Multiple Regression Including Seasonal Dummies

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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 22:37:21 +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/11/t1292106976exgqduw9ix3wrtx.htm/, Retrieved Sat, 11 Dec 2010 23:36:26 +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/t1292106976exgqduw9ix3wrtx.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}, }

Original text written by user:

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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 7 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Passengersbrussels[t] = + 884598.98401026 + 22.2814211190764DJIA[t] -76830.1862664165M1[t] -40684.7705480636M2[t] + 183955.060155536M3[t] + 359295.240713152M4[t] + 438202.516695373M5[t] + 467068.154689648M6[t] + 711310.55273661M7[t] + 600053.039546941M8[t] + 537136.647657435M9[t] + 388063.866022355M10[t] + 115275.750305964M11[t] + 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) 884598.98401026 77231.815682 11.4538 0 0 DJIA 22.2814211190764 6.190592 3.5992 0.000766 0.000383 M1 -76830.1862664165 49359.527649 -1.5565 0.126288 0.063144 M2 -40684.7705480636 49424.855971 -0.8232 0.414572 0.207286 M3 183955.060155536 49394.199955 3.7242 0.000524 0.000262 M4 359295.240713152 49343.323339 7.2815 0 0 M5 438202.516695373 49352.086974 8.8791 0 0 M6 467068.154689648 49359.429069 9.4626 0 0 M7 711310.55273661 49343.358725 14.4155 0 0 M8 600053.039546941 49349.086998 12.1594 0 0 M9 537136.647657435 49360.632498 10.8819 0 0 M10 388063.866022355 49343.704033 7.8645 0 0 M11 115275.750305964 49343.39052 2.3362 0.023797 0.011899 Multiple Linear Regression - Regression Statistics Multiple R 0.966064082276502 R-squared 0.93327981106474 Adjusted R-squared 0.916244869208929 F-TEST (value) 54.7862046706293 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 78018.6069876803 Sum Squared Residuals 286084442706.011 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 989236 1041499.56839769 -52263.5683976905 2 1008380 1083801.11795703 -75421.1179570296 3 1207763 1302592.74405951 -94829.7440595054 4 1368839 1470997.83229381 -102158.832293808 5 1469798 1556031.83064114 -86233.8306411425 6 1498721 1580608.07225578 -81887.0722557843 7 1761769 1833004.13354706 -71235.1335470604 8 1653214 1718196.96715891 -64982.9671589117 9 1599104 1657221.28704888 -58117.2870488778 10 1421179 1505282.44621525 -84103.4462152505 11 1163995 1240644.87434422 -76649.8743442185 12 1037735 1123400.11485396 -85665.1148539613 13 1015407 1049853.31880365 -34446.3188036516 14 1039210 1088863.01120686 -49653.0112068618 15 1258049 1316085.48143237 -58036.4814323732 16 1469445 1497170.25798291 -27725.2579829092 17 1552346 1571647.31900402 -19301.3190040248 18 1549144 1600109.88609026 -50965.8860902557 19 1785895 1845142.3833301 -59247.3833300997 20 1662335 1738240.21952658 -75905.2195265771 21 1629440 1681961.90861687 -52521.9086168661 22 1467430 1541838.68258847 -74408.6825884745 23 1202209 1272196.70353410 -69987.7035340972 24 1076982 1162295.67763048 -85313.6776304769 25 1039367 1088997.98786828 -49630.9878682786 26 1063449 1117276.72504633 -53827.7250463305 27 1335135 1343826.51916826 -8691.51916825695 28 1491602 1534954.42347401 -43352.4234740053 29 1591972 1626444.68640480 -34472.6864048031 30 1641248 1650430.24754558 -9182.24754557795 31 1898849 1890291.44975790 8557.55024210457 32 1798580 1782281.45369633 16298.5463036676 33 1762444 1731350.01541257 31093.9845874338 34 1622044 1583043.26903556 39000.7309644399 35 1368955 1297815.6587226 71139.3412773999 36 1262973 1180158.02449901 82814.9755009933 37 1195650 1089636.79621176 106013.203788237 38 1269530 1117226.81466302 152303.185336976 39 1479279 1341788.66039271 137490.339607294 40 1607819 1529544.94005472 78274.059945284 41 1712466 1604401.23086328 108064.769136722 42 1721766 1604561.49121564 117204.508784364 43 1949843 1849427.99186814 100415.008131857 44 1821326 1741858.72231632 79467.2776836848 45 1757802 1663503.75654761 94298.243452388 46 1590367 1480437.32478221 109929.675217787 47 1260647 1196598.29263339 64048.7073666059 48 1149235 1080149.42550551 69085.5744944893 49 1016367 986039.328718617 30327.6712813833 50 1027885 1001286.33112675 26598.6688732456 51 1262159 1238091.59494716 24067.4050528418 52 1520854 1425891.54619456 94962.4538054386 53 1544144 1512200.93308675 31943.0669132487 54 1564709 1539878.30289275 24830.6971072540 55 1821776 1800266.0414968 21509.9585031984 56 1741365 1696242.63730186 45122.3626981365 57 1623386 1638139.03237408 -14753.0323740779 58 1498658 1489076.27737850 9581.72262149837 59 1241822 1230372.47076569 11449.52923431 60 1136029 1116950.75751104 19078.2424889555 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.000516661175342282 0.00103332235068456 0.999483338824658 17 0.000358844919759532 0.000717689839519063 0.99964115508024 18 0.000258358590523796 0.000516717181047593 0.999741641409476 19 9.03878168239745e-05 0.000180775633647949 0.999909612183176 20 0.000605061711075848 0.00121012342215170 0.999394938288924 21 0.000441402044553384 0.000882804089106769 0.999558597955447 22 0.000400464815766782 0.000800929631533563 0.999599535184233 23 0.000234506701516885 0.00046901340303377 0.999765493298483 24 0.00022608397643762 0.00045216795287524 0.999773916023562 25 0.000226673453404253 0.000453346906808505 0.999773326546596 26 0.000187460681893040 0.000374921363786081 0.999812539318107 27 0.000372526549550951 0.000745053099101901 0.99962747345045 28 0.000483116562277005 0.00096623312455401 0.999516883437723 29 0.000686851726796325 0.00137370345359265 0.999313148273204 30 0.00105335655544812 0.00210671311089625 0.998946643444552 31 0.00188669842864831 0.00377339685729662 0.998113301571352 32 0.00391120040854513 0.00782240081709026 0.996088799591455 33 0.00387554933804986 0.00775109867609973 0.99612445066195 34 0.0146047075844534 0.0292094151689068 0.985395292415547 35 0.0555097037208678 0.111019407441736 0.944490296279132 36 0.156230353626925 0.312460707253850 0.843769646373075 37 0.229047232586361 0.458094465172723 0.770952767413638 38 0.487971773765329 0.975943547530659 0.512028226234671 39 0.593771118875366 0.812457762249268 0.406228881124634 40 0.624914568357521 0.750170863284958 0.375085431642479 41 0.564584369112828 0.870831261774344 0.435415630887172 42 0.577596321893169 0.844807356213662 0.422403678106831 43 0.561552620440682 0.876894759118637 0.438447379559318 44 0.458235515257073 0.916471030514146 0.541764484742927 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 18 0.620689655172414 NOK 5% type I error level 19 0.655172413793103 NOK 10% type I error level 19 0.655172413793103 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No 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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