Multiple Linear regression geg. 2000-2006 paper

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Title produced by software: Multiple Regression

Date of computation: Mon, 14 Dec 2009 01:31:37 -0700

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/14/t1260779624ocwjkkeighmha5i.htm/, Retrieved Mon, 14 Dec 2009 09:33:58 +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/Dec/14/t1260779624ocwjkkeighmha5i.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}, }

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KVN Paper

Dataseries X:

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9084 2359 9081 9700 9743 1511 9084 9081 8587 2059 9743 9084 9731 2635 8587 9743 9563 2867 9731 8587 9998 4403 9563 9731 9437 5720 9998 9563 10038 4502 9437 9998 9918 5749 10038 9437 9252 5627 9918 10038 9737 2846 9252 9918 9035 1762 9737 9252 9133 2429 9035 9737 9487 1169 9133 9035 8700 2154 9487 9133 9627 2249 8700 9487 8947 2687 9627 8700 9283 4359 8947 9627 8829 5382 9283 8947 9947 4459 8829 9283 9628 6398 9947 8829 9318 4596 9628 9947 9605 3024 9318 9628 8640 1887 9605 9318 9214 2070 8640 9605 9567 1351 9214 8640 8547 2218 9567 9214 9185 2461 8547 9567 9470 3028 9185 8547 9123 4784 9470 9185 9278 4975 9123 9470 10170 4607 9278 9123 9434 6249 10170 9278 9655 4809 9434 10170 9429 3157 9655 9434 8739 1910 9429 9655 9552 2228 8739 9429 9687 1594 9552 8739 9019 2467 9687 9552 9672 2222 9019 9687 9206 3607 9672 9019 9069 4685 9206 9672 9788 4962 9069 9206 10312 5770 9788 9069 10105 5480 10312 9788 9863 5000 10105 10312 9656 3228 9863 10105 9295 1993 9656 9863 9946 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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 4215.22359236330 -0.225551356454443X[t] + 0.328747554667996Y1[t] + 0.183946674471251Y2[t] + 717.925229804359M1[t] + 838.954319301792M2[t] -50.7580537332954M3[t] + 1197.13385489368M4[t] + 848.82920315644M5[t] + 1277.17345157679M6[t] + 1384.61060680203M7[t] + 1906.50481501236M8[t] + 1765.90738652060M9[t] + 1331.15369169222M10[t] + 999.839490864513M11[t] + 6.88760173108588t + 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) 4215.22359236330 1369.676279 3.0775 0.003206 0.001603 X -0.225551356454443 0.122016 -1.8485 0.069713 0.034857 Y1 0.328747554667996 0.142947 2.2998 0.025143 0.012572 Y2 0.183946674471251 0.130924 1.405 0.165451 0.082726 M1 717.925229804359 204.262283 3.5147 0.00087 0.000435 M2 838.954319301792 184.012259 4.5592 2.8e-05 1.4e-05 M3 -50.7580537332954 160.75109 -0.3158 0.753341 0.376671 M4 1197.13385489368 231.842391 5.1636 3e-06 2e-06 M5 848.82920315644 229.251386 3.7026 0.000483 0.000242 M6 1277.17345157679 393.433945 3.2462 0.001961 0.00098 M7 1384.61060680203 441.792523 3.1341 0.002723 0.001362 M8 1906.50481501236 436.165709 4.3711 5.3e-05 2.7e-05 M9 1765.90738652060 484.922154 3.6416 0.000586 0.000293 M10 1331.15369169222 442.263711 3.0099 0.003889 0.001944 M11 999.839490864513 220.481863 4.5348 3e-05 1.5e-05 t 6.88760173108588 2.251229 3.0595 0.003376 0.001688 Multiple Linear Regression - Regression Statistics Multiple R 0.881864393159956 R-squared 0.777684807923378 Adjusted R-squared 0.719180810008477 F-TEST (value) 13.2928489614435 F-TEST (DF numerator) 15 F-TEST (DF denominator) 57 p-value 1.71862524211974e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 268.188963309364 Sum Squared Residuals 4099743.24233424 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 9084 9177.60006033392 -93.6000603339168 2 9743 9383.9075530021 359.092446997891 3 8587 8594.6771169107 -7.67711691069607 4 9731 9460.72773123135 270.272268768654 5 9563 9230.42761337919 332.572386620813 6 9998 9474.41798642748 523.582013572517 7 9437 9403.79375190272 33.2062480972807 8 10038 10102.8865392319 -64.8865392318943 9 9918 9782.29736694962 135.702633050379 10 9252 9453.05078413684 -201.050784136838 11 9737 9514.86303499458 222.136965005417 12 9035 8803.34289507389 231.657104926107 13 9133 9236.14632559585 -103.146325595852 14 9487 9551.34442083561 -64.3444208356138 15 8700 8580.75497187464 119.245028125362 16 9627 9620.49990060863 6.50009939136454 17 8947 9340.2743068438 -393.274306843796 18 9283 9345.35451906401 -62.3545190640126 19 8829 9214.31567809545 -385.315678095445 20 9947 9863.83608284738 83.16391715262 21 9628 9576.8101518304 51.189848169589 22 9318 9656.1695151838 -338.169515183797 23 9605 9525.71891733015 79.2810826698524 24 8640 8826.54599958905 -186.545999589049 25 9214 9245.63423821196 -31.6342382119631 26 9567 9546.9149102459 20.0850897541010 27 8547 8690.1703908402 -143.170390840196 28 9185 9619.75159190682 -434.751591906821 29 9470 9172.5622547085 297.437745291494 30 9123 9422.77695431897 -299.776954318973 31 9278 9432.37080294702 -154.370802947019 32 10170 10031.2818869957 138.718113004317 33 9434 9848.9712862437 -414.971286243708 34 9655 9668.02137983353 -13.0213798335294 35 9429 9653.47407877043 -224.474078770429 36 8739 8908.13999883887 -169.139998838871 37 9552 9292.81973787038 259.180262129617 38 9687 9704.08454565094 -17.0845456509365 39 9019 8818.28300638751 200.716993612488 40 9672 9933.5520336123 -261.552033612307 41 9206 9371.54212956816 -165.542129568159 42 9069 9530.55043541614 -461.550435416143 43 9788 9451.63990134147 336.360098658525 44 10312 10009.3450126714 302.654987328573 45 10105 10245.5664568734 -140.566456873399 46 9863 9954.3023284809 -91.3023284809046 47 9656 9911.91886317635 -255.918863176347 48 9295 9084.95706022584 210.042939774162 49 9946 9586.47741275653 359.522587243473 50 9701 10021.6943729595 -320.694372959535 51 9049 9058.30796556535 -9.30796556534974 52 10190 10035.4074751616 154.592524838391 53 9706 9631.35629203207 74.643707967925 54 9765 9877.37083802837 -112.370838028368 55 9893 9874.47017405414 18.5298259458622 56 9994 10244.8429037890 -250.842903789047 57 10433 10218.4052582280 214.594741772046 58 10073 10011.6302057168 61.3697942832411 59 10112 10158.4509371938 -46.4509371937567 60 9266 9354.11600535835 -88.1160053583477 61 9820 9739.4147135869 80.585286413104 62 10097 10074.0541973059 22.9458026940939 63 9115 9274.8065484216 -159.806548421608 64 10411 10146.0612674793 264.938732520718 65 9678 9823.83740346828 -145.837403468278 66 10408 9995.52926674502 412.470733254979 67 10153 10001.4096916592 151.590308340795 68 10368 10576.8075744646 -208.807574464568 69 10581 10426.9494798749 154.050520125094 70 10597 10014.8257866482 582.174213351828 71 10680 10454.5741685347 225.425831465263 72 9738 9735.898040914 2.10195908599910 73 9556 10026.9075116445 -470.907511644462 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.172580848924557 0.345161697849114 0.827419151075443 20 0.125805258212736 0.251610516425473 0.874194741787264 21 0.632688583905155 0.73462283218969 0.367311416094845 22 0.624144023217969 0.751711953564062 0.375855976782031 23 0.527619729138908 0.944760541722185 0.472380270861092 24 0.422200180654923 0.844400361309845 0.577799819345077 25 0.556218006976011 0.887563986047977 0.443781993023989 26 0.47431080977972 0.94862161955944 0.52568919022028 27 0.382629011470775 0.76525802294155 0.617370988529225 28 0.376266520399860 0.752533040799719 0.62373347960014 29 0.557000085590874 0.885999828818251 0.442999914409126 30 0.52494849897785 0.9501030020443 0.47505150102215 31 0.463708006585591 0.927416013171183 0.536291993414409 32 0.51249552109304 0.97500895781392 0.48750447890696 33 0.502970358457174 0.994059283085652 0.497029641542826 34 0.527106890878998 0.945786218242004 0.472893109121002 35 0.49803897993954 0.99607795987908 0.50196102006046 36 0.432782416808245 0.86556483361649 0.567217583191755 37 0.537704272358133 0.924591455283734 0.462295727641867 38 0.473795115679072 0.947590231358145 0.526204884320928 39 0.51695495490879 0.96609009018242 0.48304504509121 40 0.441437082083202 0.882874164166403 0.558562917916798 41 0.364142359200942 0.728284718401884 0.635857640799058 42 0.659583750552137 0.680832498895726 0.340416249447863 43 0.693888169209406 0.612223661581188 0.306111830790594 44 0.70293524522319 0.594129509553621 0.297064754776811 45 0.618215305634227 0.763569388731546 0.381784694365773 46 0.535829072756151 0.928341854487698 0.464170927243849 47 0.570863735515301 0.858272528969398 0.429136264484699 48 0.492944608471873 0.985889216943746 0.507055391528127 49 0.749481252024191 0.501037495951618 0.250518747975809 50 0.656132659416255 0.68773468116749 0.343867340583745 51 0.569317369900472 0.861365260199055 0.430682630099528 52 0.474387819817404 0.948775639634808 0.525612180182596 53 0.613161905942593 0.773676188114813 0.386838094057407 54 0.458483667935060 0.916967335870121 0.54151633206494 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 0 0 OK

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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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