Multiple Linear Regression

*Unverified author*

R Software Module: rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Sat, 20 Dec 2008 07:28:29 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/20/t1229783645x9mvqz3hug3x4rg.htm/, Retrieved Sat, 20 Dec 2008 15:34:15 +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/2008/Dec/20/t1229783645x9mvqz3hug3x4rg.htm/}, year = {2008}, } @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 = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

IsPrivate?

No (this computation is public)

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Dataseries X:

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467 0 460 0 448 0 443 0 436 0 431 0 484 0 510 0 513 1 503 1 471 1 471 1 476 1 475 1 470 1 461 1 455 1 456 1 517 1 525 1 523 1 519 1 509 1 512 1 519 1 517 1 510 1 509 1 501 1 507 1 569 1 580 1 578 1 565 1 547 1 555 1 562 1 561 1 555 1 544 1 537 1 543 1 594 1 611 1 613 1 611 1 594 1 595 1 591 1 589 1 584 1 573 1 567 1 569 1 621 1 629 1 628 1 612 1 595 1 597 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 Werkloosheid[t] = + 448.609375 -6.76562499999992Dummy[t] + 7.47239583333319M1[t] + 1.97916666666667M2[t] -7.9140625M3[t] -18.2072916666666M4[t] -27.9005208333333M5[t] -28.7937499999999M6[t] + 24.1130208333334M7[t] + 35.2197916666667M8[t] + 33.6796875M9[t] + 21.7864583333333M10[t] + 0.0932291666666544M11[t] + 2.89322916666667t + 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) 448.609375 5.742839 78.1163 0 0 Dummy -6.76562499999992 4.637761 -1.4588 0.151412 0.075706 M1 7.47239583333319 6.122183 1.2205 0.228479 0.114239 M2 1.97916666666667 6.116324 0.3236 0.747717 0.373858 M3 -7.9140625 6.111764 -1.2949 0.20182 0.10091 M4 -18.2072916666666 6.108505 -2.9806 0.004585 0.002292 M5 -27.9005208333333 6.106548 -4.569 3.7e-05 1.8e-05 M6 -28.7937499999999 6.105896 -4.7157 2.3e-05 1.1e-05 M7 24.1130208333334 6.106548 3.9487 0.000267 0.000134 M8 35.2197916666667 6.108505 5.7657 1e-06 0 M9 33.6796875 6.064659 5.5534 1e-06 1e-06 M10 21.7864583333333 6.061375 3.5943 0.00079 0.000395 M11 0.0932291666666544 6.059403 0.0154 0.987791 0.493895 t 2.89322916666667 0.089254 32.4158 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.988336900707975 R-squared 0.976809829301045 Adjusted R-squared 0.970256085407863 F-TEST (value) 149.046078885860 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 9.57971773320205 Sum Squared Residuals 4221.46562499999 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 467 458.975000000001 8.02499999999942 2 460 456.375 3.62500000000015 3 448 449.375 -1.37499999999990 4 443 441.975 1.02500000000013 5 436 435.175 0.825000000000122 6 431 437.175 -6.17499999999999 7 484 492.975 -8.9749999999999 8 510 506.975 3.02499999999999 9 513 501.5625 11.4375 10 503 492.5625 10.4375000000000 11 471 473.7625 -2.76249999999999 12 471 476.5625 -5.56249999999999 13 476 486.928125 -10.9281249999999 14 475 484.328125 -9.32812500000007 15 470 477.328125 -7.32812500000004 16 461 469.928125 -8.92812500000003 17 455 463.128125 -8.12812500000003 18 456 465.128125 -9.128125 19 517 520.928125 -3.928125 20 525 534.928125 -9.928125 21 523 536.28125 -13.2812500000000 22 519 527.28125 -8.28124999999999 23 509 508.48125 0.518749999999995 24 512 511.28125 0.71874999999999 25 519 521.646875 -2.64687499999985 26 517 519.046875 -2.04687500000002 27 510 512.046875 -2.04687500000002 28 509 504.646875 4.35312499999997 29 501 497.846875 3.15312499999996 30 507 499.846875 7.153125 31 569 555.646875 13.3531250000000 32 580 569.646875 10.353125 33 578 571 7 34 565 562 3.00000000000001 35 547 543.2 3.79999999999999 36 555 546 9 37 562 556.365625 5.63437500000014 38 561 553.765625 7.23437499999997 39 555 546.765625 8.23437499999998 40 544 539.365625 4.63437499999997 41 537 532.565625 4.43437499999997 42 543 534.565625 8.434375 43 594 590.365625 3.63437499999997 44 611 604.365625 6.634375 45 613 605.71875 7.28125 46 611 596.71875 14.28125 47 594 577.91875 16.08125 48 595 580.71875 14.28125 49 591 591.084375 -0.0843749999998457 50 589 588.484375 0.515624999999971 51 584 581.484375 2.51562499999999 52 573 574.084375 -1.08437500000004 53 567 567.284375 -0.284375000000034 54 569 569.284375 -0.284375000000020 55 621 625.084375 -4.08437500000004 56 629 639.084375 -10.084375 57 628 640.4375 -12.4375000000000 58 612 631.4375 -19.4375 59 595 612.6375 -17.6375 60 597 615.4375 -18.4375 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0723054051321892 0.144610810264378 0.927694594867811 18 0.0582167508016257 0.116433501603251 0.941783249198374 19 0.115950064209921 0.231900128419841 0.88404993579008 20 0.0773756902777738 0.154751380555548 0.922624309722226 21 0.0514036704660467 0.102807340932093 0.948596329533953 22 0.0355545448838914 0.0711090897677828 0.964445455116109 23 0.154929654764859 0.309859309529718 0.84507034523514 24 0.23571913356335 0.4714382671267 0.76428086643665 25 0.248980855387407 0.497961710774814 0.751019144612593 26 0.270835085244779 0.541670170489558 0.72916491475522 27 0.327916956949710 0.655833913899419 0.67208304305029 28 0.365257351416116 0.730514702832233 0.634742648583884 29 0.378021524035818 0.756043048071637 0.621978475964182 30 0.466600960563364 0.933201921126728 0.533399039436636 31 0.524635775554779 0.950728448890443 0.475364224445221 32 0.476131215758758 0.952262431517517 0.523868784241242 33 0.391169131120298 0.782338262240597 0.608830868879702 34 0.362968755686261 0.725937511372523 0.637031244313739 35 0.372297757578539 0.744595515157077 0.627702242421461 36 0.338830897333671 0.677661794667342 0.661169102666329 37 0.282391092258360 0.564782184516721 0.71760890774164 38 0.231625251044181 0.463250502088362 0.768374748955819 39 0.205273007527651 0.410546015055303 0.794726992472349 40 0.191509732812195 0.383019465624389 0.808490267187805 41 0.224780713761214 0.449561427522429 0.775219286238786 42 0.246953270711583 0.493906541423166 0.753046729288417 43 0.442298727036721 0.884597454073441 0.557701272963279 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 1 0.0370370370370370 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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