miltiple regression

*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 11:03:37 -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/t1258654898uv0fh3no628ackk.htm/, Retrieved Thu, 19 Nov 2009 19:21:50 +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/t1258654898uv0fh3no628ackk.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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5.7 97.33 6.1 97.89 6 98.69 5.9 99.01 5.8 99.18 5.7 98.45 5.6 98.13 5.4 98.29 5.4 99.1 5.5 99.26 5.6 98.85 5.7 98.05 5.9 98.53 6.1 99.34 6 100.14 5.8 100.3 5.8 100.22 5.7 99.9 5.5 99.58 5.3 99.9 5.2 100.78 5.2 100.78 5 100.46 5.1 100.06 5.1 100.28 5.2 100.78 4.9 101.58 4.8 102.06 4.5 102.02 4.5 101.68 4.4 101.32 4.4 101.81 4.2 102.3 4.1 102.12 3.9 102.1 3.8 101.75 3.9 101.5 4.2 102.16 4.1 103.47 3.8 104.05 3.6 104.09 3.7 103.55 3.5 102.77 3.4 102.89 3.1 103.6 3.1 103.76 3.1 103.92 3.2 103.35 3.3 103.32 3.5 104.2 3.6 105.44 3.5 105.81 3.3 106.25 3.2 105.94 3.1 105.82 3.2 105.96 3 106.49 3 106.32 3.1 105.88 3.4 105.07

