Workshop 7: Multiple Regression 5 (2lags)

*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 12:40:19 -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/t12617700838dwhaaldgnavcm1.htm/, Retrieved Fri, 25 Dec 2009 20:41:35 +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/t12617700838dwhaaldgnavcm1.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}, }

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Dataseries X:

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22 7.5 23.7 25.6 21.3 7.6 22 23.7 20.7 7.8 21.3 22 20.4 7.8 20.7 21.3 20.3 7.8 20.4 20.7 20.4 7.5 20.3 20.4 19.8 7.5 20.4 20.3 19.5 7.1 19.8 20.4 23.1 7.5 19.5 19.8 23.5 7.5 23.1 19.5 23.5 7.6 23.5 23.1 22.9 7.7 23.5 23.5 21.9 7.7 22.9 23.5 21.5 7.9 21.9 22.9 20.5 8.1 21.5 21.9 20.2 8.2 20.5 21.5 19.4 8.2 20.2 20.5 19.2 8.2 19.4 20.2 18.8 7.9 19.2 19.4 18.8 7.3 18.8 19.2 22.6 6.9 18.8 18.8 23.3 6.6 22.6 18.8 23 6.7 23.3 22.6 21.4 6.9 23 23.3 19.9 7 21.4 23 18.8 7.1 19.9 21.4 18.6 7.2 18.8 19.9 18.4 7.1 18.6 18.8 18.6 6.9 18.4 18.6 19.9 7 18.6 18.4 19.2 6.8 19.9 18.6 18.4 6.4 19.2 19.9 21.1 6.7 18.4 19.2 20.5 6.6 21.1 18.4 19.1 6.4 20.5 21.1 18.1 6.3 19.1 20.5 17 6.2 18.1 19.1 17.1 6.5 17 18.1 17.4 6.8 17.1 17 16.8 6.8 17.4 17.1 15.3 6.4 16.8 17.4 14.3 6.1 15.3 16.8 13.4 5.8 14.3 15.3 15.3 6.1 13.4 14.3 22.1 7.2 15.3 13.4 23.7 7.3 22.1 15.3 22.2 6.9 23.7 22.1 19.5 6.1 22.2 23.7 16.6 5.8 19.5 22.2 17.3 6.2 16.6 19.5 19.8 7.1 17.3 16.6 21.2 7.7 19.8 17.3 21.5 7 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 1.65596789050869 + 0.913038167653139X[t] + 1.13860134993134Y1[t] -0.542253852886171Y2[t] -0.170092084347798M1[t] + 0.37042972215349M2[t] -0.310491195601124M3[t] -0.799607124513897M4[t] -0.99811330052441M5[t] -0.577619444079062M6[t] -1.24340332230172M7[t] + 0.0569418787685395M8[t] + 3.12930286471154M9[t] -0.909771222546327M10[t] + 0.142479922141714M11[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) 1.65596789050869 1.092255 1.5161 0.136812 0.068406 X 0.913038167653139 0.228758 3.9913 0.000252 0.000126 Y1 1.13860134993134 0.127013 8.9645 0 0 Y2 -0.542253852886171 0.100297 -5.4065 3e-06 1e-06 M1 -0.170092084347798 0.433788 -0.3921 0.696914 0.348457 M2 0.37042972215349 0.494138 0.7496 0.457548 0.228774 M3 -0.310491195601124 0.532876 -0.5827 0.563158 0.281579 M4 -0.799607124513897 0.542691 -1.4734 0.147924 0.073962 M5 -0.99811330052441 0.530749 -1.8806 0.066814 0.033407 M6 -0.577619444079062 0.527891 -1.0942 0.279958 0.139979 M7 -1.24340332230172 0.509301 -2.4414 0.018821 0.00941 M8 0.0569418787685395 0.520471 0.1094 0.913391 0.456695 M9 3.12930286471154 0.55893 5.5987 1e-06 1e-06 M10 -0.909771222546327 0.640367 -1.4207 0.162615 0.081307 M11 0.142479922141714 0.461225 0.3089 0.758877 0.379439 Multiple Linear Regression - Regression Statistics Multiple R 0.974673811936578 R-squared 0.94998903967498 Adjusted R-squared 0.933706401429624 F-TEST (value) 58.3436802660627 F-TEST (DF numerator) 14 F-TEST (DF denominator) 43 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.623972228455184 Sum Squared Residuals 16.7416777009831 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 22 21.4368154230463 0.56318457695368 2 21.3 21.1633010719134 0.136698928086635 3 20.7 20.7897983926439 -0.0897983926439313 4 20.4 19.9970993507927 0.402900649207332 5 20.3 19.7823650815345 0.517634918465548 6 20.4 19.9777635085566 0.422236491443420 7 19.8 19.4800651506157 0.319934849384338 8 19.5 19.6778088893772 -0.17780888937725 9 23.1 23.0991570491338 0.000842950866200051 10 23.5 23.3217239774946 0.178276022505369 11 23.5 22.9686056085303 0.531394391469691 12 22.9 22.7005279619994 0.199472038000559 13 21.9 21.8472750676928 