Multiple Regression

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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Thu, 19 Nov 2009 08:14:54 -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/19/t12586439007vrqp4s5qwj3z5l.htm/, Retrieved Thu, 19 Nov 2009 16:18:33 +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/19/t12586439007vrqp4s5qwj3z5l.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:

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User-defined keywords:

Dataseries X:

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1.4 1.9 1 1.6 -0.8 0 -2.9 -1.3 -0.7 -0.4 -0.7 -0.3 1.5 1.4 3 2.6 3.2 2.8 3.1 2.6 3.9 3.4 1 1.7 1.3 1.2 0.8 0 1.2 0 2.9 1.6 3.9 2.5 4.5 3.2 4.5 3.4 3.3 2.3 2 1.9 1.5 1.7 1 1.9 2.1 3.3 3 3.8 4 4.4 5.1 4.5 4.5 3.5 4.2 3 3.3 2.8 2.7 2.9 1.8 2.6 1.4 2.1 0.5 1.5 -0.4 1.1 0.8 1.5 0.7 1.7 1.9 2.3 2 2.3 1.1 1.9 0.9 2 0.4 1.6 0.7 1.2 2.1 1.9 2.8 2.1 3.9 2.4 3.5 2.9 2 2.5 2 2.3 1.5 2.5 2.5 2.6 3.1 2.4 2.7 2.5 2.8 2.1 2.5 2.2 3 2.7 3.2 3 2.8 3.2 2.4 3 2 2.7 1.8 2.5 1.1 1.6 -1.5 0.1 -3.7 -1.9

