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: Mon, 14 Dec 2009 12:39:00 -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/14/t1260819571bftv7gc3ytdpgt1.htm/, Retrieved Mon, 14 Dec 2009 20:39:43 +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/14/t1260819571bftv7gc3ytdpgt1.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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441 1919 449 1911 452 1870 462 2263 455 1802 461 1863 461 1989 463 2197 462 2409 456 2502 455 2593 456 2598 472 2053 472 2213 471 2238 465 2359 459 2151 465 2474 468 3079 467 2312 463 2565 460 1972 462 2484 461 2202 476 2151 476 1976 471 2012 453 2114 443 1772 442 1957 444 2070 438 1990 427 2182 424 2008 416 1916 406 2397 431 2114 434 1778 418 1641 412 2186 404 1773 409 1785 412 2217 406 2153 398 1895 397 2475 385 1793 390 2308 413 2051 413 1898 401 2142 397 1874 397 1560 409 1808 419 1575 424 1525 428 1997 430 1753 424 1623 433 2251 456 1890

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 wkl[t] = + 320.083426664051 + 0.0464088862435987bvg[t] + 33.8886705568584M1[t] + 37.9779189524650M2[t] + 30.5991332418776M3[t] + 17.5105061587709M4[t] + 27.4422350170458M5[t] + 25.3476416778572M6[t] + 19.2667480074425M7[t] + 25.0559262757284M8[t] + 12.9714982920935M9[t] + 13.9087390021608M10[t] + 11.7025539540255M11[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) 320.083426664051 30.443375 10.5141 0 0 bvg 0.0464088862435987 0.012061 3.8479 0.000351 0.000176 M1 33.8886705568584 15.488268 2.188 0.033568 0.016784 M2 37.9779189524650 16.37375 2.3194 0.024672 0.012336 M3 30.5991332418776 16.287027 1.8787 0.06636 0.03318 M4 17.5105061587709 15.831955 1.106 0.274229 0.137115 M5 27.4422350170458 16.960051 1.6181 0.112205 0.056102 M6 25.3476416778572 16.297661 1.5553 0.126446 0.063223 M7 19.2667480074425 15.787915 1.2203 0.228295 0.114147 M8 25.0559262757284 16.118176 1.5545 0.126631 0.063316 M9 12.9714982920935 15.754524 0.8234 0.414381 0.20719 M10 13.9087390021608 15.863625 0.8768 0.384979 0.19249 M11 11.7025539540255 15.995178 0.7316 0.467951 0.233976 Multiple Linear Regression - Regression Statistics Multiple R 0.532195561733823 R-squared 0.283232115929179 Adjusted R-squared 0.104040144911474 F-TEST (value) 1.58060717966651 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 0.129508196192337 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 24.7632969005885 Sum Squared Residuals 29434.6019225614 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 441 443.030749922375 -2.0307499223753 2 449 446.748727228033 2.25127277196703 3 452 437.467177181458 14.532822818542 4 462 442.617242392086 19.3827576079145 5 455 431.154474692061 23.8455253079386 6 461 431.890823413732 29.1091765862677 7 461 431.657449410011 29.3425505899889 8 463 447.099676016966 15.9003239830345 9 462 444.853931916974 17.1460680830264 10 456 450.107199047696 5.89280095230446 11 455 452.124222647728 2.87577735227236 12 456 440.65371312492 15.3462868750798 13 472 449.249540679017 22.7504593209827 14 472 460.7642108736 11.2357891264003 15 471 454.545647319102 16.4543526808977 16 465 447.072495471471 17.9275045285290 17 459 447.351175991077 11.6488240089226 18 465 460.246652908571 4.75334709142888 19 468 482.243135415534 -14.2431354155336 20 467 452.436697934979 14.5633020650206 21 463 452.093718170975 10.9062818290250 22 460 425.510489338588 34.4895106614118 23 462 447.065654047175 14.9343459528246 24 461 422.275794172455 38.7242058275449 25 476 453.79761153089 22.2023884691101 26 476 449.765304833867 