Paper - Multiple Regression trend dummies

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Title produced by software: Multiple Regression

Date of computation: Tue, 21 Dec 2010 11:39:48 +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/21/t1292931471nvnq6rau0vu92xu.htm/, Retrieved Tue, 21 Dec 2010 12:38:12 +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/21/t1292931471nvnq6rau0vu92xu.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}, }

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Dataseries X:

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105.31 1576.23 29.29 105.63 1546.37 28.99 106.02 1545.05 28.91 105.85 1552.34 29.29 106.57 1594.3 30.96 106.48 1605.78 30.57 106.60 1673.21 30.59 106.75 1612.94 31.39 106.69 1566.34 31.28 106.69 1530.17 31.1 106.93 1582.54 31.7 107.21 1702.16 32.57 107.88 1701.93 32.49 108.84 1811.15 32.46 108.96 1924.2 32.3 109.52 2034.25 32.97 108.45 2011.13 32.9 108.67 2013.04 32.93 108.96 2151.67 33.72 108.76 1902.09 33.33 107.85 1944.01 33.44 108.78 1916.67 33.89 107.51 1967.31 34.34 108.83 2119.88 33.56 111.54 2216.38 32.67 111.74 2522.83 32.57 112.04 2647.64 33.23 111.74 2631.23 32.85 111.81 2693.41 32.61 111.86 3021.76 32.57 114.23 2953.67 32.98 114.80 2796.8 31.33 115.17 2672.05 29.8 115.11 2251.23 28.06 114.43 2046.08 25.47 114.66 2420.04 24.65 115.11 2608.89 23.94 117.74 2660.47 23.89 118.18 2493.98 23.54 118.56 2541.7 24.28 117.63 2554.6 25.51 117.71 2699.61 27.03 117.46 2805.48 27.09 117.37 2956.66 27.3 117.34 3149.51 27.11 117.09 3372.5 26.39 116.65 3379.33 27.54 11 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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation PC&S[t] = + 113.738843991343 -0.00207277898359385PCacao[t] -0.273357797722096PSuiker[t] + 1.01210858462569M1[t] + 1.56513385120318M2[t] + 1.46759879453859M3[t] + 2.52188611260808M4[t] + 2.28336555273651M5[t] + 2.48749590591574M6[t] + 2.88797250466192M7[t] + 2.25912430046324M8[t] + 1.54326470862961M9[t] + 0.578952930550244M10[t] -0.408248904330402M11[t] + 0.373660308371163t + 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) 113.738843991343 3.048693 37.3074 0 0 PCacao -0.00207277898359385 0.000988 -2.0982 0.041942 0.020971 PSuiker -0.273357797722096 0.09366 -2.9186 0.005628 0.002814 M1 1.01210858462569 1.041455 0.9718 0.336703 0.168352 M2 1.56513385120318 1.042683 1.5011 0.14082 0.07041 M3 1.46759879453859 1.037975 1.4139 0.164759 0.08238 M4 2.52188611260808 1.033986 2.439 0.019035 0.009517 M5 2.28336555273651 1.030717 2.2153 0.032215 0.016107 M6 2.48749590591574 1.032644 2.4089 0.020467 0.010233 M7 2.88797250466192 1.033989 2.793 0.00783 0.003915 M8 2.25912430046324 1.03504 2.1826 0.034704 0.017352 M9 1.54326470862961 1.039335 1.4849 0.145049 0.072524 M10 0.578952930550244 1.091619 0.5304 0.598654 0.299327 M11 -0.408248904330402 1.098948 -0.3715 0.71214 0.35607 t 0.373660308371163 0.043792 8.5325 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.974842575525699 R-squared 0.950318047057578 Adjusted R-squared 0.933757396076771 F-TEST (value) 57.3840997047121 F-TEST (DF numerator) 14 F-TEST (DF denominator) 42 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.53589691409404 Sum Squared Residuals 99.0771318903916 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 105.31 103.850786571749 1.45921342825070 2 105.63 104.921372666465 0.708627333534891 3 106.02 105.222102610248 0.797897389752195 4 105.85 106.531063714764 -0.681063714763661 5 106.57 106.122722134916 0.447277865084235 6 106.48 106.783326834846 -0.30332683484611 7 106.6 107.412229099145 -0.812229099145282 8 106.75 107.063281354481 -0.313281354481288 9 106.69 106.847742929404 -0.157742929403725 10 106.69 106.381268279122 0.308731720877913 11 106.93 105.495160638609 