*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, 17 Dec 2010 12:30:50 +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/17/t1292588995r6y6luq7dviedk5.htm/, Retrieved Fri, 17 Dec 2010 13:30:07 +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/17/t1292588995r6y6luq7dviedk5.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:

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No (this computation is public)

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Dataseries X:

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14,1 14,8 16,8 15,4 15,2 16,9 14,1 14,7 16,5 15,2 17,6 18 16,9 16,7 19,7 15,9 17,4 17,7 15,2 15,7 17,2 17,7 17,9 16,2 17,5 16,8 19,1 16,7 18,2 18,5 17,8 16,4 18 20,3 19,5 18 20,2 19 20,2 21,5 19,7 21,1 20,2 18,2 21,3 20,4 17,2 15,8 15,1 14,5 15,8 14,3 13,9 15,5 14,3 13,6 16,3 16,8 16 16,8 16

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 20 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation HPC[t] = + 16.5858823529412 -0.274705882352939M1[t] -0.496078431372549M2[t] + 1.45352941176471M3[t] -0.116862745098041M4[t] -0.00725490196078604M5[t] + 1.04235294117647M6[t] -0.588039215686275M7[t] -1.19843137254902M8[t] + 0.931176470588234M9[t] + 1.14078431372549M10[t] + 0.690392156862744M11[t] + 0.0103921568627451t + 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) 16.5858823529412 1.077422 15.394 0 0 M1 -0.274705882352939 1.256527 -0.2186 0.827871 0.413935 M2 -0.496078431372549 1.318858 -0.3761 0.708469 0.354234 M3 1.45352941176471 1.317174 1.1035 0.275303 0.137651 M4 -0.116862745098041 1.315665 -0.0888 0.929591 0.464796 M5 -0.00725490196078604 1.314333 -0.0055 0.995619 0.497809 M6 1.04235294117647 1.313177 0.7938 0.431239 0.21562 M7 -0.588039215686275 1.312198 -0.4481 0.656073 0.328037 M8 -1.19843137254902 1.311396 -0.9139 0.365359 0.182679 M9 0.931176470588234 1.310772 0.7104 0.480892 0.240446 M10 1.14078431372549 1.310327 0.8706 0.3883 0.19415 M11 0.690392156862744 1.310059 0.527 0.600626 0.300313 t 0.0103921568627451 0.015286 0.6798 0.499868 0.249934 Multiple Linear Regression - Regression Statistics Multiple R 0.401668044404094 R-squared 0.161337217895409 Adjusted R-squared -0.0483284776307384 F-TEST (value) 0.769497449215714 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 0.677733752311882 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 2.0712441400235 Sum Squared Residuals 205.922509803921 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 14.1 16.3215686274510 -2.22156862745096 2 14.8 16.1105882352941 -1.31058823529412 3 16.8 18.0705882352941 -1.27058823529412 4 15.4 16.5105882352941 -1.11058823529412 5 15.2 16.6305882352941 -1.43058823529412 6 16.9 17.6905882352941 -0.79058823529412 7 14.1 16.0705882352941 -1.97058823529412 8 14.7 15.4705882352941 -0.770588235294119 9 16.5 17.6105882352941 -1.11058823529412 10 15.2 17.8305882352941 -2.63058823529412 11 17.6 17.3905882352941 0.209411764705882 12 18 16.7105882352941 1.28941176470588 13 16.9 16.4462745098039 0.453725490196074 14 16.7 16.2352941176471 0.464705882352941 15 19.7 18.1952941176471 1.50470588235294 16 15.9 16.6352941176471 -0.735294117647057 17 17.4 16.7552941176471 0.644705882352941 18 17.7 17.8152941176471 -0.115294117647059 19 15.2 16.1952941176471 -0.99529411764706 20 15.7 15.5952941176471 0.104705882352941 21 17.2 17.7352941176471 -0.53529411764706 22 17.7 17.9552941176471 -0.255294117647060 23 17.9 17.5152941176471 0.384705882352940 24 16.2 16.8352941176471 -0.63529411764706 25 17.5 16.5709803921569 