Q3

*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, 27 Nov 2008 05:33:19 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Nov/27/t1227789285eia7vqubga6h2px.htm/, Retrieved Thu, 27 Nov 2008 12:34:55 +0000

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/2008/Nov/27/t1227789285eia7vqubga6h2px.htm/}, year = {2008}, } @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 = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

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User-defined keywords:

Dataseries X:

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97,3 0 97,4 0 97,5 0 95,5 0 95,3 0 95,4 0 95,4 0 95,4 0 95,5 0 94,6 0 95,2 0 95,2 0 94,7 0 94,7 0 94,7 0 95,3 0 94,7 0 94,8 0 94,9 0 95,4 0 96 0 95,9 0 95,8 0 95,8 0 95,1 0 95,2 0 95,2 0 95,3 0 95,4 0 95,3 0 95,3 0 95 0 94,9 0 95,7 0 95,7 0 96,3 0 91,7 1 92,2 1 92,2 1 92,6 1 93 1 93 1 93 1 93,7 1 93,1 1 93,1 1 93,2 1 93,2 1 93 1 93,7 1 94 1 93,1 1 94,2 1 94,2 1 93,5 1 95 1 93,7 1 93,9 1 94,6 1 93,8 1 91,2 1 91,4 1 91,3 1 91,5 1 91,5 1 91,5 1 91,3 1 92,8 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 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation X[t] = + 96.4646666666667 -1.59666666666666Y[t] -1.00116666666664M1[t] -0.707666666666663M2[t] -0.630833333333333M3[t] -0.870666666666669M4[t] -0.710499999999999M5[t] -0.666999999999999M6[t] -0.773499999999998M7[t] -0.0966666666666634M8[t] -0.300499999999999M9[t] -0.273666666666665M10[t] + 0.0131666666666679M11[t] -0.0268333333333338t + 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) 96.4646666666667 0.48033 200.83 0 0 Y -1.59666666666666 0.445975 -3.5802 0.000736 0.000368 M1 -1.00116666666664 0.548947 -1.8238 0.073719 0.036859 M2 -0.707666666666663 0.547082 -1.2935 0.201334 0.100667 M3 -0.630833333333333 0.545447 -1.1565 0.252549 0.126275 M4 -0.870666666666669 0.544044 -1.6004 0.115353 0.057677 M5 -0.710499999999999 0.542875 -1.3088 0.196154 0.098077 M6 -0.666999999999999 0.541941 -1.2308 0.223746 0.111873 M7 -0.773499999999998 0.541244 -1.4291 0.158731 0.079365 M8 -0.0966666666666634 0.540784 -0.1788 0.858801 0.4294 M9 -0.300499999999999 0.565146 -0.5317 0.5971 0.29855 M10 -0.273666666666665 0.564576 -0.4847 0.629829 0.314914 M11 0.0131666666666679 0.564233 0.0233 0.981469 0.490734 t -0.0268333333333338 0.011354 -2.3633 0.02174 0.01087 Multiple Linear Regression - Regression Statistics Multiple R 0.858360551898973 R-squared 0.73678283705631 Adjusted R-squared 0.673415742273569 F-TEST (value) 11.6272150330141 F-TEST (DF numerator) 13 F-TEST (DF denominator) 54 p-value 2.14550599508812e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.891950283727735 Sum Squared Residuals 42.9610666666673 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 97.3 95.4366666666665 1.86333333333348 2 97.4 95.7033333333333 1.69666666666666 3 97.5 95.7533333333333 1.74666666666666 4 95.5 95.4866666666667 0.0133333333333264 5 95.3 95.62 -0.320000000000013 6 95.4 95.6366666666667 -0.236666666666672 7 95.4 95.5033333333333 -0.103333333333337 8 95.4 96.1533333333334 -0.753333333333338 9 95.5 95.9226666666667 -0.422666666666675 10 94.6 95.9226666666667 -1.32266666666668 11 95.2 96.1826666666667 -0.982666666666671 12 95.2 96.1426666666667 -0.94266666666667 13 94.7 95.1146666666667 -0.414666666666699 14 94.7 95.3813333333333 -0.681333333333338 15 94.7 95.4313333333333 -0.731333333333335 16 95.3 95.1646666666667 0.135333333333330 17 94.7 95.298 -0.598000000000002 18 94.8 95.3146666666667 -0.514666666666673 19 94.9 95.1813333333333 -0.281333333333332 20 95.4 95.8313333333333 -0.431333333333333 21 96 95.6006666666667 0.399333333333331 22 95.9 95.6006666666667 0.299333333333336 23 95.8 95.8606666666667 -0.0606666666666712 24 95.8 95.8206666666667 -0.0206666666666696 25 95.1 94.7926666666667 0.307333333333298 26 95.2 95.0593333333333 0.140666666666668 27 95.2 95.1093333333333 