*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: Fri, 20 Nov 2009 06:08:56 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Nov/20/t1258722608mss971w61dqdzat.htm/, Retrieved Fri, 20 Nov 2009 14:10:21 +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/2009/Nov/20/t1258722608mss971w61dqdzat.htm/}, year = {2009}, } @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 = {2009}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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280 1258 557 1199 831 1158 1081 1427 1318 934 1578 709 1859 1186 2141 986 2428 1033 2715 1257 3004 1105 3309 1179 269 1092 537 1092 813 1087 1068 2028 1411 2039 1675 2010 1958 754 2242 760 2524 715 2836 855 3143 971 3522 815 285 915 574 843 865 761 1147 1858 1516 2968 1789 4061 2087 3661 2372 3269 2669 2857 2966 2568 3270 2274 3652 1987 329 683 658 381 988 71 1303 1772 1603 3485 1929 5181 2235 4479 2544 3782 2872 3067 3198 2489 3544 1903 3903 1330 332 736 665 483 1001 242 1329 1334 1639 2423 1975 3523 2304 2986 2640 2462 2992 1908 3330 1575 3690 1237 4063 904

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 1574.61263275984 + 0.219776211344191X[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) 1574.61263275984 250.362879 6.2893 0 0 X 0.219776211344191 0.121072 1.8153 0.074655 0.037328 Multiple Linear Regression - Regression Statistics Multiple R 0.231859778739834 R-squared 0.0537589569972848 Adjusted R-squared 0.0374444562558588 F-TEST (value) 3.29516409048172 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.0746552735370818 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1055.01042915304 Sum Squared Residuals 64556726.3260573 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 280 1851.09110663083 -1571.09110663083 2 557 1838.12431016152 -1281.12431016152 3 831 1829.11348549641 -998.113485496412 4 1081 1888.233286348 -807.233286348 5 1318 1779.88361415531 -461.883614155313 6 1578 1730.43396660287 -152.433966602870 7 1859 1835.26721941405 23.7327805859509 8 2141 1791.31197714521 349.688022854789 9 2428 1801.64145907839 626.358540921612 10 2715 1850.87133041949 864.128669580513 11 3004 1817.46534629517 1186.53465370483 12 3309 1833.72878593464 1475.27121406536 13 269 1814.60825554770 -1545.60825554770 14 537 1814.60825554770 -1277.60825554770 15 813 1813.50937449097 -1000.50937449097 16 1068 2020.31878936586 -952.318789365858 17 1411 2022.73632769064 -611.736327690644 18 1675 2016.36281756166 -341.362817561662 19 1958 1740.32389611336 217.676103886641 20 2242 1741.64255338142 500.357446618576 21 2524 1731.75262387094 792.247376129065 22 2836 1762.52129345912 1073.47870654088 23 3143 1788.01533397505 1354.98466602495 24 3522 1753.73024500535 1768.26975499465 25 285 1775.70786613977 -1490.70786613977 26 574 1759.88397892299 -1185.88397892299 27 865 1741.86232959277 -876.862329592768 28 1147 1982.95683343735 -835.956833437345 29 1516 2226.90842802940 -710.908428029397 30 1789 2467.12382702860 -678.123827028597 31 2087 2379.21334249092 -292.213342490921 32 2372 2293.061067644 78.9389323560018 33 2669 2202.51326857019 466.486731429808 34 2966 2138.99794349172 827.00205650828 35 3270 2074.38373735653 1195.61626264347 36 3652 2011.30796470075 1640.69203529925 37 329 1724.71978510792 -1395.71978510792 38 658 1658.34736928198 -1000.34736928198 39 988 1590.21674376528 -602.216743765277 40 1303 1964.05607926174 -661.056079261745 41 1603 2340.53272929434 -737.532729294344 42 1929 2713.27318373409 -784.27318373409 43 2235 2558.99028337047 -323.990283370469 44 2544 2405.80626406357 138.193735936432 45 2872 2248.66627295247 623.333727047528 46 3198 2121.63562279553 1076.36437720447 47 3544 1992.84676294783 1551.15323705217 48 3903 1866.91499384761 2036.08500615239 49 332 