*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: Fri, 10 Dec 2010 08:39:53 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/10/t1291970340hntioixt6h3kw2s.htm/, Retrieved Fri, 10 Dec 2010 09:39:00 +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/2010/Dec/10/t1291970340hntioixt6h3kw2s.htm/}, year = {2010}, } @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 = {2010}, 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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9700 0 9081 0 9084 0 9743 0 8587 0 9731 0 9563 0 9998 0 9437 0 10038 0 9918 0 9252 0 9737 0 9035 0 9133 0 9487 0 8700 0 9627 0 8947 0 9283 0 8829 0 9947 0 9628 0 9318 0 9605 0 8640 0 9214 0 9567 0 8547 0 9185 0 9470 0 9123 0 9278 0 10170 0 9434 0 9655 0 9429 0 8739 0 9552 0 9687 1 9019 1 9672 1 9206 1 9069 1 9788 1 10312 1 10105 1 9863 1 9656 1 9295 1 9946 1 9701 1 9049 1 10190 1 9706 1 9765 1 9893 1 9994 1 10433 1 10073 1 10112 1 9266 1 9820 1 10097 1 9115 1 10411 1 9678 1 10408 1 10153 1 10368 1 10581 1 10597 1 10680 1 9738 1 9556 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 5 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation geboortes[t] = + 9551.32375478927 + 483.352490421457x[t] + 87.0966064586708M1[t] -645.046250684182M2[t] -286.331964969896M3[t] -79.3333333333332M4[t] -956.833333333333M5[t] + 9.66666666666673M6[t] -364.666666666667M7[t] -185.333333333333M8[t] -230M9[t] + 345.166666666667M10[t] + 223.5M11[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) 9551.32375478927 122.522937 77.9554 0 0 x 483.352490421457 66.87018 7.2282 0 0 M1 87.0966064586708 160.704363 0.542 0.589783 0.294892 M2 -645.046250684182 160.704363 -4.0139 0.000164 8.2e-05 M3 -286.331964969896 160.704363 -1.7817 0.079691 0.039845 M4 -79.3333333333332 166.69712 -0.4759 0.635809 0.317905 M5 -956.833333333333 166.69712 -5.74 0 0 M6 9.66666666666673 166.69712 0.058 0.953944 0.476972 M7 -364.666666666667 166.69712 -2.1876 0.032478 0.016239 M8 -185.333333333333 166.69712 -1.1118 0.270518 0.135259 M9 -230 166.69712 -1.3797 0.17262 0.08631 M10 345.166666666667 166.69712 2.0706 0.042564 0.021282 M11 223.5 166.69712 1.3408 0.184892 0.092446 Multiple Linear Regression - Regression Statistics Multiple R 0.850890991214296 R-squared 0.724015478929646 Adjusted R-squared 0.670599120012804 F-TEST (value) 13.5541900198922 F-TEST (DF numerator) 12 F-TEST (DF denominator) 62 p-value 3.70481423317415e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 288.727881972289 Sum Squared Residuals 5168554.96934866 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 9700 9638.42036124798 61.5796387520246 2 9081 8906.27750410509 174.722495894910 3 9084 9264.99178981938 -180.991789819376 4 9743 9471.99042145594 271.009578544062 5 8587 8594.49042145594 -7.49042145593836 6 9731 9560.99042145594 170.009578544062 7 9563 9186.6570881226 376.342911877395 8 9998 9365.99042145594 632.009578544062 9 9437 9321.32375478927 115.676245210728 10 10038 9896.49042145594 141.509578544062 11 9918 9774.82375478927 143.176245210728 12 9252 9551.32375478927 -299.323754789272 13 9737 9638.42036124794 98.5796387520576 14 9035 8906.27750410509 128.72249589491 15 9133 9264.99178981938 -131.991789819376 16 9487 9471.99042145594 15.0095785440616 17 8700 8594.49042145594 105.509578544062 18 9627 9560.99042145594 66.0095785440617 19 8947 9186.6570881226 -239.657088122605 20 9283 9365.99042145594 -82.9904214559383 21 8829 9321.32375478927 -492.323754789272 22 9947 9896.49042145594 50.5095785440617 23 9628 9774.82375478927 -146.823754789272 24 9318 9551.32375478927 -233.323754789272 25 9605 9638.42036124794 -33.4203612479425 26 8640 8906.27750410509 -266.27750410509 27 9214 9264.99178981938 -50.9917898193758 28 9567 9471.99042145594 95.0095785440616 29 8547 8594.49042145594 -47.4904214559383 30 9185 9560.99042145594 -375.990421455938 31 9470 9186.6570881226 283.342911877395 32 9123 9365.99042145594 -242.990421455938 33 9278 9321.32375478927 -43.3237547892717 34 10170 9896.49042145594 273.509578544062 35 9434 9774.82375478927 -340.823754789272 36 9655 9551.32375478927 103.676245210728 37 9429 9638.42036124794 -209.420361247942 38 8739 8906.27750410509 -167.277504105090 39 9552 9264.99178981938 287.008210180624 40 9687 9955.3429118774 -268.342911877395 41 9019 