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 manwerk[t] = + 43.1129876094284 -0.379210799699017infl[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) 43.1129876094284 2.107179 20.46 0 0 infl -0.379210799699017 0.020694 -18.3247 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.923425815020273 R-squared 0.852715235845855 Adjusted R-squared 0.850175843360438 F-TEST (value) 335.794974878042 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.409023021958564 Sum Squared Residuals 9.70339028454272 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 5.7 6.20440047472313 -0.50440047472313 2 6.1 5.99204242689168 0.107957573108319 3 6 5.68867378713247 0.311326212867535 4 5.9 5.56732633122878 0.332673668771223 5 5.8 5.50286049527994 0.297139504720056 6 5.7 5.77968437906023 -0.0796843790602275 7 5.6 5.90103183496392 -0.301031834963916 8 5.4 5.84035810701207 -0.440358107012069 9 5.4 5.53319735925587 -0.133197359255870 10 5.5 5.47252363130402 0.0274763686959769 11 5.6 5.62800005918062 -0.0280000591806245 12 5.7 5.93136869893984 -0.231368698939836 13 5.9 5.74934751508431 0.150652484915694 14 6.1 5.4421867673281 0.657813232671897 15 6 5.13881812756889 0.86118187243111 16 5.8 5.07814439961705 0.721855600382951 17 5.8 5.10848126359297 0.69151873640703 18 5.7 5.22982871949665 0.470171280503348 19 5.5 5.35117617540034 0.148823824599660 20 5.3 5.22982871949665 0.0701712805033476 21 5.2 4.89612321576152 0.303876784238481 22 5.2 4.89612321576152 0.303876784238481 23 5 5.01747067166521 -0.0174706716652073 24 5.1 5.16915499154481 -0.0691549915448111 25 5.1 5.08572861561103 0.0142713843889721 26 5.2 4.89612321576152 0.303876784238481 27 4.9 4.59275457600231 0.307245423997694 28 4.8 4.41073339214678 0.389266607853223 29 4.5 4.42590182413474 0.0740981758652598 30 4.5 4.5548334960324 -0.0548334960324018 31 4.4 4.69134938392405 -0.291349383924053 32 4.4 4.50553609207153 -0.105536092071531 33 4.2 4.31972280021902 -0.119722800219015 34 4.1 4.38798074416484 -0.287980744164836 35 3.9 4.39556496015882 -0.495564960158820 36 3.8 4.52828874005347 -0.728288740053473 37 3.9 4.62309143997823 -0.723091439978228 38 4.2 4.37281231217688 -0.172812312176877 39 4.1 3.87604616457116 0.223953835428835 40 3.8 3.65610390074574 0.143896099254264 41 3.6 3.64093546875777 -0.0409354687577725 42 3.7 3.84570930059524 -0.145709300595244 43 3.5 4.14149372436048 -0.641493724360478 44 3.4 4.09598842839659 -0.695988428396594 45 3.1 3.82674876061029 -0.726748760610294 46 3.1 3.76607503265845 -0.666075032658447 47 3.1 3.70540130470661 -0.605401304706606 48 3.2 3.92155146053505 -0.721551460535048 49 3.3 3.93292778452602 -0.63292778452602 50 3.5 3.59922228079088 -0.099222280790881 51 3.6 3.1290008891641 0.470999110835898 52 3.5 2.98869289327546 0.511307106724536 53 3.3 2.8218401414079 0.478159858592102 54 3.2 2.93939548931459 0.260604510685407 55 3.1 2.98490078527848 0.115099214721523 56 3.2 2.93181127332061 0.268188726679385 57 3 2.73082954948014 0.269170450519864 58 3 2.79529538542897 0.204704614571031 59 3.1 2.96214813729654 0.137851862703465 60 3.4 3.26930888505274 0.13069111494726 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0974700490702477 0.194940098140495 0.902529950929752 6 0.0563382891130237 0.112676578226047 0.943661710886976 7 0.044176119191087 0.088352238382174 0.955823880808913 8 0.0794825989228437 0.158965197845687 0.920517401077156 9 0.090292776651715 0.18058555330343 0.909707223348285 10 0.0581163918236122 0.116232783647224 0.941883608176388 11 0.0313494490127587 0.0626988980255175 0.968650550987241 12 0.0161693269674729 0.0323386539349459 0.983830673032527 13 0.0101370329166275 0.0202740658332550 0.989862967083373 14 0.0170157191228554 0.0340314382457107 0.982984280877145 15 0.0170474110582815 0.0340948221165631 0.982952588941719 16 0.0134382482033007 0.0268764964066013 0.9865617517967 17 0.0114814547290039 0.0229629094580078 0.988518545270996 18 0.00925893545894472 0.0185178709178894 0.990741064541055 19 0.00924181651810485 0.0184836330362097 0.990758183481895 20 0.0172189615919080 0.0344379231838161 0.982781038408092 21 0.0332615925408983 0.0665231850817967 0.966738407459102 22 0.0464504731231995 0.092900946246399 0.9535495268768 23 0.0781254730441641 0.156250946088328 0.921874526955836 24 0.094008646149719 0.188017292299438 0.905991353850281 25 0.106324859354815 0.212649718709631 0.893675140645185 26 0.136920964074502 0.273841928149005 0.863079035925498 27 0.197809917372259 0.395619834744519 0.80219008262774 28 0.311940480734322 0.623880961468645 0.688059519265678 29 0.428713867498248 0.857427734996497 0.571286132501752 30 0.544521496020952 0.910957007958095 0.455478503979048 31 0.663161129429969 0.673677741140061 0.336838870570031 32 0.751214301621983 0.497571396756035 0.248785698378017 33 0.806394578808072 0.387210842383857 0.193605421191928 34 0.847688302553229 0.304623394893542 0.152311697446771 35 0.877008879672268 0.245982240655465 0.122991120327732 36 0.91180476596221 0.17639046807558 0.08819523403779 37 0.926765088627906 0.146469822744187 0.0732349113720936 38 0.956768084408935 0.0864638311821304 0.0432319155910652 39 0.990496296684215 0.0190074066315695 0.00950370331578473 40 0.995481506794828 0.0090369864103435 0.00451849320517175 41 0.995196935296117 0.00960612940776617 0.00480306470388309 42 0.997373169245076 0.00525366150984763 0.00262683075492381 43 0.997401345830624 0.00519730833875126 0.00259865416937563 44 0.996727184382569 0.00654563123486241 0.00327281561743121 45 0.995949331181618 0.00810133763676381 0.00405066881838191 46 0.994880214462363 0.0102395710752734 0.00511978553763672 47 0.994014320943856 0.0119713581122881 0.00598567905614403 48 0.993753065964297 0.0124938680714062 0.00624693403570309 49 0.996467693712179 0.00706461257564273 0.00353230628782136 50 0.994737679002852 0.0105246419942961 0.00526232099714807 51 0.99588274806502 0.00823450386996051 0.00411725193498025 52 0.99878963295695 0.00242073408609925 0.00121036704304963 53 0.999824347112119 0.000351305775762188 0.000175652887881094 54 0.999262427128483 0.00147514574303307 0.000737572871516535 55 0.996990606891134 0.0060187862177323 0.00300939310886615 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 12 0.235294117647059 NOK 5% type I error level 26 0.509803921568627 NOK 10% type I error level 31 0.607843137254902 NOK

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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