multiple regression

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

Title produced by software: Multiple Regression

Date of computation: Sat, 21 Nov 2009 03:11:45 -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/21/t1258798399y1ya9dmn1duxl99.htm/, Retrieved Sat, 21 Nov 2009 11:13:31 +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/21/t1258798399y1ya9dmn1duxl99.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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10,9 0 10 0 9,2 0 9,2 0 9,5 0 9,6 0 9,5 0 9,1 0 8,9 0 9 0 10,1 0 10,3 0 10,2 0 9,6 0 9,2 0 9,3 0 9,4 0 9,4 0 9,2 0 9 0 9 0 9 0 9,8 0 10 0 9,8 0 9,3 0 9 0 9 0 9,1 0 9,1 0 9,1 0 9,2 0 8,8 0 8,3 0 8,4 0 8,1 0 7,7 1 7,9 1 7,9 1 8 1 7,9 1 7,6 1 7,1 1 6,8 1 6,5 1 6,9 1 8,2 1 8,7 1 8,3 1 7,9 1 7,5 1 7,8 1 8,3 1 8,4 1 8,2 1 7,7 1 7,2 1 7,3 1 8,1 1 8,5 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 Y[t] = + 10.1488888888889 -1.04722222222222X[t] + 0.0736111111111097M1[t] -0.349444444444445M2[t] -0.712500000000001M3[t] -0.595555555555556M4[t] -0.398611111111111M5[t] -0.401666666666667M6[t] -0.584722222222223M7[t] -0.827777777777779M8[t] -1.09083333333333M9[t] -1.05388888888889M10[t] -0.216944444444445M11[t] -0.0169444444444444t + 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) 10.1488888888889 0.259067 39.1747 0 0 X -1.04722222222222 0.232649 -4.5013 4.6e-05 2.3e-05 M1 0.0736111111111097 0.288788 0.2549 0.799939 0.39997 M2 -0.349444444444445 0.287144 -1.217 0.229824 0.114912 M3 -0.712500000000001 0.285648 -2.4943 0.016274 0.008137 M4 -0.595555555555556 0.284302 -2.0948 0.041725 0.020862 M5 -0.398611111111111 0.28311 -1.408 0.165862 0.082931 M6 -0.401666666666667 0.282072 -1.424 0.161199 0.0806 M7 -0.584722222222223 0.281192 -2.0794 0.043178 0.021589 M8 -0.827777777777779 0.280469 -2.9514 0.004964 0.002482 M9 -1.09083333333333 0.279905 -3.8972 0.000314 0.000157 M10 -1.05388888888889 0.279502 -3.7706 0.000463 0.000232 M11 -0.216944444444445 0.27926 -0.7769 0.441221 0.220611 t -0.0169444444444444 0.006716 -2.523 0.015157 0.007578 Multiple Linear Regression - Regression Statistics Multiple R 0.912187758983396 R-squared 0.83208650763915 Adjusted R-squared 0.784632694580648 F-TEST (value) 17.5346606312405 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 1.25899290992493e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.441421178699925 Sum Squared Residuals 8.96322222222221 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 10.9 10.2055555555556 0.694444444444441 2 10 9.76555555555555 0.234444444444445 3 9.2 9.38555555555556 -0.185555555555555 4 9.2 9.48555555555556 -0.285555555555555 5 9.5 9.66555555555555 -0.165555555555555 6 9.6 9.64555555555556 -0.045555555555555 7 9.5 9.44555555555556 0.0544444444444445 8 9.1 9.18555555555556 -0.0855555555555562 9 8.9 8.90555555555556 -0.00555555555555533 10 9 8.92555555555555 0.0744444444444446 11 10.1 9.74555555555555 0.354444444444445 12 10.3 9.94555555555556 0.354444444444445 13 10.2 10.0022222222222 0.197777777777778 14 9.6 9.56222222222222 0.0377777777777769 15 9.2 9.18222222222222 0.0177777777777773 16 9.3 9.28222222222222 0.0177777777777785 17 9.4 9.46222222222222 -0.062222222222222 18 9.4 9.44222222222222 -0.0422222222222218 19 9.2 9.24222222222222 -0.0422222222222224 20 9 8.98222222222222 0.0177777777777783 21 9 8.70222222222222 0.297777777777777 22 9 8.72222222222222 