model 3 : lineaire 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: Sun, 22 Nov 2009 04:48:44 -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/22/t1258890858fhqayv58fl3q5i7.htm/, Retrieved Sun, 22 Nov 2009 12:54:30 +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/22/t1258890858fhqayv58fl3q5i7.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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9.3 8.1 10.9 25.6 8.7 7.7 10 23.7 8.2 7.5 9.2 22 8.3 7.6 9.2 21.3 8.5 7.8 9.5 20.7 8.6 7.8 9.6 20.4 8.5 7.8 9.5 20.3 8.2 7.5 9.1 20.4 8.1 7.5 8.9 19.8 7.9 7.1 9 19.5 8.6 7.5 10.1 23.1 8.7 7.5 10.3 23.5 8.7 7.6 10.2 23.5 8.5 7.7 9.6 22.9 8.4 7.7 9.2 21.9 8.5 7.9 9.3 21.5 8.7 8.1 9.4 20.5 8.7 8.2 9.4 20.2 8.6 8.2 9.2 19.4 8.5 8.2 9 19.2 8.3 7.9 9 18.8 8 7.3 9 18.8 8.2 6.9 9.8 22.6 8.1 6.6 10 23.3 8.1 6.7 9.8 23 8 6.9 9.3 21.4 7.9 7 9 19.9 7.9 7.1 9 18.8 8 7.2 9.1 18.6 8 7.1 9.1 18.4 7.9 6.9 9.1 18.6 8 7 9.2 19.9 7.7 6.8 8.8 19.2 7.2 6.4 8.3 18.4 7.5 6.7 8.4 21.1 7.3 6.6 8.1 20.5 7 6.4 7.7 19.1 7 6.3 7.9 18.1 7 6.2 7.9 17 7.2 6.5 8 17.1 7.3 6.8 7.9 17.4 7.1 6.8 7.6 16.8 6.8 6.4 7.1 15.3 6.4 6.1 6.8 14.3 6.1 5.8 6.5 13.4 6.5 6.1 6.9 15.3 7.7 7.2 8.2 22.1 7.9 7.3 8.7 23.7 7.5 6.9 8.3 22.2 6.9 6.1 7.9 19.5 6.6 5.8 7.5 16.6 6.9 6.2 7.8 17.3 7.7 7.1 8.3 19.8 8 7.7 8.4 21.2 8 7.9 8.2 21.5 7.7 7.7 7.7 20.6 7.3 7.4 7.2 19.1 7.4 7.5 7.3 19.6 8.1 8 8.1 23.5 8.3 8. etc...

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation TW[t] = + 0.137478831210173 + 0.53055726280852WM[t] + 0.43158336885857WV[t] + 0.00605746119417265WJ[t] + 0.00172169418001913M1[t] + 0.00222234476380082M2[t] + 0.0287546597955706M3[t] + 0.0101104864524246M4[t] + 0.0303651906903672M5[t] + 0.0148706587567332M6[t] + 0.0254744210331318M7[t] + 0.0123528305229541M8[t] -0.00557343797333338M9[t] -0.0101279204730681M10[t] + 0.0287035908541254M11[t] + 0.000459327773782868t + 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) 0.137478831210173 0.10248 1.3415 0.186482 0.093241 WM 0.53055726280852 0.013579 39.073 0 0 WV 0.43158336885857 0.013553 31.8441 0 0 WJ 0.00605746119417265 0.005799 1.0446 0.301767 0.150884 M1 0.00172169418001913 0.020053 0.0859 0.93196 0.46598 M2 0.00222234476380082 0.022019 0.1009 0.920057 0.460028 M3 0.0287546597955706 0.024511 1.1731 0.246915 0.123458 M4 0.0101104864524246 0.026524 0.3812 0.704858 0.352429 M5 0.0303651906903672 0.02876 1.0558 0.296692 0.148346 M6 0.0148706587567332 0.029743 0.5 0.619535 0.309767 M7 0.0254744210331318 0.029985 0.8496 0.400056 0.200028 M8 0.0123528305229541 0.028821 0.4286 0.670252 0.335126 M9 -0.00557343797333338 0.029608 -0.1882 0.851535 0.425767 M10 -0.0101279204730681 0.027349 -0.3703 0.712882 0.356441 M11 0.0287035908541254 0.021024 1.3653 0.178957 0.089478 t 0.000459327773782868 0.000521 0.881 0.382987 0.191493 Multiple Linear Regression - Regression Statistics Multiple R 0.999124703391936 R-squared 0.998250172928024 Adjusted R-squared 0.997666897237365 F-TEST (value) 1711.45512990729 F-TEST (DF numerator) 15 F-TEST (DF denominator) 45 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0328859962513968 Sum Squared Residuals 0.0486669937251097 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 9.3 9.29650340904222 0.00349659095777864 2 8.7 8.68530627403474 0.0146937259652613 3 8.2 8.25062208516163 -0.0506220851616339 4 8.3 8.2812527430372 0.0187472569627983 5 8.5 8.5339187615517 -0.0339187615516996 6 8.6 8.56022465591945 0.0397753440805461 7 8.5 8.52752366296436 -0.027523662964361 8 8.2 8.1836666199614 0.0163333800386 9 8.1 8.07624852875068 0.0237514712493222 10 7.9 7.90127156742892 -0.00127156742892187 11 8.6 8.64933387769675 -0.0493338776967554 12 8.7 8.7098292728658 -0.00982927286579663 13 8.7 