Multiple Regression (b)

*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, 19 Nov 2009 09:58:24 -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/19/t1258650000cdltq7kxxadwcxo.htm/, Retrieved Thu, 19 Nov 2009 18:00:12 +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/19/t1258650000cdltq7kxxadwcxo.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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4 7.2 102.9 271244 4.1 7.4 97.4 269907 4 8.8 111.4 271296 3.8 9.3 87.4 270157 4.7 9.3 96.8 271322 4.3 8.7 114.1 267179 3.9 8.2 110.3 264101 4 8.3 103.9 265518 4.3 8.5 101.6 269419 4.8 8.6 94.6 268714 4.4 8.5 95.9 272482 4.3 8.2 104.7 268351 4.7 8.1 102.8 268175 4.7 7.9 98.1 270674 4.9 8.6 113.9 272764 5 8.7 80.9 272599 4.2 8.7 95.7 270333 4.3 8.5 113.2 270846 4.8 8.4 105.9 270491 4.8 8.5 108.8 269160 4.8 8.7 102.3 274027 4.2 8.7 99 273784 4.6 8.6 100.7 276663 4.8 8.5 115.5 274525 4.5 8.3 100.7 271344 4.4 8 109.9 271115 4.3 8.2 114.6 270798 3.9 8.1 85.4 273911 3.7 8.1 100.5 273985 4 8 114.8 271917 4.1 7.9 116.5 273338 3.7 7.9 112.9 270601 3.8 8 102 273547 3.8 8 106 275363 3.8 7.9 105.3 281229 3.3 8 118.8 277793 3.3 7.7 106.1 279913 3.3 7.2 109.3 282500 3.2 7.5 117.2 280041 3.4 7.3 92.5 282166 4.2 7 104.2 290304 4.9 7 112.5 283519 5.1 7 122.4 287816 5.5 7.2 113.3 285226 5.6 7.3 100 287595 6.4 7.1 110.7 289741 6.1 6.8 112.8 289148 7.1 6.4 109.8 288301 7.8 6.1 117.3 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Cons.index[t] = + 19.1792040203402 -1.12771759422269Werkl.graad[t] + 0.0611701113554312Industr.prod.[t] -4.5112205089636e-05BrutoSchuld[t] -0.135530992464506M1[t] -0.169606970781555M2[t] + 0.117225185486256M3[t] + 1.76991643625086M4[t] + 1.10643031597644M5[t] -0.190436397986985M6[t] -0.374667261299632M7[t] + 0.0442239049331357M8[t] + 1.11890712542945M9[t] + 1.0920282953553M10[t] + 0.899403442297472M11[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) 19.1792040203402 9.161293 2.0935 0.041397 0.020698 Werkl.graad -1.12771759422269 0.354742 -3.179 0.002536 0.001268 Industr.prod. 0.0611701113554312 0.037455 1.6331 0.10872 0.05436 BrutoSchuld -4.5112205089636e-05 1.2e-05 -3.6283 0.000669 0.000335 M1 -0.135530992464506 0.827324 -0.1638 0.870534 0.435267 M2 -0.169606970781555 0.852666 -0.1989 0.843137 0.421568 M3 0.117225185486256 0.772318 0.1518 0.879968 0.439984 M4 1.76991643625086 1.070611 1.6532 0.10456 0.05228 M5 1.10643031597644 0.84933 1.3027 0.198642 0.099321 M6 -0.190436397986985 0.793594 -0.24 0.811337 0.405668 M7 -0.374667261299632 0.790863 -0.4737 0.637744 0.318872 M8 0.0442239049331357 0.800673 0.0552 0.956173 0.478086 M9 1.11890712542945 0.854759 1.309 0.196508 0.098254 M10 1.0920282953553 0.845886 1.291 0.202645 0.101323 M11 0.899403442297472 0.832178 1.0808 0.284979 0.14249 Multiple Linear Regression - Regression Statistics Multiple R 0.70097736397713 R-squared 0.491369264808326 Adjusted R-squared 0.348952658954657 F-TEST (value) 3.45022451464123 F-TEST (DF numerator) 14 F-TEST (DF denominator) 50 p-value 0.000612196827674438 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.23968636711867 Sum Squared Residuals 76.841114440994 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 4 4.98209585061306 -0.982095850613056 2 4.1 4.44635575920136 -0.346355759201357 3 4 3.94810398966393 0.0518960103360667 4 3.8 3.62023657238394 0.179763427616060 5 4.7 3.47919377992114 1.22080622007886 6 4.3 4.10410041462666 0.195899585373342 7 3.9 4.39013729254062 -0.490137292540619 8 4 4.24084399206434 -0.240843992064341 9 4.3 4.77330972554396 -0.473309725543958 10 4.8 4.23727246114771 0.562727538852286 11 4.4 4.06695772349647 0.333042276503533 12 4.3 4.23052505861888 0.069474941381117 13 4.7 4.0994823620971 0.600517637902899 14 4.7 3.89071497873506 0.809285021264939 15 4.9 4.26034806982547 0.639651930174534 16 5 3.78909740027836 1.21090259972164 17 4.2 4.13315318479743 0.0668468152025671 18 4.3 4.10916437718761 0.190835622812386 19 4.8 3.60717829320941 1.19282170679059 20 4.8 4.15073536792496 0.649264632075036 21 4.8 4.38270824359518 