Regressiemodel met seizoenaliteit en lineaire trend

*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: Tue, 17 Nov 2009 14:35:13 -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/17/t1258493774danu9i5vu2hkjhh.htm/, Retrieved Tue, 17 Nov 2009 22:36:26 +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/17/t1258493774danu9i5vu2hkjhh.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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5560 36.68 3922 36.7 3759 36.71 4138 36.72 4634 36.73 3996 36.73 4308 36.87 4429 37.31 5219 37.39 4929 37.42 5755 37.51 5592 37.67 4163 37.67 4962 37.71 5208 37.78 4755 37.79 4491 37.84 5732 37.88 5731 38.34 5040 38.58 6102 38.72 4904 38.83 5369 38.9 5578 38.92 4619 38.94 4731 39.1 5011 39.14 5299 39.16 4146 39.32 4625 39.34 4736 39.44 4219 39.92 5116 40.19 4205 40.2 4121 40.27 5103 40.28 4300 40.3 4578 40.34 3809 40.4 5526 40.43 4247 40.48 3830 40.48 4394 40.63 4826 40.74 4409 40.77 4569 40.91 4106 40.92 4794 41.03 3914 41 3793 41.04 4405 41.33 4022 41.44 4100 41.46 4788 41.55 3163 41.55 3585 41.81 3903 41.78 4178 41.84 3863 41.84 4187 41.86

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 Y[t] = -20975.9090114693 + 729.663124933304X[t] -739.925932000227M1[t] -810.905937388851M2[t] -751.494489025193M3[t] -381.362579415402M4[t] -761.283258554145M5[t] -425.773370194755M6[t] -590.816319326028M7[t] -773.893453448231M8[t] -228.600657584305M9[t] -585.677294222248M10[t] -448.101342111657M11[t] -86.799782107389t + 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) -20975.9090114693 15020.830195 -1.3965 0.16928 0.08464 X 729.663124933304 411.029669 1.7752 0.08248 0.04124 M1 -739.925932000227 361.33308 -2.0478 0.046317 0.023158 M2 -810.905937388851 360.924335 -2.2467 0.029498 0.014749 M3 -751.494489025193 360.445791 -2.0849 0.042656 0.021328 M4 -381.362579415402 361.544004 -1.0548 0.297018 0.148509 M5 -761.283258554145 362.940294 -2.0975 0.041469 0.020735 M6 -425.773370194755 367.244965 -1.1594 0.252288 0.126144 M7 -590.816319326028 361.764758 -1.6332 0.109265 0.054632 M8 -773.893453448231 360.92585 -2.1442 0.037335 0.018668 M9 -228.600657584305 360.894558 -0.6334 0.529591 0.264796 M10 -585.677294222248 359.715325 -1.6282 0.11032 0.05516 M11 -448.101342111657 358.441167 -1.2501 0.217571 0.108786 t -86.799782107389 39.225203 -2.2129 0.031909 0.015955 Multiple Linear Regression - Regression Statistics Multiple R 0.618001316467757 R-squared 0.381925627155881 Adjusted R-squared 0.207252434830369 F-TEST (value) 2.18651541241740 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0.0259702954238020 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 566.34944137796 Sum Squared Residuals 14754577.7284599 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 5560 4961.40869697655 598.591303023452 2 3922 4818.22217197927 -896.222171979267 3 3759 4798.13046948487 -1039.13046948487 4 4138 5088.75922823661 -950.759228236606 5 4634 4629.3353982398 4.66460176019562 6 3996 4878.04550449181 -882.045504491806 7 4308 4728.35561074381 -420.355610743807 8 4429 4779.53046948487 -350.530469484872 9 5219 5296.39653323607 -77.3965332360716 10 4929 4874.41000823874 54.5899917612590 11 5755 4990.85585948594 764.144140514063 12 5592 5468.90351947954 123.096480520464 13 4163 4642.17780537192 -479.17780537192 14 4962 4513.58454287324 448.415457126761 15 5208 4537.27262787484 670.72737212516 16 4755 4827.90138662657 -72.9013866265736 17 4491 4397.66408162711 93.3359183728906 18 5732 4675.56071287644 1056.43928712356 19 5731 4759.3630191071 971.636980892899 20 5040 