Multiple regression - model 2

*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: Wed, 18 Nov 2009 08:54:15 -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/18/t125855975177yt93szqy3p88f.htm/, Retrieved Wed, 18 Nov 2009 16:56:03 +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/18/t125855975177yt93szqy3p88f.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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3.58 98.2 3.52 98.71 3.45 98.54 3.36 98.2 3.27 96.92 3.21 99.06 3.19 99.65 3.16 99.82 3.12 99.99 3.06 100.33 3.01 99.31 2.98 101.1 2.97 101.1 3.02 100.93 3.07 100.85 3.18 100.93 3.29 99.6 3.43 101.88 3.61 101.81 3.74 102.38 3.87 102.74 3.88 102.82 4.09 101.72 4.19 103.47 4.2 102.98 4.29 102.68 4.37 102.9 4.47 103.03 4.61 101.29 4.65 103.69 4.69 103.68 4.82 104.2 4.86 104.08 4.87 104.16 5.01 103.05 5.03 104.66 5.13 104.46 5.18 104.95 5.21 105.85 5.26 106.23 5.25 104.86 5.2 107.44 5.16 108.23 5.19 108.45 5.39 109.39 5.58 110.15 5.76 109.13 5.89 110.28 5.98 110.17 6.02 109.99 5.62 109.26 4.87 109.11 4.24 107.06 4.02 109.53 3.74 108.92 3.45 109.24 3.34 109.12 3.21 109 3.12 107.23 3.04 109.49

