Multiple regression

*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 12:11:31 -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/t1258658045iwe5ldg8b8mlsp8.htm/, Retrieved Thu, 19 Nov 2009 20:14:17 +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/t1258658045iwe5ldg8b8mlsp8.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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19 613 18 611 19 594 19 595 22 591 23 589 20 584 14 573 14 567 14 569 15 621 11 629 17 628 16 612 20 595 24 597 23 593 20 590 21 580 19 574 23 573 23 573 23 620 23 626 27 620 26 588 17 566 24 557 26 561 24 549 27 532 27 526 26 511 24 499 23 555 23 565 24 542 17 527 21 510 19 514 22 517 22 508 18 493 16 490 14 469 12 478 14 528 16 534 8 518 3 506 0 502 5 516 1 528 1 533 3 536 6 537 7 524 8 536 14 587 14 597 13 581

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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation ICONS[t] = -18.1539021701522 + 0.0602404306508848WLH[t] + 0.993570813585798M1[t] -0.110854784071067M2[t] + 0.216847847952559M3[t] + 2.87227081439044M4[t] + 3.33974186695849M5[t] + 2.79275167569221M6[t] + 3.12286746541999M7[t] + 2.02406961867441M8[t] + 3.09876244196432M9[t] + 2.36623349453238M10[t] + 0.881923445207079M11[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) -18.1539021701522 15.837948 -1.1462 0.257383 0.128691 WLH 0.0602404306508848 0.026219 2.2976 0.025985 0.012992 M1 0.993570813585798 4.572973 0.2173 0.828919 0.414459 M2 -0.110854784071067 4.805827 -0.0231 0.981693 0.490846 M3 0.216847847952559 4.869506 0.0445 0.964665 0.482333 M4 2.87227081439044 4.85743 0.5913 0.557084 0.278542 M5 3.33974186695849 4.847051 0.689 0.494124 0.247062 M6 2.79275167569221 4.867439 0.5738 0.568808 0.284404 M7 3.12286746541999 4.917883 0.635 0.528442 0.264221 M8 2.02406961867441 4.951108 0.4088 0.684496 0.342248 M9 3.09876244196432 5.03713 0.6152 0.541339 0.270669 M10 2.36623349453238 5.018993 0.4715 0.639451 0.319726 M11 0.881923445207079 4.777569 0.1846 0.854323 0.427162 Multiple Linear Regression - Regression Statistics Multiple R 0.341778622777809 R-squared 0.116812626987896 Adjusted R-squared -0.103984216265131 F-TEST (value) 0.529050258449719 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 0.885066905782545 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 7.54671563582109 Sum Squared Residuals 2733.74001062143 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 19 19.7670526324259 -0.767052632425869 2 18 18.5421461734673 -0.542146173467337 3 19 17.8457614844259 1.15423851557408 4 19 20.5614248815147 -1.56142488151468 5 22 20.7879342114792 1.21206578852080 6 23 20.1204631589112 2.87953684108885 7 20 20.1493767953845 -0.149376795384509 8 14 18.3879342114792 -4.3879342114792 9 14 19.1011844508638 -5.1011844508638 10 14 18.4891363647336 -4.48913636473362 11 15 20.1373287092543 -5.13732870925433 12 11 19.7373287092543 -8.73732870925433 13 17 20.6706590921892 -3.67065909218924 14 16 18.6023866041182 -2.60238660411822 15 20 17.9060019150768 2.09399808492320 16 24 20.6819057428165 3.31809425718355 17 23 20.9084150727810 2.09158492721904 18 20 20.1807035895620 -0.180703589562029 19 21 19.9084150727810 1.09158492721903 20 19 18.4481746421301 0.55182535786992 21 23 19.4626270347691 3.53737296523089 22 23 18.7300980873372 4.26990191266284 23 23 20.0770882786034 2.92291172139656 24 23 19.5566074173017 3.44339258269833 25 27 20.1887356469822 6.81126435301784 26 26 17.156616268497 8.84338373150301 27 17 16.1590294262011 0.840970573798854 