*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, 20 Nov 2009 00:49:33 -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/20/t1258703435v2ajabyb0g2p8vj.htm/, Retrieved Fri, 20 Nov 2009 08:50:48 +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/20/t1258703435v2ajabyb0g2p8vj.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:

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Dataseries X:

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100.49 1.9 100.16 99.6 100.25 100.03 99.72 2 100.49 100.16 99.6 100.25 100.14 2.3 99.72 100.49 100.16 99.6 98.48 2.8 100.14 99.72 100.49 100.16 100.38 2.4 98.48 100.14 99.72 100.49 101.45 2.3 100.38 98.48 100.14 99.72 98.42 2.7 101.45 100.38 98.48 100.14 98.6 2.7 98.42 101.45 100.38 98.48 100.06 2.9 98.6 98.42 101.45 100.38 98.62 3 100.06 98.6 98.42 101.45 100.84 2.2 98.62 100.06 98.6 98.42 100.02 2.3 100.84 98.62 100.06 98.6 97.95 2.8 100.02 100.84 98.62 100.06 98.32 2.8 97.95 100.02 100.84 98.62 98.27 2.8 98.32 97.95 100.02 100.84 97.22 2.2 98.27 98.32 97.95 100.02 99.28 2.6 97.22 98.27 98.32 97.95 100.38 2.8 99.28 97.22 98.27 98.32 99.02 2.5 100.38 99.28 97.22 98.27 100.32 2.4 99.02 100.38 99.28 97.22 99.81 2.3 100.32 99.02 100.38 99.28 100.6 1.9 99.81 100.32 99.02 100.38 101.19 1.7 100.6 99.81 100.32 99.02 100.47 2 101.19 100.6 99.81 100.32 101.77 2.1 100.47 101.19 100.6 99.81 102.32 1.7 101.77 100.47 101.19 100.6 102.39 1.8 102.32 101.77 100.47 101.19 101.16 1.8 102.39 102.32 1 etc...

