*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 09:23:48 -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/t12613264430kppvr64ydmgoe8.htm/, Retrieved Sun, 20 Dec 2009 17:27:35 +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/t12613264430kppvr64ydmgoe8.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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User-defined keywords:

Dataseries X:

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2756,76 10872 2645,64 2497,84 2448,05 2454,62 2407,6 2472,81 2408,64 2440,25 2350,44 2849,27 10625 2756,76 2645,64 2497,84 2448,05 2454,62 2407,6 2472,81 2408,64 2440,25 2921,44 10407 2849,27 2756,76 2645,64 2497,84 2448,05 2454,62 2407,6 2472,81 2408,64 2981,85 10463 2921,44 2849,27 2756,76 2645,64 2497,84 2448,05 2454,62 2407,6 2472,81 3080,58 10556 2981,85 2921,44 2849,27 2756,76 2645,64 2497,84 2448,05 2454,62 2407,6 3106,22 10646 3080,58 2981,85 2921,44 2849,27 2756,76 2645,64 2497,84 2448,05 2454,62 3119,31 10702 3106,22 3080,58 2981,85 2921,44 2849,27 2756,76 2645,64 2497,84 2448,05 3061,26 11353 3119,31 3106,22 3080,58 2981,85 2921,44 2849,27 2756,76 2645,64 2497,84 3097,31 11346 3061,26 3119,31 3106,22 3080,58 2981,85 2921,44 2849,27 2756,76 2645,64 3161,69 11451 3097,31 3061,26 3119,31 3106,22 3080,58 2981,85 2921,44 2849,27 2756,76 3257,16 11964 3161,69 3097,31 3061,26 3119,31 3106,22 3080,58 2981,85 2921,44 2849,27 3277,01 12574 3257,16 3161,69 3097,31 3061,26 3119,31 3106,22 3080,58 29 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 Y[t] = + 129.718988349067 + 0.0136271957734372X[t] + 1.19580953439251Y1[t] -0.238594808848829Y2[t] + 0.196588623192844Y3[t] -0.145134629903214Y4[t] + 0.185141696854423Y5[t] -0.287136374322501Y6[t] -0.0474422047632084Y7[t] + 0.214776602993101Y8[t] -0.136153330812813Y9[t] -97.833027155869M1[t] -45.5330062332913M2[t] -21.5011093797021M3[t] -32.1109665268857M4[t] -35.3444354143223M5[t] -68.1801106941438M6[t] + 91.203578826176M7[t] -75.5180147630598M8[t] -152.121790143776M9[t] -78.3595862713242M10[t] + 2.82383943069867M11[t] -3.29251587550166t + 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) 129.718988349067 320.680286 0.4045 0.688165 0.344083 X 0.0136271957734372 0.027698 0.492 0.625632 0.312816 Y1 1.19580953439251 0.163169 7.3287 0 0 Y2 -0.238594808848829 0.254168 -0.9387 0.353957 0.176979 Y3 0.196588623192844 0.258008 0.7619 0.450921 0.22546 Y4 -0.145134629903214 0.254971 -0.5692 0.572646 0.286323 Y5 0.185141696854423 0.250831 0.7381 0.465101 0.232551 Y6 -0.287136374322501 0.252906 -1.1353 0.26353 0.131765 Y7 -0.0474422047632084 0.256306 -0.1851 0.854162 0.427081 Y8 0.214776602993101 0.258388 0.8312 0.411183 0.205591 Y9 -0.136153330812813 0.187124 -0.7276 0.471431 0.235715 M1 -97.833027155869 124.418642 -0.7863 0.436687 0.218343 M2 -45.5330062332913 128.719832 -0.3537 0.725544 0.362772 M3 -21.5011093797021 127.642071 -0.1684 0.867148 0.433574 M4 -32.1109665268857 130.136862 -0.2467 0.806467 0.403234 M5 -35.3444354143223 127.303352 -0.2776 0.782835 0.391417 M6 -68.1801106941438 127.352612 -0.5354 0.595601 0.297801 M7 91.203578826176 123.383721 0.7392 0.464458 0.232229 M8 -75.5180147630598 119.309677 -0.633 0.530654 0.265327 M9 -152.121790143776 122.893405 -1.2378 0.223575 0.111787 M10 -78.3595862713242 126.196005 -0.6209 0.538449 0.269224 M11 2.82383943069867 121.091353 0.0233 0.98152 0.49076 t -3.29251587550166 6.159006 -0.5346 0.596135 0.298067 Multiple Linear Regression - Regression Statistics Multiple R 0.985935806623163 R-squared 0.972069414781667 Adjusted R-squared 0.955462039786983 F-TEST (value) 58.5323938968557 F-TEST (DF numerator) 22 F-TEST (DF denominator) 37 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 