Workshop 5

*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, 10 Dec 2010 10:26:02 +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/10/t12919768865i0d7q3hixn6hi7.htm/, Retrieved Fri, 10 Dec 2010 11:28:18 +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/2010/Dec/10/t12919768865i0d7q3hixn6hi7.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

9700 0 9081 0 9084 0 9743 0 8587 0 9731 0 9563 0 9998 0 9437 0 10038 0 9918 0 9252 0 9737 0 9035 0 9133 0 9487 0 8700 0 9627 0 8947 0 9283 0 8829 0 9947 0 9628 0 9318 0 9605 0 8640 0 9214 0 9567 0 8547 0 9185 0 9470 0 9123 0 9278 0 10170 0 9434 0 9655 0 9429 0 8739 0 9552 0 9687 0 9019 1 9672 1 9206 1 9069 1 9788 1 10312 1 10105 1 9863 1 9656 1 9295 1 9946 1 9701 1 9049 1 10190 1 9706 1 9765 1 9893 1 9994 1 10433 1 10073 1 10112 1 9266 1 9820 1 10097 1 9115 1 10411 1 9678 1 10408 1 10153 1 10368 1 10581 1 10597 1 10680 1 9738 1 9556 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 8 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Birth[t] = + 9377.45 + 488.692857142858x[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) 9377.45 69.906278 134.1432 0 0 x 488.692857142858 102.332313 4.7755 9e-06 4e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.487895901742042 R-squared 0.238042410936680 Adjusted R-squared 0.227604635744032 F-TEST (value) 22.8058572390354 F-TEST (DF numerator) 1 F-TEST (DF denominator) 73 p-value 8.99535508724902e-06 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 442.126123853076 Sum Squared Residuals 14269712.1857142 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 9700 9377.45000000006 322.54999999994 2 9081 9377.45 -296.449999999997 3 9084 9377.45 -293.449999999999 4 9743 9377.45 365.550000000001 5 8587 9377.45 -790.449999999999 6 9731 9377.45 353.550000000001 7 9563 9377.45 185.550000000001 8 9998 9377.45 620.550000000001 9 9437 9377.45 59.5500000000015 10 10038 9377.45 660.550000000001 11 9918 9377.45 540.550000000001 12 9252 9377.45 -125.449999999999 13 9737 9377.45 359.550000000001 14 9035 9377.45 -342.449999999999 15 9133 9377.45 -244.449999999999 16 9487 9377.45 109.550000000001 17 8700 9377.45 -677.449999999999 18 9627 9377.45 249.550000000001 19 8947 9377.45 -430.449999999999 20 9283 9377.45 -94.4499999999985 21 8829 9377.45 -548.449999999999 22 9947 9377.45 569.550000000001 23 9628 9377.45 250.550000000001 24 9318 9377.45 -59.4499999999985 25 9605 9377.45 227.550000000001 26 8640 9377.45 -737.449999999999 27 9214 9377.45 -163.449999999999 28 9567 9377.45 189.550000000001 29 8547 9377.45 -830.449999999999 30 9185 9377.45 -192.449999999999 31 9470 9377.45 92.5500000000015 32 9123 9377.45 -254.449999999999 33 9278 9377.45 -99.4499999999985 34 10170 9377.45 792.550000000001 35 9434 9377.45 56.5500000000015 36 9655 9377.45 277.550000000001 37 9429 9377.45 51.5500000000015 38 8739 9377.45 -638.449999999999 39 9552 9377.45 174.550000000001 40 9687 9377.45 309.550000000001 41 9019 9866.14285714286 -847.142857142857 42 9672 9866.14285714286 -194.142857142857 43 9206 9866.14285714286 -660.142857142857 44 9069 9866.14285714286 -797.142857142857 45 9788 9866.14285714286 -78.1428571428572 46 10312 9866.14285714286 445.857142857143 47 10105 9866.14285714286 238.857142857143 48 9863 9866.14285714286 -3.14285714285720 49 9656 9866.14285714286 -210.142857142857 50 9295 9866.14285714286 -571.142857142857 51 9946 9866.14285714286 79.8571428571428 52 9701 9866.14285714286 -165.142857142857 53 9049 9866.14285714286 -817.142857142857 54 10190 9866.14285714286 323.857142857143 55 9706 9866.14285714286 -160.142857142857 56 9765 9866.14285714286 -101.142857142857 57 9893 9866.14285714286 26.8571428571428 58 9994 9866.14285714286 127.857142857143 59 10433 9866.14285714286 566.857142857143 60 10073 9866.14285714286 206.857142857143 61 10112 9866.14285714286 245.857142857143 62 9266 9866.14285714286 -600.142857142857 63 9820 9866.14285714286 -46.1428571428572 64 10097 9866.14285714286 230.857142857143 65 9115 9866.14285714286 -751.142857142857 66 10411 9866.14285714286 544.857142857143 67 9678 9866.14285714286 -188.142857142857 68 10408 9866.14285714286 541.857142857143 69 10153 9866.14285714286 