0.929019607843134 26 16.8 16.36 0.440000000000001 27 19.1 18.32 0.780000000000002 28 16.7 16.76 -0.0599999999999994 29 18.2 16.88 1.32 30 18.5 17.94 0.560000000000001 31 17.8 16.32 1.48 32 16.4 15.72 0.68 33 18 17.86 0.140000000000000 34 20.3 18.08 2.22 35 19.5 17.64 1.86 36 18 16.96 1.04 37 20.2 16.6956862745098 3.50431372549019 38 19 16.4847058823529 2.51529411764706 39 20.2 18.4447058823529 1.75529411764706 40 21.5 16.8847058823529 4.61529411764706 41 19.7 17.0047058823529 2.69529411764706 42 21.1 18.0647058823529 3.03529411764706 43 20.2 16.4447058823529 3.75529411764706 44 18.2 15.8447058823529 2.35529411764706 45 21.3 17.9847058823529 3.31529411764706 46 20.4 18.2047058823529 2.19529411764706 47 17.2 17.7647058823529 -0.564705882352941 48 15.8 17.0847058823529 -1.28470588235294 49 15.1 16.8203921568627 -1.72039215686275 50 14.5 16.6094117647059 -2.10941176470588 51 15.8 18.5694117647059 -2.76941176470588 52 14.3 17.0094117647059 -2.70941176470588 53 13.9 17.1294117647059 -3.22941176470588 54 15.5 18.1894117647059 -2.68941176470588 55 14.3 16.5694117647059 -2.26941176470588 56 13.6 15.9694117647059 -2.36941176470588 57 16.3 18.1094117647059 -1.80941176470588 58 16.8 18.3294117647059 -1.52941176470588 59 16 17.8894117647059 -1.88941176470588 60 16.8 17.2094117647059 -0.409411764705882 61 16 16.9450980392157 -0.945098039215689 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0606408298448451 0.121281659689690 0.939359170155155 17 0.0177174203300013 0.0354348406600026 0.982282579669999 18 0.0095906467056371 0.0191812934112742 0.990409353294363 19 0.00407368326882022 0.00814736653764044 0.99592631673118 20 0.00154286384720155 0.00308572769440310 0.998457136152799 21 0.000786585040393174 0.00157317008078635 0.999213414959607 22 0.000455958459278312 0.000911916918556623 0.999544041540722 23 0.000332438699032767 0.000664877398065534 0.999667561300967 24 0.00399882813211151 0.00799765626422302 0.996001171867889 25 0.00195754601807304 0.00391509203614609 0.998042453981927 26 0.00109630225581004 0.00219260451162007 0.99890369774419 27 0.000563854155307832 0.00112770831061566 0.999436145844692 28 0.000425264042379703 0.000850528084759407 0.99957473595762 29 0.000178635713432628 0.000357271426865256 0.999821364286567 30 0.000100124345176611 0.000200248690353222 0.999899875654823 31 0.000132882855346331 0.000265765710692662 0.999867117144654 32 9.37031650326252e-05 0.000187406330065250 0.999906296834967 33 0.000212310359887802 0.000424620719775603 0.999787689640112 34 0.000728157730085439 0.00145631546017088 0.999271842269915 35 0.000414829161840004 0.000829658323680007 0.99958517083816 36 0.000773758513363871 0.00154751702672774 0.999226241486636 37 0.000816934133353555 0.00163386826670711 0.999183065866647 38 0.000366126798481573 0.000732253596963145 0.999633873201518 39 0.000180162125192098 0.000360324250384195 0.999819837874808 40 0.00219128710246036 0.00438257420492072 0.99780871289754 41 0.00194553783951210 0.00389107567902419 0.998054462160488 42 0.00217396619724706 0.00434793239449413 0.997826033802753 43 0.00751017559503252 0.0150203511900650 0.992489824404967 44 0.010015526817245 0.02003105363449 0.989984473182755 45 0.0691107529496977 0.138221505899395 0.930889247050302 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 24 0.8 NOK 5% type I error level 28 0.933333333333333 NOK 10% type I error level 28 0.933333333333333 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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