Multiple Regression Including Seasonal Dummies, Trend and 2 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 14:55:05 +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/t12925111990860bzzsasbk8ys.htm/, Retrieved Thu, 16 Dec 2010 15:53:19 +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/t12925111990860bzzsasbk8ys.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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1207763.00 10503.76 1008380.00 989236.00 1368839.00 10192.51 1207763.00 1008380.00 1469798.00 10467.48 1368839.00 1207763.00 1498721.00 10274.97 1469798.00 1368839.00 1761769.00 10640.91 1498721.00 1469798.00 1653214.00 10481.60 1761769.00 1498721.00 1599104.00 10568.70 1653214.00 1761769.00 1421179.00 10440.07 1599104.00 1653214.00 1163995.00 10805.87 1421179.00 1599104.00 1037735.00 10717.50 1163995.00 1421179.00 1015407.00 10864.86 1037735.00 1163995.00 1039210.00 10993.41 1015407.00 1037735.00 1258049.00 11109.32 1039210.00 1015407.00 1469445.00 11367.14 1258049.00 1039210.00 1552346.00 11168.31 1469445.00 1258049.00 1549144.00 11150.22 1552346.00 1469445.00 1785895.00 11185.68 1549144.00 1552346.00 1662335.00 11381.15 1785895.00 1549144.00 1629440.00 11679.07 1662335.00 1785895.00 1467430.00 12080.73 1629440.00 1662335.00 1202209.00 12221.93 1467430.00 1629440.00 1076982.00 12463.15 1202209.00 1467430.00 1039367.00 12621.69 1076982.00 1202209.00 1063449.00 12268.63 1039367.00 10769 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 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 129656.217228832 + 12.4358918340159DJIA[t] + 0.524824953147275Y1[t] + 0.222560731502311Y2[t] + 221392.319599041M1[t] + 273538.387148335M2[t] + 206489.247472616M3[t] + 146948.49845131M4[t] + 362848.209945268M5[t] + 116948.288333272M6[t] + 55546.0535751213M7[t] -40727.3840958933M8[t] -219577.429709325M9[t] -158224.026404345M10[t] -97473.7128923078M11[t] + 874.612157839409t + 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) 129656.217228832 57637.816627 2.2495 0.029781 0.014891 DJIA 12.4358918340159 3.079322 4.0385 0.000224 0.000112 Y1 0.524824953147275 0.145834 3.5988 0.000837 0.000418 Y2 0.222560731502311 0.132311 1.6821 0.099971 0.049985 M1 221392.319599041 21753.826268 10.1772 0 0 M2 273538.387148335 44595.776518 6.1337 0 0 M3 206489.247472616 45965.934225 4.4922 5.4e-05 2.7e-05 M4 146948.49845131 43381.094588 3.3874 0.001543 0.000771 M5 362848.209945268 41606.347274 8.721 0 0 M6 116948.288333272 67849.313808 1.7236 0.092127 0.046063 M7 55546.0535751213 49809.624871 1.1152 0.271119 0.13556 M8 -40727.3840958933 45260.145545 -0.8999 0.37333 0.186665 M9 -219577.429709325 37915.746689 -5.7912 1e-06 0 M10 -158224.026404345 35261.391135 -4.4872 5.5e-05 2.8e-05 M11 -97473.7128923078 20490.289028 -4.7571 2.3e-05 1.2e-05 t 874.612157839409 330.178509 2.6489 0.01133 0.005665 Multiple Linear Regression - Regression Statistics Multiple R 0.996234563754612 R-squared 0.992483306019343 Adjusted R-squared 0.989798772454822 F-TEST (value) 369.704189635123 F-TEST (DF numerator) 15 F-TEST (DF denominator) 42 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 26380.5966643776 Sum Squared Residuals 29229306975.4799 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1207763 1231934.84623924 -24171.8462392447 2 1368839 1389986.73089028 -21147.7308902834 3 1469798 1456143.23103228 13654.7689677218 4 1498721 1483918.05546411 14802.9445358923 5 1761769 1742892.19038527 18876.8096147344 6 1653214 1640376.99731576 12837.0026842425 7 1599104 1582504.32340551 16599.6765944936 8 1421179 1432947.51070269 -11768.5107026891 9 1163995 1154098.88550966 9896.11449033933 10 1037735 1040652.24230833 -2917.24230833081 