WS 7: Multiple 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: Thu, 26 Nov 2009 10:58:25 -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/26/t1259258681in8k28zdnsi5tyf.htm/, Retrieved Thu, 26 Nov 2009 19:04:53 +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/26/t1259258681in8k28zdnsi5tyf.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:

Aanpassing van de y-waarde

Dataseries X:

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7.8 9.5 7.6 7.5 7.7 8.1 7.8 9.6 7.8 7.6 7.5 7.7 7.8 9.5 7.8 7.8 7.6 7.5 7.5 9.1 7.8 7.8 7.8 7.6 7.5 8.9 7.5 7.8 7.8 7.8 7.1 9 7.5 7.5 7.8 7.8 7.5 10.1 7.1 7.5 7.5 7.8 7.5 10.3 7.5 7.1 7.5 7.5 7.6 10.2 7.5 7.5 7.1 7.5 7.7 9.6 7.6 7.5 7.5 7.1 7.7 9.2 7.7 7.6 7.5 7.5 7.9 9.3 7.7 7.7 7.6 7.5 8.1 9.4 7.9 7.7 7.7 7.6 8.2 9.4 8.1 7.9 7.7 7.7 8.2 9.2 8.2 8.1 7.9 7.7 8.2 9 8.2 8.2 8.1 7.9 7.9 9 8.2 8.2 8.2 8.1 7.3 9 7.9 8.2 8.2 8.2 6.9 9.8 7.3 7.9 8.2 8.2 6.6 10 6.9 7.3 7.9 8.2 6.7 9.8 6.6 6.9 7.3 7.9 6.9 9.3 6.7 6.6 6.9 7.3 7 9 6.9 6.7 6.6 6.9 7.1 9 7 6.9 6.7 6.6 7.2 9.1 7.1 7 6.9 6.7 7.1 9.1 7.2 7.1 7 6.9 6.9 9.1 7.1 7.2 7.1 7 7 9.2 6.9 7.1 7.2 7.1 6.8 8.8 7 6.9 7.1 7.2 6.4 8.3 6.8 7 6.9 7.1 6.7 8.4 6.4 6.8 7 6.9 6.6 8.1 6.7 6.4 6.8 7 6.4 7.7 6.6 6.7 6.4 6.8 6.3 7.9 6.4 6.6 6.7 6.4 6.2 7.9 6.3 6.4 6.6 6.7 6.5 8 6.2 6.3 6.4 6.6 6.8 7.9 6.5 6.2 6.3 6.4 6.8 7.6 6.8 6.5 6.2 6.3 6.4 7.1 6.8 6.8 6.5 6.2 6.1 6.8 6.4 6.8 6.8 6.5 5.8 6.5 6.1 6.4 6.8 6.8 6.1 6.9 5.8 6.1 6.4 6.8 7.2 8. 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] = -0.229569723769742 + 0.0479138564134236X[t] + 1.50956884936896Y1[t] -0.717207139860154Y2[t] -0.234146085893884Y3[t] + 0.425498249765415Y4[t] -0.0822653962756976M1[t] -0.260023264689058M2[t] -0.178609712794115M3[t] -0.118255040027475M4[t] -0.292381316010964M5[t] -0.336368271590958M6[t] + 0.144796624321569M7[t] -0.618739470511963M8[t] -0.331452155750824M9[t] -0.174684003109844M10[t] -0.253211771925874M11[t] + 0.00493334888911443t + 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) -0.229569723769742 0.673358 -0.3409 0.735032 0.367516 X 0.0479138564134236 0.070483 0.6798 0.500759 0.250379 Y1 1.50956884936896 0.154082 9.7972 0 0 Y2 -0.717207139860154 0.286072 -2.5071 0.016567 0.008284 Y3 -0.234146085893884 0.28879 -0.8108 0.42254 0.21127 Y4 0.425498249765415 0.161585 2.6333 0.012161 0.006081 M1 -0.0822653962756976 0.143541 -0.5731 0.569944 0.284972 M2 -0.260023264689058 0.148693 -1.7487 0.088414 0.044207 M3 -0.178609712794115 0.149902 -1.1915 0.240843 0.120421 M4 -0.118255040027475 0.149977 -0.7885 0.435304 0.217652 M5 -0.292381316010964 0.152041 -1.923 0.061992 0.030996 M6 -0.336368271590958 0.150705 -2.232 0.03159 0.015795 M7 0.144796624321569 0.142972 1.0128 0.317578 0.158789 M8 -0.618739470511963 0.156864 -3.9444 0.000333 0.000166 M9 -0.331452155750824 0.178646 -1.8554 0.071314 0.035657 M10 -0.174684003109844 0.158005 -1.1056 0.275867 0.137934 M11 -0.253211771925874 0.147162 -1.7206 0.093451 0.046726 t 0.00493334888911443 0.003294 1.4978 0.14245 0.071225 Multiple Linear Regression - Regression Statistics Multiple R 0.966040857003575 R-squared 0.933234937400201 Adjusted R-squared 0.903366356763449 F-TEST (value) 31.2447032133792 F-TEST (DF numerator) 17 F-TEST (DF denominator) 38 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.206139248157389 Sum Squared Residuals 1.61474880597395 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7.8 7.8855605327411 -0.0855605327411074 2 7.8 