*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, 19 Nov 2009 09:13:55 -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/19/t125864771516jz2q41uelqjvx.htm/, Retrieved Thu, 19 Nov 2009 17:22:07 +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/2009/Nov/19/t125864771516jz2q41uelqjvx.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:

Dataseries X:

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56.6 0 56 0 54.8 0 52.7 0 50.9 0 50.6 0 52.1 0 53.3 0 53.9 0 54.3 0 54.2 0 54.2 0 53.5 0 51.4 0 50.5 0 50.3 0 49.8 0 50.7 0 52.8 0 55.3 0 57.3 0 57.5 0 56.8 0 56.4 0 56.3 0 56.4 0 57 0 57.9 0 58.9 0 58.8 0 56.5 1 51.9 1 47.4 1 44.9 1 43.9 1 43.4 1 42.9 1 42.6 1 42.2 1 41.2 1 40.2 1 39.3 1 38.5 1 38.3 1 37.9 1 37.6 1 37.3 1 36 1 34.5 1 33.5 1 32.9 1 32.9 1 32.8 1 31.9 1 30.5 1 29.2 1 28.7 1 28.4 1 28 1 27.4 1 26.9 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] = + 54.3733333333333 -16.9668817204301X[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) 54.3733333333333 1.015072 53.566 0 0 X -16.9668817204301 1.423905 -11.9157 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.840503178770167 R-squared 0.706445593522755 Adjusted R-squared 0.701470095107886 F-TEST (value) 141.98488967691 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.55977894991641 Sum Squared Residuals 1823.75737634409 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 56.6 54.3733333333334 2.22666666666663 2 56 54.3733333333334 1.62666666666664 3 54.8 54.3733333333333 0.426666666666668 4 52.7 54.3733333333333 -1.67333333333333 5 50.9 54.3733333333333 -3.47333333333333 6 50.6 54.3733333333333 -3.77333333333333 7 52.1 54.3733333333333 -2.27333333333333 8 53.3 54.3733333333333 -1.07333333333333 9 53.9 54.3733333333333 -0.473333333333331 10 54.3 54.3733333333333 -0.0733333333333325 11 54.2 54.3733333333333 -0.173333333333327 12 54.2 54.3733333333333 -0.173333333333327 13 53.5 54.3733333333333 -0.87333333333333 14 51.4 54.3733333333333 -2.97333333333333 15 50.5 54.3733333333333 -3.87333333333333 16 50.3 54.3733333333333 -4.07333333333333 17 49.8 54.3733333333333 -4.57333333333333 18 50.7 54.3733333333333 -3.67333333333333 19 52.8 54.3733333333333 -1.57333333333333 20 55.3 54.3733333333333 0.926666666666668 21 57.3 54.3733333333333 2.92666666666667 22 57.5 54.3733333333333 3.12666666666667 23 56.8 54.3733333333333 2.42666666666667 24 56.4 54.3733333333333 2.02666666666667 25 56.3 54.3733333333333 1.92666666666667 26 56.4 54.3733333333333 2.02666666666667 27 57 54.3733333333333 2.62666666666667 28 57.9 54.3733333333333 3.52666666666667 29 58.9 54.3733333333333 4.52666666666667 30 58.8 54.3733333333333 4.42666666666667 31 56.5 37.4064516129032 19.0935483870968 32 51.9 37.4064516129032 14.4935483870968 33 47.4 37.4064516129032 9.99354838709677 34 44.9 37.4064516129032 7.49354838709677 35 43.9 37.4064516129032 6.49354838709677 36 43.4 37.4064516129032 5.99354838709677 37 42.9 37.4064516129032 5.49354838709677 38 42.6 37.4064516129032 5.19354838709678 39 42.2 37.4064516129032 4.79354838709678 40 41.2 37.4064516129032 3.79354838709678 41 40.2 37.4064516129032 2.79354838709678 42 39.3 37.4064516129032 1.89354838709677 43 38.5 37.4064516129032 1.09354838709677 44 38.3 37.4064516129032 0.893548387096772 45 37.9 37.4064516129032 0.493548387096774 46 37.6 37.4064516129032 0.193548387096776 47 37.3 37.4064516129032 -0.106451612903228 48 36 37.4064516129032 -1.40645161290322 49 34.5 37.4064516129032 -2.90645161290322 50 33.5 37.4064516129032 -3.90645161290323 51 32.9 37.4064516129032 -4.50645161290323 52 32.9 37.4064516129032 -4.50645161290323 53 32.8 37.4064516129032 -4.60645161290323 54 31.9 37.4064516129032 -5.50645161290323 55 30.5 37.4064516129032 -6.90645161290323 