Multiple Linear Regression 2002-2008 Paper

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Title produced by software: Multiple Regression

Date of computation: Tue, 15 Dec 2009 14:18:54 -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/15/t12609120893kiz8n940lxv1kz.htm/, Retrieved Tue, 15 Dec 2009 22:21:42 +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/15/t12609120893kiz8n940lxv1kz.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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9487 1169 8700 2154 9627 2249 8947 2687 9283 4359 8829 5382 9947 4459 9628 6398 9318 4596 9605 3024 8640 1887 9214 2070 9567 1351 8547 2218 9185 2461 9470 3028 9123 4784 9278 4975 10170 4607 9434 6249 9655 4809 9429 3157 8739 1910 9552 2228 9784 1594 9089 2467 9763 2222 9330 3607 9144 4685 9895 4962 10404 5770 10195 5480 9987 5000 9789 3228 9437 1993 10096 2288 9776 1580 9106 2111 10258 2192 9766 3601 9826 4665 9957 4876 10036 5813 10508 5589 10146 5331 10166 3075 9365 2002 9968 2306 10123 1507 9144 1992 10447 2487 9699 3490 10451 4647 10192 5594 10404 5611 10597 5788 10633 6204 10727 3013 9784 1931 9667 2549 10297 1504 9426 2090 10274 2702 9598 2939 10400 4500 9985 6208 10761 6415 11081 5657 10297 5964 10751 3163 9760 1997 10133 2422

