Workshop 7: Multiple Regression 3

*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, 25 Dec 2009 11:52:49 -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/25/t1261767259qfu9fkjqm57h8ub.htm/, Retrieved Fri, 25 Dec 2009 19:54:31 +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/25/t1261767259qfu9fkjqm57h8ub.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:

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Dataseries X:

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25.6 8.1 23.7 7.7 22 7.5 21.3 7.6 20.7 7.8 20.4 7.8 20.3 7.8 20.4 7.5 19.8 7.5 19.5 7.1 23.1 7.5 23.5 7.5 23.5 7.6 22.9 7.7 21.9 7.7 21.5 7.9 20.5 8.1 20.2 8.2 19.4 8.2 19.2 8.2 18.8 7.9 18.8 7.3 22.6 6.9 23.3 6.6 23 6.7 21.4 6.9 19.9 7 18.8 7.1 18.6 7.2 18.4 7.1 18.6 6.9 19.9 7 19.2 6.8 18.4 6.4 21.1 6.7 20.5 6.6 19.1 6.4 18.1 6.3 17 6.2 17.1 6.5 17.4 6.8 16.8 6.8 15.3 6.4 14.3 6.1 13.4 5.8 15.3 6.1 22.1 7.2 23.7 7.3 22.2 6.9 19.5 6.1 16.6 5.8 17.3 6.2 19.8 7.1 21.2 7.7 21.5 7.9 20.6 7.7 19.1 7.4 19.6 7.5 23.5 8 24 8.1