8.72190768421459 -0.0219076842145930 14 8.5 8.51333889082136 -0.0133388908213639 15 8.4 8.36163972488932 0.0383602751106846 16 8.5 8.49030168428984 0.00969831571015457 17 8.7 8.65422804455496 0.0457719554450405 18 8.7 8.6904313283177 0.00956867168229146 19 8.6 8.61033177564084 -0.0103317756408372 20 8.5 8.5101413468939 -0.0101413468938939 21 8.3 8.33108424285116 -0.0310842428511637 22 8 8.0086547304401 -0.00865473044009996 23 8.2 8.20400771204238 -0.00400771204238111 24 8.1 8.10715316672712 -0.0071531667271162 25 8.1 8.0742560028318 0.0257439971681949 26 8 7.95584381141111 0.0441561885888876 27 7.9 7.89732997804869 0.00267002195131358 28 7.9 7.92553765144659 -0.0255376514465853 29 8 8.04125425438619 -0.0412542543861857 30 8 7.97195183170665 0.0280481682933523 31 7.9 7.87811496143396 0.0218850385660404 32 8 7.9695414614167 0.0304585385833016 33 7.7 7.66908949775314 0.0309105022468588 34 7.2 7.23213378451916 -0.0321337845191584 35 7.5 7.49010528457281 0.00989471542718605 36 7.3 7.27569580783754 0.0243041921624555 37 7 6.99065158401437 0.00934841598562683 38 7 7.01881504866863 -0.0188150486686268 39 7 6.98608775787974 0.0139122421202625 40 7.2 7.1708341741582 0.0291658258417955 41 7.3 7.30937428648488 -0.0093742864848813 42 7.1 7.16122959495096 -0.0612295949509557 43 6.8 6.73519190365719 0.064808096342815 44 6.4 6.42782999022649 -0.0278299902264897 45 6.1 6.1162691449291 -0.0162691449291033 46 6.5 6.45548369285806 0.0445163071419366 47 7.7 7.68063663668493 0.0193633633150733 48 7.9 7.9309317222254 -0.030931722225397 49 7.5 7.5391702997211 -0.0391702997211052 50 6.9 6.92669597506416 -0.0266959750641583 51 6.6 6.60432045402063 -0.00432045402062672 52 6.9 6.93207374706816 -0.0320737470681631 53 7.7 7.66122465302227 0.0387753469777261 54 8 8.01616258910523 -0.0161625891052341 55 8 8.04883769630366 -0.0488376963036572 56 7.7 7.70882058150152 -0.00882058150151811 57 7.3 7.30730858571591 -0.00730858571591397 58 7.4 7.40245622475376 -0.00245622475375642 59 8.1 8.07591648900312 0.0240835109968771 60 8.3 8.27639003034415 0.0236099696558543 61 8.2 8.1775110201759 0.0224889798240978 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.722588674746764 0.554822650506471 0.277411325253236 20 0.687425213763233 0.625149572473535 0.312574786236767 21 0.684674453356894 0.630651093286211 0.315325546643106 22 0.57107065110153 0.857858697796941 0.428929348898471 23 0.465389168655737 0.930778337311473 0.534610831344263 24 0.421715763332365 0.84343152666473 0.578284236667635 25 0.333455014008192 0.666910028016383 0.666544985991808 26 0.362840352521222 0.725680705042444 0.637159647478778 27 0.264014681273175 0.52802936254635 0.735985318726825 28 0.212857152789654 0.425714305579307 0.787142847210346 29 0.365349575539329 0.730699151078659 0.634650424460671 30 0.280987758241995 0.561975516483991 0.719012241758005 31 0.214931986571461 0.429863973142922 0.785068013428539 32 0.175677993820572 0.351355987641144 0.824322006179428 33 0.184978849072291 0.369957698144582 0.815021150927709 34 0.214301494750280 0.428602989500561 0.78569850524972 35 0.153742187772963 0.307484375545926 0.846257812227037 36 0.122261141599768 0.244522283199537 0.877738858400232 37 0.096120984780465 0.19224196956093 0.903879015219535 38 0.0626358128810349 0.125271625762070 0.937364187118965 39 0.0420973604448456 0.0841947208896911 0.957902639555154 40 0.0801339828834318 0.160267965766864 0.919866017116568 41 0.0432364430688374 0.0864728861376748 0.956763556931163 42 0.232236180053360 0.464472360106719 0.76776381994664 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 0.0833333333333333 OK

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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