Paper

*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: Tue, 21 Dec 2010 10:10:56 +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/21/t12929261435yavkdq9b1t7qs5.htm/, Retrieved Tue, 21 Dec 2010 11:09:06 +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/21/t12929261435yavkdq9b1t7qs5.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:

» Textbox « » Textfile « » CSV «

695 0 638 0 762 0 635 0 721 0 854 0 418 0 367 0 824 0 687 0 601 0 676 0 740 0 691 0 683 0 594 0 729 0 731 0 386 0 331 0 707 0 715 0 657 0 653 0 642 0 643 0 718 0 654 0 632 0 731 0 392 1 344 1 792 1 852 1 649 1 629 1 685 1 617 1 715 1 715 1 629 1 916 1 531 1 357 1 917 1 828 1 708 1 858 1 775 1 785 1 1006 1 789 1 734 1 906 1 532 1 387 1 991 1 841 1 892 1 782 1 813 1 793 1 978 1 775 1 797 1 946 1 594 1 438 1 1022 1 868 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 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation MultipleLinearRegression[t] = + 612.85777262181 + 8.27552204176337X[t] + 20.3632347254447M1[t] -12.963901778809M2[t] + 100.042295050271M3[t] -19.4515081206497M4[t] -8.94531129157002M5[t] + 128.56088553751M6[t] -247.478837973705M7[t] -355.139307811292M8[t] + 146.866889017788M9[t] + 67.0397525135344M10[t] -15.3728634957463M11[t] + 2.82713650425367t + 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) 612.85777262181 32.513382 18.8494 0 0 X 8.27552204176337 30.835417 0.2684 0.789395 0.394698 M1 20.3632347254447 39.305329 0.5181 0.606445 0.303222 M2 -12.963901778809 39.291112 -0.3299 0.742673 0.371336 M3 100.042295050271 39.291313 2.5462 0.013669 0.006835 M4 -19.4515081206497 39.305932 -0.4949 0.622625 0.311312 M5 -8.94531129157002 39.334953 -0.2274 0.82093 0.410465 M6 128.56088553751 39.378344 3.2648 0.001871 0.000936 M7 -247.478837973705 39.284648 -6.2996 0 0 M8 -355.139307811292 39.272493 -9.043 0 0 M9 146.866889017788 39.274763 3.7395 0.000435 0.000218 M10 67.0397525135344 39.291457 1.7062 0.093508 0.046754 M11 -15.3728634957463 41.009957 -0.3749 0.709183 0.354591 t 2.82713650425367 0.752736 3.7558 0.000413 0.000207 Multiple Linear Regression - Regression Statistics Multiple R 0.935809520812574 R-squared 0.87573945924346 Adjusted R-squared 0.84689326228212 F-TEST (value) 30.3589225441795 F-TEST (DF numerator) 13 F-TEST (DF denominator) 56 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 64.8315113849258 Sum Squared Residuals 235374.992633411 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 695 636.048143851508 58.9518561484921 2 638 605.548143851508 32.4518561484918 3 762 721.381477184841 40.6185228151586 4 635 604.714810518175 30.2851894818252 5 721 618.048143851508 102.951856148492 6 854 758.381477184841 95.6185228151586 7 418 385.168890177881 32.8311098221192 8 367 280.335556844548 86.6644431554524 9 824 785.168890177881 38.831109822119 10 687 708.168890177881 -21.1688901778809 11 601 628.583410672854 -27.5834106728538 12 676 646.783410672854 29.2165893271462 13 740 669.973781902552 70.0262180974478 14 691 639.473781902552 51.5262180974478 15 683 755.307115235886 -72.3071152358855 16 594 638.640448569219 -44.6404485692189 17 729 651.973781902552 77.0262180974478 18 731 792.307115235886 -61.3071152358856 19 386 419.094528228925 -33.094528228925 20 331 314.261194895592 16.7388051044083 21 707 819.094528228925 -112.094528228925 22 715 742.094528228925 -27.094528228925 23 657 662.509048723898 -5.5090487238979 24 653 680.709048723898 -27.7090487238979 25 642 703.899419953596 -61.8994199535964 26 643 673.399419953596 -30.3994199535963 27 718 789.23275328693 -71.2327532869296 28 654 672.566086620263 -18.5660866202629 29 632 685.899419953596 -53.8994199535963 30 731 826.23275328693 -95.2327532869297 31 392 461.295688321732 -69.2956883217324 32 344 356.462354988399 -12.4623549883991 33 792 861.295688321732 -69.2956883217324 34 852 784.295688321732 67.7043116782676 35 649 704.710208816705 -55.7102088167054 36 