WS 7: model met seizoenaliteit

*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, 20 Nov 2009 09:34:49 -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/Nov/20/t1258735285lnubn3d46171kll.htm/, Retrieved Fri, 20 Nov 2009 17:41:37 +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/20/t1258735285lnubn3d46171kll.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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423 114 427 116 441 153 449 162 452 161 462 149 455 139 461 135 461 130 463 127 462 122 456 117 455 112 456 113 472 149 472 157 471 157 465 147 459 137 465 132 468 125 467 123 463 117 460 114 462 111 461 112 476 144 476 150 471 149 453 134 443 123 442 116 444 117 438 111 427 105 424 102 416 95 406 93 431 124 434 130 418 124 412 115 404 106 409 105 412 105 406 101 398 95 397 93 385 84 390 87 413 116 413 120 401 117 397 109 397 105 409 107 419 109 424 109 428 108 430 107

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] = + 249.278046103183 + 1.72722283205269X[t] + 0.672557628979011M1[t] -1.25466520307351M2[t] -39.6530186608123M3[t] -48.85268935236M4[t] -51.2527991218441M5[t] -37.3987925356751M6[t] -28.3992316136114M7[t] -17.6175631174533M8[t] -10.9085620197585M9[t] -6.92689352360042M10[t] -2.63622392974752M11[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) 249.278046103183 14.948085 16.6763 0 0 X 1.72722283205269 0.129112 13.3778 0 0 M1 0.672557628979011 8.259952 0.0814 0.935451 0.467725 M2 -1.25466520307351 8.254097 -0.152 0.879834 0.439917 M3 -39.6530186608123 9.145656 -4.3357 7.6e-05 3.8e-05 M4 -48.85268935236 9.544759 -5.1183 6e-06 3e-06 M5 -51.2527991218441 9.40503 -5.4495 2e-06 1e-06 M6 -37.3987925356751 8.820238 -4.2401 0.000104 5.2e-05 M7 -28.3992316136114 8.484545 -3.3472 0.001614 0.000807 M8 -17.6175631174533 8.402217 -2.0968 0.041421 0.020711 M9 -10.9085620197585 8.361048 -1.3047 0.198351 0.099176 M10 -6.92689352360042 8.30644 -0.8339 0.408545 0.204273 M11 -2.63622392974752 8.256197 -0.3193 0.750912 0.375456 Multiple Linear Regression - Regression Statistics Multiple R 0.896840328544436 R-squared 0.804322574903692 Adjusted R-squared 0.75436238126208 F-TEST (value) 16.0992685631585 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 8.59312621059871e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 13.0416734552831 Sum Squared Residuals 7994.00658616908 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 423 446.854006586170 -23.8540065861696 2 427 448.381229418222 -21.3812294182216 3 441 473.890120746433 -32.8901207464325 4 449 480.235455543359 -31.2354555433589 5 452 476.108122941822 -24.1081229418222 6 462 469.235455543359 -7.23545554335892 7 455 460.962788144896 -5.9627881448957 8 461 464.835565312843 -3.83556531284300 9 461 462.908452250274 -1.90845225027441 10 463 461.708452250274 1.29154774972559 11 462 457.363007683864 4.63699231613612 12 456 451.363117453348 4.63688254665204 13 455 443.399560922064 11.6004390779365 14 456 443.199560922064 12.8004390779363 15 472 466.981229418222 5.01877058177828 16 472 471.599341383095 0.400658616904519 17 471 469.199231613611 1.80076838638860 18 465 465.781009879254 -0.781009879253558 19 459 457.50834248079 1.49165751920967 20 465 459.653896816685 5.34610318331505 21 468 454.272338090011 13.7276619099890 22 467 454.799560922064 12.2004390779363 23 463 448.7268935236 14.2731064763996 24 460 446.18144895719 13.8185510428101 25 462 441.672338090011 20.3276619099892 26 461 441.472338090011 19.527661909989 27 476 458.345115257958 17.6548847420417 28 476 459.508781558727 16.4912184412733 29 471 455.38144895719 15.6185510428101 30 453 443.327113062569 9.6728869374314 31 443 433.327222832053 9.67277716794731 32 442 432.018331503842 9.98166849615806 33 444 440.454555433589 3.54544456641054 34 438 434.072886937431 3.9271130625686 35 427 428.000219538968 -1.00021953896817 36 424 425.454774972558 -1.45477497255762 37 416 414.036772777168 1.96322722283220 38 406 408.65510428101 -2.65510428100992 39 431 423.800658616905 7.1993413830955 40 434 424.964324917673 9.03567508232711 41 418 412.200878155873 5.79912184412733 42 412 410.509879253568 1.49012074643247 43 404 403.964434687157 0.0355653128430213 44 409 413.018880351262 -4.01888035126236 45 412 