WS 7: Multiple Regression 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: Fri, 20 Nov 2009 03:25:15 -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/20/t1258712790q29diy3477v9szt.htm/, Retrieved Fri, 20 Nov 2009 11:26:43 +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/20/t1258712790q29diy3477v9szt.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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79.8 109.87 83.4 95.74 113.6 123.06 112.9 123.39 104 120.28 109.9 115.33 99 110.4 106.3 114.49 128.9 132.03 111.1 123.16 102.9 118.82 130 128.32 87 112.24 87.5 104.53 117.6 132.57 103.4 122.52 110.8 131.8 112.6 124.55 102.5 120.96 112.4 122.6 135.6 145.52 105.1 118.57 127.7 134.25 137 136.7 91 121.37 90.5 111.63 122.4 134.42 123.3 137.65 124.3 137.86 120 119.77 118.1 130.69 119 128.28 142.7 147.45 123.6 128.42 129.6 136.9 151.6 143.95 110.4 135.64 99.2 122.48 130.5 136.83 136.2 153.04 129.7 142.71 128 123.46 121.6 144.37 135.8 146.15 143.8 147.61 147.5 158.51 136.2 147.4 156.6 165.05 123.3 154.64 104.5 126.2 139.8 157.36 136.5 154.15 112.1 123.21 118.5 113.07 94.4 110.45 102.3 113.57 111.4 122.44 99.2 114.93 87.8 111.85 115.8 126.04

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 Investgoed[t] = -19.5460387116700 + 1.05414769523628Uitvoer[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) -19.5460387116700 9.601756 -2.0357 0.04636 0.02318 Uitvoer 1.05414769523628 0.073841 14.2759 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.882302368236196 R-squared 0.7784574689952 Adjusted R-squared 0.774637770184773 F-TEST (value) 203.800746506519 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 8.46545711885406 Sum Squared Residuals 4156.5099254071 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 79.8 96.2731685639392 -16.4731685639392 2 83.4 81.3780616302512 2.02193836974879 3 113.6 110.177376664106 3.42262333589361 4 112.9 110.525245403534 2.37475459646565 5 104 107.246846071350 -3.24684607134953 6 109.9 102.02881497993 7.87118502007006 7 99 96.831866842415 2.16813315758491 8 106.3 101.143330915931 5.15666908406853 9 128.9 119.633081490376 9.2669185096242 10 111.1 110.28279143363 0.81720856636999 11 102.9 105.707790436305 -2.80779043630455 12 130 115.722193541049 14.2778064589508 13 87 98.7714986016498 -11.7714986016498 14 87.5 90.6440198713781 -3.14401987137813 15 117.6 120.202321245803 -2.60232124580339 16 103.4 109.608136908679 -6.20813690867878 17 110.8 119.390627520471 -8.59062752047147 18 112.6 111.748056730008 0.851943269991563 19 102.5 107.963666504110 -5.46366650411019 20 112.4 109.692468724298 2.70753127570232 21 135.6 133.853533899113 1.74646610088678 22 105.1 105.444253512495 -0.344253512495486 23 127.7 121.973289373800 5.72671062619966 24 137 124.555951227129 12.4440487728708 25 91 108.395867059157 -17.3958670591571 26 90.5 98.1284685075557 -7.62846850755571 27 122.4 122.152494481990 0.247505518009512 28 123.3 125.557391537604 -2.2573915376037 29 124.3 125.778762553603 -1.47876255360332 30 120 106.709230746779 13.2907692532210 31 118.1 118.220523578759 -0.12052357875919 32 119 115.680027633240 3.31997236676024 33 142.7 135.888038950919 6.81196104908078 34 123.6 115.827608310573 7.77239168942717 35 129.6 124.766780766176 4.83321923382351 36 151.6 132.198522017592 19.4014779824078 37 110.4 123.438554670179 -13.0385546701787 38 99.2 109.565971000869 -10.3659710008693 39 130.5 124.69299042751 5.80700957249005 40 136.2 141.78072456729 -5.58072456729002 41 129.7 130.891378875499 -1.19137887549928 42 128 110.599035742201 17.4009642577991 43 121.6 132.641264049591 -11.0412640495915 44 135.8 134.517646947112 1.28235305288795 45 143.8 136.056702582157 7.74329741784297 46 147.5 147.546912460232 -0.04691246023245 47 136.2 135.835331566157 