629 722.910208816705 -93.9102088167054 37 685 746.100580046404 -61.1005800464038 38 617 715.600580046404 -98.6005800464037 39 715 831.433913379737 -116.433913379737 40 715 714.76724671307 0.232753286929591 41 629 728.100580046404 -99.1005800464037 42 916 868.433913379737 47.566086620263 43 531 495.221326372777 35.7786736272235 44 357 390.387993039443 -33.3879930394431 45 917 895.221326372777 21.7786736272235 46 828 818.221326372777 9.77867362722353 47 708 738.635846867749 -30.6358468677494 48 858 756.835846867749 101.164153132251 49 775 780.026218097448 -5.02621809744784 50 785 749.526218097448 35.4737819025522 51 1006 865.359551430781 140.640448569219 52 789 748.692884764115 40.3071152358855 53 734 762.026218097448 -28.0262180974478 54 906 902.359551430781 3.6404485692189 55 532 529.14696442382 2.85303557617941 56 387 424.313631090487 -37.3136310904872 57 991 929.14696442382 61.8530355761795 58 841 852.14696442382 -11.1469644238206 59 892 772.561484918793 119.438515081207 60 782 790.761484918793 -8.76148491879352 61 813 813.951856148492 -0.951856148491914 62 793 783.451856148492 9.54814385150814 63 978 899.285189481825 78.7148105181748 64 775 782.618522815159 -7.61852281515855 65 797 795.951856148492 1.04814385150811 66 946 936.285189481825 9.71481051817482 67 594 563.072602474865 30.9273975251353 68 438 458.239269141531 -20.2392691415313 69 1022 963.072602474865 58.9273975251354 70 868 886.072602474865 -18.0726024748647 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.447804527529935 0.89560905505987 0.552195472470065 18 0.583990078133595 0.83201984373281 0.416009921866405 19 0.435277228320462 0.870554456640924 0.564722771679538 20 0.340945084771609 0.681890169543217 0.659054915228391 21 0.387464418534525 0.77492883706905 0.612535581465475 22 0.339258891877328 0.678517783754655 0.660741108122672 23 0.34262851015636 0.68525702031272 0.65737148984364 24 0.249368607272741 0.498737214545482 0.750631392727259 25 0.19405674825848 0.38811349651696 0.80594325174152 26 0.140556183108285 0.28111236621657 0.859443816891715 27 0.105180997608933 0.210361995217865 0.894819002391067 28 0.105265382236193 0.210530764472385 0.894734617763807 29 0.112765941262931 0.225531882525863 0.887234058737069 30 0.0785894365074258 0.157178873014852 0.921410563492574 31 0.0529762688540275 0.105952537708055 0.947023731145972 32 0.0376950739447328 0.0753901478894655 0.962304926055267 33 0.0295504517020805 0.059100903404161 0.97044954829792 34 0.126861125154994 0.253722250309989 0.873138874845006 35 0.0993098049970323 0.198619609994065 0.900690195002968 36 0.130063495900371 0.260126991800742 0.869936504099629 37 0.0934100273988496 0.186820054797699 0.90658997260115 38 0.105800376035348 0.211600752070697 0.894199623964651 39 0.543594463567004 0.912811072865993 0.456405536432996 40 0.544556423333668 0.910887153332664 0.455443576666332 41 0.670487564426634 0.659024871146732 0.329512435573366 42 0.770336670687697 0.459326658624607 0.229663329312303 43 0.806124069769376 0.387751860461248 0.193875930230624 44 0.730008392690143 0.539983214619713 0.269991607309857 45 0.769764609107357 0.460470781785286 0.230235390892643 46 0.707136664336812 0.585726671326376 0.292863335663188 47 0.956503098953876 0.0869938020922485 0.0434969010461243 48 0.996275705744743 0.0074485885105133 0.00372429425525665 49 0.990541808518594 0.0189163829628118 0.00945819148140588 50 0.98110546553521 0.0377890689295791 0.0188945344647895 51 0.99365775329158 0.0126844934168385 0.00634224670841924 52 0.998071787552428 0.00385642489514316 0.00192821244757158 53 0.993052752671301 0.0138944946573979 0.00694724732869897 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 2 0.0540540540540541 NOK 5% type I error level 6 0.162162162162162 NOK 10% type I error level 9 0.243243243243243 NOK

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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