mutiple 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, 23 Nov 2009 06:24:59 -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/23/t12589835368m5a9uyv6bxze8s.htm/, Retrieved Mon, 23 Nov 2009 14:39:09 +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/23/t12589835368m5a9uyv6bxze8s.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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7.0 519 6.9 517 6.7 510 6.7 509 6.5 501 6.4 507 6.5 569 6.5 580 6.5 578 6.7 565 6.8 547 7.2 555 7.6 562 7.6 561 7.2 555 6.4 544 6.1 537 6.3 543 7.1 594 7.5 611 7.4 613 7.1 611 6.8 594 6.9 595 7.2 591 7.4 589 7.3 584 6.9 573 6.9 567 6.8 569 7.1 621 7.2 629 7.1 628 7.0 612 6.9 595 7.1 597 7.3 593 7.5 590 7.5 580 7.5 574 7.3 573 7.0 573 6.7 620 6.5 626 6.5 620 6.5 588 6.6 566 6.8 557 6.9 561 6.9 549 6.8 532 6.8 526 6.5 511 6.1 499 6.1 555 5.9 565 5.7 542 5.9 527 5.9 510 6.1 514 6.3 517 6.2 508 5.9 493 5.7 490 5.4 469 5.6 478 6.2 528 6.3 534 6.0 518 5.6 506 5.5 502 5.9 516

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 wzo[t] = + 1.49104668013750 + 0.0092863652921957werklbr[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) 1.49104668013750 0.681802 2.1869 0.032092 0.016046 werklbr 0.0092863652921957 0.001223 7.595 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.672138091618934 R-squared 0.451769614205143 Adjusted R-squared 0.443937751550931 F-TEST (value) 57.6835465777947 F-TEST (DF numerator) 1 F-TEST (DF denominator) 70 p-value 1.01636032923125e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.421437113764678 Sum Squared Residuals 12.4326468600812 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7 6.31067026678707 0.689329733212931 2 6.9 6.29209753620267 0.607902463797328 3 6.7 6.2270929791573 0.472907020842697 4 6.7 6.21780661386511 0.482193386134892 5 6.5 6.14351569152754 0.356484308472458 6 6.4 6.19923388328072 0.200766116719284 7 6.5 6.77498853139685 -0.274988531396850 8 6.5 6.877138549611 -0.377138549611003 9 6.5 6.85856581902661 -0.358565819026612 10 6.7 6.73784307022807 -0.0378430702280673 11 6.8 6.57068849496854 0.229311505031455 12 7.2 6.64497941730611 0.55502058269389 13 7.6 6.70998397435148 0.89001602564852 14 7.6 6.70069760905928 0.899302390940715 15 7.2 6.64497941730611 0.55502058269389 16 6.4 6.54282939909196 -0.142829399091957 17 6.1 6.47782484204659 -0.377824842046588 18 6.3 6.53354303379976 -0.233543033799762 19 7.1 7.00714766370174 0.0928523362982567 20 7.5 7.16501587366907 0.33498412633093 21 7.4 7.18358860425346 0.216411395746539 22 7.1 7.16501587366907 -0.0650158736690703 23 6.8 7.00714766370174 -0.207147663701743 24 6.9 7.01643402899394 -0.116434028993938 25 7.2 6.97928856782516 0.220711432174844 26 7.4 6.96071583724076 0.439284162759236 27 7.3 6.91428401077979 0.385715989220214 28 6.9 6.81213399256563 0.0878660074343672 29 6.9 6.75641580081246 0.143584199187541 30 6.8 6.77498853139685 0.0250114686031495 31 7.1 7.25787952659103 -0.157879526591027 32 7.2 7.3321704489286 -0.132170448928593 33 7.1 7.3228840836364 -0.222884083636397 34 7 7.17430223896127 -0.174302238961266 35 6.9 7.01643402899394 -0.116434028993938 36 7.1 7.03500675957833 0.0649932404216696 37 7.3 6.99786129840955 0.302138701590453 38 7.5 6.97000220253296 0.52999779746704 39 7.5 6.877138549611 0.622861450388997 40 7.5 6.82142035785783 0.678579642142171 41 7.3 6.81213399256563 0.487866007434367 42 7 6.81213399256563 0.187866007434367 43 6.7 7.24859316129883 -0.548593161298831 44 6.5 7.304311353052 -0.804311353052006 45 6.5 7.24859316129883 -0.748593161298831 46 6.5 6.95142947194857 -0.451429471948569 47 6.6 6.74712943552026 -0.147129435520264 48 6.8 6.6635521478905 0.136447852109498 49 6.9 6.70069760905928 0.199302390940716 50 6.9 6.58926122555294 0.310738774447064 51 6.8 6.43139301558561 0.368606984414391 52 6.8 6.37567482383244 0.424325176167565 53 6.5 6.2363793444495 0.263620655550501 54 6.1 6.12494296094315 -0.0249429609431512 55 6.1 6.64497941730611 -0.544979417306111 56 5.9 6.73784307022807 -0.837843070228067 57 5.7 6.52425666850757 -0.824256668507566 58 5.9 6.38496118912463 -0.48496118912463 59 5.9 6.2270929791573 -0.327092979157303 60 6.1 6.26423844032609 -0.164238440326087 61 6.3 6.29209753620267 0.00790246379732623 62 6.2 6.20852024857291 -0.00852024857291209 63 5.9 6.06922476918998 -0.169224769189976 64 5.7 6.04136567331339 -0.341365673313389 65 5.4 5.84635200217728 -0.446352002177279 66 5.6 5.92992928980704 -0.329929289807041 67 6.2 6.39424755441683 -0.194247554416826 68 6.3 6.44996574617 -0.149965746170001 69 6 6.30138390149487 -0.301383901494869 70 5.6 6.18994751798852 -0.589947517988521 71 5.5 6.15280205681974 -0.652802056819738 72 5.9 6.28281117091048 -0.382811170910478 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.000393218765861203 0.000786437531722405 0.999606781234139 6 0.0101738971951061 0.0203477943902121 0.989826102804894 7 0.116760305804653 0.233520611609306 0.883239694195347 8 0.0614672885026928 0.122934577005386 0.938532711497307 9 0.0299123794759347 0.0598247589518694 0.970087620524065 10 0.0154544642859594 0.0309089285719189 0.98454553571404 11 0.00931325168649184 0.0186265033729837 0.990686748313508 12 0.0461311331484279 0.0922622662968558 0.953868866851572 13 0.347471588206537 0.694943176413073 0.652528411793463 14 0.63840574548791 0.72318850902418 0.36159425451209 15 0.64369667144103 0.71260665711794 0.35630332855897 16 0.643504805676568 0.712990388646863 0.356495194323432 17 0.740491732400706 0.519016535198588 0.259508267599294 18 0.74037366016705 0.519252679665899 0.259626339832950 19 0.680736374735897 0.638527250528205 0.319263625264103 20 0.674445819484021 0.651108361031958 0.325554180515979 21 0.623384965195407 0.753230069609186 0.376615034804593 22 0.551995355241641 0.896009289516718 0.448004644758359 23 0.504488995881383 0.991022008237235 0.495511004118617 24 0.438620578237202 0.877241156474405 0.561379421762798 25 0.385797945933576 0.771595891867151 0.614202054066424 26 0.39249512523715 0.7849902504743 0.60750487476285 27 0.378976924149799 0.757953848299599 0.621023075850201 28 0.319598479444737 0.639196958889474 0.680401520555263 29 0.267968300473667 0.535936600947334 0.732031699526333 30 0.219995951133029 0.439991902266058 0.780004048866971 31 0.175559303806408 0.351118607612816 0.824440696193592 32 0.134724469562111 0.269448939124221 0.86527553043789 33 0.105094581042979 0.210189162085958 0.894905418957021 34 0.079776521865368 0.159553043730736 0.920223478134632 35 0.0590226643414463 0.118045328682893 0.940977335658554 36 0.0420648426399391 0.0841296852798782 0.95793515736006 37 0.0376785278125363 0.0753570556250727 0.962321472187464 38 0.0568788182842651 0.113757636568530 0.943121181715735 39 0.109170467347749 0.218340934695498 0.890829532652251 40 0.238524772722861 0.477049545445722 0.761475227277139 41 0.348511179871508 0.697022359743016 0.651488820128492 42 0.363912756965847 0.727825513931694 0.636087243034153 43 0.366013552283183 0.732027104566367 0.633986447716817 44 0.457689458260887 0.915378916521773 0.542310541739113 45 0.551270402458467 0.897459195083065 0.448729597541533 46 0.560241913097575 0.87951617380485 0.439758086902425 47 0.504867536979893 0.990264926040213 0.495132463020107 48 0.464626792760907 0.929253585521814 0.535373207239093 49 0.455446695622134 0.910893391244268 0.544553304377866 50 0.527960570263094 0.944078859473813 0.472039429736906 51 0.669418283494306 0.661163433011388 0.330581716505694 52 0.87177959988519 0.256440800229620 0.128220400114810 53 0.947488167646647 0.105023664706706 0.0525118323533531 54 0.95289340290635 0.0942131941873 0.04710659709365 55 0.947363491833027 0.105273016333945 0.0526365081669726 56 0.974927127698363 0.0501457446032747 0.0250728723016374 57 0.99714557932839 0.00570884134321892 0.00285442067160946 58 0.997620327944335 0.00475934411132919 0.00237967205566459 59 0.995497149298946 0.00900570140210862 0.00450285070105431 60 0.991251806048025 0.0174963879039508 0.0087481939519754 61 0.990946349998073 0.0181073000038535 0.00905365000192676 62 0.993940398129654 0.012119203740692 0.006059601870346 63 0.993260382356454 0.0134792352870929 0.00673961764354645 64 0.984982926015665 0.0300341479686693 0.0150170739843347 65 0.968445008615455 0.0631099827690906 0.0315549913845453 66 0.99973647873878 0.000527042522439965 0.000263521261219983 67 0.997442005107376 0.00511598978524716 0.00255799489262358 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 6 0.0952380952380952 NOK 5% type I error level 14 0.222222222222222 NOK 10% type I error level 21 0.333333333333333 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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