0.0527249323071645 14 21.5 21.7571554695251 -0.257155469525107 15 20.5 21.3456554982148 -0.845655498214755 16 20.2 20.0261435772904 0.173856422709579 17 19.4 20.0283108491867 -0.628310849186675 18 19.2 19.7005997815528 -0.500599781552799 19 18.8 18.9669872653569 -0.166987265356861 20 18.8 19.3725197964399 -0.572519796439939 21 22.6 22.2965670564761 0.303432943523851 22 23.3 22.3102666486615 0.989733351338549 23 23 22.1902779140993 0.809722085900703 24 21.4 21.5092475234885 -0.109247523488489 25 19.9 19.7713732518817 0.128626748118296 26 18.8 19.5629030148692 -0.762903014869161 27 18.6 18.5342052082846 0.0657947917153587 28 18.4 18.3225444307951 0.077455569204924 29 18.6 17.8221611218449 0.777838878155106 30 19.9 18.6701298356191 1.22987016438093 31 19.2 19.1934693081993 0.00653069180071754 32 18.4 18.6266482885043 -0.226648288504332 33 21.1 21.4416173418185 -0.341617341818513 34 20.5 20.8192661649189 -0.319266164918905 35 19.1 19.5416634633248 -0.441663463324846 36 18.1 18.0391901462456 0.0608098537543599 37 17 17.3983482892418 -0.398348289241824 38 17.1 17.5025739140007 -0.402573914000742 39 17.4 17.80590381971 -0.405903819709998 40 16.8 17.604142910488 -0.804142910488006 41 15.3 16.1945845015916 -0.894584501591583 42 14.3 14.9586171945757 -0.658617194575673 43 13.4 13.6937012954550 -0.29370129545498 44 15.3 14.7854705847691 0.514529415230851 45 22.1 21.5135445875977 0.58645541240229 46 23.7 24.2779811761546 -0.577981176154579 47 22.2 23.0994530140455 -0.899453014045548 48 19.5 19.6510343682664 -0.151034368266431 49 16.6 16.9461879681373 -0.346187968137317 50 17.3 16.0140665296916 1.28593347030837 51 19.8 18.5244370811467 1.27556291885333 52 21.2 21.0500697306338 0.149930269366171 53 21.5 21.2725784458424 0.227421554157604 54 20.6 21.0928896796959 -0.49288967969588 55 19.1 18.9657769803732 0.134223019626786 56 19.6 19.1375524409093 0.462447559090670 57 23.5 24.0491139649738 -0.549113964973829 58 24 24.2707620327704 -0.270762032770434 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.0646149345824356 0.129229869164871 0.935385065417564 19 0.0190470452125269 0.0380940904250537 0.980952954787473 20 0.00594319978651863 0.0118863995730373 0.994056800213481 21 0.00450870122898065 0.0090174024579613 0.99549129877102 22 0.00235445930502681 0.00470891861005362 0.997645540694973 23 0.00466928467928549 0.00933856935857098 0.995330715320714 24 0.0119280130975684 0.0238560261951368 0.988071986902432 25 0.00500543701713592 0.0100108740342718 0.994994562982864 26 0.00491997545722641 0.00983995091445282 0.995080024542774 27 0.010498156562456 0.020996313124912 0.989501843437544 28 0.00537353878492025 0.0107470775698405 0.99462646121508 29 0.00795898357881545 0.0159179671576309 0.992041016421185 30 0.0661694607636518 0.132338921527304 0.933830539236348 31 0.0642893525435106 0.128578705087021 0.93571064745649 32 0.0454133791841852 0.0908267583683704 0.954586620815815 33 0.0413120379067199 0.0826240758134397 0.95868796209328 34 0.0444882455017079 0.0889764910034158 0.955511754498292 35 0.0362337427612301 0.0724674855224603 0.96376625723877 36 0.0268068921465362 0.0536137842930725 0.973193107853464 37 0.0276660743015567 0.0553321486031135 0.972333925698443 38 0.620107468691911 0.759785062616178 0.379892531308089 39 0.919796135195981 0.160407729608038 0.0802038648040189 40 0.848503637892778 0.302992724214444 0.151496362107222 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.173913043478261 NOK 5% type I error level 11 0.478260869565217 NOK 10% type I error level 17 0.739130434782609 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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