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: Fri, 24 Dec 2010 10:19:14 +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/24/t1293185880kucvcih4i9xe346.htm/, Retrieved Fri, 24 Dec 2010 11:18:03 +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/24/t1293185880kucvcih4i9xe346.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 «

25 0 29 0 28 0 25 0 26 0 24 0 28 0 28 0 28 0 28 0 32 0 31 0 22 0 29 0 31 0 29 0 32 0 32 0 31 0 29 0 28 0 28 0 29 0 22 0 26 0 24 0 27 0 27 0 23 0 21 0 19 0 17 0 19 1 21 1 13 1 8 1 5 1 10 1 6 1 6 1 8 1 11 1 12 1 13 1 19 1 19 1 18 1 20 1 15 1 15 1 15 1 17 1 22 1 17 1 21 1 23 1 26 1 26 1 28 1 30 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 Y[t] = + 26.8125 -10.2767857142857X[t] + 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) 26.8125 0.958694 27.9677 0 0 X -10.2767857142857 1.403385 -7.3229 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.693108686292623 R-squared 0.480399651014285 Adjusted R-squared 0.471441024307635 F-TEST (value) 53.6242514332772 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 8.403331364093e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.42319298637691 Sum Squared Residuals 1705.83928571429 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 25 26.8125 -1.81250000000003 2 29 26.8125 2.18750000000001 3 28 26.8125 1.1875 4 25 26.8125 -1.8125 5 26 26.8125 -0.812499999999999 6 24 26.8125 -2.8125 7 28 26.8125 1.1875 8 28 26.8125 1.1875 9 28 26.8125 1.1875 10 28 26.8125 1.1875 11 32 26.8125 5.1875 12 31 26.8125 4.1875 13 22 26.8125 -4.8125 14 29 26.8125 2.1875 15 31 26.8125 4.1875 16 29 26.8125 2.1875 17 32 26.8125 5.1875 18 32 26.8125 5.1875 19 31 26.8125 4.1875 20 29 26.8125 2.1875 21 28 26.8125 1.1875 22 28 26.8125 1.1875 23 29 26.8125 2.1875 24 22 26.8125 -4.8125 25 26 26.8125 -0.812499999999999 26 24 26.8125 -2.8125 27 27 26.8125 0.187500000000001 28 27 26.8125 0.187500000000001 29 23 26.8125 -3.8125 30 21 26.8125 -5.8125 31 19 26.8125 -7.8125 32 17 26.8125 -9.8125 33 19 16.5357142857143 2.46428571428571 34 21 16.5357142857143 4.46428571428571 35 13 16.5357142857143 -3.53571428571429 36 8 16.5357142857143 -8.53571428571429 37 5 16.5357142857143 -11.5357142857143 38 10 16.5357142857143 -6.53571428571429 39 6 16.5357142857143 -10.5357142857143 40 6 16.5357142857143 -10.5357142857143 41 8 16.5357142857143 -8.53571428571429 42 11 16.5357142857143 -5.53571428571429 43 12 16.5357142857143 -4.53571428571429 44 13 16.5357142857143 -3.53571428571429 45 19 16.5357142857143 2.46428571428571 46 19 16.5357142857143 2.46428571428571 47 18 16.5357142857143 1.46428571428571 48 20 16.5357142857143 3.46428571428571 49 15 16.5357142857143 -1.53571428571429 50 15 16.5357142857143 -1.53571428571429 51 15 16.5357142857143 -1.53571428571429 52 17 16.5357142857143 0.464285714285714 53 22 16.5357142857143 5.46428571428571 54 17 16.5357142857143 0.464285714285714 55 21 16.5357142857143 4.46428571428571 56 23 16.5357142857143 6.46428571428571 57 26 16.5357142857143 9.46428571428571 58 26 16.5357142857143 9.46428571428571 59 28 16.5357142857143 11.4642857142857 60 30 16.5357142857143 13.4642857142857 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.063005415896032 0.126010831792064 0.936994584103968 6 0.035760914901183 0.0715218298023661 0.964239085098817 7 0.0155757620529261 0.0311515241058522 0.984424237947074 8 0.00624208019889612 0.0124841603977922 0.993757919801104 9 0.00232623729068713 0.00465247458137426 0.997673762709313 10 0.000812055623307434 0.00162411124661487 0.999187944376693 11 0.0030151703755294 0.00603034075105881 0.99698482962447 12 0.00284378774556889 0.00568757549113778 0.997156212254431 13 0.00640102034055652 0.012802040681113 0.993598979659443 14 0.00330533321904937 0.00661066643809874 0.99669466678095 15 0.00280788517048332 0.00561577034096665 0.997192114829517 16 0.00137641468237527 0.00275282936475053 0.998623585317625 17 0.00157847630414347 0.00315695260828694 0.998421523695857 18 0.00165313192287955 0.00330626384575909 0.99834686807712 19 0.00121427285101439 0.00242854570202877 0.998785727148986 20 0.00061382297575724 0.00122764595151448 0.999386177024243 21 0.000291218827602133 0.000582437655204267 0.999708781172398 22 0.000136864109847651 0.000273728219695302 0.999863135890152 23 6.99821977393661e-05 0.000139964395478732 0.99993001780226 24 0.000182160752444697 0.000364321504889394 0.999817839247555 25 0.000100675883756273 0.000201351767512545 0.999899324116244 26 8.40566353535033e-05 0.000168113270707007 0.999915943364647 27 4.39044309983566e-05 8.78088619967132e-05 0.999956095569002 28 2.49016488668296e-05 4.98032977336593e-05 0.999975098351133 29 2.922088835233e-05 5.844177670466e-05 0.999970779111648 30 6.42600279423556e-05 0.000128520055884711 0.999935739972058 31 0.000230155506537021 0.000460311013074042 0.999769844493463 32 0.00103444480257762 0.00206888960515524 0.998965555197422 33 0.000551866566893975 0.00110373313378795 0.999448133433106 34 0.000333236890420668 0.000666473780841337 0.99966676310958 35 0.000340502789909968 0.000681005579819936 0.99965949721009 36 0.0010146996479211 0.0020293992958422 0.998985300352079 37 0.00514898003326687 0.0102979600665337 0.994851019966733 38 0.00495829377570153 0.00991658755140306 0.995041706224298 39 0.0136717119971305 0.0273434239942611 0.98632828800287 40 0.0403333404353633 0.0806666808707266 0.959666659564637 41 0.081896383365469 0.163792766730938 0.918103616634531 42 0.110148854397526 0.220297708795053 0.889851145602474 43 0.144898821810487 0.289797643620974 0.855101178189513 44 0.186143719972373 0.372287439944747 0.813856280027627 45 0.183036255514708 0.366072511029416 0.816963744485292 46 0.16888291322515 0.3377658264503 0.83111708677485 47 0.149273774682469 0.298547549364939 0.85072622531753 48 0.127970883766255 0.255941767532509 0.872029116233745 49 0.14260849643723 0.285216992874461 0.85739150356277 50 0.182275356153714 0.364550712307428 0.817724643846286 51 0.283996007414285 0.567992014828569 0.716003992585715 52 0.380291406893206 0.760582813786412 0.619708593106794 53 0.325923099008489 0.651846198016979 0.67407690099151 54 0.607233227982029 0.785533544035942 0.392766772017971 55 0.716646766037404 0.566706467925192 0.283353233962596 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 28 0.549019607843137 NOK 5% type I error level 33 0.647058823529412 NOK 10% type I error level 35 0.686274509803922 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No 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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