56 29.2 37.4064516129032 -8.20645161290322 57 28.7 37.4064516129032 -8.70645161290322 58 28.4 37.4064516129032 -9.00645161290323 59 28 37.4064516129032 -9.40645161290322 60 27.4 37.4064516129032 -10.0064516129032 61 26.9 37.4064516129032 -10.5064516129032 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.121299662546934 0.242599325093867 0.878700337453066 6 0.0898063497553399 0.179612699510680 0.91019365024466 7 0.0414271438322237 0.0828542876644475 0.958572856167776 8 0.0159369439833210 0.0318738879666421 0.98406305601668 9 0.0057933867662332 0.0115867735324664 0.994206613233767 10 0.00206298953393168 0.00412597906786335 0.997937010466068 11 0.000682080619245045 0.00136416123849009 0.999317919380755 12 0.000213174742952776 0.000426349485905551 0.999786825257047 13 6.11106642435872e-05 0.000122221328487174 0.999938889335756 14 3.26899552568827e-05 6.53799105137653e-05 0.999967310044743 15 2.79180701328672e-05 5.58361402657344e-05 0.999972081929867 16 2.32641688351796e-05 4.65283376703593e-05 0.999976735831165 17 2.35247904665053e-05 4.70495809330106e-05 0.999976475209533 18 1.33556335352777e-05 2.67112670705553e-05 0.999986644366465 19 4.54074728775392e-06 9.08149457550784e-06 0.999995459252712 20 2.64244200755979e-06 5.28488401511957e-06 0.999997357557992 21 4.51230246042079e-06 9.02460492084159e-06 0.99999548769754 22 6.32424340335339e-06 1.26484868067068e-05 0.999993675756597 23 4.98905659150479e-06 9.97811318300958e-06 0.999995010943408 24 3.06602478413377e-06 6.13204956826754e-06 0.999996933975216 25 1.72277026116238e-06 3.44554052232475e-06 0.99999827722974 26 9.56218870911934e-07 1.91243774182387e-06 0.999999043781129 27 6.25272822517773e-07 1.25054564503555e-06 0.999999374727177 28 5.5297751214059e-07 1.10595502428118e-06 0.999999447022488 29 6.95959703083132e-07 1.39191940616626e-06 0.999999304040297 30 7.06685207995336e-07 1.41337041599067e-06 0.999999293314792 31 4.64545896062296e-06 9.29091792124591e-06 0.99999535454104 32 2.84073038122633e-05 5.68146076245266e-05 0.999971592696188 33 0.000171469558112041 0.000342939116224083 0.999828530441888 34 0.00073032210733163 0.00146064421466326 0.999269677892668 35 0.00208228967877776 0.00416457935755552 0.997917710321222 36 0.00476356440876406 0.0095271288175281 0.995236435591236 37 0.00987574193148844 0.0197514838629769 0.990124258068512 38 0.0198158592648723 0.0396317185297447 0.980184140735128 39 0.0396881018060233 0.0793762036120467 0.960311898193977 40 0.074777743166435 0.14955548633287 0.925222256833565 41 0.129055642368454 0.258111284736908 0.870944357631546 42 0.202458856797026 0.404917713594051 0.797541143202974 43 0.289884923799045 0.57976984759809 0.710115076200955 44 0.398451703033844 0.796903406067688 0.601548296966156 45 0.523728608705491 0.952542782589019 0.476271391294509 46 0.665173738542768 0.669652522914464 0.334826261457232 47 0.812154358308304 0.375691283383393 0.187845641691696 48 0.900561870487456 0.198876259025088 0.0994381295125438 49 0.93784103396383 0.12431793207234 0.06215896603617 50 0.953810435187989 0.0923791296240223 0.0461895648120111 51 0.961466316034898 0.0770673679302036 0.0385336839651018 52 0.972080137768937 0.0558397244621259 0.0279198622310630 53 0.98645269788143 0.0270946042371391 0.0135473021185695 54 0.9948271103703 0.0103457792594003 0.00517288962970013 55 0.996776271054875 0.00644745789025077 0.00322372894512538 56 0.992856784777805 0.0142864304443904 0.00714321522219518 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 28 0.538461538461538 NOK 5% type I error level 35 0.673076923076923 NOK 10% type I error level 40 0.769230769230769 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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