Multiple Linear Regression (M1)

*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: Sun, 20 Dec 2009 04:56:52 -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/Dec/20/t1261310406jpkzigp8zvkwaij.htm/, Retrieved Sun, 20 Dec 2009 13:00:18 +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/Dec/20/t1261310406jpkzigp8zvkwaij.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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921365 0 987921 0 1132614 0 1332224 0 1418133 0 1411549 0 1695920 0 1636173 0 1539653 0 1395314 0 1127575 0 1036076 0 989236 0 1008380 0 1207763 0 1368839 0 1469798 0 1498721 0 1761769 0 1653214 0 1599104 0 1421179 0 1163995 0 1037735 0 1015407 0 1039210 0 1258049 0 1469445 0 1552346 0 1549144 0 1785895 0 1662335 0 1629440 0 1467430 0 1202209 0 1076982 0 1039367 1 1063449 1 1335135 1 1491602 1 1591972 1 1641248 1 1898849 1 1798580 1 1762444 1 1622044 1 1368955 1 1262973 1 1195650 1 1269530 1 1479279 1 1607819 1 1712466 1 1721766 1 1949843 1 1821326 1 1757802 1 1590367 1 1260647 1 1149235 1

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 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 1347837.27777778 + 168510.555555556X[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) 1347837.27777778 43662.663384 30.8693 0 0 X 168510.555555556 69036.732501 2.4409 0.017725 0.008863 Multiple Linear Regression - Regression Statistics Multiple R 0.305210702601077 R-squared 0.0931535729822429 Adjusted R-squared 0.0775182897577987 F-TEST (value) 5.95790761478546 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.0177254198313308 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 261975.980301977 Sum Squared Residuals 3980622026800.56 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 921365 1347837.27777778 -426472.277777779 2 987921 1347837.27777778 -359916.277777778 3 1132614 1347837.27777778 -215223.277777778 4 1332224 1347837.27777778 -15613.2777777777 5 1418133 1347837.27777778 70295.7222222223 6 1411549 1347837.27777778 63711.7222222223 7 1695920 1347837.27777778 348082.722222222 8 1636173 1347837.27777778 288335.722222222 9 1539653 1347837.27777778 191815.722222222 10 1395314 1347837.27777778 47476.7222222223 11 1127575 1347837.27777778 -220262.277777778 12 1036076 1347837.27777778 -311761.277777778 13 989236 1347837.27777778 -358601.277777778 14 1008380 1347837.27777778 -339457.277777778 15 1207763 1347837.27777778 -140074.277777778 16 1368839 1347837.27777778 21001.7222222223 17 1469798 1347837.27777778 121960.722222222 18 1498721 1347837.27777778 150883.722222222 19 1761769 1347837.27777778 413931.722222222 20 1653214 1347837.27777778 305376.722222222 21 1599104 1347837.27777778 251266.722222222 22 1421179 1347837.27777778 73341.7222222223 23 1163995 1347837.27777778 -183842.277777778 24 1037735 1347837.27777778 -310102.277777778 25 1015407 1347837.27777778 -332430.277777778 26 1039210 1347837.27777778 -308627.277777778 27 1258049 1347837.27777778 -89788.2777777777 28 1469445 1347837.27777778 121607.722222222 29 1552346 1347837.27777778 204508.722222222 30 1549144 1347837.27777778 201306.722222222 31 1785895 1347837.27777778 438057.722222222 32 1662335 1347837.27777778 314497.722222222 33 1629440 1347837.27777778 281602.722222222 34 1467430 1347837.27777778 119592.722222222 35 1202209 1347837.27777778 -145628.277777778 36 1076982 1347837.27777778 -270855.277777778 37 1039367 1516347.83333333 -476980.833333333 38 1063449 1516347.83333333 -452898.833333333 39 1335135 1516347.83333333 -181212.833333333 40 1491602 1516347.83333333 -24745.8333333333 41 1591972 1516347.83333333 75624.1666666667 42 1641248 1516347.83333333 124900.166666667 43 1898849 1516347.83333333 382501.166666667 44 1798580 1516347.83333333 282232.166666667 45 1762444 1516347.83333333 246096.166666667 46 1622044 