11 1015407 980606.263243803 34800.7367561969 12 1039210 1040734.41267586 -1524.41267585885 13 1258049 1271965.861002 -13916.8610020010 14 1469445 1448342.54335553 21102.4566444735 15 1552346 1539346.25118104 12999.7488189553 16 1549144 1571012.11087183 -21868.1108718256 17 1785895 1804997.42895035 -19102.4289503532 18 1662335 1685943.15629329 -23608.1562932917 19 1629440 1616964.53912120 12475.4608788035 20 1467430 1481796.99310387 -14366.9931038669 21 1202209 1213229.48165308 -11020.4816530790 22 1076982 1103205.61993474 -26223.6199347369 23 1039367 1042052.09771843 -2685.09771843052 24 1063449 1088397.90346119 -24948.9034611854 25 1335135 1315998.05247231 19136.9475276891 26 1491602 1525777.60745416 -34175.6074541639 27 1591972 1609210.42197474 -17238.4219747393 28 1641248 1635320.66660515 5927.3333948493 29 1898849 1897849.41585780 999.58414220199 30 1798580 1800798.97299965 -2218.97299964819 31 1762444 1751668.68602654 10775.3139734618 32 1622044 1615416.38998068 6627.61001931874 33 1368955 1348770.24445763 20184.7555423690 34 1262973 1245593.91381338 17379.0861866209 35 1195650 1187627.81022728 8022.18977271624 36 1269530 1222280.90412311 47249.095876888 37 1479279 1468294.92167016 10984.0783298362 38 1607819 1654769.67368396 -46950.6736839629 39 1712466 1700477.06702117 11988.9329788306 40 1721766 1709318.95964833 12447.0403516682 41 1949843 1954612.79756419 -4769.79756419246 42 1821326 1833416.31692726 -12090.3169272630 43 1757802 1747584.04468930 10217.9553106956 44 1590367 1571270.58294535 19096.4170546472 45 1260647 1285115.30627868 -24468.3062786766 46 1149235 1136378.83240563 12856.1675943738 47 1016367 1056504.82881048 -40137.8288104827 48 1027885 1048660.77973984 -20775.7797398437 49 1262159 1254191.31861628 7967.68138372035 50 1520854 1439682.44461606 81171.5553839367 51 1544144 1565549.02879077 -21405.0287907684 52 1564709 1576018.20741058 -11309.2074105842 53 1821776 1817780.16724239 3995.83275760925 54 1741365 1716284.55646404 25080.4435359603 55 1623386 1673454.40675745 -50068.4067574546 56 1498658 1498246.52326741 411.476732590057 57 1241822 1236414.08210095 5407.9178990472 58 1136029 1137123.39153793 -1094.39153792697 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.127475651195992 0.254951302391985 0.872524348804008 20 0.109202513177519 0.218405026355039 0.89079748682248 21 0.065791358952582 0.131582717905164 0.934208641047418 22 0.0292884674890476 0.0585769349780952 0.970711532510952 23 0.0153251702974933 0.0306503405949865 0.984674829702507 24 0.00679884630722808 0.0135976926144562 0.993201153692772 25 0.0098190405776762 0.0196380811553524 0.990180959422324 26 0.0158512285403045 0.0317024570806089 0.984148771459695 27 0.02505444097409 0.05010888194818 0.97494555902591 28 0.0318888589537789 0.0637777179075578 0.968111141046221 29 0.0232122243361216 0.0464244486722431 0.976787775663878 30 0.0268890288559927 0.0537780577119853 0.973110971144007 31 0.0157151084924536 0.0314302169849072 0.984284891507546 32 0.0292799888803841 0.0585599777607682 0.970720011119616 33 0.0234225320350922 0.0468450640701843 0.976577467964908 34 0.164931240911555 0.32986248182311 0.835068759088445 35 0.193152001268936 0.386304002537871 0.806847998731064 36 0.194751374570672 0.389502749141345 0.805248625429328 37 0.278883599078344 0.557767198156687 0.721116400921656 38 0.480762534704821 0.961525069409642 0.519237465295179 39 0.341184747234269 0.682369494468537 0.658815252765731 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 7 0.333333333333333 NOK 10% type I error level 12 0.571428571428571 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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