0.0906666666666705 28 95.3 94.8426666666667 0.457333333333335 29 95.4 94.976 0.424000000000006 30 95.3 94.9926666666667 0.307333333333332 31 95.3 94.8593333333333 0.440666666666664 32 95 95.5093333333333 -0.509333333333334 33 94.9 95.2786666666667 -0.378666666666658 34 95.7 95.2786666666667 0.421333333333339 35 95.7 95.5386666666667 0.161333333333340 36 96.3 95.4986666666667 0.801333333333335 37 91.7 92.874 -1.17400000000003 38 92.2 93.1406666666667 -0.94066666666667 39 92.2 93.1906666666667 -0.990666666666667 40 92.6 92.924 -0.324000000000005 41 93 93.0573333333333 -0.0573333333333363 42 93 93.074 -0.0740000000000018 43 93 92.9406666666667 0.0593333333333308 44 93.7 93.5906666666667 0.109333333333333 45 93.1 93.36 -0.260000000000006 46 93.1 93.36 -0.260000000000006 47 93.2 93.62 -0.419999999999997 48 93.2 93.58 -0.379999999999995 49 93 92.552 0.447999999999972 50 93.7 92.8186666666667 0.881333333333336 51 94 92.8686666666667 1.13133333333334 52 93.1 92.602 0.498000000000001 53 94.2 92.7353333333333 1.46466666666667 54 94.2 92.752 1.44800000000001 55 93.5 92.6186666666667 0.881333333333336 56 95 93.2686666666667 1.73133333333333 57 93.7 93.038 0.662000000000008 58 93.9 93.038 0.86200000000001 59 94.6 93.298 1.302 60 93.8 93.258 0.542000000000004 61 91.2 92.23 -1.03000000000002 62 91.4 92.4966666666667 -1.09666666666666 63 91.3 92.5466666666667 -1.24666666666666 64 91.5 92.28 -0.779999999999988 65 91.5 92.4133333333333 -0.913333333333326 66 91.5 92.43 -0.92999999999999 67 91.3 92.2966666666667 -0.996666666666661 68 92.8 92.9466666666667 -0.146666666666662 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.789767667846344 0.420464664307312 0.210232332153656 18 0.741862491168164 0.516275017663673 0.258137508831836 19 0.683812652017327 0.632374695965346 0.316187347982673 20 0.685353008907293 0.629293982185413 0.314646991092707 21 0.716947142294806 0.566105715410388 0.283052857705194 22 0.810407609026144 0.379184781947712 0.189592390973856 23 0.791630466306099 0.416739067387802 0.208369533693901 24 0.761963026988498 0.476073946023004 0.238036973011502 25 0.687577099515117 0.624845800969766 0.312422900484883 26 0.598013845351579 0.803972309296842 0.401986154648421 27 0.504189442731066 0.991621114537867 0.495810557268934 28 0.44505119008604 0.89010238017208 0.55494880991396 29 0.412722398726576 0.825444797453152 0.587277601273424 30 0.353033162956614 0.706066325913228 0.646966837043386 31 0.297633937144752 0.595267874289504 0.702366062855248 32 0.244579059876004 0.489158119752008 0.755420940123996 33 0.189709805465898 0.379419610931797 0.810290194534101 34 0.155959110229856 0.311918220459711 0.844040889770144 35 0.122439102617311 0.244878205234621 0.87756089738269 36 0.105707101499630 0.211414202999259 0.89429289850037 37 0.0806902290686476 0.161380458137295 0.919309770931352 38 0.0629420421117607 0.125884084223521 0.93705795788824 39 0.05223643003701 0.10447286007402 0.94776356996299 40 0.0406567171791443 0.0813134343582886 0.959343282820856 41 0.0397345506940245 0.079469101388049 0.960265449305975 42 0.0356404526412553 0.0712809052825105 0.964359547358745 43 0.0264361083745335 0.0528722167490671 0.973563891625467 44 0.042775056281484 0.085550112562968 0.957224943718516 45 0.0478488503632959 0.0956977007265918 0.952151149636704 46 0.0791049452706157 0.158209890541231 0.920895054729384 47 0.400647410108575 0.80129482021715 0.599352589891425 48 0.978744199808866 0.0425116003822676 0.0212558001911338 49 0.968277731143217 0.0634445377135652 0.0317222688567826 50 0.924029631194296 0.151940737611407 0.0759703688057037 51 0.863687115509872 0.272625768980256 0.136312884490128 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 1 0.0285714285714286 OK 10% type I error level 8 0.228571428571429 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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