1516347.83333333 105696.166666667 47 1368955 1516347.83333333 -147392.833333333 48 1262973 1516347.83333333 -253374.833333333 49 1195650 1516347.83333333 -320697.833333333 50 1269530 1516347.83333333 -246817.833333333 51 1479279 1516347.83333333 -37068.8333333333 52 1607819 1516347.83333333 91471.1666666667 53 1712466 1516347.83333333 196118.166666667 54 1721766 1516347.83333333 205418.166666667 55 1949843 1516347.83333333 433495.166666667 56 1821326 1516347.83333333 304978.166666667 57 1757802 1516347.83333333 241454.166666667 58 1590367 1516347.83333333 74019.1666666667 59 1260647 1516347.83333333 -255700.833333333 60 1149235 1516347.83333333 -367112.833333333 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.554457886942039 0.891084226115923 0.445542113057961 6 0.50505414127507 0.98989171744986 0.49494585872493 7 0.725395368905702 0.549209262188596 0.274604631094298 8 0.758443702524847 0.483112594950307 0.241556297475153 9 0.711127758052574 0.577744483894853 0.288872241947426 10 0.611339959345204 0.777320081309592 0.388660040654796 11 0.568414631578358 0.863170736843284 0.431585368421642 12 0.579494961572586 0.841010076854828 0.420505038427414 13 0.616299525200195 0.76740094959961 0.383700474799805 14 0.632088498272051 0.735823003455898 0.367911501727949 15 0.557055458656248 0.885889082687504 0.442944541343752 16 0.479515164323883 0.959030328647767 0.520484835676117 17 0.431652096149643 0.863304192299286 0.568347903850357 18 0.393060625334499 0.786121250668997 0.606939374665501 19 0.535483813802373 0.929032372395255 0.464516186197627 20 0.56489322504535 0.8702135499093 0.43510677495465 21 0.5552466496208 0.8895067007584 0.4447533503792 22 0.480988053099068 0.961976106198135 0.519011946900932 23 0.438699139508305 0.87739827901661 0.561300860491695 24 0.466403062301546 0.932806124603091 0.533596937698454 25 0.516370080941338 0.967259838117324 0.483629919058662 26 0.561845130308733 0.876309739382535 0.438154869691267 27 0.509874903706299 0.980250192587401 0.490125096293701 28 0.450198808380308 0.900397616760616 0.549801191619692 29 0.411308462002591 0.822616924005181 0.588691537997409 30 0.369900152295033 0.739800304590066 0.630099847704967 31 0.466499551836599 0.932999103673197 0.533500448163401 32 0.485799511246833 0.971599022493666 0.514200488753167 33 0.508647337042729 0.982705325914542 0.491352662957271 34 0.480477255189741 0.960954510379482 0.519522744810259 35 0.419638247112229 0.839276494224457 0.580361752887771 36 0.377423004643332 0.754846009286663 0.622576995356668 37 0.453745387760086 0.907490775520172 0.546254612239914 38 0.553772229162595 0.89245554167481 0.446227770837405 39 0.540029999861488 0.919940000277024 0.459970000138512 40 0.500180750269688 0.999638499460625 0.499819249730312 41 0.457765486423171 0.915530972846342 0.542234513576829 42 0.413420675179528 0.826841350359055 0.586579324820472 43 0.511508877532116 0.976982244935768 0.488491122467884 44 0.519425511061089 0.961148977877822 0.480574488938911 45 0.502922377059357 0.994155245881286 0.497077622940643 46 0.424994343937825 0.84998868787565 0.575005656062175 47 0.359859599437750 0.719719198875499 0.64014040056225 48 0.350689688439085 0.70137937687817 0.649310311560915 49 0.410652126374472 0.821304252748943 0.589347873625528 50 0.437467364764757 0.874934729529515 0.562532635235243 51 0.350243532428436 0.700487064856871 0.649756467571564 52 0.250710094258610 0.501420188517221 0.749289905741390 53 0.176441639480952 0.352883278961904 0.823558360519048 54 0.115955214840112 0.231910429680224 0.884044785159888 55 0.169937949983239 0.339875899966478 0.83006205001676 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 0 0 OK 10% type I error level 0 0 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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