4664.6052528615 375.394747138503 21 6102 5225.2511041087 876.748895891302 22 4904 4861.63762910603 42.362370893971 23 5369 4963.49021785456 405.509782145438 24 5578 5339.3850403575 238.614959642502 25 4619 4527.25258874855 91.7474112514548 26 4731 4486.21890124186 244.781098758136 27 5011 4488.01709249547 522.982907504535 28 5299 4785.94248249653 513.05751750347 29 4146 4435.96812123973 -289.968121239729 30 4625 4699.2714899904 -74.2714899903987 31 4736 4520.39507124506 215.604928754937 32 4219 4600.75645498346 -381.75645498346 33 5116 5256.25851247199 -140.258512471986 34 4205 4819.67872497599 -614.678724975991 35 4121 4921.53131372452 -800.531313724524 36 5103 5290.12950497812 -187.129504978124 37 4300 4477.99705336917 -177.997053369170 38 4578 4349.40379087049 228.596209129506 39 3809 4365.79524462276 -556.795244622759 40 5526 4671.01726587316 854.982734126839 41 4247 4240.77996087369 6.22003912630821 42 3830 4489.49006712569 -659.490067125693 43 4394 4347.09680462703 46.9031953729692 44 4826 4157.4828321401 668.517167859899 45 4409 4637.86573964464 -228.865739644638 46 4569 4296.14215838996 272.857841610036 47 4106 4354.2149596425 -248.214959642503 48 4794 4795.77946338943 -1.77946338943394 49 3914 3947.16385553382 -33.1638555338176 50 3793 3818.57059303514 -25.5705930351362 51 4405 4002.78456552206 402.215434477936 52 4022 4366.37963676713 -344.379636767129 53 4100 3914.25243801966 185.747561980335 54 4788 4228.63222551566 559.36777448434 55 3163 3976.789494277 -813.789494276999 56 3585 3896.62499053007 -311.624990530069 57 3903 4333.22811053861 -430.228110538606 58 4178 3933.13147928927 244.868520710726 59 3863 3983.90764929248 -120.907649292476 60 4187 4359.80247179541 -172.802471795408 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.98447911526464 0.0310417694707174 0.0155208847353587 18 0.968436798177031 0.0631264036459373 0.0315632018229686 19 0.97943742237151 0.0411251552569789 0.0205625776284895 20 0.963642395011113 0.072715209977775 0.0363576049888875 21 0.9556007764401 0.088798447119798 0.044399223559899 22 0.962878854435964 0.0742422911280714 0.0371211455640357 23 0.973835190909451 0.0523296181810976 0.0261648090905488 24 0.959666541109701 0.080666917780598 0.040333458890299 25 0.953061604122692 0.0938767917546158 0.0469383958773079 26 0.924148315312925 0.151703369374151 0.0758516846870754 27 0.890450684965556 0.219098630068887 0.109549315034444 28 0.840238573247776 0.319522853504448 0.159761426752224 29 0.854130157300236 0.291739685399529 0.145869842699764 30 0.818723775648812 0.362552448702376 0.181276224351188 31 0.784461335444151 0.431077329111698 0.215538664555849 32 0.767194392649825 0.46561121470035 0.232805607350175 33 0.735922078705622 0.528155842588757 0.264077921294378 34 0.73000246762626 0.53999506474748 0.26999753237374 35 0.760093123776092 0.479813752447817 0.239906876223909 36 0.675853741146533 0.648292517706935 0.324146258853467 37 0.588578515058161 0.822842969883678 0.411421484941839 38 0.473494684126184 0.946989368252367 0.526505315873816 39 0.553191451186967 0.893617097626066 0.446808548813033 40 0.646821966975791 0.706356066048418 0.353178033024209 41 0.520872463986411 0.958255072027178 0.479127536013589 42 0.89492939118921 0.210141217621580 0.105070608810790 43 0.913996362646163 0.172007274707673 0.0860036373538365 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 2 0.0740740740740741 NOK 10% type I error level 9 0.333333333333333 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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