Multiple Regression1

*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: Sat, 11 Dec 2010 22:18:38 +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/t1292106621fuv5238oylxr45o.htm/, Retrieved Sat, 11 Dec 2010 23:30:21 +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/t1292106621fuv5238oylxr45o.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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No (this computation is public)

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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 5 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation Passengersbrussels[t] = + 1071749.92409519 + 33.106785413492DJIA[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) 1071749.92409519 234577.707801 4.5688 2.6e-05 1.3e-05 DJIA 33.106785413492 20.976377 1.5783 0.119938 0.059969 Multiple Linear Regression - Regression Statistics Multiple R 0.202927626023469 R-squared 0.0411796214035209 Adjusted R-squared 0.0246482355656507 F-TEST (value) 2.49099632707054 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.119938212265450 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 266239.885706474 Sum Squared Residuals 4111253250977.78 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 989236 1419038.1166756 -429802.116675601 2 1008380 1428185.19041749 -419805.190417493 3 1207763 1419495.65245001 -211732.652450014 4 1368839 1409191.16549006 -40352.1654900646 5 1469798 1418294.53827521 51503.4617247875 6 1498721 1411921.15101526 86799.848984739 7 1761769 1424036.24806947 337732.751930526 8 1653214 1418762.00608525 234451.993914749 9 1599104 1421645.60709477 177458.392905234 10 1421179 1417387.08128703 3791.91871297136 11 1163995 1429497.54339128 -265502.543391284 12 1037735 1426571.89676429 -388836.896764294 13 1015407 1431450.51266283 -416043.512662826 14 1039210 1435706.38992773 -396496.38992773 15 1258049 1439543.79742501 -181494.797425008 16 1469445 1448079.38884031 21365.6111596853 17 1552346 1441496.76669655 110849.23330345 18 1549144 1440897.86494842 108246.13505158 19 1785895 1442071.83155918 343823.168440818 20 1662335 1448543.21490396 213791.785096042 21 1629440 1458406.38841435 171033.611585655 22 1467430 1471704.05984353 -4274.05984352845 23 1202209 1476378.73794391 -274169.737943914 24 1076982 1484364.75672136 -407382.756721356 25 1039367 1489613.50648081 -450246.506480811 26 1063449 1477924.82482272 -414475.824822724 27 1335135 1480762.73846837 -145627.738468368 28 1491602 1504220.88234095 -12618.8823409521 29 1591972 1522917.27726751 69054.7227324866 30 1641248 1515666.22912625 125581.770873750 31 1898849 1509156.44191040 389692.558089605 32 1798580 1513981.75588441 284598.244115588 33 1762444 1531789.56469048 230654.435309525 34 1622044 1532927.77597299 89116.224027009 35 1368955 1514444.58874449 -145489.588744493 36 1262973 1510905.47338379 -247932.47338379 37 1195650 1490562.67801862 -294912.678018616 38 1269530 1477850.66562340 -208320.665623397 39 1479279 1477734.79187445 1544.20812554985 40 1607819 1496183.21697826 111635.783021736 41 1712466 1490164.07232224 222301.927677763 42 1721766 1447512.26960618 274253.730393818 43 1949843 1448439.59066561 501403.409334387 44 1821326 1453919.75685511 367406.243144891 45 1757802 1430980.39630995 326821.603690046 46 1590367 1380471.02914386 209895.970856140 47 1260647 1364051.05678233 -103404.056782331 48 1149235 1362307.98453031 -213072.98453031 49 1016367 1336632.67923858 -320265.679238585 50 1027885 1305580.83199571 -277695.831995708 51 1262159 1323656.80576362 -61497.8057636207 52 1520854 1342170.12016685 178683.879833155 53 1544144 1353168.52534906 190975.474650938 54 1564709 1351402.94048296 213306.05951704 55 1821776 1375392.44826143 446383.551738569 56 1741365 1386141.22828163 355223.771718371 57 1623386 1393292.29393094 230093.706069057 58 1498658 1393307.19198438 105350.808015621 59 1241822 1414234.32211210 -172412.322112102 60 1136029 1416989.13772636 -280960.137726358 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.336551027408893 0.673102054817787 0.663448972591107 6 0.225726299053095 0.451452598106191 0.774273700946905 7 0.709651963684601 0.580696072630797 0.290348036315398 8 0.713602570087373 0.572794859825254 0.286397429912627 9 0.676385494217328 0.647229011565344 0.323614505782672 10 0.56882177546268 0.86235644907464 0.43117822453732 11 0.496113926992465 0.99222785398493 0.503886073007535 12 0.497629573011568 0.995259146023137 0.502370426988431 13 0.469575741297462 0.939151482594924 0.530424258702538 14 0.424606942472294 0.849213884944589 0.575393057527706 15 0.408019262923499 0.816038525846998 0.591980737076501 16 0.490220512854605 0.98044102570921 0.509779487145395 17 0.498703878797394 0.997407757594787 0.501296121202606 18 0.472868262894514 0.945736525789028 0.527131737105486 19 0.578510303874273 0.842979392251454 0.421489696125727 20 0.55367015367938 0.89265969264124 0.44632984632062 21 0.492748459800135 0.98549691960027 0.507251540199865 22 0.418996391177156 0.837992782354313 0.581003608822844 23 0.436906665335915 0.87381333067183 0.563093334664085 24 0.511864569098319 0.976270861803362 0.488135430901681 25 0.597851173648182 0.804297652703636 0.402148826351818 26 0.668808710018814 0.662382579962372 0.331191289981186 27 0.630037026115978 0.739925947768044 0.369962973884022 28 0.599586837984897 0.800826324030206 0.400413162015103 29 0.574531123860656 0.850937752278688 0.425468876139344 30 0.538690941259919 0.922618117480161 0.461309058740081 31 0.62197678478271 0.75604643043458 0.37802321521729 32 0.614584456213431 0.770831087573137 0.385415543786569 33 0.573122756125457 0.853754487749086 0.426877243874543 34 0.497610750976743 0.995221501953485 0.502389249023257 35 0.46201995971998 0.92403991943996 0.53798004028002 36 0.498471476946992 0.996942953893983 0.501528523053008 37 0.603629937168227 0.792740125663545 0.396370062831773 38 0.677417958295361 0.645164083409277 0.322582041704639 39 0.661177444549415 0.67764511090117 0.338822555450585 40 0.644149363555708 0.711701272888583 0.355850636444292 41 0.616949259166243 0.766101481667514 0.383050740833757 42 0.574373249157682 0.851253501684637 0.425626750842318 43 0.631754108055115 0.73649178388977 0.368245891944885 44 0.604629433798416 0.790741132403168 0.395370566201584 45 0.580037076327375 0.83992584734525 0.419962923672625 46 0.52964701994663 0.940705960106741 0.470352980053371 47 0.451814345531793 0.903628691063586 0.548185654468207 48 0.425717600958218 0.851435201916436 0.574282399041782 49 0.486965044616028 0.973930089232056 0.513034955383972 50 0.641207066478689 0.717585867042622 0.358792933521311 51 0.773782546131827 0.452434907736346 0.226217453868173 52 0.752717987869609 0.494564024260782 0.247282012130391 53 0.7316504951113 0.536699009777401 0.268349504888700 54 0.994608676681441 0.0107826466371171 0.00539132331855853 55 0.984046005240724 0.0319079895185526 0.0159539947592763 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 2 0.0392156862745098 OK 10% type I error level 2 0.0392156862745098 OK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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