Model 2

*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: Fri, 20 Nov 2009 17:17:37 -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/t1258762744v44lol5xfqlv4l2.htm/, Retrieved Sat, 21 Nov 2009 01:19:15 +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/t1258762744v44lol5xfqlv4l2.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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104.08 99.2 103.86 93.6 107.47 104.2 111.1 95.3 117.33 102.7 119.04 103.1 123.68 100 125.9 107.2 124.54 107 119.39 119 118.8 110.4 114.81 101.7 117.9 102.4 120.53 98.8 125.15 105.6 126.49 104.4 131.85 106.3 127.4 107.2 131.08 108.5 122.37 106.9 124.34 114.2 119.61 125.9 119.97 110.6 116.46 110.5 117.03 106.7 120.96 104.7 124.71 107.4 127.08 109.8 131.91 103.4 137.69 114.8 142.46 114.3 144.32 109.6 138.06 118.3 124.45 127.3 126.71 112.3 121.83 114.9 122.51 108.2 125.48 105.4 127.77 122.1 128.03 113.5 132.84 110 133.41 125.3 139.99 114.3 138.53 115.6 136.12 127.1 124.75 123 122.88 122.2 121.46 126.4 118.4 112.7 122.45 105.8 128.94 120.9 133.25 116.3 137.94 115.7 140.04 127.9 130.74 108.3 131.55 121.1 129.47 128.6 125.45 123.1 127.87 127.7 124.68 126.6

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 44.6238087674328 + 0.648372618794754X[t] + 2.73643325933069M1[t] + 8.11863080589267M2[t] + 5.54052302280312M3[t] + 10.6327205693652M4[t] + 15.9723299978759M5[t] + 11.9014141427661M6[t] + 18.2417059744356M7[t] + 15.2405881180513M8[t] + 8.69991469123985M9[t] -2.07156680759192M10[t] + 2.99600897634725M11[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) 44.6238087674328 11.967091 3.7289 0.000517 0.000258 X 0.648372618794754 0.101102 6.413 0 0 M1 2.73643325933069 3.507264 0.7802 0.439174 0.219587 M2 8.11863080589267 3.653685 2.222 0.031132 0.015566 M3 5.54052302280312 3.376909 1.6407 0.107534 0.053767 M4 10.6327205693652 3.452856 3.0794 0.003459 0.00173 M5 15.9723299978759 3.458734 4.618 3e-05 1.5e-05 M6 11.9014141427661 3.353047 3.5494 0.000889 0.000444 M7 18.2417059744356 3.42548 5.3253 3e-06 1e-06 M8 15.2405881180513 3.37643 4.5138 4.3e-05 2.1e-05 M9 8.69991469123985 3.366723 2.5841 0.012931 0.006466 M10 -2.07156680759192 3.440674 -0.6021 0.550013 0.275007 M11 2.99600897634725 3.353436 0.8934 0.376187 0.188094 Multiple Linear Regression - Regression Statistics Multiple R 0.837478632407121 R-squared 0.701370459738502 Adjusted R-squared 0.625124619671736 F-TEST (value) 9.19880296583182 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 9.2879609558949e-09 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.30132070747676 Sum Squared Residuals 1320.88805844553 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 104.08 111.678805811202 -7.59880581120246 2 103.86 113.430116692514 -9.57011669251427 3 107.47 117.724758668649 -10.2547586686491 4 111.1 117.046439907938 -5.94643990793789 5 117.33 127.184006715530 -9.85400671552983 6 119.04 123.372439907938 -4.33243990793787 7 123.68 127.702776621344 -4.02277662134366 8 125.9 129.369941620282 -3.4699416202816 9 124.54 122.699593669711 1.84040633028884 10 119.39 119.708583596416 -0.318583596416448 11 118.8 119.200154858721 -0.400154858720747 12 114.81 110.563304098859 4.24669590114087 13 117.9 113.753598191346 4.14640180865386 14 120.53 116.801654310247 3.72834568975299 15 125.15 118.632480334962 6.51751966503823 16 126.49 122.946630738970 3.54336926102984 17 131.85 129.518148143191 2.33185185680907 18 127.4 126.030767644996 1.36923235500362 19 131.08 133.213943881099 -2.13394388109904 20 122.37 129.175429834643 -6.80542983464318 21 124.34 127.367876525033 -3.0278765250334 22 119.61 124.182354666100 -4.57235466610025 23 