*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: Thu, 19 Nov 2009 00:41:31 -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/19/t1258616946azq5c1b71o0rjm9.htm/, Retrieved Thu, 19 Nov 2009 08:49:17 +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/19/t1258616946azq5c1b71o0rjm9.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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Dataseries X:

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2.085 0 2.053 0 2.077 0 2.058 0 2.057 0 2.076 0 2.07 0 2.062 0 2.073 0 2.061 0 2.094 0 2.067 0 2.086 0 2.276 0 2.326 0 2.349 0 2.52 0 2.628 0 2.577 0 2.698 0 2.814 0 2.968 0 3.041 0 3.278 0 3.328 0 3.5 0 3.563 0 3.569 0 3.69 0 3.819 0 3.79 0 3.956 0 4.063 0 4.047 0 4.029 0 3.941 0 4.022 0 3.879 0 4.022 0 4.028 0 4.091 0 3.987 0 4.01 0 4.007 0 4.191 0 4.299 0 4.273 0 3.82 0 3.15 1 2.486 1 1.812 1 1.257 1 1.062 1 0.842 1 0.782 1 0.698 1 0.358 1 0.347 1 0.363 1 0.359 1 0.355 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation intb[t] = + 1.39724318181818 -3.83335227272727x[t] + 0.608895202020203M1[t] + 0.718696464646465M2[t] + 0.582606818181819M3[t] + 0.417517171717172M4[t] + 0.392027525252525M5[t] + 0.321137878787879M6[t] + 0.239248232323232M7[t] + 0.220358585858586M8[t] + 0.178668939393939M9[t] + 0.165979292929293M10[t] + 0.124289646464646M11[t] + 0.0572896464646465t + 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) 1.39724318181818 0.309584 4.5133 4.3e-05 2.1e-05 x -3.83335227272727 0.254746 -15.0477 0 0 M1 0.608895202020203 0.344213 1.7689 0.083391 0.041696 M2 0.718696464646465 0.35916 2.001 0.051181 0.02559 M3 0.582606818181819 0.358211 1.6264 0.110545 0.055272 M4 0.417517171717172 0.35736 1.1683 0.248563 0.124282 M5 0.392027525252525 0.356608 1.0993 0.277226 0.138613 M6 0.321137878787879 0.355954 0.9022 0.371557 0.185779 M7 0.239248232323232 0.355401 0.6732 0.50413 0.252065 M8 0.220358585858586 0.354947 0.6208 0.537716 0.268858 M9 0.178668939393939 0.354593 0.5039 0.616707 0.308353 M10 0.165979292929293 0.354341 0.4684 0.641652 0.320826 M11 0.124289646464646 0.354189 0.3509 0.727222 0.363611 t 0.0572896464646465 0.005985 9.5726 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.911664152524966 R-squared 0.831131526999065 Adjusted R-squared 0.784423225956253 F-TEST (value) 17.7940860284614 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 6.49480469405717e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.559942248133108 Sum Squared Residuals 14.7361600984849 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2.085 2.06342803030303 0.0215719696969742 2 2.053 2.23051893939394 -0.17751893939394 3 2.077 2.15171893939394 -0.0747189393939398 4 2.058 2.04391893939394 0.0140810606060597 5 2.057 2.07571893939394 -0.0187189393939412 6 2.076 2.06211893939394 0.0138810606060598 7 2.07 2.03751893939394 0.0324810606060605 8 2.062 2.07591893939394 -0.0139189393939394 9 2.073 2.09151893939394 -0.0185189393939398 10 2.061 2.13611893939394 -0.0751189393939398 11 2.094 2.15171893939394 -0.0577189393939396 12 2.067 2.08471893939394 -0.0177189393939393 13 2.086 2.75090378787879 -0.664903787878789 14 2.276 2.9179946969697 -0.641994696969698 15 2.326 2.8391946969697 -0.513194696969697 16 2.349 2.73139469696970 -0.382394696969696 17 2.52 2.76319469696970 -0.243194696969697 18 2.628 2.74959469696970 -0.121594696969697 19 2.577 2.7249946969697 -0.147994696969697 20 2.698 2.7633946969697 -0.065394696969697 21 2.814 2.7789946969697 0.0350053030303032 22 2.968 2.82359469696970 0.144405303030303 23 3.041 2.83919469696970 0.201805303030303 24 3.278 2.77219469696970 0.505805303030303 25 3.328 