286.857142857143 70 10368 9866.14285714286 501.857142857143 71 10581 9866.14285714286 714.857142857143 72 10597 9866.14285714286 730.857142857143 73 10680 9866.14285714286 813.857142857143 74 9738 9866.14285714286 -128.142857142857 75 9556 9866.14285714286 -310.142857142857 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.805535796914764 0.388928406170472 0.194464203085236 6 0.775819617918916 0.448360764162167 0.224180382081084 7 0.682980951161742 0.634038097676515 0.317019048838258 8 0.742108826096925 0.515782347806149 0.257891173903075 9 0.638577886661657 0.722844226676685 0.361422113338343 10 0.69395381057941 0.61209237884118 0.30604618942059 11 0.68041410398289 0.63917179203422 0.31958589601711 12 0.614991370418494 0.770017259163011 0.385008629581506 13 0.551638393024509 0.896723213950981 0.448361606975491 14 0.550671413208913 0.898657173582175 0.449328586791088 15 0.507653761615978 0.984692476768045 0.492346238384022 16 0.422586457908033 0.845172915816067 0.577413542091967 17 0.558632908001248 0.882734183997504 0.441367091998752 18 0.495789621329061 0.991579242658121 0.50421037867094 19 0.498384983714987 0.996769967429973 0.501615016285013 20 0.424410577964068 0.848821155928135 0.575589422035932 21 0.465077321120631 0.930154642241263 0.534922678879369 22 0.508060748573465 0.98387850285307 0.491939251426535 23 0.45458715859782 0.90917431719564 0.54541284140218 24 0.384046666706994 0.768093333413989 0.615953333293006 25 0.332309853513382 0.664619707026764 0.667690146486618 26 0.454226381368186 0.908452762736372 0.545773618631814 27 0.39325547805152 0.78651095610304 0.60674452194848 28 0.338298686109552 0.676597372219105 0.661701313890448 29 0.504112553991349 0.991774892017301 0.495887446008651 30 0.449247531803018 0.898495063606035 0.550752468196982 31 0.384328166994882 0.768656333989763 0.615671833005118 32 0.344958772442631 0.689917544885261 0.655041227557369 33 0.290810579657195 0.581621159314391 0.709189420342805 34 0.412404751164964 0.824809502329927 0.587595248835036 35 0.348225719917888 0.696451439835775 0.651774280082112 36 0.30963538392078 0.61927076784156 0.69036461607922 37 0.253643502950853 0.507287005901705 0.746356497049147 38 0.325590214620705 0.651180429241409 0.674409785379295 39 0.273184661004500 0.546369322009001 0.7268153389955 40 0.232732670694917 0.465465341389833 0.767267329305083 41 0.276426689901581 0.552853379803162 0.723573310098419 42 0.257649245918644 0.515298491837287 0.742350754081357 43 0.277272572940092 0.554545145880183 0.722727427059908 44 0.355405527150662 0.710811054301325 0.644594472849338 45 0.332912088500434 0.665824177000869 0.667087911499566 46 0.404671546955937 0.809343093911874 0.595328453044063 47 0.384101714454233 0.768203428908466 0.615898285545767 48 0.327499079870491 0.654998159740983 0.672500920129508 49 0.281097855386957 0.562195710773914 0.718902144613043 50 0.318234498404962 0.636468996809924 0.681765501595038 51 0.268538801964469 0.537077603928939 0.731461198035531 52 0.225939995207576 0.451879990415152 0.774060004792424 53 0.404744711153196 0.809489422306393 0.595255288846804 54 0.376844610701456 0.753689221402912 0.623155389298544 55 0.333566384108755 0.66713276821751 0.666433615891245 56 0.286197110794098 0.572394221588196 0.713802889205902 57 0.233417967159873 0.466835934319746 0.766582032840127 58 0.18479726053474 0.36959452106948 0.81520273946526 59 0.195682508946008 0.391365017892017 0.804317491053992 60 0.149695934173567 0.299391868347134 0.850304065826433 61 0.111889482502049 0.223778965004098 0.888110517497951 62 0.186272113217015 0.372544226434029 0.813727886782985 63 0.145951422285647 0.291902844571294 0.854048577714353 64 0.102661472717383 0.205322945434767 0.897338527282617 65 0.369220037199921 0.738440074399842 0.630779962800079 66 0.313718524181942 0.627437048363884 0.686281475818058 67 0.349057005013606 0.698114010027213 0.650942994986394 68 0.26924812793008 0.53849625586016 0.73075187206992 69 0.175684270250418 0.351368540500835 0.824315729749582 70 0.105105959470602 0.210211918941205 0.894894040529398 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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