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 Y[t] = + 52.9237129088855 -0.456561313045817X[t] + 0.529784539853996Y1[t] + 0.00162859374509381Y2[t] + 0.211227162008023Y3[t] -0.268925741424594Y4[t] + 0.779715739410484M1[t] + 0.72084508465959M2[t] + 1.39416708889035M3[t] -0.251540990310942M4[t] + 1.76147716754322M5[t] + 1.11916991858972M6[t] -0.0997453594323863M7[t] + 0.207964494807340M8[t] + 1.42410413266541M9[t] + 1.02998559517508M10[t] + 1.50184215778441M11[t] -0.00838297611828369t + 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) 52.9237129088855 13.880394 3.8128 0.00049 0.000245 X -0.456561313045817 0.136277 -3.3502 0.001834 0.000917 Y1 0.529784539853996 0.150924 3.5103 0.001171 0.000585 Y2 0.00162859374509381 0.169852 0.0096 0.9924 0.4962 Y3 0.211227162008023 0.165712 1.2747 0.210167 0.105083 Y4 -0.268925741424594 0.139164 -1.9324 0.060785 0.030393 M1 0.779715739410484 0.631015 1.2357 0.224172 0.112086 M2 0.72084508465959 0.604075 1.1933 0.240147 0.120074 M3 1.39416708889035 0.62547 2.229 0.031805 0.015902 M4 -0.251540990310942 0.599557 -0.4195 0.677181 0.33859 M5 1.76147716754322 0.654977 2.6894 0.010573 0.005287 M6 1.11916991858972 0.607961 1.8409 0.07346 0.03673 M7 -0.0997453594323863 0.66993 -0.1489 0.882428 0.441214 M8 0.207964494807340 0.660939 0.3147 0.754748 0.377374 M9 1.42410413266541 0.661141 2.154 0.037645 0.018823 M10 1.02998559517508 0.684813 1.504 0.140838 0.070419 M11 1.50184215778441 0.655248 2.292 0.027535 0.013767 t -0.00838297611828369 0.008107 -1.0341 0.30764 0.15382 Multiple Linear Regression - Regression Statistics Multiple R 0.904112399567176 R-squared 0.817419231051118 Adjusted R-squared 0.735738360731881 F-TEST (value) 10.0074745513406 F-TEST (DF numerator) 17 F-TEST (DF denominator) 38 p-value 2.72499856013297e-09 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.871677777536568 Sum Squared Residuals 28.8732416183414 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 100.49 100.327887702781 0.162112297219314 2 99.72 100.194257532837 -0.474257532837242 3 100.14 100.607920449935 -0.467920449934754 4 98.48 98.865910775912 -0.385910775912100 5 100.38 99.9230217466653 0.456978253334675 6 101.45 100.617663041944 0.832336958055842 7 98.42 99.3141181480133 -0.894118148013252 8 98.6 98.8576888042644 -0.257688804264411 9 100.06 99.779613936163 0.280386063837036 10 98.62 99.1774660221021 -0.5574660221021 11 100.84 100.098542554186 0.741457445814154 12 100.02 99.9764228155368 0.043577184463196 13 97.95 99.3918683819685 -1.44186838196845 14 98.32 99.0828026740398 -0.762802674039766 15 98.27 99.1701693740367 -0.900169374036691 16 97.22 97.5474073418492 -0.32740734184915 17 99.28 99.4478931365246 -0.167893136524621 18 100.38 99.685472895083 0.694527104916929 19 99.02 98.9729186987735 0.0470813012264834 20 100.32 99.31668616935 1.00331383064994 21 99.81 100.934966827585 -1.12496682758545 22 100.6 99.8639296398403 0.736070360159702 23 101.19 101.476749011563 -0.286749011563036 24 100.47 99.686085634843 0.783914365156985 25 101.77 100.335299854558 1.43470014544176 26 102.32 101.050390753080 1.26960924691957 27 102.39 101.652422574590 0.737577425409654 28 101.16 100.504534008056 0.655465991943504 29 100.63 101.624219682587 -0.99421968258655 30 101.48 100.785897881166 0.694102118834376 31 101.44 99.7294191200466 1.71058087995335 32 100.09 100.227767187345 -0.137767187345252 33 100.7 101.087979438499 -0.387979438498780 34 100.78 100.678099663345 0.101900336655159 35 99.81 100.545300765718 -0.735300765718213 36 98.45 98.6936203162728 -0.24362031627277 37 98.49 98.5044075775133 -0.0144075775132797 38 97.48 98.0471015090648 -0.567101509064845 39 97.91 98.104956193221 -0.194956193220964 40 96.94 96.6859666547372 0.25403334526283 41 98.53 98.0902830587537 0.439716941246282 42 96.82 98.1862518815735 -1.36625188157353 43 95.76 95.415490193245 0.344509806754938 44 95.27 95.7015135892872 -0.431513589287205 45 97.32 96.0874397977528 1.23256020224719 46 96.68 96.9605046747128 -0.280504674712759 47 97.87 97.5894076685329 0.280592331467096 48 97.42 98.0038712333474 -0.583871233347412 49 97.94 98.0805364831793 -0.140536483179338 50 99.52 98.9854475309777 0.534552469022281 51 100.99 100.164531408217 0.825468591782755 52 99.92 100.116181219445 -0.196181219445085 53 101.97 101.704582375470 0.265417624530215 54 101.58 102.434714300234 -0.854714300233616 55 99.54 100.748053839922 -1.20805383992152 56 100.83 101.006344249753 -0.176344249753068 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.593475660422072 0.813048679155856 0.406524339577928 22 0.473498161661901 0.946996323323802 0.526501838338099 23 0.445663568857683 0.891327137715367 0.554336431142317 24 0.416825557649025 0.83365111529805 0.583174442350975 25 0.583074056802449 0.833851886395102 0.416925943197551 26 0.58369702741661 0.832605945166779 0.416302972583389 27 0.482413061559585 0.96482612311917 0.517586938440415 28 0.418984965646474 0.837969931292949 0.581015034353526 29 0.609556527386277 0.780886945227447 0.390443472613723 30 0.614564307178687 0.770871385642627 0.385435692821313 31 0.725467682453386 0.549064635093227 0.274532317546614 32 0.777811566458133 0.444376867083734 0.222188433541867 33 0.7781744549891 0.4436510900218 0.2218255450109 34 0.674735376359363 0.650529247281274 0.325264623640637 35 0.495918592245693 0.991837184491385 0.504081407754307 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 0 0 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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