26.2346951661332 27 471 444.057239028049 26.942760971951 28 453 435.702318341789 17.2976816582107 29 443 429.762208104753 13.2377918952465 30 442 436.253258720631 5.74674127936941 31 444 435.416569195743 8.58343080425745 32 438 437.493036564541 0.50696343545938 33 427 434.319114739677 -7.31911473967667 34 424 427.181209243358 -3.18120924335777 35 416 420.705406660811 -4.70540666081134 36 406 431.325526989957 -25.3255269899568 37 431 452.080482739877 -21.0804827398768 38 434 440.576345357634 -6.5763453576343 39 418 426.839542231674 -8.83954223167388 40 412 439.043758151328 -27.0437581513285 41 404 429.808616990997 -25.8086169909971 42 409 428.270930286732 -19.2709302867316 43 412 442.238675473552 -30.2386754735516 44 406 445.057685022247 -39.0576850222472 45 398 420.999764387764 -22.9997643877639 46 397 448.854159119118 -51.8541591191183 47 385 414.997113652849 -29.9971136528487 48 390 427.195136114277 -37.1951361142766 49 413 449.15672290653 -36.1567229065300 50 413 446.145411706866 -33.1454117068662 51 401 450.090394239717 -49.0903942397168 52 397 424.564185643326 -27.5641856433257 53 397 419.923524221111 -22.9235242211106 54 409 429.338334670334 -20.3383346703344 55 419 412.444170505161 6.55582949483881 56 424 415.912904461267 8.08709553873278 57 428 425.733470784611 2.26652921538908 58 430 415.34694325124 14.6530567487599 59 424 407.107602991437 16.8923970085631 60 433 424.549829598391 8.45017040160858 61 456 441.684892221311 14.3151077786893 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.055207313297288 0.110414626594576 0.944792686702712 17 0.0368852401085492 0.0737704802170984 0.96311475989145 18 0.0285916079667333 0.0571832159334665 0.971408392033267 19 0.0160703809994514 0.0321407619989029 0.983929619000549 20 0.006347941828962 0.012695883657924 0.993652058171038 21 0.00244743576167115 0.0048948715233423 0.997552564238329 22 0.00145220182447963 0.00290440364895926 0.99854779817552 23 0.00084024978734797 0.00168049957469594 0.999159750212652 24 0.000619657541242624 0.00123931508248525 0.999380342458757 25 0.000941413194160665 0.00188282638832133 0.99905858680584 26 0.00163231187970040 0.00326462375940079 0.9983676881203 27 0.00322228769219382 0.00644457538438763 0.996777712307806 28 0.00481321199147992 0.00962642398295984 0.99518678800852 29 0.0084738702361803 0.0169477404723606 0.99152612976382 30 0.0182025034246722 0.0364050068493445 0.981797496575328 31 0.0265348163950994 0.0530696327901988 0.9734651836049 32 0.0521583721279377 0.104316744255875 0.947841627872062 33 0.105946003042839 0.211892006085678 0.89405399695716 34 0.134899034876980 0.269798069753961 0.86510096512302 35 0.159240850878901 0.318481701757801 0.8407591491211 36 0.340173820807381 0.680347641614762 0.659826179192619 37 0.355000387637574 0.710000775275149 0.644999612362426 38 0.333406787947648 0.666813575895296 0.666593212052352 39 0.309281952369152 0.618563904738304 0.690718047630848 40 0.400651755521143 0.801303511042286 0.599348244478857 41 0.403264866883003 0.806529733766007 0.596735133116997 42 0.331250687414773 0.662501374829546 0.668749312585227 43 0.305294579566953 0.610589159133905 0.694705420433047 44 0.274703794420139 0.549407588840277 0.725296205579861 45 0.293452791806144 0.586905583612287 0.706547208193856 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 8 0.266666666666667 NOK 5% type I error level 12 0.4 NOK 10% type I error level 15 0.5 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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