3.43837954545455 -0.110379545454546 26 3.5 3.60547045454545 -0.105470454545454 27 3.563 3.52667045454545 0.0363295454545457 28 3.569 3.41887045454545 0.150129545454546 29 3.69 3.45067045454545 0.239329545454546 30 3.819 3.43707045454545 0.381929545454546 31 3.79 3.41247045454545 0.377529545454546 32 3.956 3.45087045454545 0.505129545454546 33 4.063 3.46647045454545 0.596529545454546 34 4.047 3.51107045454545 0.535929545454545 35 4.029 3.52667045454545 0.502329545454546 36 3.941 3.45967045454545 0.481329545454545 37 4.022 4.1258553030303 -0.103855303030303 38 3.879 4.29294621212121 -0.413946212121212 39 4.022 4.21414621212121 -0.192146212121212 40 4.028 4.10634621212121 -0.0783462121212124 41 4.091 4.13814621212121 -0.0471462121212118 42 3.987 4.12454621212121 -0.137546212121212 43 4.01 4.09994621212121 -0.0899462121212124 44 4.007 4.13834621212121 -0.131346212121212 45 4.191 4.15394621212121 0.0370537878787875 46 4.299 4.19854621212121 0.100453787878788 47 4.273 4.21414621212121 0.0588537878787878 48 3.82 4.14714621212121 -0.327146212121212 49 3.15 0.979978787878788 2.17002121212121 50 2.486 1.14706969696970 1.33893030303030 51 1.812 1.06826969696970 0.743730303030303 52 1.257 0.960469696969697 0.296530303030303 53 1.062 0.992269696969697 0.0697303030303033 54 0.842 0.978669696969697 -0.136669696969697 55 0.782 0.954069696969697 -0.172069696969697 56 0.698 0.992469696969697 -0.294469696969697 57 0.358 1.00806969696970 -0.650069696969697 58 0.347 1.05266969696970 -0.705669696969697 59 0.363 1.06826969696970 -0.705269696969697 60 0.359 1.00126969696970 -0.642269696969697 61 0.355 1.66745454545455 -1.31245454545455 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0138090384074867 0.0276180768149735 0.986190961592513 18 0.00697425237043502 0.0139485047408700 0.993025747629565 19 0.00233623491840571 0.00467246983681142 0.997663765081594 20 0.00125773284874501 0.00251546569749003 0.998742267151255 21 0.000910789265612725 0.00182157853122545 0.999089210734387 22 0.00109651589578275 0.00219303179156551 0.998903484104217 23 0.0010491707397629 0.0020983414795258 0.998950829260237 24 0.0019715473177545 0.003943094635509 0.998028452682245 25 0.00177240442503592 0.00354480885007184 0.998227595574964 26 0.00194693927514134 0.00389387855028268 0.998053060724859 27 0.00174708079651353 0.00349416159302706 0.998252919203486 28 0.00128491017837773 0.00256982035675546 0.998715089821622 29 0.000893828424572538 0.00178765684914508 0.999106171575427 30 0.000582415850594801 0.00116483170118960 0.999417584149405 31 0.00036905377081785 0.0007381075416357 0.999630946229182 32 0.00025747165054591 0.00051494330109182 0.999742528349454 33 0.000170560320246441 0.000341120640492882 0.999829439679754 34 9.4106707221614e-05 0.000188213414443228 0.999905893292778 35 5.42081511758471e-05 0.000108416302351694 0.999945791848824 36 4.44490942367047e-05 8.88981884734094e-05 0.999955550905763 37 0.00115958729599567 0.00231917459199134 0.998840412704004 38 0.0553517845993116 0.110703569198623 0.944648215400688 39 0.277496479240772 0.554992958481544 0.722503520759228 40 0.430358975676105 0.86071795135221 0.569641024323895 41 0.499285654980658 0.998571309961316 0.500714345019342 42 0.576350048152286 0.847299903695427 0.423649951847714 43 0.649492105326299 0.701015789347402 0.350507894673701 44 0.779768842367338 0.440462315265324 0.220231157632662 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 19 0.678571428571429 NOK 5% type I error level 21 0.75 NOK 10% type I error level 21 0.75 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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