Multiple Regression Including Seasonal Dummies, Trend and 3 Time Lags

*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: Thu, 16 Dec 2010 14:59:00 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/16/t129251143186kyhf4qgmlt2tw.htm/, Retrieved Thu, 16 Dec 2010 15:57:15 +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/16/t129251143186kyhf4qgmlt2tw.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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1368839.00 10192.51 1207763.00 1008380.00 989236.00 1469798.00 10467.48 1368839.00 1207763.00 1008380.00 1498721.00 10274.97 1469798.00 1368839.00 1207763.00 1761769.00 10640.91 1498721.00 1469798.00 1368839.00 1653214.00 10481.60 1761769.00 1498721.00 1469798.00 1599104.00 10568.70 1653214.00 1761769.00 1498721.00 1421179.00 10440.07 1599104.00 1653214.00 1761769.00 1163995.00 10805.87 1421179.00 1599104.00 1653214.00 1037735.00 10717.50 1163995.00 1421179.00 1599104.00 1015407.00 10864.86 1037735.00 1163995.00 1421179.00 1039210.00 10993.41 1015407.00 1037735.00 1163995.00 1258049.00 11109.32 1039210.00 1015407.00 1037735.00 1469445.00 11367.14 1258049.00 1039210.00 1015407.00 1552346.00 11168.31 1469445.00 1258049.00 1039210.00 1549144.00 11150.22 1552346.00 1469445.00 1258049.00 1785895.00 11185.68 1549144.00 1552346.00 1469445.00 1662335.00 11381.15 1785895.00 1549144.00 1552346.00 1629440.00 11679.07 1662335.00 1785895.00 1549144.00 1467430.00 12080.73 1629440.00 1662335.00 1785895.00 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 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation Y[t] = + 345982.443779569 + 12.86590127775DJIA[t] + 0.488162532437531Y1[t] + 0.148976790925935Y2[t] + 0.113189507793498Y3[t] + 63514.0036435645M1[t] + 16416.9706865163M2[t] -52070.7870921669M3[t] + 149906.134853546M4[t] -94662.8881851629M5[t] -144095.110759461M6[t] -278527.444988467M7[t] -455213.199363434M8[t] -408260.643328069M9[t] -355094.6489226M10[t] -237124.718321815M11[t] + 809.480894379494t + 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) 345982.443779569 59244.552722 5.8399 1e-06 0 DJIA 12.86590127775 3.241656 3.9689 0.000292 0.000146 Y1 0.488162532437531 0.151773 3.2164 0.002572 0.001286 Y2 0.148976790925935 0.167025 0.8919 0.377758 0.188879 Y3 0.113189507793498 0.139894 0.8091 0.42324 0.21162 M1 63514.0036435645 39904.304961 1.5917 0.119335 0.059667 M2 16416.9706865163 60049.946825 0.2734 0.785961 0.392981 M3 -52070.7870921669 53850.510152 -0.967 0.339379 0.16969 M4 149906.134853546 44648.020452 3.3575 0.001735 0.000868 M5 -94662.8881851629 64018.889479 -1.4787 0.147061 0.07353 M6 -144095.110759461 65355.155099 -2.2048 0.033281 0.01664 M7 -278527.444988467 47294.613622 -5.8892 1e-06 0 M8 -455213.199363434 43807.024028 -10.3913 0 0 M9 -408260.643328069 52094.223395 -7.837 0 0 M10 -355094.6489226 45035.142405 -7.8848 0 0 M11 -237124.718321815 25272.447981 -9.3827 0 0 t 809.480894379494 338.0139 2.3948 0.021403 0.010701 Multiple Linear Regression - Regression Statistics Multiple R 0.996336264351105 R-squared 0.992685951661115 Adjusted R-squared 0.989760332325561 F-TEST (value) 339.307967922339 F-TEST (DF numerator) 16 F-TEST (DF denominator) 40 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 26454.3175036557 Sum Squared Residuals 27993236583.3689 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1368839 1393222.75277984 -24383.7527798437 2 1469798 1460974.54510881 8823.45489118923 3 1498721 1486668.50388947 12052.496110535 4 1761769 1741554.84056126 20214.1594387354 5 1653214 1639892.16475828 13321.8352417208 6 1599104 1581859.38640429 17244.613595707 7 