Multiple Regression Including Seasonal Dummies, Trend and 4 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 15:01:16 +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/t1292511562athtjbplsy1p3cq.htm/, Retrieved Thu, 16 Dec 2010 15:59:32 +0100

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@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/t1292511562athtjbplsy1p3cq.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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1469798.00 10467.48 1368839.00 1207763.00 1008380.00 989236.00 1498721.00 10274.97 1469798.00 1368839.00 1207763.00 1008380.00 1761769.00 10640.91 1498721.00 1469798.00 1368839.00 1207763.00 1653214.00 10481.60 1761769.00 1498721.00 1469798.00 1368839.00 1599104.00 10568.70 1653214.00 1761769.00 1498721.00 1469798.00 1421179.00 10440.07 1599104.00 1653214.00 1761769.00 1498721.00 1163995.00 10805.87 1421179.00 1599104.00 1653214.00 1761769.00 1037735.00 10717.50 1163995.00 1421179.00 1599104.00 1653214.00 1015407.00 10864.86 1037735.00 1163995.00 1421179.00 1599104.00 1039210.00 10993.41 1015407.00 1037735.00 1163995.00 1421179.00 1258049.00 11109.32 1039210.00 1015407.00 1037735.00 1163995.00 1469445.00 11367.14 1258049.00 1039210.00 1015407.00 1037735.00 1552346.00 11168.31 1469445.00 1258049.00 1039210.00 1015407.00 1549144.00 11150.22 1552346.00 1469445.00 1258049.00 1039210.00 1785895.00 11185.68 1549144.00 1552346.00 1469445.00 1258049.00 1662335.00 11381.15 1785895.00 1549144.00 1552346. 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 9 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 405212.376212214 + 14.2876090146007DJIA[t] + 0.433366643601294Y1[t] + 0.117304164406053Y2[t] + 0.00146315681682760Y3[t] + 0.198193232954481Y4[t] -20549.0296296705M1[t] -58844.700821326M2[t] + 122216.654744061M3[t] -135295.969565926M4[t] -196839.08413439M5[t] -314012.44835748M6[t] -562394.662160664M7[t] -520562.784357975M8[t] -489669.430133837M9[t] -377128.669217277M10[t] -98765.5693057837M11[t] + 737.570519139868t + 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) 405212.376212214 74469.218399 5.4413 3e-06 2e-06 DJIA 14.2876090146007 3.482476 4.1027 0.000208 0.000104 Y1 0.433366643601294 0.153249 2.8279 0.007438 0.003719 Y2 0.117304164406053 0.168455 0.6964 0.490445 0.245223 Y3 0.00146315681682760 0.166695 0.0088 0.993043 0.496521 Y4 0.198193232954481 0.148377 1.3357 0.189578 0.094789 M1 -20549.0296296705 41758.628709 -0.4921 0.625486 0.312743 M2 -58844.700821326 61514.799407 -0.9566 0.344819 0.172409 M3 122216.654744061 53218.606555 2.2965 0.027252 0.013626 M4 -135295.969565926 46548.744748 -2.9065 0.006067 0.003033 M5 -196839.08413439 63011.720563 -3.1238 0.003409 0.001705 M6 -314012.44835748 67733.758776 -4.636 4.1e-05 2.1e-05 M7 -562394.662160664 66408.528594 -8.4687 0 0 M8 -520562.784357975 85788.088171 -6.068 0 0 M9 -489669.430133837 86000.973134 -5.6938 1e-06 1e-06 M10 -377128.669217277 72659.905333 -5.1903 7e-06 4e-06 M11 -98765.5693057837 44491.573327 -2.2199 0.032469 0.016235 t 737.570519139868 341.719637 2.1584 0.037278 0.018639 Multiple Linear Regression - Regression Statistics Multiple R 0.996597096381513 R-squared 0.993205772516062 Adjusted R-squared 0.9901662496943 F-TEST (value) 326.763716135021 F-TEST (DF numerator) 17 F-TEST (DF denominator) 38 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 26131.9690148321 Sum Squared Residuals 25949432574.5016 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1469798 1467376.27035185 2421.72964815017 2 1498721 1493800.75047243 4920.24952756512 3 