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 Y[t] = + 9399.84683756305 -0.182112565551713X[t] + 118.360565193085M1[t] -606.177135135852M2[t] + 337.499352090133M3[t] + 14.2390095400075M4[t] + 483.092621236134M5[t] + 581.298817068296M6[t] + 1180.67299145633M7[t] + 1190.75708559729M8[t] + 838.529102410993M9[t] + 489.508087197206M10[t] -530.339992476914M11[t] + 18.8712121859727t + 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) 9399.84683756305 210.2483 44.7083 0 0 X -0.182112565551713 0.08818 -2.0652 0.04338 0.02169 M1 118.360565193085 152.904506 0.7741 0.442027 0.221013 M2 -606.177135135852 136.374073 -4.445 4e-05 2e-05 M3 337.499352090133 136.640864 2.47 0.016475 0.008238 M4 14.2390095400075 160.644036 0.0886 0.929676 0.464838 M5 483.092621236134 247.380441 1.9528 0.05567 0.027835 M6 581.298817068296 302.356525 1.9226 0.059451 0.029725 M7 1180.67299145633 310.714704 3.7999 0.000349 0.000175 M8 1190.75708559729 343.370979 3.4678 0.000995 0.000498 M9 838.529102410993 299.434553 2.8004 0.006923 0.003461 M10 489.508087197206 153.386073 3.1913 0.002287 0.001144 M11 -530.339992476914 139.045375 -3.8142 0.000334 0.000167 t 18.8712121859727 1.479658 12.7538 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.925793868889783 R-squared 0.857094287673913 Adjusted R-squared 0.825063696980135 F-TEST (value) 26.7586163448556 F-TEST (DF numerator) 13 F-TEST (DF denominator) 58 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 234.803793045944 Sum Squared Residuals 3197703.63126823 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 9487 9324.1890258121 162.810974187893 2 8700 8439.14166060075 260.858339399249 3 9627 9384.3886662853 242.611333714706 4 8947 9000.23423220949 -53.2342322094903 5 9283 9183.46684648913 99.5331535108737 6 8829 9114.24309994786 -285.243099947860 7 9947 9900.5783845261 46.4216154738984 8 9628 9576.41742624826 51.5825737517444 9 9318 9571.22749837212 -253.227498372122 10 9605 9527.3586483916 77.6413516084 11 8640 8733.44376793575 -93.4437679357492 12 9214 9249.32837310267 -35.3283731026727 13 9567 9517.49908511341 49.500914886589 14 8547 8653.94100263711 -106.941002637112 15 9185 9572.23534862 -387.235348620003 16 9470 9164.58839358803 305.411606411971 17 9123 9332.52355236132 -209.523552361321 18 9278 9414.81746035908 -136.817460359079 19 10170 10100.0802710561 69.9197289438807 20 9434 9830.00674474713 -396.006744747134 21 9655 9758.89206814128 -103.892068141278 22 9429 9729.5922234049 -300.592223404894 23 8739 8955.70972515973 -216.709725159732 24 9552 9447.00913397717 104.990866022826 25 9784 9699.70027791602 84.2997220839826 26 9089 8835.04952004641 253.950479953592 27 9763 9842.21479801853 -79.2147980185354 28 9330 9285.59976436526 44.4002356347399 29 9144 9577.00724258261 -433.007242582613 30 9895 9643.63946994292 251.360530057076 31 10404 10114.7379035511 289.26209644885 32 10195 10196.5058538881 -1.50585388807323 33 9987 9950.56311435257 36.4368856474263 34 9789 9943.1167774824 -154.116777482395 35 9437 9167.04892845061 269.951071549388 36 10096 9662.53692627574 433.463073724256 37 9776 9928.70440006541 -152.704400065414 38 9106 9126.33613961449 -20.3361396144906 39 10258 10074.1327212168 183.867278783241 40 9766 9513.14698599024 252.853014009757 41 9826 9807.10404012532 18.8959598746803 42 9957 9885.75569681204 71.2443031879565 43 10036 10333.3616094641 -297.361609464099 44 10508 10403.1101304746 104.889869525391 45 10146 10116.7384013866 29.2615986133708 46 10166 10197.4345462435 -31.4345462434795 47 9365 9391.86446159232 -26.8644615923193 48 9968 9885.71344632749 82.2865536725145 49 10123 10168.4531635824 -45.4531635823612 50 9144 9374.46208114682 -230.462081146817 51 10447 10246.8640606107 200.135939389323 52 9699 9759.81602699816 -60.8160269981556 53 10451 10036.8366125369 414.163387463077 54 10192 9981.45342097759 210.546579022414 55 10404 10596.6028939372 -192.602893937217 56 10597 10593.3242761615 3.6757238385094 57 10633 10184.2086778917 448.791322108344 58 10727 10435.1800715394 291.819928460642 59 9784 9631.24899997816 152.751000021836 60 9667 10067.9146391301 -400.914639130092 61 10297 10395.4540475107 -98.454047510689 62 9426 9583.06959595442 -157.069595954422 63 10274 10434.1644052487 -160.164405248731 64 9598 10086.6145968488 -488.614596848822 65 10400 10290.0617059047 109.938294095303 66 9985 10096.0908519605 -111.090851960507 67 10761 10676.6389374653 84.3610625346872 68 11081 10843.6355684804 237.364431519562 69 10297 10454.3702398557 -157.37023985574 70 10751 10634.3177329383 116.682267061726 71 9760 9845.68411688342 -85.6841168834232 72 10133 10317.4974811868 -184.497481186832 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.715340947890739 0.569318104218522 0.284659052109261 18 0.700015188352808 0.599969623294384 0.299984811647192 19 0.588766885678467 0.822466228643067 0.411233114321533 20 0.626947029995355 0.74610594000929 0.373052970004645 21 0.586048771562259 0.827902456875482 0.413951228437741 22 0.565149855007191 0.869700289985619 0.434850144992809 23 0.502582808323879 0.994834383352242 0.497417191676121 24 0.457844030525237 0.915688061050475 0.542155969474763 25 0.380039590193317 0.760079180386634 0.619960409806683 26 0.387197857911955 0.774395715823911 0.612802142088045 27 0.320843894867286 0.641687789734571 0.679156105132714 28 0.239984877007389 0.479969754014777 0.760015122992611 29 0.415326818727208 0.830653637454415 0.584673181272792 30 0.560165348177311 0.879669303645378 0.439834651822689 31 0.597689058986644 0.804621882026712 0.402310941013356 32 0.561767015184392 0.876465969631216 0.438232984815608 33 0.511326770475656 0.977346459048689 0.488673229524344 34 0.560082004477075 0.87983599104585 0.439917995522925 35 0.579376728696302 0.841246542607397 0.420623271303698 36 0.709460043999358 0.581079912001284 0.290539956000642 37 0.704183488500019 0.591633022999962 0.295816511499981 38 0.644541916940573 0.710916166118854 0.355458083059427 39 0.602005560241926 0.795988879516147 0.397994439758074 40 0.627626464162329 0.744747071675341 0.372373535837671 41 0.639055395924168 0.721889208151665 0.360944604075832 42 0.569656340213522 0.860687319572956 0.430343659786478 43 0.642769933266566 0.714460133466868 0.357230066733434 44 0.569621022699942 0.860757954600116 0.430378977300058 45 0.492463137527849 0.984926275055698 0.507536862472151 46 0.565812987921507 0.868374024156985 0.434187012078493 47 0.578159844939341 0.843680310121318 0.421840155060659 48 0.516382270128185 0.96723545974363 0.483617729871815 49 0.428926951884711 0.857853903769422 0.571073048115289 50 0.424047429741677 0.848094859483353 0.575952570258323 51 0.379678486501675 0.75935697300335 0.620321513498325 52 0.296137915135341 0.592275830270681 0.703862084864659 53 0.265376183881049 0.530752367762098 0.734623816118951 54 0.288685110325019 0.577370220650038 0.711314889674981 55 0.18319782556138 0.36639565112276 0.81680217443862 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

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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