*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: Tue, 21 Dec 2010 18:40:03 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/21/t1292956680iock9y0n567cle1.htm/, Retrieved Tue, 21 Dec 2010 19:38:11 +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/2010/Dec/21/t1292956680iock9y0n567cle1.htm/}, year = {2010}, } @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 = {2010}, 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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627 0 696 0 825 0 677 0 656 0 785 0 412 0 352 0 839 0 729 0 696 0 641 0 695 0 638 0 762 0 635 0 721 0 854 0 418 0 367 0 824 0 687 0 601 0 676 0 740 0 691 0 683 0 594 0 729 0 731 0 386 0 331 0 707 0 715 0 657 0 653 0 642 0 643 0 718 0 654 0 632 0 731 0 392 1 344 1 792 1 852 1 649 1 629 1 685 1 617 1 715 1 715 1 629 1 916 1 531 1 357 1 917 1 828 1 708 1 858 1 775 1 785 1 1006 1 789 1 734 1 906 1 532 1 387 1 991 1 841 1

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 7 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 661.39268292683 + 75.0182926829268X[t] + 7.60121951219552M1[t] -8.06544715447153M2[t] + 98.4345528455284M3[t] -9.06544715447148M4[t] -2.89878048780486M5[t] + 134.101219512195M6[t] -253.735162601626M7[t] -342.568495934959M8[t] + 146.098170731707M9[t] + 76.4315040650406M10[t] -29.2M11[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) 661.39268292683 28.884941 22.8975 0 0 X 75.0182926829268 15.573716 4.817 1.1e-05 6e-06 M1 7.60121951219552 38.204132 0.199 0.842999 0.4215 M2 -8.06544715447153 38.204132 -0.2111 0.833551 0.416775 M3 98.4345528455284 38.204132 2.5765 0.012594 0.006297 M4 -9.06544715447148 38.204132 -0.2373 0.813283 0.406641 M5 -2.89878048780486 38.204132 -0.0759 0.939783 0.469892 M6 134.101219512195 38.204132 3.5101 0.000883 0.000441 M7 -253.735162601626 38.221763 -6.6385 0 0 M8 -342.568495934959 38.221763 -8.9627 0 0 M9 146.098170731707 38.221763 3.8224 0.000329 0.000165 M10 76.4315040650406 38.221763 1.9997 0.050308 0.025154 M11 -29.2 39.888177 -0.732 0.46714 0.23357 Multiple Linear Regression - Regression Statistics Multiple R 0.930216186377923 R-squared 0.865302153399486 Adjusted R-squared 0.836944712009904 F-TEST (value) 30.5141123810057 F-TEST (DF numerator) 12 F-TEST (DF denominator) 57 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 63.0687447979535 Sum Squared Residuals 226726.994512195 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 627 668.993902439022 -41.9939024390224 2 696 653.327235772358 42.6727642276422 3 825 759.827235772358 65.1727642276424 4 677 652.327235772358 24.6727642276422 5 656 658.493902439024 -2.49390243902429 6 785 795.493902439025 -10.4939024390246 7 412 407.657520325203 4.34247967479673 8 352 318.82418699187 33.1758130081301 9 839 807.490853658536 31.5091463414637 10 729 737.82418699187 -8.82418699187037 11 696 632.192682926829 63.8073170731707 12 641 661.392682926829 -20.3926829268292 13 695 668.993902439025 26.0060975609752 14 638 653.327235772358 -15.3272357723577 15 762 759.827235772358 2.17276422764226 16 635 652.327235772358 -17.3272357723577 17 721 658.493902439024 62.5060975609756 18 854 795.493902439024 58.5060975609756 19 418 407.657520325203 10.3424796747967 20 367 318.82418699187 48.1758130081300 21 824 807.490853658537 16.5091463414633 22 687 737.82418699187 -50.8241869918699 23 601 632.192682926829 -31.1926829268293 24 676 661.392682926829 14.6073170731707 25 740 668.993902439025 71.0060975609752 26 691 653.327235772358 37.6727642276423 27 683 759.827235772358 -76.8272357723577 28 594 652.327235772358 -58.3272357723577 29 729 658.493902439024 70.5060975609756 30 731 795.493902439024 -64.4939024390244 31 386 407.657520325203 -21.6575203252033 32 331 318.82418699187 12.1758130081300 33 707 807.490853658537 -100.490853658537 34 715 737.82418699187 -22.8241869918699 35 657 632.192682926829 24.8073170731707 36 653 661.392682926829 -8.39268292682928 37 642 668.993902439025 -26.9939024390248 38 643 653.327235772358 -10.3272357723577 39 718 759.827235772358 -41.8272357723578 40 654 652.327235772358 1.67276422764227 41 632 658.493902439024 -26.4939024390245 42 731 795.493902439024 -64.4939024390244 43 392 482.67581300813 -90.67581300813 44 344 393.842479674797 -49.8424796747967 45 792 882.509146341463 -90.5091463414634 46 852 812.842479674797 39.1575203252033 47 649 707.210975609756 -58.2109756097561 48 629 736.410975609756 -107.410975609756 49 685 744.012195121952 -59.0121951219516 50 617 728.345528455284 -111.345528455284 51 715 834.845528455285 -119.845528455285 52 715 727.345528455284 -12.3455284552845 53 629 733.512195121951 -104.512195121951 54 916 870.512195121951 45.4878048780488 55 531 482.67581300813 48.3241869918699 56 357 393.842479674797 -36.8424796747967 57 917 882.509146341464 34.4908536585365 58 828 812.842479674797 15.1575203252034 59 708 707.210975609756 0.789024390243946 60 858 736.410975609756 121.589024390244 61 775 744.012195121952 30.9878048780484 62 785 728.345528455284 56.6544715447155 63 1006 834.845528455285 171.154471544715 64 789 727.345528455284 61.6544715447155 65 734 733.512195121951 0.487804878048803 66 906 870.512195121951 35.4878048780488 67 532 482.67581300813 49.3241869918699 68 387 393.842479674797 -6.84247967479674 69 991 882.509146341463 108.490853658537 70 841 812.842479674797 28.1575203252033 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.283994948779162 0.567989897558324 0.716005051220838 17 0.225272970620628 0.450545941241257 0.774727029379372 18 0.189595830558681 0.379191661117362 0.810404169441319 19 0.104801763121337 0.209603526242674 0.895198236878663 20 0.0587301108500135 0.117460221700027 0.941269889149986 21 0.0299888351723092 0.0599776703446183 0.97001116482769 22 0.0184325135810061 0.0368650271620122 0.981567486418994 23 0.0261332836250375 0.052266567250075 0.973866716374962 24 0.0149217914654033 0.0298435829308066 0.985078208534597 25 0.0208588884519247 0.0417177769038494 0.979141111548075 26 0.0127749201709348 0.0255498403418695 0.987225079829065 27 0.0286287560252879 0.0572575120505758 0.971371243974712 28 0.0246289237725529 0.0492578475451058 0.975371076227447 29 0.0229282514103949 0.0458565028207898 0.977071748589605 30 0.0273216716961565 0.054643343392313 0.972678328303843 31 0.0167567237978379 0.0335134475956759 0.983243276202162 32 0.0113592598925707 0.0227185197851414 0.98864074010743 33 0.0267879254099278 0.0535758508198556 0.973212074590072 34 0.0161453400743632 0.0322906801487264 0.983854659925637 35 0.0105986107544418 0.0211972215088836 0.989401389245558 36 0.0058082255841414 0.0116164511682828 0.994191774415859 37 0.00372967643800005 0.00745935287600011 0.996270323562 38 0.00230395755264915 0.00460791510529829 0.997696042447351 39 0.00136074582422031 0.00272149164844063 0.99863925417578 40 0.000675784261014619 0.00135156852202924 0.999324215738985 41 0.000728127876403002 0.00145625575280600 0.999271872123597 42 0.000475030468647181 0.000950060937294363 0.999524969531353 43 0.000499514600915729 0.000999029201831458 0.999500485399084 44 0.000229270405653065 0.00045854081130613 0.999770729594347 45 0.000388730782884489 0.000777461565768979 0.999611269217116 46 0.000769685901222576 0.00153937180244515 0.999230314098777 47 0.000418259657201288 0.000836519314402575 0.999581740342799 48 0.00232952952782195 0.00465905905564389 0.997670470472178 49 0.00159435587518208 0.00318871175036416 0.998405644124818 50 0.00439131111480057 0.00878262222960114 0.9956086888852 51 0.488826292470925 0.97765258494185 0.511173707529075 52 0.537228702366755 0.92554259526649 0.462771297633245 53 0.817798065158885 0.364403869682230 0.182201934841115 54 0.720668173247186 0.558663653505629 0.279331826752814 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 14 0.358974358974359 NOK 5% type I error level 25 0.641025641025641 NOK 10% type I error level 30 0.769230769230769 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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