1761769 1744957.29958051 16811.7004194935 4 1653214 1635366.99605114 17847.0039488613 5 1599104 1579670.12208000 19433.8779200041 6 1421179 1431330.31392723 -10151.3139272284 7 1163995 1157433.49017591 6561.509824087 8 1037735 1044819.99435049 -7084.99435048725 9 1015407 982686.146506649 32720.853493351 10 1039210 1037674.28436873 1535.71563126705 11 1258049 1274970.42379117 -16921.423791174 12 1469445 1450730.36195842 18714.6380415762 13 1552346 1540970.36758902 11375.6324109812 14 1549144 1568915.75262806 -19771.7526280553 15 1785895 1803240.02427383 -17345.0242738306 16 1662335 1693500.70155881 -31165.7015588062 17 1629440 1627602.54990807 1837.45009193418 18 1467430 1487867.60805326 -20437.6080532646 19 1202209 1214913.58318302 -12704.5831830183 20 1076982 1102450.219886 -25468.2198860006 21 1039367 1044208.95726232 -4841.95726232361 22 1063449 1088954.82497912 -25505.8249791178 23 1335135 1322555.93444543 12579.0655545659 24 1491602 1527873.09070757 -36271.090707575 25 1591972 1608387.94815593 -16415.9481559265 26 1641248 1634722.22472636 6525.7752736366 27 1898849 1900915.43440801 -2066.43440801189 28 1798580 1794796.31802505 3783.68197495404 29 1762444 1748405.11936344 14038.8806365614 30 1622044 1615181.50576964 6862.49423035942 31 1368955 1345384.71893368 23570.281066323 32 1262973 1240351.47679698 22621.5232030184 33 1195650 1180218.44213451 15431.5578654944 34 1269530 1218206.44903519 51323.5509648063 35 1479279 1471061.37678225 8217.623217747 36 1607819 1656987.37632816 -49168.3763281587 37 1712466 1701652.80157463 10813.1984253682 38 1721766 1721066.03950806 699.960491935596 39 1949843 1961329.90679539 -11486.9067953871 40 1821326 1832480.64676173 -11154.6467617347 41 1757802 1753588.6968792 4213.30312079985 42 1590367 1574927.16000081 15439.8399991883 43 1260647 1285199.19503061 -24552.1950306141 44 1149235 1138921.78294903 10313.2170509651 45 1016367 1059677.45409652 -43310.4540965218 46 1027885 1055238.44161696 -27353.4416169555 47 1262159 1266034.26498114 -3875.26498113887 48 1520854 1454129.17100584 66724.8289941574 49 1544144 1552338.61232857 -8194.61232857302 50 1564709 1557083.23266508 7625.76733491793 51 1821776 1807689.33494226 14086.6650577361 52 1741365 1720675.33760327 20689.6623967256 53 1623386 1662909.5117693 -39523.5117692996 54 1498658 1490371.41224905 8286.58775094537 55 1241822 1234697.01267678 7124.98732322246 56 1136029 1136410.52601750 -381.526017495626 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.0236986683364031 0.0473973366728061 0.976301331663597 22 0.0059233635521012 0.0118467271042024 0.994076636447899 23 0.0210508896911284 0.0421017793822569 0.978949110308872 24 0.0130742884427536 0.0261485768855072 0.986925711557246 25 0.0219285548727814 0.0438571097455627 0.978071445127219 26 0.0565886331844631 0.113177266368926 0.943411366815537 27 0.0696811703255266 0.139362340651053 0.930318829674473 28 0.0533617053747738 0.106723410749548 0.946638294625226 29 0.0263116783419665 0.052623356683933 0.973688321658034 30 0.0357112444771493 0.0714224889542986 0.96428875552285 31 0.0240352988435742 0.0480705976871484 0.975964701156426 32 0.112290892529011 0.224581785058022 0.887709107470989 33 0.236166916885292 0.472333833770584 0.763833083114708 34 0.296619422405108 0.593238844810216 0.703380577594892 35 0.498793083801226 0.997586167602452 0.501206916198774 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 6 0.4 NOK 10% type I error level 8 0.533333333333333 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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