WS 7:Multiple Regression

*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, 26 Nov 2009 10:35:05 -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/26/t1259257056g4o2hani37cuo34.htm/, Retrieved Thu, 26 Nov 2009 18:37:48 +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/26/t1259257056g4o2hani37cuo34.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:

Linear Trend

Dataseries X:

» Textbox « » Textfile « » CSV «

8.1 1.3 7.7 1.3 7.5 1.2 7.6 1.1 7.8 1.4 7.8 1.2 7.8 1.5 7.5 1.1 7.5 1.3 7.1 1.5 7.5 1.1 7.5 1.4 7.6 1.3 7.7 1.5 7.7 1.6 7.9 1.7 8.1 1.1 8.2 1.6 8.2 1.3 8.2 1.7 7.9 1.6 7.3 1.7 6.9 1.9 6.6 1.8 6.7 1.9 6.9 1.6 7.0 1.5 7.1 1.6 7.2 1.6 7.1 1.7 6.9 2.0 7.0 2.0 6.8 1.9 6.4 1.7 6.7 1.8 6.6 1.9 6.4 1.7 6.3 2.0 6.2 2.1 6.5 2.4 6.8 2.5 6.8 2.5 6.4 2.6 6.1 2.2 5.8 2.5 6.1 2.8 7.2 2.8 7.3 2.9 6.9 3.0 6.1 3.1 5.8 2.9 6.2 2.7 7.1 2.2 7.7 2.5 7.9 2.3 7.7 2.6 7.4 2.3 7.5 2.2 8.0 1.8 8.1 1.8

