paper 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, 18 Dec 2008 11:59:33 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/18/t12296268294eug8zr6hc7epab.htm/, Retrieved Thu, 18 Dec 2008 20:00:38 +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/2008/Dec/18/t12296268294eug8zr6hc7epab.htm/}, year = {2008}, } @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 = {2008}, 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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111078 0 150739 0 159129 0 157928 0 147768 0 137507 0 136919 0 136151 0 133001 0 125554 0 119647 0 114158 0 116193 0 152803 0 161761 0 160942 0 149470 0 139208 0 134588 0 130322 0 126611 0 122401 0 117352 0 112135 0 112879 0 148729 0 157230 0 157221 0 146681 0 136524 0 132111 1 125326 1 122716 1 116615 1 113719 1 110737 1 112093 1 143565 1 149946 1 149147 1 134339 1 122683 1 115614 1 116566 1 111272 1 104609 1 101802 1 94542 1 93051 1 124129 1 130374 1 123946 1 114971 1 105531 1 104919 1 104782 1 101281 1 94545 1 93248 1 84031 1 87486 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Yt[t] = + 124865.4 -577.333333333319D[t] -686.999999999996M1[t] + 34812.9333333333M2[t] + 43102.3333333334M3[t] + 41845.5333333334M4[t] + 31248.9333333333M5[t] + 21488.1333333333M6[t] + 18737.6M7[t] + 17131.2M8[t] + 14072.4M9[t] + 8435.4M10[t] + 5438.59999999999M11[t] -594.4t + 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) 124865.4 3137.468953 39.7981 0 0 D -577.333333333319 3028.94565 -0.1906 0.849656 0.424828 M1 -686.999999999996 3521.1267 -0.1951 0.846149 0.423075 M2 34812.9333333333 3697.623706 9.4149 0 0 M3 43102.3333333334 3690.90785 11.678 0 0 M4 41845.5333333334 3686.173036 11.352 0 0 M5 31248.9333333333 3683.426904 8.4837 0 0 M6 21488.1333333333 3682.673903 5.8349 0 0 M7 18737.6 3695.129511 5.0709 7e-06 3e-06 M8 17131.2 3686.173036 4.6474 2.7e-05 1.4e-05 M9 14072.4 3679.191815 3.8249 0.000385 0.000192 M10 8435.4 3674.197107 2.2958 0.026191 0.013096 M11 5438.59999999999 3671.19702 1.4814 0.145167 0.072584 t -594.4 85.70651 -6.9353 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.967436908828479 R-squared 0.935934172563603 Adjusted R-squared 0.918213837315237 F-TEST (value) 52.8169563072993 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5803.09011593107 Sum Squared Residuals 1582765179.99999 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 111078 123584.000000000 -12505.9999999998 2 150739 158489.533333333 -7750.53333333337 3 159129 166184.533333333 -7055.53333333327 4 157928 164333.333333333 -6405.33333333330 5 147768 153142.333333333 -5374.3333333334 6 137507 142787.133333333 -5280.13333333338 7 136919 139442.2 -2523.20000000002 8 136151 137241.4 -1090.39999999999 9 133001 133588.2 -587.200000000042 10 125554 127356.8 -1802.80 11 119647 123765.6 -4118.60000000001 12 114158 117732.6 -3574.60000000002 13 116193 116451.2 -258.20000000005 14 152803 151356.733333333 1446.26666666668 15 161761 159051.733333333 2709.26666666665 16 160942 157200.533333333 3741.46666666664 17 149470 146009.533333333 3460.46666666668 18 139208 135654.333333333 3553.66666666667 19 134588 132309.4 2278.6 20 130322 130108.6 213.399999999997 21 126611 126455.4 155.600000000010 22 122401 120224 2177.00000000000 23 117352 116632.8 719.200000000004 24 112135 110599.8 1535.19999999999 25 112879 109318.4 3560.59999999995 26 148729 144223.933333333 4505.06666666667 27 157230 151918.933333333 5311.06666666666 28 157221 150067.733333333 7153.26666666665 29 146681 138876.733333333 7804.26666666669 30 136524 128521.533333333 8002.46666666669 31 132111 124599.266666667 7511.73333333333 32 125326 122398.466666667 2927.53333333333 33 122716 118745.266666667 3970.73333333334 34 116615 112513.866666667 4101.13333333333 35 113719 108922.666666667 4796.33333333333 36 110737 102889.666666667 7847.33333333332 37 112093 101608.266666667 10484.7333333333 38 143565 136513.8 7051.20000000001 39 149946 144208.8 5737.19999999998 40 149147 142357.6 6789.4 41 134339 131166.6 3172.40000000001 42 122683 120811.4 1871.60000000001 43 115614 117466.466666667 -1852.46666666666 44 116566 115265.666666667 1300.33333333333 45 111272 111612.466666667 -340.466666666654 46 104609 105381.066666667 -772.066666666665 47 101802 101789.866666667 12.1333333333389 48 94542 95756.8666666667 -1214.86666666667 49 93051 94475.4666666667 -1424.46666666671 50 124129 129381 -5251.99999999998 51 130374 137076 -6702.00000000001 52 123946 135224.8 -11278.8 53 114971 124033.8 -9062.79999999998 54 105531 113678.6 -8147.59999999998 55 104919 110333.666666667 -5414.66666666665 56 104782 108132.866666667 -3350.86666666666 57 101281 104479.666666667 -3198.66666666665 58 94545 98248.2666666667 -3703.26666666666 59 93248 94657.0666666667 -1409.06666666665 60 84031 88624.0666666667 -4593.06666666666 61 87486 87342.6666666667 143.333333333301 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0255987668181911 0.0511975336363823 0.974401233181809 18 0.00757024906610458 0.0151404981322092 0.992429750933895 19 0.0304811530082522 0.0609623060165044 0.969518846991748 20 0.133618599648734 0.267237199297468 0.866381400351266 21 0.222532222434466 0.445064444868932 0.777467777565534 22 0.174248973883942 0.348497947767883 0.825751026116058 23 0.149942254257784 0.299884508515568 0.850057745742216 24 0.132368808830683 0.264737617661366 0.867631191169317 25 0.153141363207055 0.30628272641411 0.846858636792945 26 0.132400108184611 0.264800216369222 0.86759989181539 27 0.105797547824318 0.211595095648637 0.894202452175682 28 0.067040862758514 0.134081725517028 0.932959137241486 29 0.0393809455143428 0.0787618910286856 0.960619054485657 30 0.0218412780674768 0.0436825561349536 0.978158721932523 31 0.0125041994737718 0.0250083989475435 0.987495800526228 32 0.0126723559871408 0.0253447119742817 0.98732764401286 33 0.00805603322834052 0.0161120664566810 0.99194396677166 34 0.00489005036802115 0.0097801007360423 0.995109949631979 35 0.00384595013723635 0.0076919002744727 0.996154049862764 36 0.00290430832811650 0.00580861665623299 0.997095691671883 37 0.00302017071033052 0.00604034142066104 0.99697982928967 38 0.00260177408512206 0.00520354817024412 0.997398225914878 39 0.00392757576171386 0.00785515152342771 0.996072424238286 40 0.0907612087791865 0.181522417558373 0.909238791220814 41 0.444281805883553 0.888563611767107 0.555718194116447 42 0.892354293071494 0.215291413857012 0.107645706928506 43 0.90250415320641 0.194991693587179 0.0974958467935894 44 0.876740365054625 0.246519269890750 0.123259634945375 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 6 0.214285714285714 NOK 5% type I error level 11 0.392857142857143 NOK 10% type I error level 14 0.5 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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