9077.8429118774 -58.8429118773951 42 9672 10044.3429118774 -372.342911877395 43 9206 9670.00957854406 -464.009578544062 44 9069 9849.3429118774 -780.342911877395 45 9788 9804.67624521073 -16.6762452107284 46 10312 10379.8429118774 -67.842911877395 47 10105 10258.1762452107 -153.176245210728 48 9863 10034.6762452107 -171.676245210728 49 9656 10121.7728516694 -465.772851669399 50 9295 9389.62999452655 -94.6299945265467 51 9946 9748.34428024083 197.655719759168 52 9701 9955.3429118774 -254.342911877395 53 9049 9077.8429118774 -28.8429118773951 54 10190 10044.3429118774 145.657088122605 55 9706 9670.00957854406 35.9904214559383 56 9765 9849.3429118774 -84.342911877395 57 9893 9804.67624521073 88.3237547892717 58 9994 10379.8429118774 -385.842911877395 59 10433 10258.1762452107 174.823754789272 60 10073 10034.6762452107 38.3237547892717 61 10112 10121.7728516694 -9.77285166939906 62 9266 9389.62999452655 -123.629994526547 63 9820 9748.34428024083 71.6557197591676 64 10097 9955.3429118774 141.657088122605 65 9115 9077.8429118774 37.157088122605 66 10411 10044.3429118774 366.657088122605 67 9678 9670.00957854406 7.99042145593837 68 10408 9849.3429118774 558.657088122605 69 10153 9804.67624521073 348.323754789272 70 10368 10379.8429118774 -11.8429118773950 71 10581 10258.1762452107 322.823754789272 72 10597 10034.6762452107 562.323754789272 73 10680 10121.7728516694 558.227148330601 74 9738 9389.62999452655 348.370005473453 75 9556 9748.34428024083 -192.344280240832 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0628864086217765 0.125772817243553 0.937113591378224 17 0.0252538380525609 0.0505076761051218 0.97474616194744 18 0.00970958652843406 0.0194191730568681 0.990290413471566 19 0.148579319870377 0.297158639740754 0.851420680129623 20 0.375555160510833 0.751110321021666 0.624444839489167 21 0.508910414703815 0.98217917059237 0.491089585296185 22 0.406094554376384 0.812189108752768 0.593905445623616 23 0.346973812461059 0.693947624922119 0.653026187538941 24 0.272384861908357 0.544769723816715 0.727615138091643 25 0.201453921323331 0.402907842646662 0.798546078676669 26 0.220629084902142 0.441258169804285 0.779370915097858 27 0.162010428414125 0.324020856828250 0.837989571585875 28 0.117628050260768 0.235256100521536 0.882371949739232 29 0.0802646370649753 0.160529274129951 0.919735362935025 30 0.116285925472923 0.232571850945845 0.883714074527077 31 0.107301041391532 0.214602082783063 0.892698958608468 32 0.134188895600302 0.268377791200604 0.865811104399698 33 0.09905893413577 0.19811786827154 0.90094106586423 34 0.0947900699005712 0.189580139801142 0.905209930099429 35 0.0951337328751895 0.190267465750379 0.90486626712481 36 0.086249305825813 0.172498611651626 0.913750694174187 37 0.0712702639584215 0.142540527916843 0.928729736041578 38 0.0589649249372218 0.117929849874444 0.941035075062778 39 0.0571857836021641 0.114371567204328 0.942814216397836 40 0.0393145980716531 0.0786291961433063 0.960685401928347 41 0.0282828166572107 0.0565656333144214 0.97171718334279 42 0.0314177911054975 0.062835582210995 0.968582208894503 43 0.0347914111105862 0.0695828222211725 0.965208588889414 44 0.176210807685340 0.352421615370679 0.82378919231466 45 0.201813126610689 0.403626253221378 0.798186873389311 46 0.154250353596875 0.308500707193749 0.845749646403125 47 0.158510708037767 0.317021416075535 0.841489291962233 48 0.179424776637648 0.358849553275295 0.820575223362352 49 0.350709779801318 0.701419559602636 0.649290220198682 50 0.303553555207902 0.607107110415804 0.696446444792098 51 0.310627320986465 0.62125464197293 0.689372679013535 52 0.295266014005051 0.590532028010103 0.704733985994949 53 0.219145807559674 0.438291615119348 0.780854192440326 54 0.194213920440546 0.388427840881093 0.805786079559454 55 0.132094428020570 0.264188856041139 0.86790557197943 56 0.207467473951453 0.414934947902906 0.792532526048547 57 0.16515156893878 0.33030313787756 0.83484843106122 58 0.151190529244099 0.302381058488199 0.8488094707559 59 0.0957868379932259 0.191573675986452 0.904213162006774 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 1 0.0227272727272727 OK 10% type I error level 6 0.136363636363636 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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