119.97 119.329829382480 0.640170617520318 24 116.46 116.268983144253 0.191016855747034 25 117.03 116.541600452164 0.488399547836416 26 120.96 120.627052761136 0.332947238863934 27 124.71 119.799551048792 4.91044895120766 28 127.08 126.447842880462 0.632157119538181 29 131.91 127.637867548686 4.27213245131386 30 137.69 130.958399547837 6.73160045216349 31 142.46 136.974505070109 5.4854949298914 32 144.32 130.926035905389 13.393964094611 33 138.06 130.026204262092 8.03379573790812 34 124.45 125.090076332413 -0.6400763324129 35 126.71 120.432062834431 6.27793716556923 36 121.83 119.121822666950 2.70817733305012 37 122.51 117.514159380356 4.99584061964429 38 125.48 121.080913594292 4.39908640570762 39 127.77 129.330628545075 -1.56062854507522 40 128.03 128.846821570002 -0.816821570002408 41 132.84 131.917126832731 0.922873167268494 42 133.41 137.766312045181 -4.35631204518143 43 139.99 136.974505070109 3.01549492989139 44 138.53 134.816271618158 3.71372838184247 45 136.12 135.731883307486 0.388116692514287 46 124.75 122.302074071595 2.44792592840454 47 122.88 126.850951760499 -3.97095176049883 48 121.46 126.578107783090 -5.11810778308955 49 118.4 120.431836164932 -2.0318361649321 50 122.45 121.340262641810 1.10973735818972 51 128.94 128.552581402522 0.387418597478484 52 133.25 130.662264902628 2.58773509737228 53 137.94 135.612850759862 2.32714924013839 54 140.04 139.452080854048 0.587919145952199 55 130.74 133.08426935734 -2.34426935734009 56 131.55 138.382321021529 -6.83232102152867 57 129.47 136.704442235678 -7.23444223567785 58 125.45 122.366911333475 3.08308866652507 59 127.87 130.41700116387 -2.54700116386997 60 124.68 126.707782306848 -2.02778230684848 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.926941902908301 0.146116194183398 0.073058097091699 17 0.88706768483201 0.225864630335981 0.112932315167990 18 0.820285115720262 0.359429768559475 0.179714884279738 19 0.88741086406543 0.225178271869141 0.112589135934571 20 0.926191656197284 0.147616687605432 0.0738083438027159 21 0.977242972912232 0.0455140541755353 0.0227570270877676 22 0.983702225703043 0.0325955485939144 0.0162977742969572 23 0.97533256924043 0.0493348615191416 0.0246674307595708 24 0.980256324381556 0.0394873512368874 0.0197436756184437 25 0.967969453864816 0.0640610922703673 0.0320305461351836 26 0.949502796781631 0.100994406436738 0.050497203218369 27 0.938573426100455 0.122853147799090 0.0614265738995452 28 0.916445907100514 0.167108185798973 0.0835540928994865 29 0.921915856853262 0.156168286293476 0.0780841431467381 30 0.886176666031354 0.227646667937291 0.113823333968646 31 0.89719698997548 0.205606020049040 0.102803010024520 32 0.98034360451614 0.0393127909677223 0.0196563954838611 33 0.979664171349014 0.0406716573019721 0.0203358286509860 34 0.963332827478874 0.0733343450422513 0.0366671725211256 35 0.951236319335284 0.0975273613294315 0.0487636806647158 36 0.929058228502285 0.141883542995429 0.0709417714977147 37 0.924736307629298 0.150527384741404 0.0752636923707021 38 0.887401984353007 0.225196031293987 0.112598015646993 39 0.861532119475264 0.276935761049472 0.138467880524736 40 0.80558319263871 0.38883361472258 0.19441680736129 41 0.706496574189177 0.587006851621646 0.293503425810823 42 0.692489542451993 0.615020915096013 0.307510457548007 43 0.751785812822064 0.496428374355873 0.248214187177936 44 0.770929676560268 0.458140646879465 0.229070323439732 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 6 0.206896551724138 NOK 10% type I error level 9 0.310344827586207 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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