*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: Wed, 18 Nov 2009 11:07:50 -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/18/t1258567734hudaqb0wp67ekp1.htm/, Retrieved Wed, 18 Nov 2009 19:09:06 +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/18/t1258567734hudaqb0wp67ekp1.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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3030.29 101.2 2803.47 101.1 2767.63 100.7 2882.6 100.1 2863.36 99.9 2897.06 99.7 3012.61 99.5 3142.95 99.2 3032.93 99 3045.78 99 3110.52 99.3 3013.24 99.5 2987.1 99.7 2995.55 100 2833.18 100.4 2848.96 100.6 2794.83 100.7 2845.26 100.7 2915.02 100.6 2892.63 100.5 2604.42 100.6 2641.65 100.5 2659.81 100.4 2638.53 100.3 2720.25 100.4 2745.88 100.4 2735.7 100.4 2811.7 100.4 2799.43 100.4 2555.28 100.5 2304.98 100.6 2214.95 100.6 2065.81 100.5 1940.49 100.5 2042.00 100.7 1995.37 101.1 1946.81 101.5 1765.9 101.9 1635.25 102.1 1833.42 102.1 1910.43 102.1 1959.67 102.4 1969.6 102.8 2061.41 103.1 2093.48 103.1 2120.88 102.9 2174.56 102.4 2196.72 101.9 2350.44 101.3 2440.25 100.7 2408.64 100.6 2472.81 101 2407.6 101.5 2454.62 101.9 2448.05 102.1 2497.84 102.3 2645.64 102.5 2756.76 102.9 2849.27 103.6 2921.44 104.3

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 G.indx[t] = + 105.02459764215 -0.00156561284976838Bel20[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) 105.02459764215 0.847463 123.9282 0 0 Bel20 -0.00156561284976838 0.000329 -4.7532 1.4e-05 7e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.52946860542578 R-squared 0.28033700413152 Adjusted R-squared 0.267929021444133 F-TEST (value) 22.5932781495961 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 1.36235912630012e-05 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.01658362472473 Sum Squared Residuals 59.9396514313913 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 101.2 100.280336679626 0.919663320374281 2 101.1 100.635448986210 0.464551013790185 3 100.7 100.691560550746 0.00843944925449507 4 100.1 100.511562041408 -0.411562041407643 5 99.9 100.541684432637 -0.641684432637175 6 99.7 100.4889232796 -0.788923279599984 7 99.5 100.308016714809 -0.80801671480925 8 99.2 100.103954735970 -0.903954735970438 9 99 100.276203461702 -1.27620346170196 10 99 100.256085336582 -1.25608533658243 11 99.3 100.154727560688 -0.854727560688432 12 99.5 100.307030378714 -0.807030378713897 13 99.7 100.347955498607 -0.64795549860684 14 100 100.334726070026 -0.334726070026299 15 100.4 100.588934628443 -0.188934628443185 16 100.6 100.564229257674 0.0357707423261485 17 100.7 100.648975881232 0.0510241187681947 18 100.7 100.570022025218 0.129977974782014 19 100.6 100.460804872818 0.139195127181847 20 100.5 100.495858944524 0.00414105547553943 21 100.6 100.947084223956 -0.347084223956210 22 100.5 100.888796457559 -0.388796457559328 23 100.4 100.860364928208 -0.460364928207529 24 100.3 100.893681169651 -0.593681169650608 25 100.4 100.765739287568 -0.365739287567528 26 100.4 100.725612630228 -0.325612630227964 27 100.4 100.741550569039 -0.341550569038607 28 100.4 100.622563992456 -0.222563992456210 29 100.4 100.641774062123 -0.241774062122868 30 100.5 101.024018439394 -0.524018439393823 31 100.6 101.415891335691 -0.815891335690853 32 100.6 101.556843460555 -0.9568434605555 33 100.5 101.79033896097 -1.29033896096995 34 100.5 101.986541563303 -1.48654156330292 35 100.7 101.827616202923 -1.12761620292293 36 101.1 101.900620730108 -0.800620730107641 37 101.5 101.976646890092 -0.476646890092388 38 101.9 102.259881910744 -0.359881910743979 39 102.1 102.464429229566 -0.364429229566229 40 102.1 102.154171731128 -0.0541717311276294 41 102.1 102.033603885567 0.0663961144330334 42 102.4 101.956513108844 0.44348689115564 43 102.8 101.940966573246 0.859033426753831 44 103.1 101.797227657509 1.30277234249106 45 103.1 101.747018453417 1.35298154658313 46 102.9 101.704120661333 1.1958793386668 47 102.4 101.620078563558 0.779921436442366 48 