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: Sat, 21 Nov 2009 03:00:03 -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/t1258797790pchucdr3dut0rp9.htm/, Retrieved Sat, 21 Nov 2009 11:03:22 +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/t1258797790pchucdr3dut0rp9.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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No (this computation is public)

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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] = + 9.74222222222222 -1.55555555555556X[t] + 0.259999999999998M1[t] -0.180000000000001M2[t] -0.560000000000001M3[t] -0.460000000000001M4[t] -0.28M5[t] -0.3M6[t] -0.500000000000001M7[t] -0.760000000000001M8[t] -1.04M9[t] -1.02M10[t] -0.200000000000000M11[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) 9.74222222222222 0.214083 45.5067 0 0 X -1.55555555555556 0.122785 -12.6689 0 0 M1 0.259999999999998 0.294684 0.8823 0.382105 0.191052 M2 -0.180000000000001 0.294684 -0.6108 0.544259 0.27213 M3 -0.560000000000001 0.294684 -1.9003 0.063532 0.031766 M4 -0.460000000000001 0.294684 -1.561 0.125234 0.062617 M5 -0.28 0.294684 -0.9502 0.346887 0.173443 M6 -0.3 0.294684 -1.018 0.313871 0.156936 M7 -0.500000000000001 0.294684 -1.6967 0.096362 0.048181 M8 -0.760000000000001 0.294684 -2.579 0.013098 0.006549 M9 -1.04 0.294684 -3.5292 0.000944 0.000472 M10 -1.02 0.294684 -3.4613 0.001155 0.000577 M11 -0.200000000000000 0.294684 -0.6787 0.500661 0.250331 Multiple Linear Regression - Regression Statistics Multiple R 0.89936121167632 R-squared 0.8088505890679 Adjusted R-squared 0.760046484149065 F-TEST (value) 16.5734130441095 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 5.09037256790634e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.465936612334712 Sum Squared Residuals 10.2035555555555 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 10.9 10.0022222222222 0.897777777777774 2 10 9.56222222222222 0.437777777777779 3 9.2 9.18222222222222 0.0177777777777778 4 9.2 9.28222222222222 -0.0822222222222222 5 9.5 9.46222222222222 0.0377777777777778 6 9.6 9.44222222222222 0.157777777777778 7 9.5 9.24222222222222 0.257777777777778 8 9.1 8.98222222222222 0.117777777777777 9 8.9 8.70222222222222 0.197777777777778 10 9 8.72222222222222 0.277777777777778 11 10.1 9.54222222222222 0.557777777777778 12 10.3 9.74222222222222 0.557777777777778 13 10.2 10.0022222222222 0.197777777777778 14 9.6 9.56222222222222 0.0377777777777772 15 9.2 9.18222222222222 0.0177777777777771 16 9.3 9.28222222222222 0.0177777777777784 17 9.4 9.46222222222222 -0.0622222222222221 18 9.4 9.44222222222222 -0.042222222222222 19 9.2 9.24222222222222 -0.0422222222222225 20 9 8.98222222222222 0.0177777777777782 21 9 8.70222222222222 0.297777777777777 22 9 8.72222222222222 0.277777777777778 23 9.8 9.54222222222222 0.257777777777778 24 10 9.74222222222222 0.257777777777777 25 9.8 10.0022222222222 -0.202222222222221 26 9.3 9.56222222222222 -0.262222222222222 27 9 9.18222222222222 -0.182222222222222 28 9 9.28222222222222 -0.282222222222222 29 9.1 9.46222222222222 -0.362222222222223 30 9.1 9.44222222222222 -0.342222222222222 31 9.1 9.24222222222222 -0.142222222222222 32 9.2 8.98222222222222 0.217777777777777 33 8.8 8.70222222222222 0.097777777777778 34 8.3 8.72222222222222 -0.422222222222222 35 8.4 9.54222222222222 -1.14222222222222 36 8.1 9.74222222222222 -1.64222222222222 37 7.7 8.44666666666667 -0.746666666666666 38 7.9 8.00666666666667 -0.106666666666667 39 7.9 7.62666666666667 0.273333333333334 40 8 7.72666666666667 0.273333333333333 41 7.9 7.90666666666667 -0.00666666666666656 42 7.6 7.88666666666667 -0.286666666666667 43 7.1 7.68666666666667 -0.586666666666667 44 6.8 7.42666666666667 -0.626666666666666 45 6.5 7.14666666666667 -0.646666666666667 46 6.9 7.16666666666667 -0.266666666666667 47 8.2 7.98666666666667 0.213333333333332 48 8.7 8.18666666666667 0.513333333333332 49 8.3 8.44666666666667 -0.146666666666665 50 7.9 8.00666666666667 -0.106666666666667 51 7.5 7.62666666666667 -0.126666666666667 52 7.8 7.72666666666667 0.073333333333333 53 8.3 7.90666666666667 0.393333333333334 54 8.4 7.88666666666667 0.513333333333333 55 8.2 7.68666666666667 0.513333333333333 56 7.7 7.42666666666667 0.273333333333334 57 7.2 7.14666666666667 0.0533333333333332 58 7.3 7.16666666666667 0.133333333333333 59 8.1 7.98666666666667 0.113333333333333 60 8.5 8.18666666666667 0.313333333333333 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.296795934498721 0.593591868997443 0.703204065501279 17 0.157223714595178 0.314447429190356 0.842776285404822 18 0.0832762897115845 0.166552579423169 0.916723710288415 19 0.0493058207308733 0.0986116414617466 0.950694179269127 20 0.0223113841723108 0.0446227683446216 0.97768861582769 21 0.0107371767169032 0.0214743534338064 0.989262823283097 22 0.00492995525732523 0.00985991051465047 0.995070044742675 23 0.00364665574174569 0.00729331148349138 0.996353344258254 24 0.00291428712782469 0.00582857425564938 0.997085712872175 25 0.0171368642807347 0.0342737285614693 0.982863135719265 26 0.0185023627501214 0.0370047255002428 0.981497637249879 27 0.0106660351132213 0.0213320702264426 0.989333964886779 28 0.00617719985943318 0.0123543997188664 0.993822800140567 29 0.00416471086765328 0.00832942173530657 0.995835289132347 30 0.00304849331772107 0.00609698663544213 0.99695150668228 31 0.00188902697749817 0.00377805395499633 0.998110973022502 32 0.00245889535961065 0.0049177907192213 0.99754110464039 33 0.00988674278497403 0.0197734855699481 0.990113257215026 34 0.0605196399718852 0.121039279943770 0.939480360028115 35 0.383818706174033 0.767637412348067 0.616181293825967 36 0.726581286625815 0.546837426748371 0.273418713374185 37 0.688176748787356 0.623646502425288 0.311823251212644 38 0.611986587153915 0.776026825692171 0.388013412846086 39 0.569965753270924 0.860068493458153 0.430034246729076 40 0.480333018036403 0.960666036072806 0.519666981963597 41 0.386507786179606 0.773015572359211 0.613492213820394 42 0.378783406159027 0.757566812318053 0.621216593840973 43 0.54977526315652 0.90044947368696 0.45022473684348 44 0.69769186437076 0.604616271258481 0.302308135629240 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 7 0.241379310344828 NOK 5% type I error level 14 0.482758620689655 NOK 10% type I error level 15 0.517241379310345 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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