Multiple Regression Including Seasonal Dummies and Trend

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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Sat, 11 Dec 2010 22:40:30 +0000

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/11/t1292107191ulljfe1j8aoi6dh.htm/, Retrieved Sat, 11 Dec 2010 23:40:01 +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/11/t1292107191ulljfe1j8aoi6dh.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}, }

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Dataseries X:

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989236.00 10489.94 1008380.00 10766.23 1207763.00 10503.76 1368839.00 10192.51 1469798.00 10467.48 1498721.00 10274.97 1761769.00 10640.91 1653214.00 10481.60 1599104.00 10568.70 1421179.00 10440.07 1163995.00 10805.87 1037735.00 10717.50 1015407.00 10864.86 1039210.00 10993.41 1258049.00 11109.32 1469445.00 11367.14 1552346.00 11168.31 1549144.00 11150.22 1785895.00 11185.68 1662335.00 11381.15 1629440.00 11679.07 1467430.00 12080.73 1202209.00 12221.93 1076982.00 12463.15 1039367.00 12621.69 1063449.00 12268.63 1335135.00 12354.35 1491602.00 13062.91 1591972.00 13627.64 1641248.00 13408.62 1898849.00 13211.99 1798580.00 13357.74 1762444.00 13895.63 1622044.00 13930.01 1368955.00 13371.72 1262973.00 13264.82 1195650.00 12650.36 1269530.00 12266.39 1479279.00 12262.89 1607819.00 12820.13 1712466.00 12638.32 1721766.00 11350.01 1949843.00 11378.02 1821326.00 11543.55 1757802.00 10850.66 1590367.00 9325.01 1260647.00 8829.04 1149235.00 8776.39 1016367.00 8000.86 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 12 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation Passengersbrussels[t] = + 643506.889523468 + 33.0554647235123DJIA[t] -37601.179288474M1[t] -2084.93078149814M2[t] + 218151.003437736M3[t] + 386307.756645739M4[t] + 460143.964267434M5[t] + 489460.295892718M6[t] + 728273.668754439M7[t] + 612201.854211425M8[t] + 544958.341672279M9[t] + 395143.631685458M10[t] + 118806.647585149M11[t] + 3366.03286395014t + 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) 643506.889523468 55597.882389 11.5743 0 0 DJIA 33.0554647235123 4.053388 8.155 0 0 M1 -37601.179288474 31091.216256 -1.2094 0.232695 0.116347 M2 -2084.93078149814 31121.009248 -0.067 0.946877 0.473438 M3 218151.003437736 31033.333354 7.0296 0 0 M4 386307.756645739 30907.22407 12.4989 0 0 M5 460143.964267434 30859.091968 14.9111 0 0 M6 489460.295892718 30867.966075 15.8566 0 0 M7 728273.668754439 30811.808138 23.6362 0 0 M8 612201.854211425 30785.045101 19.8863 0 0 M9 544958.341672279 30773.524087 17.7087 0 0 M10 395143.631685458 30760.584075 12.8458 0 0 M11 118806.647585149 30752.229412 3.8634 0.000348 0.000174 t 3366.03286395014 388.608455 8.6618 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.987238960106177 R-squared 0.974640764351525 Adjusted R-squared 0.967474023842174 F-TEST (value) 135.994984481405 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 48619.1823506069 Sum Squared Residuals 108735945052.312 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 989236 956021.584720707 33214.4152792926 2 1008380 1004036.76044009 4343.23955991047 3 1207763 1218962.65969729 -11199.6596972938 4 1368839 1380196.93237405 -11357.9323740538 5 1469798 1466488.43399472 3309.56600527653 6 1498721 1492807.29097003 5913.70902996611 7 1761769 1747083.01345663 14685.9865433729 8 1653214 1629111.16569246 24102.8343075397 9 1599104 1568112.81699468 30991.1830053173 10 1421179 1417412.21544443 3766.78455557408 11 1163995 1156532.95320393 7462.04679607066 12 1037735 1038171.22706511 -436.227065113023 13 1015407 1008807.13392225 6599.86607775436 14 1039210 1051938.69528338 -12728.6952833793 15 1258049 1279372.12128267 -21323.1212826655 16 1469445 1459417.26726964 10027.7327303650 17 1552346 1530047.08970430 22298.9102956958 18 1549144 1562131.48083669 -12987.4808366895 19 1785895 1805483.03334146 -19588.0333414567 20 1662335 1699238.60335190 -36903.6033518976 21 1629440 1645209.00772713 -15769.0077271301 22 1467430 1512037.38856511 -44607.3885651054 23 1202209 1243733.86894771 -41524.8689477072 24 1076982 1136266.89342711 -59284.8934271138 25 1039367 1107272.36037986 -67905.3603798555 26 1063449 1134484.07937550 -71035.0793754983 27 1335135 1360919.56089478 -25784.5608947817 28 1491602 1555864.12705123 -64262.1270512271 29 1591972 1651733.78013018 -59761.7801301816 30 1641248 1677176.33673567 -35928.3367356715 31 1898849 1912856.04643276 -14007.0464327586 32 1798580 1804968.09873715 -6388.09873714658 33 1762444 1758870.82298208 3573.17701791966 34 1622044 1613558.59273640 8485.40726359599 35 1368955 1322133.10609956 46821.8939004439 36 1262973 1203158.86219941 59814.1378005865 37 1195650 1148612.45492088 47037.5450791198 38 1269530 1174802.42950192 94727.5704980807 39 1479279 1398288.70245857 80990.297541429 40 1607819 1588231.31569305 19587.6843069455 41 1712466 1659423.74213732 53042.2578626821 42 1721766 1649520.42086860 72245.5791313965 43 1949843 1892625.71016118 57217.2898388197 44 1821326 1785391.5995578 35934.4004422005 45 1757802 1698610.31893033 59191.6810696714 46 1590367 1501730.57205203 88636.4279479687 47 1260647 1212365.10197675 48281.8980232471 48 1149235 1095184.11703786 54050.8829621392 49 1016367 1035313.46605631 -18946.4660563113 50 1027885 1043192.03539911 -15307.0353991136 51 1262159 1284841.95566669 -22682.955666688 52 1520854 1474849.35761203 46004.6423879704 53 1544144 1563032.95403347 -18888.9540334729 54 1564709 1593952.47058900 -29243.4705890016 55 1821776 1860084.19660798 -38308.1966079773 56 1741365 1758110.53266070 -16745.5326606960 57 1623386 1701373.03336578 -77987.0333657784 58 1498658 1554939.23120203 -56281.2312020333 59 1241822 1302862.96977205 -61040.9697720544 60 1136029 1190172.9002705 -54143.9002704989 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.00587755359110589 0.0117551071822118 0.994122446408894 18 0.00357038066369307 0.00714076132738613 0.996429619336307 19 0.00141650247659 0.00283300495318 0.99858349752341 20 0.00507617235053519 0.0101523447010704 0.994923827649465 21 0.00284292498918308 0.00568584997836615 0.997157075010817 22 0.00111709418141948 0.00223418836283896 0.99888290581858 23 0.000383523561019023 0.000767047122038047 0.99961647643898 24 0.00014805048839746 0.00029610097679492 0.999851949511603 25 0.000109808546066683 0.000219617092133366 0.999890191453933 26 8.6238630538108e-05 0.000172477261076216 0.999913761369462 27 0.000139771707031106 0.000279543414062212 0.999860228292969 28 0.000142706439452857 0.000285412878905714 0.999857293560547 29 0.000169866324540954 0.000339732649081908 0.99983013367546 30 0.00055434663858781 0.00110869327717562 0.999445653361412 31 0.00180745806162467 0.00361491612324934 0.998192541938375 32 0.00923550108382033 0.0184710021676407 0.99076449891618 33 0.0189438027098674 0.0378876054197348 0.981056197290133 34 0.136645162767549 0.273290325535097 0.863354837232451 35 0.390811885137579 0.781623770275158 0.609188114862421 36 0.736873932598083 0.526252134803833 0.263126067401917 37 0.702227461456783 0.595545077086435 0.297772538543217 38 0.79221125729994 0.415577485400118 0.207788742700059 39 0.818465003667327 0.363069992665346 0.181534996332673 40 0.95398264705338 0.09203470589324 0.04601735294662 41 0.90360358776734 0.192792824465320 0.0963964122326599 42 0.817030298259474 0.365939403481053 0.182969701740526 43 0.671832693773745 0.65633461245251 0.328167306226255 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 13 0.481481481481481 NOK 5% type I error level 17 0.62962962962963 NOK 10% type I error level 18 0.666666666666667 NOK

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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