0.277777777777778 23 9.8 9.54222222222222 0.257777777777779 24 10 9.74222222222222 0.257777777777777 25 9.8 9.79888888888889 0.00111111111111274 26 9.3 9.35888888888889 -0.058888888888888 27 9 8.97888888888889 0.0211111111111114 28 9 9.07888888888889 -0.078888888888889 29 9.1 9.25888888888889 -0.158888888888889 30 9.1 9.23888888888889 -0.138888888888889 31 9.1 9.03888888888889 0.0611111111111115 32 9.2 8.77888888888889 0.421111111111111 33 8.8 8.49888888888889 0.301111111111111 34 8.3 8.51888888888889 -0.218888888888888 35 8.4 9.33888888888889 -0.938888888888888 36 8.1 9.53888888888889 -1.43888888888889 37 7.7 8.54833333333333 -0.848333333333332 38 7.9 8.10833333333333 -0.208333333333333 39 7.9 7.72833333333333 0.171666666666667 40 8 7.82833333333333 0.171666666666667 41 7.9 8.00833333333333 -0.108333333333333 42 7.6 7.98833333333333 -0.388333333333334 43 7.1 7.78833333333333 -0.688333333333333 44 6.8 7.52833333333333 -0.728333333333333 45 6.5 7.24833333333333 -0.748333333333333 46 6.9 7.26833333333333 -0.368333333333333 47 8.2 8.08833333333333 0.111666666666666 48 8.7 8.28833333333333 0.411666666666666 49 8.3 8.345 -0.0449999999999986 50 7.9 7.905 -0.00499999999999963 51 7.5 7.525 -0.0249999999999998 52 7.8 7.625 0.175000000000000 53 8.3 7.805 0.495 54 8.4 7.785 0.615 55 8.2 7.585 0.615 56 7.7 7.325 0.375000000000000 57 7.2 7.045 0.155 58 7.3 7.065 0.235000000000000 59 8.1 7.885 0.215000000000000 60 8.5 8.085 0.414999999999999 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.165919387815706 0.331838775631411 0.834080612184294 18 0.0685061751915904 0.137012350383181 0.93149382480841 19 0.0262290532772150 0.0524581065544299 0.973770946722785 20 0.00979198653032462 0.0195839730606492 0.990208013469675 21 0.00614245053337344 0.0122849010667469 0.993857549466627 22 0.0029528367940419 0.0059056735880838 0.997047163205958 23 0.00156780423797609 0.00313560847595219 0.998432195762024 24 0.00105647946626048 0.00211295893252096 0.99894352053374 25 0.00212283929047071 0.00424567858094142 0.99787716070953 26 0.000924386387254971 0.00184877277450994 0.999075613612745 27 0.000434944552306247 0.000869889104612494 0.999565055447694 28 0.000162381389302793 0.000324762778605587 0.999837618610697 29 5.22380539453475e-05 0.000104476107890695 0.999947761946055 30 1.61578184231980e-05 3.23156368463960e-05 0.999983842181577 31 6.24553203688844e-06 1.24910640737769e-05 0.999993754467963 32 5.59775451129938e-05 0.000111955090225988 0.999944022454887 33 0.000496309892376454 0.00099261978475291 0.999503690107624 34 0.00605514100883994 0.0121102820176799 0.99394485899116 35 0.180552988747298 0.361105977494595 0.819447011252702 36 0.550808313321141 0.898383373357718 0.449191686678859 37 0.45819456186791 0.91638912373582 0.54180543813209 38 0.425613248313819 0.851226496627638 0.574386751686181 39 0.553941814790109 0.892116370419782 0.446058185209891 40 0.609837768876502 0.780324462246996 0.390162231123498 41 0.478165992107335 0.95633198421467 0.521834007892665 42 0.364951088790473 0.729902177580946 0.635048911209527 43 0.412886536950994 0.825773073901988 0.587113463049006 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 12 0.444444444444444 NOK 5% type I error level 15 0.555555555555556 NOK 10% type I error level 16 0.592592592592593 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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