Paper: Y(t-2) met 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: Sun, 13 Dec 2009 05:27:32 -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/13/t1260707479d8tvfh3cwnaw5x1.htm/, Retrieved Sun, 13 Dec 2009 13:31:36 +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/13/t1260707479d8tvfh3cwnaw5x1.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:

Paper: Y(t-2) met lineaire trend

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No (this computation is public)

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Dataseries X:

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0.7905 0.313 0.7744 0.779 0.7719 0.364 0.7905 0.7744 0.7811 0.363 0.7719 0.7905 0.7557 -0.155 0.7811 0.7719 0.7637 0.052 0.7557 0.7811 0.7595 0.568 0.7637 0.7557 0.7471 0.668 0.7595 0.7637 0.7615 1.378 0.7471 0.7595 0.7487 0.252 0.7615 0.7471 0.7389 -0.402 0.7487 0.7615 0.7337 -0.05 0.7389 0.7487 0.751 0.555 0.7337 0.7389 0.7382 0.05 0.751 0.7337 0.7159 0.15 0.7382 0.751 0.7542 0.45 0.7159 0.7382 0.7636 0.299 0.7542 0.7159 0.7433 0.199 0.7636 0.7542 0.7658 0.496 0.7433 0.7636 0.7627 0.444 0.7658 0.7433 0.748 -0.393 0.7627 0.7658 0.7692 -0.444 0.748 0.7627 0.785 0.198 0.7692 0.748 0.7913 0.494 0.785 0.7692 0.772 0.133 0.7913 0.785 0.788 0.388 0.772 0.7913 0.807 0.484 0.788 0.772 0.8268 0.278 0.807 0.788 0.8244 0.369 0.8268 0.807 0.8487 0.165 0.8244 0.8268 0.8572 0.155 0.8487 0.8244 0.8214 0.087 0.8572 0.8487 0.8827 0.414 0.8214 0.8572 0.9216 0.36 0.8827 0.8214 0.8865 0.975 0.9216 0.8827 0.8816 0.27 0.8865 0.9216 0.8884 0.359 0.8816 0.8865 0.9466 0.169 0.8884 0.8816 0.918 0. 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 USDOLLAR[t] = + 0.111935700362069 + 0.0182913192531364Amerikaanse_inflatie[t] + 1.05085697392121`Y[t-1]`[t] -0.205313543557759`Y[t-2]`[t] + 0.000281672269563672t + 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) 0.111935700362069 0.04581 2.4435 0.01812 0.00906 Amerikaanse_inflatie 0.0182913192531364 0.011845 1.5442 0.128849 0.064425 `Y[t-1]` 1.05085697392121 0.161194 6.5192 0 0 `Y[t-2]` -0.205313543557759 0.149834 -1.3703 0.176724 0.088362 t 0.000281672269563672 0.000288 0.9795 0.332044 0.166022 Multiple Linear Regression - Regression Statistics Multiple R 0.924271488211047 R-squared 0.854277783919864 Adjusted R-squared 0.842620006633453 F-TEST (value) 73.2796452472692 F-TEST (DF numerator) 4 F-TEST (DF denominator) 50 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0317330932234768 Sum Squared Residuals 0.0503494602764936 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 0.7905 0.771786945730956 0.0187130542690440 2 0.7719 0.790864714862925 -0.0189647148629254 3 0.7811 0.768276608047022 0.0128233919529781 4 0.7557 0.77257009301371 -0.0168700930137101 5 0.7637 0.748057416630343 0.0156425833696571 6 0.7595 0.771399229432262 -0.0118992294322618 7 0.7471 0.767453925988208 -0.0203539259882079 8 0.7615 0.768554125333818 -0.00705412533381808 9 0.7487 0.765918000488932 -0.0172180004889317 10 0.7389 0.737829665673521 0.00107033432647899 11 0.7337 0.7368794973333 -0.0031794973333001 12 0.751 0.744775034213487 0.00622496578651287 13 0.7382 0.755067046335554 -0.0168670463355542 14 0.7159 0.74017495696069 -0.0242749569606907 15 0.7542 0.725137927845292 0.0290620721547083 16 0.7636 0.767483925030152 -0.00388392503015215 17 0.7433 0.767951012210999 -0.0246510122109993 18 0.7658 0.750402862418701 0.0153971375812990 19 0.7627 0.777545532934551 -0.0148455329345513 20 0.748 0.754640159640035 -0.00664015964003453 21 0.7692 0.739177849096075 0.0300221509039245 22 0.785 0.776498825263581 0.0085011747364186 23 0.7913 0.794445621096604 -0.00314562109660412 24 0.772 0.791500572063277 -0.0195005720632765 25 0.788 0.774871515821297 0.0131284841787032 26 0.807 0.797685417712566 0.00931458228743438 27 0.8268 0.810880344023562 0.0159196559764379 28 0.8244 0.829732537101204 -0.00533253710120359 29 0.8487 0.819695515343273 0.0290044846567270 30 0.8572 0.84582285139113 