28 24 18.2722885167811 5.72771148321894 29 26 18.9807212919527 7.01927870804735 30 24 17.7108459328758 6.28915406712425 31 27 17.0168744015385 9.9831255984615 32 27 15.5566339708876 11.4433660291124 33 26 15.7277203344143 10.2722796655857 34 24 14.2723062191717 9.72769378082831 35 23 16.1614602862959 6.83853971370407 36 23 15.8819411475977 7.1180588524023 37 24 15.4899820562132 8.51001794378684 38 17 13.4819499987930 3.51805000120698 39 21 12.7855653097516 8.2144346902484 40 19 15.681949998793 3.31805000120698 41 22 16.3301423433137 5.66985765668628 42 22 15.2409882761895 6.75901172381052 43 18 14.667497606154 3.33250239384601 44 16 13.3879784674558 2.61202153254424 45 14 13.1976222470771 0.802377752922909 46 12 13.0072571755031 -1.00725717550311 47 14 14.5349686587220 -0.534968658722045 48 16 14.0144877974203 1.98551220257972 49 8 14.0442117205919 -6.04421172059192 50 3 12.2169009551244 -9.21690095512444 51 0 12.3036418645445 -12.3036418645445 52 5 15.8024308600948 -10.8024308600948 53 1 16.9927870804735 -15.9927870804735 54 1 16.7469990424616 -15.7469990424616 55 3 17.2578361241420 -14.2578361241420 56 6 16.2192787080473 -10.2192787080473 57 7 16.5108459328758 -9.51084593287576 58 8 16.5012021532544 -8.50120215325442 59 14 18.0891540671242 -4.08915406712425 60 14 17.8096349284260 -3.80963492842602 61 13 17.8393588515977 -4.83935885159765 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0247686257163056 0.0495372514326111 0.975231374283694 17 0.00563558177978008 0.0112711635595602 0.99436441822022 18 0.00185453401391643 0.00370906802783287 0.998145465986084 19 0.000368160417849792 0.000736320835699584 0.99963183958215 20 0.000326957538578018 0.000653915077156037 0.999673042461422 21 0.00118363994407974 0.00236727988815947 0.99881636005592 22 0.00157040889608225 0.0031408177921645 0.998429591103918 23 0.00161013364883733 0.00322026729767466 0.998389866351163 24 0.00388316825851215 0.00776633651702429 0.996116831741488 25 0.00459043768534291 0.00918087537068582 0.995409562314657 26 0.00345893735193995 0.0069178747038799 0.99654106264806 27 0.0025009781385619 0.0050019562771238 0.997499021861438 28 0.00130498446176677 0.00260996892353354 0.998695015538233 29 0.000818972893142173 0.00163794578628435 0.999181027106858 30 0.000484723034320581 0.000969446068641161 0.99951527696568 31 0.000425328975447299 0.000850657950894598 0.999574671024553 32 0.000545393788117125 0.00109078757623425 0.999454606211883 33 0.000543529923087983 0.00108705984617597 0.999456470076912 34 0.000467190979448097 0.000934381958896195 0.999532809020552 35 0.000286292972347416 0.000572585944694831 0.999713707027653 36 0.00014696530191611 0.00029393060383222 0.999853034698084 37 0.000166461968781079 0.000332923937562158 0.999833538031219 38 0.000527430562009913 0.00105486112401983 0.99947256943799 39 0.00211074520636553 0.00422149041273107 0.997889254793634 40 0.004391656610691 0.008783313221382 0.99560834338931 41 0.0281726808625868 0.0563453617251737 0.971827319137413 42 0.215971333988345 0.43194266797669 0.784028666011655 43 0.52502545856053 0.94994908287894 0.47497454143947 44 0.685688158301389 0.628623683397222 0.314311841698611 45 0.765490867997522 0.469018264004957 0.234509132002478 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 23 0.766666666666667 NOK 5% type I error level 25 0.833333333333333 NOK 10% type I error level 26 0.866666666666667 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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