7.82435037201857 -0.0243503720185753 3 7.8 7.65395020064679 0.146049799353210 4 7.5 7.69579328753494 -0.195793287534939 5 7.5 7.14924658430028 0.350753415699724 6 7.1 7.33014650520879 -0.230146505208786 7 7.5 7.33536627808577 0.164633721914228 8 7.5 7.34940722418606 0.150592775813939 9 7.6 7.44361208060846 0.156387919391535 10 7.7 7.46366441896368 0.236335581036319 11 7.7 7.62033992732844 0.0796600726715576 12 7.9 7.78814111120937 0.111858888790630 13 8.1 8.03664943572507 0.0633505642749256 14 8.2 8.06484708307913 0.135152916920870 15 8.2 8.10229745236659 0.0977025476334083 16 8.2 8.12455242152795 0.0754475784720476 17 7.9 8.01704453579727 -0.117044535797272 18 7.3 7.56767009927225 -0.267670099272247 19 6.9 7.4015202615413 -0.501520261541295 20 6.6 6.54924085681624 0.0507591431837624 21 6.7 6.67872912692389 0.0212708730761152 22 6.9 7.02095221164052 -0.120952211640516 23 7 7.06322121653935 -0.063221216539349 24 7.1 7.17781771080019 -0.0778177108001891 25 7.2 7.18023382780359 0.0197661721964073 26 7.1 7.14833052059392 -0.0483305205939237 27 6.9 7.03113503884222 -0.131135038842221 28 7 6.8901566066387 0.109843393361305 29 6.8 7.06216088345381 -0.262160883453809 30 6.4 6.62979525689864 -0.229795256898644 31 6.7 6.5517845170236 0.148215482976396 32 6.6 6.60794016706523 -0.00794016706522784 33 6.4 6.52343504565964 -0.123435045659639 34 6.3 6.22408313691031 0.0759168630896873 35 6.2 6.29403734353754 -0.0940373435375424 36 6.5 6.48201707124523 0.0179829287547712 37 6.8 6.86279996565031 -0.0627999656503122 38 6.8 6.89417458566753 -0.0941745856675275 39 6.4 6.62860876554212 -0.228608765542120 40 6.1 6.13310073968772 -0.0331007396877239 41 5.8 5.91119533173232 -0.11119533173232 42 6.1 5.74725718911172 0.352742810888278 43 7.2 6.86372076988155 0.336279230118454 44 7.3 7.51703289533019 -0.217032895330191 45 6.9 6.95422374680801 -0.0542237468080115 46 6.1 6.29130023248549 -0.191300232485491 47 5.8 5.72240151259467 0.0775984874053338 48 6.2 6.25202410674521 -0.0520241067452122 49 7.1 7.03475623807991 0.0652437619200868 50 7.7 7.66829743864084 0.0317025613591568 51 7.9 7.78400854260228 0.115991457397723 52 7.7 7.65639694461069 0.0436030553893108 53 7.4 7.26035266471632 0.139647335283677 54 7.5 7.1251309495086 0.374869050491399 55 8 8.14760817346778 -0.147608173467783 56 8.1 8.07637885660228 0.0236211433977181 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.32541475443312 0.65082950886624 0.67458524556688 22 0.820253800721795 0.359492398556409 0.179746199278205 23 0.766792812232145 0.466414375535709 0.233207187767855 24 0.711626308549592 0.576747382900816 0.288373691450408 25 0.599737295415911 0.800525409168177 0.400262704584089 26 0.470884944898182 0.941769889796364 0.529115055101818 27 0.344281758253805 0.68856351650761 0.655718241746195 28 0.360391591639626 0.720783183279253 0.639608408360373 29 0.266292630237592 0.532585260475185 0.733707369762408 30 0.61284970530034 0.774300589399321 0.387150294699660 31 0.900614048123803 0.198771903752395 0.0993859518761975 32 0.962177842371953 0.0756443152560946 0.0378221576280473 33 0.930751491949068 0.138497016101864 0.069248508050932 34 0.937233358996566 0.125533282006868 0.0627666410034339 35 0.866462961408853 0.267074077182294 0.133537038591147 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 0.0666666666666667 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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