 111.522087044196 -12.3220870441959 38 130.5 122.427527351784 8.07247264821574 39 136.2 139.022762602462 -2.82276260246225 40 129.7 128.591471873465 1.10852812653533 41 128 121.321178335072 6.67882166492759 42 121.6 117.053832127755 4.54616787224478 43 135.8 116.737732408695 19.0622675913053 44 143.8 121.637278054133 22.1627219458671 45 147.5 132.542718361721 14.9572816382787 46 136.2 108.519139713121 27.6808602868792 47 156.6 127.801222575813 28.7987774241867 48 123.3 113.102585639499 10.1974143605014 49 104.5 96.5073503888206 7.99264961117944 50 139.8 137.916413585750 1.88358641424962 51 136.5 135.229565973736 1.27043402626416 52 112.1 110.415738027484 1.68426197251598 53 118.5 129.539771030646 -11.0397710306463 54 94.4 101.090796315198 -6.69079631519827 55 102.3 100.142497158017 2.15750284198331 56 111.4 123.217776649436 -11.8177766494356 57 99.2 105.990341960637 -6.79034196063655 58 87.8 99.0361481413048 -11.2361481413048 59 115.8 118.476280863528 -2.6762808635276 60 79.7 96.3493005292903 -16.6493005292903 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.188313690313087 0.376627380626174 0.811686309686913 6 0.184392448024232 0.368784896048463 0.815607551975768 7 0.298298918311352 0.596597836622704 0.701701081688648 8 0.61703498135506 0.76593003728988 0.38296501864494 9 0.549007327853092 0.901985344293817 0.450992672146908 10 0.434505080052105 0.869010160104209 0.565494919947895 11 0.577013838606838 0.845972322786324 0.422986161393162 12 0.490087407882785 0.980174815765571 0.509912592117215 13 0.504575941228833 0.990848117542334 0.495424058771167 14 0.452594770956136 0.905189541912271 0.547405229043864 15 0.407792054075276 0.815584108150551 0.592207945924724 16 0.343572908754650 0.6871458175093 0.65642709124535 17 0.281402417394331 0.562804834788662 0.718597582605669 18 0.215985838509176 0.431971677018353 0.784014161490824 19 0.184934541811452 0.369869083622904 0.815065458188548 20 0.187396841429861 0.374793682859723 0.812603158570139 21 0.141578576824194 0.283157153648388 0.858421423175806 22 0.170583369828039 0.341166739656077 0.829416630171961 23 0.221624371582684 0.443248743165368 0.778375628417316 24 0.16731638327254 0.33463276654508 0.83268361672746 25 0.243883720113915 0.48776744022783 0.756116279886085 26 0.225894586381009 0.451789172762018 0.774105413618991 27 0.233136957214068 0.466273914428136 0.766863042785932 28 0.205886010590490 0.411772021180981 0.79411398940951 29 0.168719876383575 0.337439752767149 0.831280123616425 30 0.140855832363932 0.281711664727865 0.859144167636068 31 0.131949352348175 0.263898704696350 0.868050647651825 32 0.133762760075017 0.267525520150034 0.866237239924983 33 0.1784485415122 0.3568970830244 0.8215514584878 34 0.206690012199683 0.413380024399367 0.793309987800317 35 0.319560563285587 0.639121126571173 0.680439436714413 36 0.255730278997320 0.511460557994639 0.74426972100268 37 0.268112306419876 0.536224612839751 0.731887693580124 38 0.231216878011835 0.46243375602367 0.768783121988165 39 0.202912479944124 0.405824959888247 0.797087520055876 40 0.161025521708948 0.322051043417895 0.838974478291052 41 0.125789253198104 0.251578506396208 0.874210746801896 42 0.0917574670176308 0.183514934035262 0.90824253298237 43 0.128731793872939 0.257463587745877 0.871268206127061 44 0.211210727695389 0.422421455390778 0.788789272304611 45 0.207573117388364 0.415146234776728 0.792426882611636 46 0.546433774184764 0.907132451630473 0.453566225815236 47 0.928886240398431 0.142227519203138 0.071113759601569 48 0.952993142161838 0.0940137156763237 0.0470068578381618 49 0.9794010010082 0.0411979979835982 0.0205989989917991 50 0.963952278568594 0.0720954428628114 0.0360477214314057 51 0.948712436050982 0.102575127898035 0.0512875639490177 52 0.949352952866937 0.101294094266126 0.0506470471330632 53 0.906572350026486 0.186855299947027 0.0934276499735137 54 0.819259815123841 0.361480369752317 0.180740184876159 55 0.90458100982527 0.190837980349461 0.0954189901747304 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 3 0.0588235294117647 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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