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 bbp[t] = -1.43203385352092 + 1.28719395449612dnst[t] + 0.257300688479588M1[t] + 0.488499680895607M2[t] + 0.810643425568731M3[t] + 0.935266767208264M4[t] + 1.16062146088837M5[t] + 1.07210921906822M6[t] + 0.954463274539534M7[t] + 0.95702448364031M8[t] + 0.888512241820156M9[t] + 0.857231637269767M10[t] + 0.345536725460465M11[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) -1.43203385352092 0.395309 -3.6226 0.000672 0.000336 dnst 1.28719395449612 0.084519 15.2296 0 0 M1 0.257300688479588 0.463534 0.5551 0.581263 0.290631 M2 0.488499680895607 0.464021 1.0528 0.297417 0.148708 M3 0.810643425568731 0.467838 1.7327 0.089185 0.044592 M4 0.935266767208264 0.476423 1.9631 0.055099 0.02755 M5 1.16062146088837 0.485353 2.3913 0.020516 0.010258 M6 1.07210921906822 0.485612 2.2077 0.031786 0.015893 M7 0.954463274539534 0.48416 1.9714 0.054118 0.027059 M8 0.95702448364031 0.484101 1.9769 0.05347 0.026735 M9 0.888512241820156 0.484065 1.8355 0.072262 0.036131 M10 0.857231637269767 0.48408 1.7708 0.082562 0.041281 M11 0.345536725460465 0.48416 0.7137 0.478677 0.239338 Multiple Linear Regression - Regression Statistics Multiple R 0.914494865846938 R-squared 0.836300859660409 Adjusted R-squared 0.797783414874622 F-TEST (value) 21.7122621791626 F-TEST (DF numerator) 12 F-TEST (DF denominator) 51 p-value 6.66133814775094e-16 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.765355857745887 Sum Squared Residuals 29.8742490382830 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.4 1.27093534850128 0.129064651498718 2 1 1.11597615456847 -0.115976154568473 3 -0.8 -0.62139042795219 -0.17860957204781 4 -2.9 -2.17011922715761 -0.729880772842387 5 -0.7 -0.786289974430998 0.0862899744309977 6 -0.7 -0.74608282080154 0.0460828208015405 7 1.5 1.32450095731318 0.175499042686818 8 3 2.8716949118093 0.128305088190698 9 3.2 3.06062146088837 0.139378539111630 10 3.1 2.77190206543876 0.328097934561242 11 3.9 3.28996231722635 0.610037682773647 12 1 0.756195869122484 0.243804130877516 13 1.3 0.369899580354012 0.930100419645988 14 0.8 -0.943534172625313 1.74353417262531 15 1.2 -0.62139042795219 1.82139042795219 16 2.9 1.56274324088113 1.33725675911887 17 3.9 2.94657249360775 0.95342750639225 18 4.5 3.75909601993488 0.740903980065122 19 4.5 3.89888886630542 0.601111133694579 20 3.3 2.48553672546047 0.814463274539534 21 2 1.90214690184186 0.0978530981581372 22 1.5 1.61342750639225 -0.113427506392250 23 1 1.35917138548217 -0.359171385482173 24 2.1 2.81570619631627 -0.715706196316275 25 3 3.71660386204392 -0.716603862043924 26 4 4.72011922715761 -0.720119227157614 27 5.1 5.17098236728035 -0.0709823672803501 28 4.5 4.00841175442376 0.491588245576238 29 4.2 3.59016947085581 0.609830529144191 30 3.3 3.24421843813643 0.0557815618635703 31 2.7 3.25529188905736 -0.555291889057361 32 1.8 2.8716949118093 -1.07169491180930 33 1.4 2.15958569274109 -0.759585692741087 34 0.5 1.35598871549303 -0.855988715493027 35 -0.4 0.329416221885277 -0.729416221885277 36 0.8 0.498757078223259 0.301242921776741 37 0.7 1.01349655760207 -0.313496557602072 38 1.9 2.01701192271576 -0.117011922715762 39 2 2.33915566738889 -0.339155667388886 40 1.1 1.94890142722997 -0.84890142722997 41 0.9 2.30297551635969 -1.40297551635969 42 0.4 1.69958569274109 -1.29958569274109 43 0.7 1.06706216641396 -0.367062166413958 44 2.1 1.97065914366202 0.129340856337983 45 2.8 2.15958569274109 0.640414307258913 46 3.9 2.51446327453953 1.38553672546047 47 3.5 2.64636533997829 0.853634660021707 48 2 1.78595103271938 0.214048967280621 49 2 1.78581293029974 0.214187069700257 50 1.5 2.27445071361499 -0.774450713614986 51 2.5 2.72531385373772 -0.225313853737722 52 3.1 2.59249840447803 0.50750159552197 53 2.7 2.94657249360775 -0.246572493607749 54 2.8 2.34318266998915 0.456817330010854 55 2.5 2.35425612091008 0.145743879089922 56 3 3.00041430725891 -0.000414307258913817 57 3.2 3.31806025178759 -0.118060251787595 58 2.8 3.54421843813643 -0.744218438136431 59 2.4 2.77508473542790 -0.375084735427905 60 2 2.04338982361860 -0.0433898236186033 61 1.8 2.04325172119897 -0.243251721198968 62 1.1 1.11597615456848 -0.0159761545684783 63 -1.5 -0.492671032502578 -1.00732896749742 64 -3.7 -2.94243559985529 -0.757564400144715 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.987471609069475 0.0250567818610493 0.0125283909305246 17 0.978342628471996 0.0433147430560087 0.0216573715280044 18 0.960368275299167 0.0792634494016656 0.0396317247008328 19 0.931430572520243 0.137138854959515 0.0685694274797574 20 0.920773573835056 0.158452852329888 0.079226426164944 21 0.872547294211325 0.25490541157735 0.127452705788675 22 0.80909422070735 0.381811558585299 0.190905779292650 23 0.749184564171713 0.501630871656574 0.250815435828287 24 0.785149647483812 0.429700705032376 0.214850352516188 25 0.854631829009194 0.290736341981612 0.145368170990806 26 0.901859014111742 0.196281971776516 0.098140985888258 27 0.865433768675897 0.269132462648207 0.134566231324103 28 0.81573394322486 0.368532113550281 0.184266056775141 29 0.811996350972434 0.376007298055132 0.188003649027566 30 0.748607318808301 0.502785362383398 0.251392681191699 31 0.72792522094445 0.544149558111101 0.272074779055550 32 0.805341941272284 0.389316117455432 0.194658058727716 33 0.80207037289327 0.395859254213461 0.197929627106731 34 0.805510238992043 0.388979522015915 0.194489761007957 35 0.785395489893449 0.429209020213101 0.214604510106551 36 0.724118971831311 0.551762056337377 0.275881028168689 37 0.647610517938013 0.704778964123974 0.352389482061987 38 0.562888029868181 0.874223940263639 0.437111970131819 39 0.490035809364935 0.98007161872987 0.509964190635065 40 0.511770330532654 0.976459338934692 0.488229669467346 41 0.5864178907571 0.827164218485799 0.413582109242899 42 0.71542533150924 0.56914933698152 0.28457466849076 43 0.623904616165484 0.752190767669033 0.376095383834516 44 0.509041138107178 0.981917723785644 0.490958861892822 45 0.455393552352725 0.91078710470545 0.544606447647275 46 0.846467298101931 0.307065403796138 0.153532701898069 47 0.918598917613298 0.162802164773405 0.0814010823867023 48 0.828472039220802 0.343055921558395 0.171527960779198 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 0.0606060606060606 NOK 10% type I error level 3 0.090909090909091 OK

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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