0.0113771486088707 31 0.8214 0.848803879121356 -0.0274038791213563 32 0.8827 0.815700968000076 0.0669990319999246 33 0.9216 0.886762666390708 0.0348373336092923 34 0.8865 0.926586116066395 -0.0400861160663946 35 0.8816 0.869100631633466 0.0124993683665343 36 0.8884 0.873067537523222 0.0153324624767778 37 0.9466 0.878025722920787 0.068574277079213 38 0.918 0.941948898658037 -0.0239488986580372 39 0.9337 0.896074683767931 0.0376253162320691 40 0.9559 0.92479949586585 0.0311005041341493 41 0.9626 0.953216659474735 0.0093833405252648 42 0.9434 0.952377722909721 -0.00897772290972091 43 0.8639 0.917535181652333 -0.0536351816523331 44 0.7996 0.839313223686658 -0.0397132236866579 45 0.668 0.772525228091967 -0.104525228091967 46 0.6572 0.632076705482831 0.0251232945171685 47 0.6928 0.664234493624526 0.0285655063754739 48 0.6438 0.723660898079205 -0.079860898079205 49 0.6454 0.66728150082859 -0.0218815008285904 50 0.6873 0.669482469451824 0.0178175305481762 51 0.7265 0.715698088207876 0.0108019117921236 52 0.7912 0.750729092051951 0.0404709079480485 53 0.8114 0.82271423790652 -0.0113142379065199 54 0.8281 0.817458732491543 0.0106412675084565 55 0.8393 0.839245437074864 5.45629251361077e-05 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.00292014863661846 0.00584029727323691 0.997079851363382 9 0.00601524298895622 0.0120304859779124 0.993984757011044 10 0.00129259332101904 0.00258518664203808 0.99870740667898 11 0.000260022456611605 0.000520044913223211 0.999739977543388 12 0.000184354005669952 0.000368708011339903 0.99981564599433 13 0.000114113469553773 0.000228226939107545 0.999885886530446 14 3.75948640625916e-05 7.51897281251832e-05 0.999962405135937 15 6.59619591920553e-05 0.000131923918384111 0.999934038040808 16 0.000913499898764166 0.00182699979752833 0.999086500101236 17 0.000543927068423557 0.00108785413684711 0.999456072931576 18 0.000543319597212376 0.00108663919442475 0.999456680402788 19 0.000251251456803674 0.000502502913607348 0.999748748543196 20 9.7170494400496e-05 0.000194340988800992 0.9999028295056 21 0.000121900428489044 0.000243800856978087 0.999878099571511 22 9.60531530960808e-05 0.000192106306192162 0.999903946846904 23 3.69625055449042e-05 7.39250110898084e-05 0.999963037494455 24 2.81546867707756e-05 5.63093735415511e-05 0.999971845313229 25 1.02041602877537e-05 2.04083205755075e-05 0.999989795839712 26 5.96968606940429e-06 1.19393721388086e-05 0.99999403031393 27 4.0002148869588e-06 8.0004297739176e-06 0.999995999785113 28 1.64106479442602e-06 3.28212958885205e-06 0.999998358935206 29 8.1752653745584e-07 1.63505307491168e-06 0.999999182473463 30 2.83704112820456e-07 5.67408225640912e-07 0.999999716295887 31 1.86515285208087e-06 3.73030570416173e-06 0.999998134847148 32 5.82415558789093e-06 1.16483111757819e-05 0.999994175844412 33 2.31356658958941e-05 4.62713317917883e-05 0.999976864334104 34 0.00011154118668687 0.00022308237337374 0.999888458813313 35 6.05805119773678e-05 0.000121161023954736 0.999939419488023 36 2.43840497138314e-05 4.87680994276628e-05 0.999975615950286 37 0.000260013765089429 0.000520027530178858 0.99973998623491 38 0.000179275924111041 0.000358551848222082 0.999820724075889 39 0.000197747831133697 0.000395495662267394 0.999802252168866 40 0.000286569783096201 0.000573139566192401 0.999713430216904 41 0.000494055700118796 0.000988111400237591 0.999505944299881 42 0.00370775363525324 0.00741550727050648 0.996292246364747 43 0.0264369237469829 0.0528738474939658 0.973563076253017 44 0.728132274565953 0.543735450868094 0.271867725434047 45 0.749234982229061 0.501530035541878 0.250765017770939 46 0.637998077966581 0.724003844066838 0.362001922033419 47 0.817284563729866 0.365430872540268 0.182715436270134 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 34 0.85 NOK 5% type I error level 35 0.875 NOK 10% type I error level 36 0.9 NOK

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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