179.230088334974 Sum Squared Residuals 1188566.70888882 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2756.76 2694.99724081164 61.7627591883607 2 2849.27 2854.36269349885 -5.09269349885296 3 2921.44 2984.53477025541 -63.094770255414 4 2981.85 3022.15013511431 -40.3001351143129 5 3080.58 3106.32647221096 -25.7464722109604 6 3106.22 3143.7939178847 -37.5739178846978 7 3119.31 3298.95101900772 -179.641019007723 8 3061.26 3164.47722173451 -103.217221734515 9 3097.31 2992.47239411682 104.837605883177 10 3161.69 3122.43324606821 39.2567539317895 11 3257.16 3238.82495849518 18.3350415048225 12 3277.01 3348.8623081975 -71.8523081974984 13 3295.32 3259.60420117132 35.7157988286754 14 3363.99 3360.62432010985 3.36567989014842 15 3494.17 3462.77528969113 31.3947103088707 16 3667.03 3577.38075401742 89.649245982579 17 3813.06 3745.58545629987 67.4745437001274 18 3917.96 3875.7921470718 42.1678529282009 19 3895.51 4170.2552809412 -274.745280941203 20 3801.06 3920.92953489374 -119.869534893738 21 3570.12 3737.48905010517 -167.369050105171 22 3701.61 3515.30195230011 186.308047699893 23 3862.27 3768.55507720033 93.7149227996692 24 3970.1 3867.19548366755 102.904516332445 25 4138.52 3917.31679772707 221.203202272926 26 4199.75 4137.89699321978 61.8530067802159 27 4290.89 4269.31858062881 21.5714193711878 28 4443.91 4363.44862879933 80.4613712006736 29 4502.64 4430.88351008449 71.7564899155094 30 4356.98 4493.099148005 -136.119148005004 31 4591.27 4449.1259214508 142.1440785492 32 4696.96 4572.90360947257 124.056390527433 33 4621.4 4545.45870708939 75.941292910611 34 4562.84 4521.59952920488 41.2404707951171 35 4202.52 4504.70165455865 -302.181654558650 36 4296.49 4162.45531168665 134.034688313352 37 4435.23 4219.15759654057 216.072403459430 38 4105.18 4255.25033673187 -150.070336731873 39 4116.68 4017.44179023862 99.2382097613795 40 3844.49 4064.4403117008 -219.950311700802 41 3720.98 3737.13634150576 -16.1563415057587 42 3674.4 3661.16060013745 13.2393998625486 43 3857.62 3559.07598778486 298.544012215144 44 3801.06 3793.37170559368 7.68829440631748 45 3504.37 3600.58445965303 -96.2144596530258 46 3032.6 3322.53367218556 -289.933672185561 47 3047.03 2962.02192431968 85.0080756803184 48 2962.34 3044.55733776145 -82.2173377614527 49 2197.82 2732.57416374939 -534.754163749392 50 2014.45 1924.50565643964 89.9443435603621 51 1862.83 1951.93956918602 -89.1095691860238 52 1905.41 1815.27017036814 90.139829631862 53 1810.99 1908.31821989892 -97.3282198989177 54 1670.07 1551.78418690105 118.285813098952 55 1864.44 1850.74179081542 13.6982091845815 56 2052.02 1960.67792830550 91.3420716945026 57 2029.6 1946.79538903559 82.804610964409 58 2070.83 2047.70160024124 23.1283997587616 59 2293.41 2188.28638542616 105.12361457384 60 2443.27 2526.13955868685 -82.8695586868457 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 26 0.0725081383075598 0.145016276615120 0.92749186169244 27 0.0211506686709099 0.0423013373418198 0.97884933132909 28 0.0110215265847632 0.0220430531695265 0.988978473415237 29 0.010270172445703 0.020540344891406 0.989729827554297 30 0.00421912252619449 0.00843824505238899 0.995780877473806 31 0.00703012579914803 0.0140602515982961 0.992969874200852 32 0.00950411593953977 0.0190082318790795 0.99049588406046 33 0.0034766880795137 0.0069533761590274 0.996523311920486 34 0.00411976933283398 0.00823953866566796 0.995880230667166 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 3 0.333333333333333 NOK 5% type I error level 8 0.888888888888889 NOK 10% type I error level 8 0.888888888888889 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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