*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: Sun, 20 Dec 2009 03:33:23 -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/20/t1261307293t9o8fehvesuf764.htm/, Retrieved Sun, 20 Dec 2009 12:08:24 +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/20/t1261307293t9o8fehvesuf764.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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2849,27 10872 2921,44 10625 2981,85 10407 3080,58 10463 3106,22 10556 3119,31 10646 3061,26 10702 3097,31 11353 3161,69 11346 3257,16 11451 3277,01 11964 3295,32 12574 3363,99 13031 3494,17 13812 3667,03 14544 3813,06 14931 3917,96 14886 3895,51 16005 3801,06 17064 3570,12 15168 3701,61 16050 3862,27 15839 3970,1 15137 4138,52 14954 4199,75 15648 4290,89 15305 4443,91 15579 4502,64 16348 4356,98 15928 4591,27 16171 4696,96 15937 4621,4 15713 4562,84 15594 4202,52 15683 4296,49 16438 4435,23 17032 4105,18 17696 4116,68 17745 3844,49 19394 3720,98 20148 3674,4 20108 3857,62 18584 3801,06 18441 3504,37 18391 3032,6 19178 3047,03 18079 2962,34 18483 2197,82 19644 2014,45 19195 1862,83 19650 1905,41 20830 1810,99 23595 1670,07 22937 1864,44 21814 2052,02 21928 2029,6 21777 2070,83 21383 2293,41 21467 2443,27 22052 2513,17 22680

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 4956.68494757646 -0.0944174386294636X[t] -206.665378833773M1[t] -162.867354864276M2[t] -63.2297797597224M3[t] + 43.2200006714761M4[t] -17.5093311952292M5[t] + 80.4289009723294M6[t] + 113.359632514790M7[t] -36.0877919874512M8[t] -73.0366645904004M9[t] -65.9604239235216M10[t] + 20.7673994902416M11[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) 4956.68494757646 668.029121 7.4199 0 0 X -0.0944174386294636 0.031208 -3.0255 0.004017 0.002009 M1 -206.665378833773 555.538543 -0.372 0.711558 0.355779 M2 -162.867354864276 555.04636 -0.2934 0.770485 0.385243 M3 -63.2297797597224 553.027403 -0.1143 0.90946 0.45473 M4 43.2200006714761 551.771414 0.0783 0.937898 0.468949 M5 -17.5093311952292 551.917498 -0.0317 0.974826 0.487413 M6 80.4289009723294 552.176089 0.1457 0.884814 0.442407 M7 113.359632514790 551.981421 0.2054 0.838171 0.419086 M8 -36.0877919874512 552.411094 -0.0653 0.94819 0.474095 M9 -73.0366645904004 552.094385 -0.1323 0.89532 0.44766 M10 -65.9604239235216 552.374595 -0.1194 0.905458 0.452729 M11 20.7673994902416 551.981025 0.0376 0.970147 0.485074 Multiple Linear Regression - Regression Statistics Multiple R 0.408125227374325 R-squared 0.166566201219344 Adjusted R-squared -0.0462254069799719 F-TEST (value) 0.782766776513697 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0.664948367821439 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 872.31795811634 Sum Squared Residuals 35764115.1424563 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2849.27 3723.51317596317 -874.24317596317 2 2921.44 3790.63230727414 -869.192307274135 3 2981.85 3910.85288399991 -929.002883999913 4 3080.58 4012.01528786786 -931.435287867861 5 3106.22 3942.50513420862 -836.285134208616 6 3119.31 4031.94579689952 -912.635796899523 7 3061.26 4059.58915187873 -998.329151878732 8 3097.31 3848.67597482871 -751.365974828712 9 3161.69 3812.38802429617 -650.698024296168 10 3257.16 3809.55043390695 -552.390433906953 11 3277.01 3847.8421113038 -570.832111303802 12 3295.32 3769.48007424959 -474.160074249587 13 3363.99 3519.66592596215 -155.675925962150 14 3494.17 3489.72393036204 4.44606963796447 15 3667.03 3520.24794038982 146.782059610178 16 3813.06 3590.15817207142 222.901827928582 17 3917.96 3533.67762494304 384.282375056961 18 3895.51 3525.96274328423 369.547256715773 19 3801.06 3458.90540731809 342.154592681914 20 3570.12 3488.47344645731 81.646553542692 21 3701.61 3368.24839298317 333.361607016828 22 3862.27 3395.24671320087 467.023286799133 23 3970.1 3548.25557853251 