0.417291756404821 22 4.2 4.16493031188489 0.035069688115112 23 4.6 4.0591883691005 0.5408116308995 24 4.8 4.27432422876732 0.52567577123268 25 4.5 3.6025210314771 0.8974789685229 26 4.4 4.47985605086235 -0.0798560508623518 27 4.3 4.84294478066957 -0.542944780669568 28 3.9 4.68180624483381 -0.781806244833815 29 3.7 4.93865050284977 -1.23865050284977 30 4 4.72258018081665 -0.722580180816648 31 4.1 4.69100582279813 -0.591005822798131 32 3.7 5.01315669348168 -1.31315669348168 33 3.8 5.17541338458746 -1.37541338458746 34 3.8 5.31129123549225 -1.51129123549225 35 3.8 4.92399086885209 -1.12399086885209 36 3.3 4.89261770711866 -1.59261770711866 37 3.3 4.22290370391695 -0.922903703916952 38 3.3 4.83172560448174 -1.53172560448174 39 3.2 5.37441727450607 -2.17441727450607 40 3.4 5.64588685782058 -2.24588685782058 41 4.2 5.66928319365205 -1.46928319365205 42 4.9 5.18621471547189 -0.286214715471888 43 5.1 5.41372080930785 -0.313720809307846 44 5.5 5.16726105454381 0.332738945456192 45 5.6 5.20873922073327 0.39126077926673 46 6.4 5.96511330888441 0.434886691115585 47 6.1 6.26601250555795 -0.166012505557954 48 7.1 5.67239580459418 1.42760419540582 49 7.8 6.25031789732603 1.54968210267397 50 7.9 5.28643185618582 2.61356814381418 51 7.4 4.70015432444589 2.69984567555411 52 7.5 4.85922049746194 2.64077950253806 53 6.8 4.63638372582463 2.16361627417537 54 5.2 4.57794031189719 0.622059688102808 55 4.7 4.49795778214399 0.202042217856006 56 4.1 3.52800289198521 0.571997108014792 57 3.9 2.85982942554014 1.04017057445986 58 2.6 2.12139268259073 0.478607317409269 59 2.7 2.28385053299299 0.416149467007008 60 1.8 2.23013720090096 -0.430137200900958 61 1 2.14267915456976 -1.14267915456976 62 0.3 1.76491575053367 -1.46491575053367 63 1.3 1.97403156088908 -0.674031560889081 64 1 2.00375242722136 -1.00375242722136 65 1.1 1.84333561295498 -0.74333561295498 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.0792089914756419 0.158417982951284 0.920791008524358 19 0.0320198819230387 0.0640397638460775 0.967980118076961 20 0.0139194468396791 0.0278388936793583 0.98608055316032 21 0.00440657784675354 0.00881315569350707 0.995593422153247 22 0.00292412155611089 0.00584824311222178 0.99707587844389 23 0.00101566210522894 0.00203132421045788 0.998984337894771 24 0.000404787967067448 0.000809575934134897 0.999595212032933 25 0.000240445988530577 0.000480891977061155 0.99975955401147 26 8.05596697315464e-05 0.000161119339463093 0.999919440330268 27 2.33890915395731e-05 4.67781830791462e-05 0.99997661090846 28 7.44302671943692e-06 1.48860534388738e-05 0.99999255697328 29 2.42187628770629e-06 4.84375257541257e-06 0.999997578123712 30 6.1859377294455e-07 1.2371875458891e-06 0.999999381406227 31 1.93170057115475e-07 3.86340114230951e-07 0.999999806829943 32 5.6337905387488e-08 1.12675810774976e-07 0.999999943662095 33 2.12725187478656e-08 4.25450374957312e-08 0.999999978727481 34 4.79914517265863e-09 9.59829034531725e-09 0.999999995200855 35 1.25261912380757e-09 2.50523824761513e-09 0.99999999874738 36 2.42476266505483e-09 4.84952533010966e-09 0.999999997575237 37 4.56092156675827e-09 9.12184313351654e-09 0.999999995439078 38 3.51276315100169e-09 7.02552630200337e-09 0.999999996487237 39 1.07481618862785e-08 2.1496323772557e-08 0.999999989251838 40 4.78535660465935e-08 9.5707132093187e-08 0.999999952146434 41 3.16233249703565e-06 6.3246649940713e-06 0.999996837667503 42 3.76368488209598e-06 7.52736976419196e-06 0.999996236315118 43 9.97195822912086e-06 1.99439164582417e-05 0.99999002804177 44 2.74746389234141e-05 5.49492778468281e-05 0.999972525361077 45 0.000388823893489553 0.000777647786979106 0.99961117610651 46 0.0278824009955598 0.0557648019911197 0.97211759900444 47 0.78528849796929 0.429423004061418 0.214711502030709 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 25 0.833333333333333 NOK 5% type I error level 26 0.866666666666667 NOK 10% type I error level 28 0.933333333333333 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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