Linear Trend

*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, 18 Dec 2010 10:46:37 +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/18/t12926691146tej38ekabs65j5.htm/, Retrieved Sat, 18 Dec 2010 11:45:17 +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/18/t12926691146tej38ekabs65j5.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:

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94.6 116.1 95.9 107.5 104.7 116.7 102.8 112.5 98.1 113 113.9 126.4 80.9 114.1 95.7 112.5 113.2 112.4 105.9 113.1 108.8 116.3 102.3 111.7 99 118.8 100.7 116.5 115.5 125.1 100.7 113.1 109.9 119.6 114.6 114.4 85.4 114 100.5 117.8 114.8 117 116.5 120.9 112.9 115 102 117.3 106 119.4 105.3 114.9 118.8 125.8 106.1 117.6 109.3 117.6 117.2 114.9 92.5 121.9 104.2 117 112.5 106.4 122.4 110.5 113.3 113.6 100 114.2 110.7 125.4 112.8 124.6 109.8 120.2 117.3 120.8 109.1 111.4 115.9 124.1 96 120.2 99.8 125.5 116.8 116 115.7 117 99.4 105.7 94.3 102 91 106.4 93.2 96.9 103.1 107.6 94.1 98.8 91.8 101.1 102.7 105.7 82.6 104.6 89.1 103.2 104.5 101.6 105.1 106.7 95.1 99.5 88.7 101

