*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: Fri, 20 Nov 2009 07:45:44 -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/Nov/20/t12587284405fzyceng6on73ft.htm/, Retrieved Fri, 20 Nov 2009 15:47:32 +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/Nov/20/t12587284405fzyceng6on73ft.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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20 0 21 20 22 22 21 0 20 21 20 22 21 0 21 20 21 20 21 0 21 21 20 21 19 0 21 21 21 20 21 0 19 21 21 21 21 0 21 19 21 21 22 0 21 21 19 21 19 0 22 21 21 19 24 0 19 22 21 21 22 0 24 19 22 21 22 0 22 24 19 22 22 0 22 22 24 19 24 0 22 22 22 24 22 0 24 22 22 22 23 0 22 24 22 22 24 0 23 22 24 22 21 0 24 23 22 24 20 0 21 24 23 22 22 0 20 21 24 23 23 0 22 20 21 24 23 0 23 22 20 21 22 0 23 23 22 20 20 0 22 23 23 22 21 1 20 22 23 23 21 1 21 20 22 23 20 1 21 21 20 22 20 1 20 21 21 20 17 1 20 20 21 21 18 1 17 20 20 21 19 1 18 17 20 20 19 1 19 18 17 20 20 1 19 19 18 17 21 1 20 19 19 18 20 1 21 20 19 19 21 1 20 21 20 19 19 1 21 20 21 20 22 1 19 21 20 21 20 1 22 19 21 20 18 1 20 22 19 21 16 1 18 20 22 19 17 1 16 18 20 22 18 1 17 16 18 20 19 1 18 17 16 18 18 1 19 18 17 16 20 1 18 19 18 17 21 1 20 18 19 18 18 1 21 20 18 19 19 1 18 21 20 18 19 1 19 18 21 20 19 1 19 19 18 21 21 1 19 19 19 18 19 1 21 19 19 19 19 1 19 21 19 19 17 1 19 19 21 19 16 1 17 19 19 21 16 1 16 17 19 19 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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 8.0159995930798 -0.396793923176626X[t] + 0.429040951968163Y1[t] + 0.117021660229381Y2[t] + 0.0321909561234374Y3[t] + 0.0338671313400513Y4[t] + 0.522084696539833M1[t] + 1.60881904762785M2[t] + 0.429102182903779M3[t] + 1.21952238869631M4[t] -0.0690080080649838M5[t] + 0.0599839230344053M6[t] + 0.21231296050147M7[t] + 0.870779207535017M8[t] + 0.429799214978465M9[t] + 2.35819701640652M10[t] + 0.84752047137127M11[t] -0.027329651821742t + 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) 8.0159995930798 3.519316 2.2777 0.027053 0.013527 X -0.396793923176626 0.640905 -0.6191 0.538651 0.269326 Y1 0.429040951968163 0.142421 3.0125 0.004058 0.002029 Y2 0.117021660229381 0.150743 0.7763 0.44123 0.220615 Y3 0.0321909561234374 0.152593 0.211 0.833776 0.416888 Y4 0.0338671313400513 0.143065 0.2367 0.813837 0.406918 M1 0.522084696539833 0.907111 0.5755 0.567503 0.283752 M2 1.60881904762785 0.891956 1.8037 0.077305 0.038652 M3 0.429102182903779 0.865743 0.4956 0.622316 0.311158 M4 1.21952238869631 0.829226 1.4707 0.147644 0.073822 M5 -0.0690080080649838 0.884278 -0.078 0.938109 0.469054 M6 0.0599839230344053 0.850947 0.0705 0.944084 0.472042 M7 0.21231296050147 0.899826 0.2359 0.814436 0.407218 M8 0.870779207535017 0.886825 0.9819 0.330874 0.165437 M9 0.429799214978465 0.905179 0.4748 0.636981 0.31849 M10 2.35819701640652 0.877821 2.6864 0.009778 0.004889 M11 0.84752047137127 0.930227 0.9111 0.36662 0.18331 t -0.027329651821742 0.018785 -1.4549 0.151948 0.075974 Multiple Linear Regression - Regression Statistics Multiple R 0.86441380380356 R-squared 0.747211224206141 Adjusted R-squared 0.661263040436229 F-TEST (value) 8.69374071017563 F-TEST (DF numerator) 17 F-TEST (DF denominator) 50 p-value 1.00810626513237e-09 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.35086548652037 Sum Squared Residuals 91.2418781335953 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 20 21.3143257579137 -1.31432575791368 2 21 21.9973292531943 -0.997329253194314 3 21 21.0667587218306 -0.0667587218305936 4 21 21.9485471112474 -0.948547111247382 5 19 20.6310108874477 -1.63101088744773 6 21 19.9084583941291 1.09154160587090 7 21 20.657496363252 0.342503636748009 8 22 21.4582943666757 0.541705633324316 9 19 21.4156733238323 -2.41567332383232 10 24 22.2143745404436 1.78562545955636 11 22 22.5026990788628 -0.502699078862749 12 22 21.2921696158501 0.707830384149943 13 22 21.6122347267064 0.387765273293582 14 24 22.7765931704261 1.22340682957392 15 22 22.3598942951365 -0.359894295136486 16 23 22.4989462656297 0.501053734370291 17 24 21.4424657608030 2.55753423919705 18 21 22.0935430027114 -1.09354300271137 19 20 21.0128978861249 -1.01289788612492 20 22 20.9299866361439 1.07001336385609 21 23 21.1400314984423 1.85996850155770 22 23 23.5703915703319 -0.570391570331944 23 22 22.1799218146112 -0.179921814611156 24 20 20.9759559582535 -0.97595595825352 25 21 20.1326806469693 0.867319353030671 26 21 21.3548920216216 -0.35489202162157 27 20 20.1666181217182 -0.166618121718209 28 20 20.4651244171642 -0.465124417164170 29 17 19.0661098396918 -2.06610983969180 30 18 17.8484583069415 0.151541693058476 31 19 18.0175665325268 0.982433467473184 32 19 19.0981928715659 -0.098192871565853 33 20 18.6774944495202 1.32250555047978 34 21 21.0736616385582 -0.0736616385581894 35 20 20.1155851852388 -0.115585185238791 36 21 18.9609067264304 2.03909327356957 37 19 19.8337391503508 -0.833739150350797 38 22 20.1537597811267 1.84624021887326 39 20 19.9981166248100 0.00188337518995838 40 18 20.2236754746258 -2.22367547462582 41 16 17.8445288073379 -1.84452880733791 42 17 16.8912853439937 0.108714656006254 43 18 17.0791661862215 0.920833813778507 44 19 18.1242492187039 0.875750781296134 45 18 18.1664588799664 -0.166458879966450 46 20 19.8215658252975 0.17843417470253 47 21 19.0906779596109 1.90932204038909 48 18 18.8805882840614 -0.880588284061438 49 19 18.2357569140112 0.764243085988755 50 19 19.4730628033611 -0.473062803361084 51 19 18.3203322100144 0.679667789985612 52 21 19.0140123260885 1.98598767391154 53 19 18.5901013127818 0.409898687218198 54 19 18.0677250085819 0.932274991418116 55 17 18.0230629860153 -1.02306298601532 56 16 17.7994700277240 -1.79947002772403 57 16 16.6003418482387 -0.600341848238704 58 17 18.3200064253688 -1.32000642536876 59 16 17.1111159616764 -1.11111596167639 60 15 15.8903794154045 -0.890379415404548 61 16 15.8712628040485 0.128737195951467 62 16 17.2443629702702 -1.24436297027021 63 16 16.0882800264903 -0.0882800264902834 64 18 16.8496944052445 1.15030559475554 65 19 16.4257833919378 2.5742166080622 66 16 17.1905299436424 -1.19052994364237 67 16 16.2098100458595 -0.209810045859461 68 16 16.5898068791867 -0.589806879186663 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.83601316536837 0.327973669263261 0.163986834631631 22 0.901851186682881 0.196297626634237 0.0981488133171186 23 0.826218909914364 0.347562180171272 0.173781090085636 24 0.880699403610114 0.238601192779772 0.119300596389886 25 0.821918100302104 0.356163799395791 0.178081899697896 26 0.736375071406229 0.527249857187542 0.263624928593771 27 0.643959049079458 0.712081901841083 0.356040950920542 28 0.547278452790011 0.905443094419978 0.452721547209989 29 0.655619766842733 0.688760466314534 0.344380233157267 30 0.555932586749274 0.888134826501452 0.444067413250726 31 0.488856418410922 0.977712836821843 0.511143581589078 32 0.392139168691886 0.784278337383773 0.607860831308114 33 0.394277599306861 0.788555198613722 0.605722400693139 34 0.301268986327637 0.602537972655274 0.698731013672363 35 0.228081144184266 0.456162288368532 0.771918855815734 36 0.290500054001188 0.581000108002376 0.709499945998812 37 0.257595714790649 0.515191429581298 0.742404285209351 38 0.335071763784689 0.670143527569377 0.664928236215311 39 0.291072515642083 0.582145031284165 0.708927484357917 40 0.476422235401031 0.952844470802063 0.523577764598969 41 0.826460581151675 0.347078837696650 0.173539418848325 42 0.825332886951502 0.349334226096995 0.174667113048498 43 0.732528246636192 0.534943506727615 0.267471753363808 44 0.669570581778412 0.660858836443175 0.330429418221588 45 0.533483626895273 0.933032746209453 0.466516373104727 46 0.387785654916564 0.775571309833127 0.612214345083437 47 0.508406167616343 0.983187664767314 0.491593832383657 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 0 0 OK

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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