421.844421467486 24 4138.52 3544.76657031146 593.753429688536 25 4199.75 3272.57548906884 927.174510931157 26 4290.89 3348.75869448825 942.131305511754 27 4443.91 3422.52589140833 1021.38410859167 28 4502.64 3456.36866153347 1046.27133846653 29 4356.98 3435.29465389114 921.685346108862 30 4591.27 3510.28944847174 1080.98055152826 31 4696.96 3565.31386065349 1131.64613934651 32 4621.4 3437.01594240425 1184.38405759575 33 4562.84 3411.30274499821 1151.53725500179 34 4202.52 3409.97583362706 792.544166372937 35 4296.49 3425.41849087558 871.071509124418 36 4435.23 3348.56713283944 1086.66286716056 37 4105.18 3079.2085747557 1025.97142524430 38 4116.68 3118.38014423236 998.299855767645 39 3844.49 3062.32336303692 782.166636963076 40 3720.98 3097.58239474151 623.397605258494 41 3674.4 3040.62976041998 633.77023958002 42 3857.62 3282.46016905884 575.159830941159 43 3801.06 3328.89259432531 472.167405674685 44 3504.37 3184.16604175455 320.203958245453 45 3032.6 3072.91064495021 -40.3106449502098 46 3047.03 3183.75165067087 -136.721650670869 47 2962.34 3232.33482887833 -269.994828878329 48 2197.82 3101.94878313928 -904.12878313928 49 2014.45 2937.67683425014 -923.226834250136 50 1862.83 2938.51492364323 -1075.68492364323 51 1905.41 2926.73992116501 -1021.32992116501 52 1810.99 2772.12548378575 -961.135483785746 53 1670.07 2773.52282653723 -1103.45282653723 54 1864.44 2977.49184228567 -1113.05184228567 55 2052.02 2999.65898582438 -947.638985824375 56 2029.6 2864.46859455518 -834.868594555183 57 2070.83 2864.72019277224 -793.890192772243 58 2293.41 2863.86536859425 -570.455368594246 59 2443.27 2895.35899040977 -452.088990409773 60 2513.17 2815.29743946023 -302.127439460229 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.00197395535181374 0.00394791070362749 0.998026044648186 17 0.000250140115061029 0.000500280230122057 0.99974985988494 18 0.00010990212647081 0.00021980425294162 0.99989009787353 19 0.000212569208304142 0.000425138416608284 0.999787430791696 20 6.70559067973575e-05 0.000134111813594715 0.999932944093203 21 2.32788329008024e-05 4.65576658016047e-05 0.999976721167099 22 4.18621813678232e-06 8.37243627356463e-06 0.999995813781863 23 3.29587635284704e-06 6.59175270569408e-06 0.999996704123647 24 3.83749912654468e-05 7.67499825308936e-05 0.999961625008735 25 0.000188318403781130 0.000376636807562260 0.999811681596219 26 0.000581769720639734 0.00116353944127947 0.99941823027936 27 0.00112871274530795 0.00225742549061591 0.998871287254692 28 0.000960930114007974 0.00192186022801595 0.999039069885992 29 0.000659062635789068 0.00131812527157814 0.99934093736421 30 0.000763753109588089 0.00152750621917618 0.999236246890412 31 0.00271336344978438 0.00542672689956877 0.997286636550216 32 0.00443226280327115 0.0088645256065423 0.995567737196729 33 0.00522649422696856 0.0104529884539371 0.994773505773031 34 0.00280449855247827 0.00560899710495655 0.997195501447522 35 0.00128394277436240 0.00256788554872481 0.998716057225638 36 0.000554601979292821 0.00110920395858564 0.999445398020707 37 0.00118899912111138 0.00237799824222276 0.998811000878889 38 0.00378876169663109 0.00757752339326218 0.99621123830337 39 0.0366270788008075 0.073254157601615 0.963372921199192 40 0.0800925253974365 0.160185050794873 0.919907474602564 41 0.181242568564346 0.362485137128693 0.818757431435654 42 0.275425075448044 0.550850150896088 0.724574924551956 43 0.357318519744257 0.714637039488515 0.642681480255743 44 0.478991250033525 0.95798250006705 0.521008749966475 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 22 0.758620689655172 NOK 5% type I error level 23 0.793103448275862 NOK 10% type I error level 24 0.827586206896552 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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