1421179 1433769.21566521 -12590.2156652135 8 1163995 1155394.54911255 8600.45088744576 9 1037735 1043840.65381177 -6105.6538117671 10 1015407 979622.956806689 35784.0431933114 11 1039210 1041236.24689217 -2026.24689217068 12 1258049 1274663.80444328 -16614.8044432789 13 1469445 1450152.17530915 19292.8246908541 14 1552346 1539798.28460604 12547.7153939645 15 1549144 1568619.80205982 -19475.8020598213 16 1785895 1806577.46746442 -20682.4674644227 17 1662335 1689812.29046102 -27477.2904610172 18 1629440 1619613.16700634 9826.8329936645 19 1467430 1483490.08194721 -16060.0819472071 20 1202209 1211456.97472636 -9247.97472636082 21 1076982 1104992.47058993 -28010.4705899322 22 1039367 1042026.97080501 -2659.97080500856 23 1063449 1089225.56249363 -25776.5624936311 24 1335135 1320240.41239038 14894.5876096201 25 1491602 1525637.12146894 -34035.1214689353 26 1591972 1606197.39494394 -14225.39494394 27 1641248 1638760.05790273 2487.94209726907 28 1898849 1895734.55874413 3114.44125586782 29 1798580 1798303.18947638 276.81052362487 30 1762444 1751607.41497511 10836.5850248893 31 1622044 1615006.62660201 7037.37339798946 32 1368955 1346676.60546899 22278.3945310066 33 1262973 1244508.15288044 18464.8471195628 34 1195650 1185245.4110355 10404.5889644959 35 1269530 1221784.2586519 47745.7413481024 36 1479279 1473713.26019963 5565.73980037216 37 1607819 1650983.89066225 -43164.8906622513 38 1712466 1704715.7747635 7750.22523649741 39 1721766 1714437.83591185 7328.16408815093 40 1949843 1952263.8777702 -2420.87777020246 41 1821326 1835203.10075281 -13877.1007528114 42 1757802 1749959.36252177 7842.63747822778 43 1590367 1572367.48332177 17999.5166782263 44 1260647 1284364.23752591 -23717.237525907 45 1149235 1138357.75527632 10877.2447236826 46 1016367 1059895.6613528 -43528.6613527988 47 1027885 1047827.9319623 -19942.9319623007 48 1262159 1266004.52296671 -3845.52296671334 49 1520854 1438563.05977982 82290.9402201763 50 1544144 1559040.00057771 -14896.0005777111 51 1564709 1567101.80023613 -2392.80023613371 52 1821776 1822001.25545998 -225.25545997812 53 1741365 1713609.25455152 27755.7454484828 54 1623386 1669136.66909249 -45750.6690924886 55 1498658 1495044.5924638 3613.40753620485 56 1241822 1239735.63316618 2086.36683381545 57 1136029 1131254.96744155 4774.03255845393 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 20 0.231840940907782 0.463681881815564 0.768159059092218 21 0.117149898073335 0.23429979614667 0.882850101926665 22 0.051636586349961 0.103273172699922 0.948363413650039 23 0.0260142100951644 0.0520284201903289 0.973985789904836 24 0.0291210423024228 0.0582420846048456 0.970878957697577 25 0.0266254014302951 0.0532508028605903 0.973374598569705 26 0.0213013302041266 0.0426026604082533 0.978698669795873 27 0.0636890209643227 0.127378041928645 0.936310979035677 28 0.0401297809112819 0.0802595618225638 0.959870219088718 29 0.0333410043931464 0.0666820087862928 0.966658995606854 30 0.0163364836895256 0.0326729673790511 0.983663516310474 31 0.0227175297803399 0.0454350595606799 0.97728247021966 32 0.0131731566427025 0.0263463132854051 0.986826843357297 33 0.0832763147134839 0.166552629426968 0.916723685286516 34 0.158611076723062 0.317222153446124 0.841388923276938 35 0.395962383729124 0.791924767458248 0.604037616270876 36 0.31648991625988 0.63297983251976 0.68351008374012 37 0.335609180242717 0.671218360485434 0.664390819757283 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 4 0.222222222222222 NOK 10% type I error level 9 0.5 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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