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] = + 8.6179974317817 -0.784077046548957X[t] -0.131680096308189M1[t] -0.288490850722311M2[t] -0.423709309791332M3[t] -0.176201605136436M4[t] + 0.0501722311396466M5[t] + 0.276087640449438M6[t] + 0.223595345104334M7[t] + 0.0640584269662919M8[t] -0.159796950240770M9[t] -0.316607704654896M10[t] -0.0188707865168539M11[t] + 0.00385537720706263t + 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) 8.6179974317817 0.389086 22.1494 0 0 X -0.784077046548957 0.249202 -3.1463 0.002898 0.001449 M1 -0.131680096308189 0.364768 -0.361 0.719754 0.359877 M2 -0.288490850722311 0.365305 -0.7897 0.43374 0.21687 M3 -0.423709309791332 0.362866 -1.1677 0.248955 0.124477 M4 -0.176201605136436 0.362764 -0.4857 0.629472 0.314736 M5 0.0501722311396466 0.360405 0.1392 0.889891 0.444946 M6 0.276087640449438 0.360898 0.765 0.448177 0.224089 M7 0.223595345104334 0.360859 0.6196 0.538565 0.269283 M8 0.0640584269662919 0.359909 0.178 0.859516 0.429758 M9 -0.159796950240770 0.359486 -0.4445 0.658754 0.329377 M10 -0.316607704654896 0.359682 -0.8802 0.383303 0.191651 M11 -0.0188707865168539 0.35939 -0.0525 0.958351 0.479176 t 0.00385537720706263 0.007897 0.4882 0.627716 0.313858 Multiple Linear Regression - Regression Statistics Multiple R 0.650277121162806 R-squared 0.422860334307787 Adjusted R-squared 0.259755646177378 F-TEST (value) 2.59257007971282 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0.00877358101141823 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.567813318395598 Sum Squared Residuals 14.8309503691814 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.1 7.47087255216695 0.629127447833052 2 7.7 7.31791717495987 0.382082825040129 3 7.5 7.26496179775281 0.235038202247192 4 7.6 7.59473258426966 0.00526741573033712 5 7.8 7.58973868378812 0.210261316211878 6 7.8 7.97632487961477 -0.176324879614767 7 7.8 7.69246484751204 0.107535152487962 8 7.5 7.85041412520064 -0.350414125200642 9 7.5 7.47359871589085 0.0264012841091502 10 7.1 7.163827929374 -0.0638279293739963 11 7.5 7.77905104333868 -0.279051043338684 12 7.5 7.56655409309791 -0.0665540930979128 13 7.6 7.51713707865168 0.0828629213483181 14 7.7 7.20736629213483 0.492633707865169 15 7.7 6.99759550561798 0.702404494382023 16 7.9 7.17055088282504 0.729449117174961 17 8.1 7.87122632423756 0.22877367576244 18 8.2 7.70895858747993 0.491041412520064 19 8.2 7.89554478330658 0.304455216693418 20 8.2 7.42623242375602 0.773767576243981 21 7.9 7.28464012841091 0.615359871589086 22 7.3 7.05327704654896 0.246722953451044 23 6.9 7.19805393258427 -0.298053932584269 24 6.6 7.29918780096308 -0.699187800963082 25 6.7 7.09295537720706 -0.392955377207059 26 6.9 7.17522311396469 -0.275223113964687 27 7 7.12226773675762 -0.122267736757625 28 7.1 7.29522311396469 -0.195223113964687 29 7.2 7.52545232744783 -0.325452327447833 30 7.1 7.67681540930979 -0.576815409309792 31 6.9 7.39295537720706 -0.492955377207062 32 7 7.23727383627608 -0.237273836276083 33 6.8 7.09568154093098 -0.295681540930979 34 6.4 7.0995415730337 -0.699541573033708 35 6.7 7.32272616372392 -0.622726163723916 36 6.6 7.26704462279294 -0.667044622792938 37 6.4 7.2960353130016 -0.896035313001602 38 6.3 6.90785682182986 -0.607856821829856 39 6.2 6.698086035313 -0.498086035313002 40 6.5 6.71422600321027 -0.214226003210273 41 6.8 6.86604751203852 -0.0660475120385234 42 6.8 7.09581829855538 -0.295818298555377 43 6.4 6.96877367576244 -0.56877367576244 44 6.1 7.12672295345104 -1.02672295345104 45 5.8 6.67149983948636 -0.871499839486357 46 6.1 6.2833213483146 -0.183321348314607 47 7.2 6.58491364365971 0.615086356340289 48 7.3 6.52923210272873 0.770767897271267 49 6.9 6.32299967897271 0.57700032102729 50 6.1 6.09163659711075 0.00836340288924503 51 5.8 6.11708892455859 -0.317088924558588 52 6.2 6.52526741573034 -0.325267415730337 53 7.1 7.14753515248796 -0.0475351524879622 54 7.7 7.14208282504013 0.557917174959871 55 7.9 7.25026131621188 0.649738683788121 56 7.7 6.85935666131621 0.840643338683788 57 7.4 6.8745797752809 0.525420224719100 58 7.5 6.80003210272873 0.699967897271267 59 8 7.41525521669342 0.58474478330658 60 8.1 7.43798138041734 0.662018619582663 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.100879655468819 0.201759310937637 0.899120344531181 18 0.0495432551261383 0.0990865102522766 0.950456744873862 19 0.0305354518330083 0.0610709036660167 0.969464548166992 20 0.0308579667733648 0.0617159335467296 0.969142033226635 21 0.0241002327187309 0.0482004654374618 0.975899767281269 22 0.0150526659620523 0.0301053319241047 0.984947334037948 23 0.0552117741279051 0.110423548255810 0.944788225872095 24 0.132976293495268 0.265952586990536 0.867023706504732 25 0.293839719833587 0.587679439667174 0.706160280166413 26 0.323498417700895 0.646996835401789 0.676501582299105 27 0.351346968588563 0.702693937177126 0.648653031411437 28 0.371630920516040 0.743261841032079 0.62836907948396 29 0.352037740300088 0.704075480600175 0.647962259699912 30 0.308245660700952 0.616491321401905 0.691754339299048 31 0.292534597222568 0.585069194445135 0.707465402777432 32 0.306519034381953 0.613038068763907 0.693480965618047 33 0.396021320454333 0.792042640908666 0.603978679545667 34 0.333822386026490 0.667644772052981 0.66617761397351 35 0.244090549992733 0.488181099985465 0.755909450007267 36 0.174289210024020 0.348578420048039 0.82571078997598 37 0.141085635300336 0.282171270600672 0.858914364699664 38 0.102570291113581 0.205140582227162 0.897429708886419 39 0.145399497863562 0.290798995727124 0.854600502136438 40 0.560507913361408 0.878984173277184 0.439492086638592 41 0.854487258376502 0.291025483246996 0.145512741623498 42 0.95840180725519 0.0831963854896197 0.0415981927448099 43 0.902002258140743 0.195995483718514 0.0979977418592572 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 2 0.0740740740740741 NOK 10% type I error level 6 0.222222222222222 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = 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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