Paper - multiple regression analysis (1)

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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: Fri, 04 Dec 2009 03:59:15 -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/Dec/04/t1259924447cpp12f66q707nuz.htm/, Retrieved Fri, 04 Dec 2009 12:01:00 +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/Dec/04/t1259924447cpp12f66q707nuz.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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No (this computation is public)

User-defined keywords:

Dataseries X:

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216234 213587 209465 204045 200237 203666 241476 260307 243324 244460 233575 237217 235243 230354 227184 221678 217142 219452 256446 265845 248624 241114 229245 231805 219277 219313 212610 214771 211142 211457 240048 240636 230580 208795 197922 194596 194581 185686 178106 172608 167302 168053 202300 202388 182516 173476 166444 171297 169701 164182 161914 159612 151001 158114 186530 187069 174330 169362 166827 178037 186412 189226 191563 188906 186005 195309 223532 226899 214126

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 Werkl[t] = + 202590.4 + 984.266666666718M1[t] -2199.06666666668M2[t] -5783.40000000002M3[t] -8987.06666666668M4[t] -13785.5666666667M5[t] -9915.23333333333M6[t] + 22464.9333333333M7[t] + 27933.6M8[t] + 12992.9333333333M9[t] + 4850.99999999998M10[t] -3787.80000000001M11[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) 202590.4 12502.325321 16.2042 0 0 M1 984.266666666718 16928.22851 0.0581 0.953838 0.476919 M2 -2199.06666666668 16928.22851 -0.1299 0.897099 0.448549 M3 -5783.40000000002 16928.22851 -0.3416 0.733876 0.366938 M4 -8987.06666666668 16928.22851 -0.5309 0.597556 0.298778 M5 -13785.5666666667 16928.22851 -0.8144 0.418832 0.209416 M6 -9915.23333333333 16928.22851 -0.5857 0.560375 0.280187 M7 22464.9333333333 16928.22851 1.3271 0.189777 0.094889 M8 27933.6 16928.22851 1.6501 0.104419 0.052209 M9 12992.9333333333 16928.22851 0.7675 0.445934 0.222967 M10 4850.99999999998 17680.958031 0.2744 0.784798 0.392399 M11 -3787.80000000001 17680.958031 -0.2142 0.831132 0.415566 Multiple Linear Regression - Regression Statistics Multiple R 0.445760423272586 R-squared 0.198702354956155 Adjusted R-squared 0.0440659673161151 F-TEST (value) 1.28496505892708 F-TEST (DF numerator) 11 F-TEST (DF denominator) 57 p-value 0.256654159789030 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 27956.0492954451 Sum Squared Residuals 44547819455.9333 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 216234 203574.666666666 12659.3333333337 2 213587 200391.333333333 13195.6666666667 3 209465 196807 12658 4 204045 193603.333333333 10441.6666666667 5 200237 188804.833333333 11432.1666666667 6 203666 192675.166666667 10990.8333333333 7 241476 225055.333333333 16420.6666666666 8 260307 230524 29783.0000000001 9 243324 215583.333333333 27740.6666666667 10 244460 207441.4 37018.6 11 233575 198802.6 34772.4 12 237217 202590.4 34626.6 13 235243 203574.666666667 31668.3333333333 14 230354 200391.333333333 29962.6666666667 15 227184 196807 30377 16 221678 193603.333333333 28074.6666666667 17 217142 188804.833333333 28337.1666666667 18 219452 192675.166666667 26776.8333333333 19 256446 225055.333333333 31390.6666666667 20 265845 230524 35321 21 248624 215583.333333333 33040.6666666667 22 241114 207441.4 33672.6 23 229245 198802.6 30442.4 24 231805 202590.4 29214.6 25 219277 203574.666666667 15702.3333333333 26 219313 200391.333333333 18921.6666666667 27 212610 196807 15803 28 214771 193603.333333333 21167.6666666667 29 211142 188804.833333333 22337.1666666667 30 211457 192675.166666667 18781.8333333333 31 240048 225055.333333333 14992.6666666667 32 240636 230524 10112 33 230580 215583.333333333 14996.6666666667 34 208795 207441.4 1353.60000000001 35 197922 198802.6 -880.600000000004 36 194596 202590.4 -7994.4 37 