0.364668433842576 48 156.6 154.441038387078 2.15896161292226 49 123.3 143.467360879668 -20.1673608796681 50 104.5 113.487400427148 -8.9874004271483 51 139.8 146.334642610711 -6.53464261071074 52 136.5 142.950828509002 -6.45082850900229 53 112.1 110.335498818392 1.76450118160818 54 118.5 99.646441188696 18.8535588113041 55 94.4 96.8845742271769 -2.4845742271769 56 102.3 100.173515036314 2.12648496368591 57 111.4 109.523805093060 1.87619490694012 58 99.2 101.607155901835 -2.40715590183544 59 87.8 98.3603810005077 -10.5603810005077 60 115.8 113.318736795910 2.4812632040895 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.689885278567949 0.620229442864101 0.310114721432051 6 0.681280591178815 0.637438817642369 0.318719408821185 7 0.557853505457964 0.884292989084071 0.442146494542036 8 0.46768592606841 0.93537185213682 0.53231407393159 9 0.399840024239486 0.799680048478972 0.600159975760514 10 0.296422276171837 0.592844552343674 0.703577723828163 11 0.229082988929358 0.458165977858715 0.770917011070642 12 0.290025986739472 0.580051973478944 0.709974013260528 13 0.375954310006026 0.751908620012052 0.624045689993974 14 0.290329644564593 0.580659289129185 0.709670355435407 15 0.279025033639344 0.558050067278689 0.720974966360656 16 0.266623588752707 0.533247177505414 0.733376411247293 17 0.317468876119549 0.634937752239098 0.682531123880451 18 0.243615231711748 0.487230463423495 0.756384768288252 19 0.207272221034355 0.414544442068711 0.792727778965645 20 0.156511783428864 0.313023566857727 0.843488216571136 21 0.113187967986784 0.226375935973569 0.886812032013216 22 0.0782749386674313 0.156549877334863 0.921725061332569 23 0.0591284458295114 0.118256891659023 0.940871554170489 24 0.0762141566088126 0.152428313217625 0.923785843391187 25 0.238749423316274 0.477498846632548 0.761250576683726 26 0.225107098918204 0.450214197836408 0.774892901081796 27 0.173189614723744 0.346379229447487 0.826810385276256 28 0.139132437392661 0.278264874785321 0.86086756260734 29 0.105838802953897 0.211677605907793 0.894161197046103 30 0.167833692206277 0.335667384412554 0.832166307793723 31 0.124697767284261 0.249395534568522 0.87530223271574 32 0.0923048414151722 0.184609682830344 0.907695158584828 33 0.0753405815146284 0.150681163029257 0.924659418485372 34 0.0672075458736362 0.134415091747272 0.932792454126364 35 0.0496798373860346 0.0993596747720693 0.950320162613965 36 0.174703620274923 0.349407240549846 0.825296379725077 37 0.283145157395556 0.566290314791113 0.716854842604444 38 0.330913694433938 0.661827388867876 0.669086305566062 39 0.290799284432918 0.581598568865836 0.709200715567082 40 0.274429910639939 0.548859821279878 0.725570089360061 41 0.216468784821717 0.432937569643434 0.783531215178283 42 0.441744747567693 0.883489495135385 0.558255252432307 43 0.485429092088717 0.970858184177434 0.514570907911283 44 0.410541332288996 0.821082664577993 0.589458667711004 45 0.436439921266179 0.872879842532358 0.563560078733821 46 0.381297853438221 0.762595706876441 0.61870214656178 47 0.319345881355341 0.638691762710682 0.68065411864466 48 0.387928716638451 0.775857433276902 0.612071283361549 49 0.56987634063798 0.86024731872404 0.43012365936202 50 0.565849172425393 0.868301655149213 0.434150827574607 51 0.462293633261849 0.924587266523699 0.537706366738151 52 0.387737733149169 0.775475466298338 0.612262266850831 53 0.271544410271027 0.543088820542053 0.728455589728973 54 0.910201444906525 0.179597110186950 0.0897985550934752 55 0.820390270275883 0.359219459448234 0.179609729724117 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 0 0 OK 10% type I error level 1 0.0196078431372549 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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