1736.36792430916 -1404.36792430916 50 665 1680.76454283908 -1015.76454283908 51 1001 1627.79847590513 -626.798475905133 52 1329 1867.79409869299 -538.794098692989 53 1639 2107.13039284681 -468.130392846813 54 1975 2348.88422532542 -373.884225325423 55 2304 2230.86439983359 73.1356001664077 56 2640 2115.70166508924 524.298334910764 57 2992 1993.94564400455 998.054355995445 58 3330 1920.76016562694 1409.23983437306 59 3690 1846.47580619260 1843.52419380740 60 4063 1773.29032781499 2289.70967218501 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0903790621533375 0.180758124306675 0.909620937846662 6 0.0343802613815953 0.0687605227631905 0.965619738618405 7 0.0691322101110813 0.138264420222163 0.930867789888919 8 0.0746845737398752 0.149369147479750 0.925315426260125 9 0.102427150080362 0.204854300160724 0.897572849919638 10 0.213299036988905 0.426598073977809 0.786700963011095 11 0.306909259327257 0.613818518654513 0.693090740672743 12 0.453906568866853 0.907813137733706 0.546093431133147 13 0.540176510056011 0.919646979887978 0.459823489943989 14 0.551160996526842 0.897678006946317 0.448839003473158 15 0.515502391640444 0.968995216719111 0.484497608359556 16 0.446125600147835 0.89225120029567 0.553874399852165 17 0.374567234186011 0.749134468372022 0.625432765813989 18 0.307614997704785 0.61522999540957 0.692385002295215 19 0.241761700038992 0.483523400077984 0.758238299961008 20 0.194905005837665 0.389810011675331 0.805094994162335 21 0.166986929353390 0.333973858706779 0.83301307064661 22 0.166346130922583 0.332692261845165 0.833653869077417 23 0.201144254700230 0.402288509400461 0.79885574529977 24 0.286904073240100 0.573808146480199 0.7130959267599 25 0.383532762100558 0.767065524201117 0.616467237899442 26 0.424011861951469 0.848023723902938 0.575988138048531 27 0.418791815635115 0.83758363127023 0.581208184364885 28 0.37397717682331 0.74795435364662 0.62602282317669 29 0.337146471171769 0.674292942343538 0.662853528828231 30 0.307611941574912 0.615223883149824 0.692388058425088 31 0.263194864623799 0.526389729247599 0.7368051353762 32 0.220044081998782 0.440088163997564 0.779955918001218 33 0.188826080525171 0.377652161050342 0.81117391947483 34 0.176452741593108 0.352905483186217 0.823547258406892 35 0.191464749258424 0.382929498516848 0.808535250741576 36 0.264360930508538 0.528721861017077 0.735639069491462 37 0.319620917214577 0.639241834429153 0.680379082785423 38 0.331402366731119 0.662804733462238 0.668597633268881 39 0.313380695969987 0.626761391939974 0.686619304030013 40 0.290903333979374 0.581806667958748 0.709096666020626 41 0.259732461016266 0.519464922032532 0.740267538983734 42 0.226648597458369 0.453297194916737 0.773351402541631 43 0.184502445292342 0.369004890584684 0.815497554707658 44 0.140358551131250 0.280717102262499 0.85964144886875 45 0.104742263719236 0.209484527438472 0.895257736280764 46 0.0884193343763509 0.176838668752702 0.91158066562365 47 0.102359089456146 0.204718178912292 0.897640910543854 48 0.188372087578019 0.376744175156038 0.811627912421981 49 0.270475166007540 0.540950332015079 0.72952483399246 50 0.376881822184584 0.753763644369168 0.623118177815416 51 0.698581928455804 0.602836143088391 0.301418071544196 52 0.981847567982066 0.0363048640358671 0.0181524320179335 53 0.999950922385854 9.81552282914263e-05 4.90776141457132e-05 54 0.9997987162157 0.00040256756859842 0.00020128378429921 55 0.999257038039036 0.00148592392192808 0.000742961960964042 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 3 0.0588235294117647 NOK 5% type I error level 4 0.0784313725490196 NOK 10% type I error level 5 0.0980392156862745 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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