419.727881448957 -7.72788144895719 46 406 416.800658616905 -10.8006586169045 47 398 410.727991218441 -12.7279912184413 48 397 409.909769484083 -12.9097694840834 49 385 395.037321624588 -10.0373216245882 50 390 398.291767288694 -8.29176728869379 51 413 409.982875960483 3.01712403951701 52 413 407.692096597146 5.30790340285399 53 401 400.110318331504 0.889681668496137 54 397 400.146542261251 -3.1465422612514 55 397 402.237211855104 -5.2372118551043 56 409 416.473326015368 -7.47332601536774 57 419 426.636772777168 -7.63677277716795 58 424 430.618441273326 -6.61844127332602 59 428 433.181888035126 -5.18188803512624 60 430 434.090889132821 -4.09088913282107 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.822774092680115 0.354451814639770 0.177225907319885 17 0.843066406760073 0.313867186479855 0.156933593239927 18 0.910536822597323 0.178926354805353 0.0894631774026766 19 0.95543775907487 0.089124481850259 0.0445622409251295 20 0.981281106216034 0.0374377875679309 0.0187188937839655 21 0.993762031933417 0.0124759361331656 0.00623796806658282 22 0.994632738048926 0.0107345239021483 0.00536726195107413 23 0.998509907921172 0.00298018415765596 0.00149009207882798 24 0.99907506990515 0.00184986018970020 0.000924930094850099 25 0.99977529973006 0.000449400539878131 0.000224700269939066 26 0.999952265890444 9.54682191113312e-05 4.77341095556656e-05 27 0.999892641387798 0.000214717224403220 0.000107358612201610 28 0.99993517937495 0.000129641250098083 6.48206250490413e-05 29 0.999975397971088 4.92040578248489e-05 2.46020289124245e-05 30 0.999998930119707 2.13976058567414e-06 1.06988029283707e-06 31 0.999999229378057 1.54124388674602e-06 7.70621943373008e-07 32 0.999999768055438 4.6388912463921e-07 2.31944562319605e-07 33 0.999999558119687 8.83760626570498e-07 4.41880313285249e-07 34 0.999999915768556 1.68462887994463e-07 8.42314439972317e-08 35 0.99999995227226 9.5455478169989e-08 4.77277390849945e-08 36 0.999999977080932 4.58381359462264e-08 2.29190679731132e-08 37 0.99999997705549 4.58890186888941e-08 2.29445093444471e-08 38 0.99999989280515 2.1438970039435e-07 1.07194850197175e-07 39 0.999999052144334 1.89571133181989e-06 9.47855665909943e-07 40 0.999993927436394 1.21451272126162e-05 6.0725636063081e-06 41 0.999954768448811 9.0463102376881e-05 4.52315511884405e-05 42 0.999703912921787 0.000592174156425752 0.000296087078212876 43 0.999409790047642 0.00118041990471695 0.000590209952358475 44 0.999534872564838 0.00093025487032365 0.000465127435161825 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 22 0.758620689655172 NOK 5% type I error level 25 0.862068965517241 NOK 10% type I error level 26 0.896551724137931 NOK

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/1046831258734885.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/1046831258734885.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/1k0wy1258734885.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/1k0wy1258734885.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/24ghc1258734885.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/24ghc1258734885.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/3qjhg1258734885.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/3qjhg1258734885.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/46iaz1258734885.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/46iaz1258734885.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/5oz221258734885.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/5oz221258734885.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/6e7v81258734885.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/6e7v81258734885.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/7sf0f1258734885.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/7sf0f1258734885.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/8bok91258734885.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/8bok91258734885.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/9qjch1258734885.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258735285lnubn3d46171kll/9qjch1258734885.ps (open in new window)

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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