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 08:52:15 -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/19/t1258646152iyrmjile4cx9m5l.htm/, Retrieved Thu, 19 Nov 2009 16:56:04 +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/t1258646152iyrmjile4cx9m5l.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}, }

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Dataseries X:

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

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] = -0.729216595290699 + 0.771547876570157dnst[t] + 0.259590756245638y1[t] -0.324288162170647y2[t] -0.107507909542879y3[t] + 0.144420397190849y4[t] + 0.443964966435733y5[t] + 0.0983821269698033M1[t] + 0.165757665776796M2[t] + 0.616461506665139M3[t] + 0.527372688535687M4[t] + 0.806866310733002M5[t] + 0.530453908871542M6[t] + 0.516266033363936M7[t] + 0.463812304639643M8[t] + 0.469250583288777M9[t] + 0.639151419026464M10[t] + 0.504141234618875M11[t] -0.00444541894845899t + 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) -0.729216595290699 0.444382 -1.641 0.108645 0.054323 dnst 0.771547876570157 0.128777 5.9914 0 0 y1 0.259590756245638 0.10678 2.4311 0.019629 0.009814 y2 -0.324288162170647 0.15824 -2.0493 0.047026 0.023513 y3 -0.107507909542879 0.137156 -0.7838 0.437754 0.218877 y4 0.144420397190849 0.143499 1.0064 0.320264 0.160132 y5 0.443964966435733 0.127899 3.4712 0.001257 0.000629 M1 0.0983821269698033 0.426031 0.2309 0.818549 0.409274 M2 0.165757665776796 0.422258 0.3926 0.696736 0.348368 M3 0.616461506665139 0.429299 1.436 0.158785 0.079393 M4 0.527372688535687 0.433325 1.217 0.230722 0.115361 M5 0.806866310733002 0.421667 1.9135 0.062858 0.031429 M6 0.530453908871542 0.432543 1.2264 0.227234 0.113617 M7 0.516266033363936 0.430148 1.2002 0.237119 0.11856 M8 0.463812304639643 0.435468 1.0651 0.293223 0.146611 M9 0.469250583288777 0.436026 1.0762 0.288287 0.144144 M10 0.639151419026464 0.425767 1.5012 0.141163 0.070582 M11 0.504141234618875 0.42189 1.195 0.239141 0.11957 t -0.00444541894845899 0.005431 -0.8185 0.417908 0.208954 Multiple Linear Regression - Regression Statistics Multiple R 0.94277800551632 R-squared 0.88883036768533 Adjusted R-squared 0.838804033143728 F-TEST (value) 17.7672495062812 F-TEST (DF numerator) 18 F-TEST (DF denominator) 40 p-value 1.05693231944315e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.616571628980764 Sum Squared Residuals 15.2064229465597 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.4 1.51615084871885 -0.116150848718851 2 1 0.579779094667316 0.420220905332684 3 -0.8 -1.14708846400720 0.347088464007197 4 -2.9 -1.93794350746888 -0.962056492531122 5 -0.7 -0.902855913860021 0.202855913860021 6 -0.7 0.305394414236960 -1.00539441423696 7 1.5 1.29173110714735 0.208268892852651 8 3 3.16734693446111 -0.167346934461113 9 3.2 3.47848030148526 -0.27848030148526 10 3.1 3.65970697497630 -0.559706974976296 11 3.9 3.25866036489077 0.641339635109231 12 1 1.16471932922785 -0.164719329227850 13 1.3 0.357370619976431 0.942629380023569 14 0.8 -0.384536857095754 1.18453685709575 15 1.2 0.58486875175523 0.61513124824477 16 2.9 2.30810613342526 0.59189386657474 17 3.9 3.96471512959917 -0.0647151295991742 18 4.5 4.22274533112759 0.277254668872406 19 4.5 4.13936500364154 0.360634996358457 20 3.3 2.51506719378454 0.784932806215458 21 2 1.76272093083875 0.237279069161248 22 1.5 1.57525127711698 -0.0752512771169794 23 1 1.58846572794806 -0.588465727948059 24 2.1 2.70247193849348 -0.602471938493483 25 3 3.71316950190142 -0.71316950190142 26 4 4.80909483753591 -0.809094837535908 27 5.1 5.08050128338402 0.0194987166159812 28 4.5 4.20477911151452 0.295220888485478 29 4.2 3.60067337120741 0.599326628792594 30 3.3 2.87037946689991 0.429620533100088 31 2.7 3.1716897034693 -0.471689703469303 32 1.8 2.25741409713870 -0.457414097138697 33 1.4 1.55801442355075 -0.158014423550746 34 0.5 0.681841193208814 -0.181841193208814 35 -0.4 0.357012370938903 -0.757012370938903 36 0.8 0.515143251735245 0.284856748264755 37 0.7 1.34240890741249 -0.642408907412486 38 1.9 2.04420878225251 -0.144208782252515 39 2 2.38190921982621 -0.381909219826208 40 1.1 1.51421036970784 -0.41421036970784 41 0.9 1.59579404154382 -0.695794041543819 42 0.4 0.805886298537762 -0.405886298537762 43 0.7 0.823349514499889 -0.123349514499889 44 2.1 2.30560720091432 -0.205607200914316 45 2.8 2.92296942632349 -0.122969426323488 46 3.9 3.34740746105416 0.552592538945842 47 3.5 2.81299859267776 0.687001407322235 48 2 1.51766548054342 0.482334519456579 49 2 1.47090012199081 0.529099878009188 50 1.5 2.15145414264002 -0.651454142640015 51 2.5 3.09980920904174 -0.59980920904174 52 3.1 2.61084789282126 0.489152107178743 53 2.7 2.74167337150962 -0.041673371509623 54 2.8 2.09559448919777 0.704405510802226 55 2.5 2.47386467124192 0.0261353287580837 56 3 2.95456457370133 0.0454354262986676 57 3.2 2.87781491780176 0.322185082198245 58 2.8 2.53579309364375 0.264206906356247 59 2.4 2.38286294354450 0.017137056455496 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 22 0.599929646119163 0.800140707761673 0.400070353880837 23 0.812483496298936 0.375033007402127 0.187516503701064 24 0.832692517892448 0.334614964215103 0.167307482107552 25 0.98157864006673 0.0368427198665406 0.0184213599332703 26 0.996586219089001 0.00682756182199829 0.00341378091099914 27 0.991648134103495 0.0167037317930101 0.00835186589650504 28 0.9868762301915 0.0262475396170011 0.0131237698085005 29 0.985492770238583 0.0290144595228337 0.0145072297614169 30 0.976213140029853 0.047573719940293 0.0237868599701465 31 0.988968632529727 0.0220627349405458 0.0110313674702729 32 0.979270208184443 0.0414595836311131 0.0207297918155566 33 0.957265069358884 0.0854698612822327 0.0427349306411164 34 0.934245289439733 0.131509421120534 0.0657547105602668 35 0.887059501082386 0.225880997835229 0.112940498917614 36 0.84393590693589 0.312128186128220 0.156064093064110 37 0.773083450355551 0.453833099288899 0.226916549644449 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.0625 NOK 5% type I error level 8 0.5 NOK 10% type I error level 9 0.5625 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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