1.43483936139147 12 107.21 105.791302745274 1.41869725472561 13 107.88 107.199417001255 0.680582998744768 14 108.84 107.907914389547 0.932085610452576 15 108.96 107.993449224794 0.966550775205752 16 109.52 109.010137799617 0.509862200383406 17 108.45 109.212335244057 -0.762335244057418 18 108.67 109.777966163817 -1.10796616381749 19 108.96 110.048801060239 -1.08880106023876 20 108.76 110.417546884248 -1.65754688424821 21 107.85 109.958387348044 -2.10838734804407 22 108.78 109.301394646772 -0.521394646772368 23 107.51 108.459876583559 -0.949876583558746 24 108.83 109.138760988957 -0.308760988956638 25 111.54 110.567795150009 0.972204849990654 26 111.74 110.886613385208 0.853386614792113 27 112.04 110.723618945476 1.31638105452449 28 111.74 112.289456838171 -0.549456838171347 29 111.81 112.361317060924 -0.551317060924372 30 111.86 112.269445055121 -0.409445055120619 31 114.23 113.072640786165 1.15735921383520 32 114.8 113.593650095735 1.20634990426489 33 115.17 113.928267420991 1.24173257900922 34 115.11 114.685525371195 0.424474628805016 35 114.43 115.20521114927 -0.775211149269998 36 114.66 115.436137327399 -0.776137327398937 37 115.11 116.624545945727 -1.51454594572678 38 117.74 117.457985470588 0.282014529412228 39 118.18 118.174882924476 0.00511707552440113 40 118.56 119.301632767505 -0.741632767504808 41 117.63 119.073803575918 -1.44380357591787 42 117.71 118.935516704520 -1.22551670451974 43 117.46 119.473807032781 -2.01380703278067 44 117.37 118.847851272692 -1.47785127269179 45 117.34 118.157854543810 -0.817854543810452 46 117.09 117.301811702911 -0.211811702910562 47 116.65 116.359751628563 0.290248371437276 48 116.71 117.04379893837 -0.333798938370038 49 116.82 118.417455331259 -1.59745533125934 50 117.33 120.106114088192 -2.77611408819181 51 117.95 121.035946295007 -3.08594629500683 52 123.53 122.067708879944 1.46229112005641 53 124.91 122.599821984185 2.31017801581542 54 125.99 122.943745241696 3.04625475830396 55 126.29 123.532522021670 2.75747797832951 56 125.68 123.437670392844 2.2423296071564 57 125.52 123.677747757751 1.84225224224902 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.000427636591851069 0.000855273183702139 0.999572363408149 19 0.000641814386018117 0.00128362877203623 0.999358185613982 20 0.000106423130361723 0.000212846260723447 0.999893576869638 21 0.000140218500404509 0.000280437000809017 0.999859781499596 22 2.98093109354873e-05 5.96186218709745e-05 0.999970190689065 23 0.000360824139641472 0.000721648279282943 0.999639175860359 24 0.000188570031061334 0.000377140062122668 0.99981142996894 25 0.00251878874971727 0.00503757749943453 0.997481211250283 26 0.00137001151270789 0.00274002302541579 0.998629988487292 27 0.000993019592025953 0.00198603918405191 0.999006980407974 28 0.000340111735060157 0.000680223470120315 0.99965988826494 29 0.000108433775660968 0.000216867551321936 0.99989156622434 30 3.55216148736772e-05 7.10432297473544e-05 0.999964478385126 31 0.000141011978844738 0.000282023957689476 0.999858988021155 32 0.000541760234733191 0.00108352046946638 0.999458239765267 33 0.00670430986934816 0.0134086197386963 0.993295690130652 34 0.0769226108689802 0.153845221737960 0.92307738913102 35 0.0526062713657006 0.105212542731401 0.9473937286343 36 0.0452682554796738 0.0905365109593475 0.954731744520326 37 0.137625700551100 0.275251401102201 0.8623742994489 38 0.096033325456793 0.192066650913586 0.903966674543207 39 0.879213992491607 0.241572015016785 0.120786007508393 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 15 0.681818181818182 NOK 5% type I error level 16 0.727272727272727 NOK 10% type I error level 17 0.772727272727273 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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