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 6 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation I.P.C.N.[t] = + 35.3607050481265 + 0.80254709385946T.I.P.[t] + 4.40769055462782M1[t] -1.67120091994249M2[t] -1.61314465658159M3[t] -3.05293292720598M4[t] -2.50303586532053M5[t] -5.2220493813806M6[t] + 13.1270665500967M7[t] + 5.15709840515961M8[t] -10.8793637664784M9[t] -8.40882886848745M10[t] -6.113968161498M11[t] -0.120470689324151t + 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) 35.3607050481265 8.439546 4.1899 0.000125 6.3e-05 T.I.P. 0.80254709385946 0.0814 9.8593 0 0 M1 4.40769055462782 2.479298 1.7778 0.082048 0.041024 M2 -1.67120091994249 2.48515 -0.6725 0.504646 0.252323 M3 -1.61314465658159 2.657938 -0.6069 0.546891 0.273445 M4 -3.05293292720598 2.513266 -1.2147 0.230669 0.115334 M5 -2.50303586532053 2.503426 -0.9998 0.322617 0.161308 M6 -5.2220493813806 2.74159 -1.9048 0.063076 0.031538 M7 13.1270665500967 2.600069 5.0487 7e-06 4e-06 M8 5.15709840515961 2.45626 2.0996 0.041281 0.020641 M9 -10.8793637664784 2.72884 -3.9868 0.000237 0.000119 M10 -8.40882886848745 2.758882 -3.0479 0.003812 0.001906 M11 -6.113968161498 2.546047 -2.4014 0.020433 0.010216 t -0.120470689324151 0.030364 -3.9675 0.000252 0.000126 Multiple Linear Regression - Regression Statistics Multiple R 0.889237640042933 R-squared 0.790743580469125 Adjusted R-squared 0.731605896688661 F-TEST (value) 13.3712301517351 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 1.52530210684176e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.87919616466738 Sum Squared Residuals 692.215492662624 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 116.1 115.568879992535 0.531120007464823 2 107.5 110.412829050658 -2.91282905065791 3 116.7 117.412829050658 -0.712829050657896 4 112.5 114.327730612376 -1.82773061237639 5 113 110.985185643798 2.01481435620178 6 126.4 120.825945521393 5.57405447860653 7 114.1 112.570536666184 1.52946333381561 8 112.5 116.357794821043 -3.8577948210432 9 112.4 114.245436102622 -1.84543610262158 10 113.1 110.736906526114 2.36309347388566 11 116.3 115.238683115972 1.06131688402793 12 111.7 116.015624478059 -4.31562447805942 13 118.8 117.654438933627 1.14556106637312 14 116.5 112.819406829293 3.68059317070651 15 125.1 124.63468939245 0.465310607549741 16 113.1 111.196733443382 1.90326655661829 17 119.6 119.00959307945 0.590406920549952 18 114.4 119.942080215205 -5.54208021520527 19 114 114.736350316662 -0.736350316662142 20 117.8 118.764372599679 -0.964372599678794 21 117 114.083863180907 2.9161368190931 22 120.9 117.798257449135 3.10174255086521 23 115 117.083477928906 -2.08347792890604 24 117.3 114.329212078012 2.97078792198823 25 119.4 121.826620318753 -2.42662031875328 26 114.9 115.065475189157 -0.16547518915718 27 125.8 125.837446530297 -0.0374465302966544 28 117.6 114.084839478333 3.51516052166702 29 117.6 117.082416551245 0.517583448755453 30 114.9 120.58305438735 -5.68305438735006 31 121.9 118.988786411174 2.91121358882552 32 117 120.288148575069 -3.28814857506898 33 106.4 110.79235659314 -4.39235659314032 34 110.5 121.087637031016 -10.5876370310158 35 113.6 115.95884849456 -2.35884849456001 36 114.2 111.278469618403 2.92153038159697 37 125.4 124.152943388003 1.24705661199707 38 124.6 119.638930121213 4.96106987878667 39 120.2 117.168874413672 3.03112558632831 40 120.8 121.627718657669 -0.827718657669112 41 111.4 115.476258860583 -4.07625886058283 42 124.1 118.094094893443 6.00590510655705 43 120.2 120.352052967793 -0.152052967792781 44 125.5 115.311293090198 10.1887069098025 45 116 112.797660824846 3.20233917515381 46 117 114.264923230268 2.73507676973241 47 105.7 103.357795618024 2.34220438197631 48 102 105.258302911514 -3.25830291151429 49 106.4 106.897117367082 -0.49711736708174 50 96.9 102.463358809678 -5.56335880967809 51 107.6 110.346160612923 -2.7461606129235 52 98.8 101.56297780824 -2.76297780823982 53 101.1 100.146545864924 0.953454135075638 54 105.7 106.054824982608 -0.354824982608251 55 104.6 108.152273638186 -3.5522736381862 56 103.2 105.278390914011 -2.07839091401149 57 101.6 101.480683298485 0.119316701514987 58 106.7 104.312275763467 2.38772423653251 59 99.5 98.4611948425382 1.03880515746181 60 101 99.3183909140115 1.6816090859885 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.117915118832618 0.235830237665237 0.882084881167382 18 0.669119897642093 0.661760204715814 0.330880102357907 19 0.534201641346124 0.931596717307752 0.465798358653876 20 0.426092123603887 0.852184247207774 0.573907876396113 21 0.368988038765255 0.73797607753051 0.631011961234745 22 0.286366660834793 0.572733321669587 0.713633339165207 23 0.219543857156315 0.43908771431263 0.780456142843685 24 0.227455230415668 0.454910460831336 0.772544769584332 25 0.181112730615597 0.362225461231195 0.818887269384403 26 0.119980692282967 0.239961384565934 0.880019307717033 27 0.0737994389073879 0.147598877814776 0.926200561092612 28 0.0693029613401551 0.13860592268031 0.930697038659845 29 0.0467686664687125 0.093537332937425 0.953231333531288 30 0.0657919213349359 0.131583842669872 0.934208078665064 31 0.0725837543393373 0.145167508678675 0.927416245660663 32 0.0543034651069258 0.108606930213852 0.945696534893074 33 0.0429249078545564 0.0858498157091128 0.957075092145444 34 0.452841518433908 0.905683036867816 0.547158481566092 35 0.520089485042161 0.959821029915678 0.479910514957839 36 0.43276116899049 0.865522337980979 0.56723883100951 37 0.353614586807037 0.707229173614073 0.646385413192963 38 0.431911901468353 0.863823802936707 0.568088098531647 39 0.355621736083964 0.711243472167927 0.644378263916036 40 0.24788637414395 0.4957727482879 0.75211362585605 41 0.498751580339638 0.997503160679275 0.501248419660362 42 0.404151730578282 0.808303461156565 0.595848269421718 43 0.276446229730008 0.552892459460015 0.723553770269992 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 0 0 OK 10% type I error level 2 0.0740740740740741 OK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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