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 5.26942072945178 + 2.50007398091293X[t] -0.217758070084566M1[t] -1.26885563857858M2[t] -2.64996060516387M3[t] -3.47108924564129M4[t] -4.11222676382827M5[t] -4.4033480062144M6[t] -4.57445445241795M7[t] -4.35555645976671M8[t] -4.61665254864245M9[t] -3.84775011713644M10[t] -0.628890594559936M11[t] -0.00888763532341999t + 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) 5.26942072945178 2.038176 2.5854 0.012962 0.006481 X 2.50007398091293 0.25038 9.9851 0 0 M1 -0.217758070084566 0.681393 -0.3196 0.750735 0.375368 M2 -1.26885563857858 0.686395 -1.8486 0.070952 0.035476 M3 -2.64996060516387 0.689209 -3.8449 0.000369 0.000184 M4 -3.47108924564129 0.679409 -5.109 6e-06 3e-06 M5 -4.11222676382827 0.674816 -6.0939 0 0 M6 -4.4033480062144 0.675837 -6.5154 0 0 M7 -4.57445445241795 0.674328 -6.7837 0 0 M8 -4.35555645976671 0.673047 -6.4714 0 0 M9 -4.61665254864245 0.674394 -6.8456 0 0 M10 -3.84775011713644 0.678941 -5.6673 1e-06 0 M11 -0.628890594559936 0.672335 -0.9354 0.354476 0.177238 t -0.00888763532341999 0.009131 -0.9733 0.335495 0.167748 Multiple Linear Regression - Regression Statistics Multiple R 0.929033635433186 R-squared 0.863103495766201 Adjusted R-squared 0.824415353265345 F-TEST (value) 22.3092513616311 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 1.33226762955019e-15 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.06293959262310 Sum Squared Residuals 51.9726665680255 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 25.6 25.2933742694384 0.306625730561593 2 23.7 23.2333594732559 0.466640526744101 3 22 21.3433520751646 0.656647924835388 4 21.3 20.7633431974551 0.53665680254494 5 20.7 20.6133328401272 0.086667159872749 6 20.4 20.3133239624177 0.0866760375822986 7 20.3 20.1333298808907 0.166670119109268 8 20.4 19.5933180439447 0.806681956055333 9 19.8 19.3233343197455 0.476665680254492 10 19.5 19.0833195235629 0.416680476437076 11 23.1 23.2933210031812 -0.193321003181179 12 23.5 23.9133239624177 -0.4133239624177 13 23.5 23.936685655101 -0.436685655101007 14 22.9 23.1267078493749 -0.226707849374868 15 21.9 21.7367152474662 0.163284752533841 16 21.5 21.4067137678479 0.0932862321520993 17 20.5 21.2567034105201 -0.756703410520089 18 20.2 21.2067019309018 -1.00670193090183 19 19.4 21.0267078493749 -1.62670784937486 20 19.2 21.2367182067027 -2.03671820670267 21 18.8 20.2167122882296 -1.41671228822964 22 18.8 19.4766826958645 -0.67668269586447 23 22.6 21.6866249907524 0.913375009247613 24 23.3 21.5566057557150 1.74339424428498 25 23 21.5799674483983 1.42003255160167 26 21.4 21.0199970407635 0.380002959236514 27 19.9 19.8800118369461 0.0199881630539311 28 18.8 19.3000029592365 -0.500002959236516 29 18.6 18.8999852038174 -0.299985203817415 30 18.4 18.3499689280166 0.0500310719834287 31 18.6 17.6699600503070 0.930039949692981 32 19.9 18.1299778057261 1.77002219427388 33 19.2 17.3599792853444 1.84002071465562 34 18.4 17.1199644891618 1.28003551083820 35 21.1 21.0799585706888 0.0200414293112387 36 20.5 21.449954131834 -0.949954131833985 37 19.1 20.7232936302434 -1.62329363024341 38 18.1 19.4133010283347 -1.31330102833469 39 17 17.7733010283347 -0.773301028334686 40 17.1 17.6933069468077 -0.59330694680772 41 17.4 17.7933039875712 -0.393303987571205 42 16.8 17.4932951098617 -0.69329510986165 43 15.3 16.3132714359695 -1.01327143596952 44 14.3 15.7732595990234 -1.47325959902345 45 13.4 14.7532536805504 -1.35325368055041 46 15.3 16.2632906710069 -0.963290671006876 47 22.1 22.2233439372642 -0.123343937264185 48 23.7 23.093354294592 0.606645705408005 49 22.2 21.8666789968188 0.333321003181161 50 19.5 18.8066346082711 0.693365391728939 51 16.6 16.6666198120885 -0.0666198120884735 52 17.3 16.8366331286528 0.463366871347197 53 19.8 18.4366745579640 1.36332544203596 54 21.2 19.6367100688022 1.56328993119775 55 21.5 19.9567307834579 1.54326921654213 56 20.6 19.6667263446031 0.93327365539691 57 19.1 18.6467204261301 0.453279573869943 58 19.6 19.6567426204039 -0.0567426204039348 59 23.5 24.1167514981135 -0.616751498113488 60 24 24.9867618554413 -0.986761855441295 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.00149813755970024 0.00299627511940049 0.9985018624403 18 0.000940880321118032 0.00188176064223606 0.999059119678882 19 0.00382295135590395 0.00764590271180789 0.996177048644096 20 0.0167463317756523 0.0334926635513045 0.983253668224348 21 0.0113303628704334 0.0226607257408669 0.988669637129567 22 0.00518405661662778 0.0103681132332556 0.994815943383372 23 0.00357852180249166 0.00715704360498332 0.996421478197508 24 0.00403512106569359 0.00807024213138719 0.995964878934306 25 0.00264220930081047 0.00528441860162095 0.99735779069919 26 0.00155547042551221 0.00311094085102443 0.998444529574488 27 0.00105376402844648 0.00210752805689296 0.998946235971554 28 0.00187815879130963 0.00375631758261926 0.99812184120869 29 0.00124498596512014 0.00248997193024029 0.99875501403488 30 0.000568274911144342 0.00113654982228868 0.999431725088856 31 0.000342664191008392 0.000685328382016784 0.999657335808992 32 0.00263649544369284 0.00527299088738567 0.997363504556307 33 0.00855431618949417 0.0171086323789883 0.991445683810506 34 0.0262397096091103 0.0524794192182206 0.97376029039089 35 0.0423305739630435 0.0846611479260871 0.957669426036956 36 0.0757824248213741 0.151564849642748 0.924217575178626 37 0.251302783157935 0.502605566315871 0.748697216842065 38 0.340447592723313 0.680895185446625 0.659552407276687 39 0.294497453282392 0.588994906564783 0.705502546717608 40 0.245929843128037 0.491859686256074 0.754070156871963 41 0.416251124902051 0.832502249804102 0.583748875097949 42 0.889639836467716 0.220720327064569 0.110360163532284 43 0.890280765118967 0.219438469762066 0.109719234881033 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 13 0.481481481481481 NOK 5% type I error level 17 0.62962962962963 NOK 10% type I error level 19 0.703703703703704 NOK

Charts produced by software:

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Parameters (Session):

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

Parameters (R input):

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