101.9 101.585384582807 0.314615417193233 49 101.3 101.344718575540 -0.0447185755403801 50 100.7 101.204110885503 -0.504110885502676 51 100.6 101.253599907684 -0.653599907683864 52 101 101.153134531114 -0.153134531114221 53 101.5 101.255228145048 0.244771854952383 54 101.9 101.181613028852 0.718386971148498 55 102.1 101.191899105274 0.908100894725509 56 102.3 101.113947241485 1.18605275851548 57 102.5 100.882549662289 1.61745033771125 58 102.9 100.708578762422 2.19142123757752 59 103.6 100.563743917690 3.03625608230958 60 104.3 100.450753638323 3.84924636167736 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.241371938313620 0.482743876627239 0.75862806168638 6 0.230782135270807 0.461564270541614 0.769217864729193 7 0.211859225093072 0.423718450186143 0.788140774906928 8 0.151111097477294 0.302222194954588 0.848888902522706 9 0.142734548599929 0.285469097199858 0.857265451400071 10 0.118501650320201 0.237003300640403 0.881498349679799 11 0.0752196796923327 0.150439359384665 0.924780320307667 12 0.0477231306173293 0.0954462612346585 0.95227686938267 13 0.0286674066341794 0.0573348132683588 0.97133259336582 14 0.0172999826618428 0.0345999653236855 0.982700017338157 15 0.00919060643075037 0.0183812128615007 0.99080939356925 16 0.00496068992844707 0.00992137985689413 0.995039310071553 17 0.0024307928184867 0.0048615856369734 0.997569207181513 18 0.00127655751368873 0.00255311502737746 0.998723442486311 19 0.00081127043081526 0.00162254086163052 0.999188729569185 20 0.000419628049802666 0.000839256099605332 0.999580371950197 21 0.00048549053006713 0.00097098106013426 0.999514509469933 22 0.000363009663422359 0.000726019326844719 0.999636990336578 23 0.000264264092686305 0.000528528185372609 0.999735735907314 24 0.000232238422872829 0.000464476845745657 0.999767761577127 25 0.000143209128701927 0.000286418257403854 0.999856790871298 26 9.13773895548201e-05 0.000182754779109640 0.999908622610445 27 6.44892410824026e-05 0.000128978482164805 0.999935510758918 28 5.3757833202351e-05 0.000107515666404702 0.999946242166798 29 5.82856594215919e-05 0.000116571318843184 0.999941714340578 30 8.40921699437727e-05 0.000168184339887545 0.999915907830056 31 0.00016874474640859 0.00033748949281718 0.999831255253591 32 0.000281920505697742 0.000563841011395483 0.999718079494302 33 0.00056039776539084 0.00112079553078168 0.99943960223461 34 0.000957142174098245 0.00191428434819649 0.999042857825902 35 0.00112153532362934 0.00224307064725869 0.99887846467637 36 0.000916082525558293 0.00183216505111659 0.999083917474442 37 0.000688791484274926 0.00137758296854985 0.999311208515725 38 0.000491906852837908 0.000983813705675817 0.999508093147162 39 0.000323713292406975 0.00064742658481395 0.999676286707593 40 0.000247764043012029 0.000495528086024057 0.999752235956988 41 0.000190679505071837 0.000381359010143675 0.999809320494928 42 0.000248257165568781 0.000496514331137562 0.999751742834431 43 0.000799200496923161 0.00159840099384632 0.999200799503077 44 0.00615913748616246 0.0123182749723249 0.993840862513838 45 0.0414775033243909 0.0829550066487818 0.95852249667561 46 0.206638776971325 0.413277553942651 0.793361223028675 47 0.500016355460788 0.999967289078424 0.499983644539212 48 0.840288248641335 0.31942350271733 0.159711751358665 49 0.796719205637204 0.406561588725591 0.203280794362796 50 0.831179505644961 0.337640988710079 0.168820494355039 51 0.899466823446679 0.201066353106642 0.100533176553321 52 0.984680284111943 0.0306394317761135 0.0153197158880568 53 0.968826411484034 0.0623471770319311 0.0311735885159656 54 0.926958061517207 0.146083876965586 0.073041938482793 55 0.86410377180552 0.271792456388961 0.135896228194481 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 28 0.549019607843137 NOK 5% type I error level 32 0.627450980392157 NOK 10% type I error level 36 0.705882352941177 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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