194581 203574.666666667 -8993.66666666673 38 185686 200391.333333333 -14705.3333333333 39 178106 196807 -18701 40 172608 193603.333333333 -20995.3333333333 41 167302 188804.833333333 -21502.8333333333 42 168053 192675.166666667 -24622.1666666667 43 202300 225055.333333333 -22755.3333333333 44 202388 230524 -28136 45 182516 215583.333333333 -33067.3333333333 46 173476 207441.4 -33965.4 47 166444 198802.6 -32358.6 48 171297 202590.4 -31293.4 49 169701 203574.666666667 -33873.6666666667 50 164182 200391.333333333 -36209.3333333333 51 161914 196807 -34893 52 159612 193603.333333333 -33991.3333333333 53 151001 188804.833333333 -37803.8333333333 54 158114 192675.166666667 -34561.1666666666 55 186530 225055.333333333 -38525.3333333333 56 187069 230524 -43455 57 174330 215583.333333333 -41253.3333333333 58 169362 207441.4 -38079.4 59 166827 198802.6 -31975.6 60 178037 202590.4 -24553.4 61 186412 203574.666666667 -17162.6666666667 62 189226 200391.333333333 -11165.3333333333 63 191563 196807 -5244 64 188906 193603.333333333 -4697.33333333334 65 186005 188804.833333333 -2799.83333333334 66 195309 192675.166666667 2633.83333333335 67 223532 225055.333333333 -1523.33333333332 68 226899 230524 -3625.00000000000 69 214126 215583.333333333 -1457.33333333332 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 15 0.109019261627383 0.218038523254766 0.890980738372617 16 0.0675551523601192 0.135110304720238 0.93244484763988 17 0.0423156532245387 0.0846313064490773 0.957684346775461 18 0.0258775437638876 0.0517550875277752 0.974122456236112 19 0.0164839116154130 0.0329678232308261 0.983516088384587 20 0.00885084331598356 0.0177016866319671 0.991149156684016 21 0.00474567688538691 0.00949135377077382 0.995254323114613 22 0.00289038180724094 0.00578076361448189 0.99710961819276 23 0.00179938158782429 0.00359876317564858 0.998200618412176 24 0.00121954124003322 0.00243908248006644 0.998780458759967 25 0.000716308159936708 0.00143261631987342 0.999283691840063 26 0.000422845286078060 0.000845690572156119 0.999577154713922 27 0.000262036799188852 0.000524073598377704 0.999737963200811 28 0.000180246327134337 0.000360492654268675 0.999819753672866 29 0.000149024734891683 0.000298049469783366 0.999850975265108 30 0.000111772835262297 0.000223545670524595 0.999888227164738 31 0.000121485969035239 0.000242971938070478 0.999878514030965 32 0.000627565118245735 0.00125513023649147 0.999372434881754 33 0.00165774991414539 0.00331549982829079 0.998342250085855 34 0.0203014103510398 0.0406028207020797 0.97969858964896 35 0.0791467689200517 0.158293537840103 0.920853231079948 36 0.198636208471009 0.397272416942017 0.801363791528992 37 0.255734732508873 0.511469465017747 0.744265267491127 38 0.331177273541467 0.662354547082934 0.668822726458533 39 0.400572030343642 0.801144060687284 0.599427969656358 40 0.464359964107648 0.928719928215297 0.535640035892352 41 0.511595361845367 0.976809276309266 0.488404638154633 42 0.552054069513039 0.895891860973923 0.447945930486961 43 0.566449355970686 0.867101288058628 0.433550644029314 44 0.606247014616337 0.787505970767327 0.393752985383663 45 0.652511715792575 0.694976568414851 0.347488284207425 46 0.660026100756905 0.67994779848619 0.339973899243095 47 0.636622052958007 0.726755894083987 0.363377947041993 48 0.593592760849925 0.81281447830015 0.406407239150075 49 0.551687278401917 0.896625443196166 0.448312721598083 50 0.526164665994816 0.947670668010368 0.473835334005184 51 0.502520389984824 0.994959220030352 0.497479610015176 52 0.464163687856552 0.928327375713104 0.535836312143448 53 0.448621816158423 0.897243632316847 0.551378183841577 54 0.427480195679160 0.854960391358319 0.57251980432084 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 13 0.325 NOK 5% type I error level 16 0.4 NOK 10% type I error level 18 0.45 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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