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] = -13.1065644125238 + 0.163823860231794X[t] + 0.542126094040477M1[t] + 0.564658423824252M2[t] + 0.498071355737762M3[t] + 0.378794878533125M4[t] + 0.537377157333334M5[t] + 0.118459313143055M6[t] + 0.0718516204310666M7[t] + 0.00687503074762187M8[t] + 0.0105743611306001M9[t] -0.0227774790022489M10[t] + 0.252466448716831M11[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) -13.1065644125238 2.947952 -4.446 5.3e-05 2.7e-05 X 0.163823860231794 0.02766 5.9229 0 0 M1 0.542126094040477 0.507807 1.0676 0.291162 0.145581 M2 0.564658423824252 0.507556 1.1125 0.271579 0.13579 M3 0.498071355737762 0.507457 0.9815 0.33137 0.165685 M4 0.378794878533125 0.507388 0.7466 0.459048 0.229524 M5 0.537377157333334 0.514547 1.0444 0.301656 0.150828 M6 0.118459313143055 0.505045 0.2346 0.815575 0.407788 M7 0.0718516204310666 0.50475 0.1424 0.887411 0.443706 M8 0.00687503074762187 0.504116 0.0136 0.989177 0.494588 M9 0.0105743611306001 0.503795 0.021 0.983343 0.491671 M10 -0.0227774790022489 0.50358 -0.0452 0.964115 0.482057 M11 0.252466448716831 0.505606 0.4993 0.619873 0.309937 Multiple Linear Regression - Regression Statistics Multiple R 0.660076701238409 R-squared 0.435701251517779 Adjusted R-squared 0.291624975309553 F-TEST (value) 3.02410128151897 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0.00325924447922588 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.795919341471842 Sum Squared Residuals 29.7739171120616 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 3.58 3.52306475627885 0.0569352437211541 2 3.52 3.62914725478083 -0.109147254780831 3 3.45 3.53471013045494 -0.0847101304549378 4 3.36 3.35973354077149 0.000266459228508711 5 3.27 3.308621278475 -0.0386212784750035 6 3.21 3.24028649518076 -0.0302864951807638 7 3.19 3.29033488000553 -0.100334880005534 8 3.16 3.25320834656149 -0.0932083465614925 9 3.12 3.28475773318388 -0.164757733183876 10 3.06 3.30710600552984 -0.247106005529838 11 3.01 3.41524959581249 -0.405249595812488 12 2.98 3.45602785691057 -0.476027856910568 13 2.97 3.99815395095104 -1.02815395095104 14 3.02 3.99283622449542 -0.972836224495416 15 3.07 3.91314324759038 -0.84314324759038 16 3.18 3.80697267920429 -0.62697267920429 17 3.29 3.74766922389621 -0.45766922389621 18 3.43 3.70226978103442 -0.272269781034422 19 3.61 3.64419441810621 -0.0341944181062093 20 3.74 3.67259742875489 0.0674025712451143 21 3.87 3.73527334882131 0.134726651178690 22 3.88 3.71502741750700 0.164972582492996 23 4.09 3.81006509897111 0.279934901028889 24 4.19 3.84429040565992 0.34570959434008 25 4.2 4.30614280818682 -0.106142808186819 26 4.29 4.27952797990106 0.0104720200989442 27 4.37 4.24898216106556 0.121017838934440 28 4.47 4.15100278569106 0.318997214308944 29 4.61 4.02453154768794 0.585468452312056 30 4.65 3.99879096805397 0.651209031946031 31 4.69 3.95054503673966 0.739454963260336 32 4.82 3.97075685437675 0.849243145623248 33 4.86 3.95479732153191 0.905202678468086 34 4.87 3.93455139021761 0.935448609782392 35 5.01 4.0279508330794 0.982049166920603 36 5.03 4.03924079933575 0.990759200664246 37 5.13 4.54860212132987 0.581397878670127 38 5.18 4.65140814262723 0.528591857372772 39 5.21 4.73226254874935 0.477737451250649 40 5.26 4.6752391384328 0.584760861567202 41 5.25 4.60938272871545 0.640617271284552 42 5.2 4.6131304439232 0.586869556076803 43 5.16 4.69594360079433 0.464056399205673 44 5.19 4.66700826036188 0.522991739638123 45 5.39 4.82470201936274 0.565297980637258 46 5.58 4.91585631300606 0.664143686993944 47 5.76 5.0239999032887 0.736000096711295 48 5.89 4.95993089383844 0.930069106161562 49 5.98 5.48403636325342 0.495963636746582 50 6.02 5.47708039819547 0.542919601804531 51 5.62 5.29090191213977 0.32909808786023 52 4.87 5.14705185590036 -0.277051855900364 53 4.24 4.9697952212254 -0.729795221225395 54 4.02 4.95552231180765 -0.935522311807648 55 3.74 4.80898206435427 -1.06898206435427 56 3.45 4.79642910994499 -1.34642910994499 57 3.34 4.78046957710016 -1.44046957710016 58 3.21 4.72745887373949 -1.51745887373949 59 3.12 4.7127345688483 -1.59273456884830 60 3.04 4.83051004425532 -1.79051004425532 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.00527527695255471 0.0105505539051094 0.994724723047445 17 0.00420186001881021 0.00840372003762043 0.99579813998119 18 0.00404762235844829 0.00809524471689659 0.995952377641552 19 0.00363567912070182 0.00727135824140363 0.996364320879298 20 0.00383440669606214 0.00766881339212429 0.996165593303938 21 0.00417126613380783 0.00834253226761566 0.995828733866192 22 0.00381670453490627 0.00763340906981254 0.996183295465094 23 0.00478124551018615 0.0095624910203723 0.995218754489814 24 0.00570076031191847 0.0114015206238369 0.994299239688082 25 0.0042026971219444 0.0084053942438888 0.995797302878056 26 0.00357886915321905 0.0071577383064381 0.99642113084678 27 0.00275093796763955 0.0055018759352791 0.99724906203236 28 0.00180672456627512 0.00361344913255025 0.998193275433725 29 0.00135083976095259 0.00270167952190517 0.998649160239047 30 0.000838147943771824 0.00167629588754365 0.999161852056228 31 0.000492526210117892 0.000985052420235783 0.999507473789882 32 0.000293972759255253 0.000587945518510506 0.999706027240745 33 0.000187483585604295 0.000374967171208591 0.999812516414396 34 0.000125626153332733 0.000251252306665467 0.999874373846667 35 9.59782993896246e-05 0.000191956598779249 0.99990402170061 36 8.44377364319742e-05 0.000168875472863948 0.999915562263568 37 3.48081442941482e-05 6.96162885882965e-05 0.999965191855706 38 1.22393270553689e-05 2.44786541107377e-05 0.999987760672945 39 3.72699226052745e-06 7.45398452105491e-06 0.99999627300774 40 1.32168603237017e-06 2.64337206474034e-06 0.999998678313968 41 1.03351387754281e-06 2.06702775508561e-06 0.999998966486122 42 8.39229974004462e-06 1.67845994800892e-05 0.99999160770026 43 5.1533859281466e-05 0.000103067718562932 0.999948466140719 44 0.0108594892130677 0.0217189784261353 0.989140510786932 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 26 0.896551724137931 NOK 5% type I error level 29 1 NOK 10% type I error level 29 1 NOK

Charts produced by software:

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Parameters (Session):

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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