Basic Structural Time Series Model

*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: Mon, 29 Nov 2010 14:58:28 +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/Nov/29/t12910438954hfcp5r7ftvm7al.htm/, Retrieved Mon, 29 Nov 2010 16:18:29 +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/Nov/29/t12910438954hfcp5r7ftvm7al.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}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

13328 12873 14000 13477 14237 13674 13529 14058 12975 14326 14008 16193 14483 14011 15057 14884 15414 14440 14900 15074 14442 15307 14938 17193 15528 14765 15838 15723 16150 15486 15986 15983 15692 16490 15686 18897 16316 15636 17163 16534 16518 16375 16290 16352 15943 16362 16393 19051 16747 16320 17910 16961 17480 17049 16879 17473 16998 17307 17418 20169 17871 17226 19062 17804 19100 18522 18060 18869 18127 18871 18890 21263 19547 18450 20254 19240 20216 19420 19415 20018 18652 19978 19509 21971

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 9 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation HPC[t] = + 15515.9166666667 -2132.84102182539M1[t] -2859.02430555556M2[t] -1507.77901785714M3[t] -2251.39087301588M4[t] -1687.43129960317M5[t] -2357.90029761905M6[t] -2422.36929563492M7[t] -2104.69543650794M8[t] -2896.45014880952M9[t] -2143.91914682540M10[t] -2478.67385912699M11[t] + 77.7547123015873t + 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) 15515.9166666667 129.27789 120.0199 0 0 M1 -2132.84102182539 159.023856 -13.4121 0 0 M2 -2859.02430555556 158.904073 -17.9921 0 0 M3 -1507.77901785714 158.79562 -9.4951 0 0 M4 -2251.39087301588 158.69852 -14.1866 0 0 M5 -1687.43129960317 158.612794 -10.6387 0 0 M6 -2357.90029761905 158.538461 -14.8727 0 0 M7 -2422.36929563492 158.475536 -15.2854 0 0 M8 -2104.69543650794 158.424034 -13.2852 0 0 M9 -2896.45014880952 158.383965 -18.2875 0 0 M10 -2143.91914682540 158.355338 -13.5387 0 0 M11 -2478.67385912699 158.338159 -15.6543 0 0 t 77.7547123015873 1.346645 57.7396 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.99130582514897 R-squared 0.98268723897428 Adjusted R-squared 0.979761138519229 F-TEST (value) 335.835100014381 F-TEST (DF numerator) 12 F-TEST (DF denominator) 71 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 296.212858097257 Sum Squared Residuals 6229686.06845236 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 13328 13460.8303571428 -132.830357142841 2 12873 12812.4017857143 60.59821428571 3 14000 14241.4017857143 -241.401785714278 4 13477 13575.5446428571 -98.544642857145 5 14237 14217.2589285714 19.7410714285763 6 13674 13624.5446428571 49.4553571428511 7 13529 13637.8303571429 -108.830357142853 8 14058 14033.2589285714 24.7410714285705 9 12975 13319.2589285714 -344.25892857143 10 14326 14149.5446428571 176.455357142857 11 14008 13892.5446428571 115.455357142854 12 16193 16448.9732142857 -255.973214285714 13 14483 14393.8869047619 89.1130952380873 14 14011 13745.4583333333 265.541666666667 15 15057 15174.4583333333 -117.458333333334 16 14884 14508.6011904762 375.398809523809 17 15414 15150.3154761905 263.684523809523 18 14440 14557.6011904762 -117.60119047619 19 14900 14570.8869047619 329.113095238095 20 15074 14966.3154761905 107.684523809524 21 14442 14252.3154761905 189.684523809524 22 15307 15082.6011904762 224.398809523809 23 14938 14825.6011904762 112.398809523810 24 17193 17382.0297619048 -189.029761904763 25 15528 15326.9434523810 201.056547619046 26 14765 14678.5148809524 86.4851190476194 27 15838 16107.5148809524 -269.514880952383 28 15723 15441.6577380952 281.342261904762 29 16150 16083.3720238095 66.6279761904752 30 15486 15490.6577380952 -4.65773809523746 31 15986 15503.9434523810 482.056547619047 32 15983 15899.3720238095 83.6279761904762 33 15692 15185.3720238095 506.627976190476 34 16490 16015.6577380952 474.342261904761 35 15686 15758.6577380952 -72.6577380952376 36 18897 18315.0863095238 581.913690476189 37 16316 16260 55.9999999999982 38 15636 15611.5714285714 24.4285714285718 39 17163 17040.5714285714 122.428571428570 40 16534 16374.7142857143 159.285714285714 41 16518 17016.4285714286 -498.428571428572 42 16375 16423.7142857143 -48.7142857142847 43 16290 16437 -147.000000000001 44 16352 16832.4285714286 -480.428571428571 45 15943 16118.4285714286 -175.428571428571 46 16362 16948.7142857143 -586.714285714285 47 16393 16691.7142857143 -298.714285714285 48 19051 19248.1428571429 -197.142857142859 49 16747 17193.0565476190 -446.056547619049 50 16320 16544.6279761905 -224.627976190475 51 17910 17973.6279761905 -63.6279761904773 52 16961 17307.7708333333 -346.770833333333 53 17480 17949.4851190476 -469.48511904762 54 17049 17356.7708333333 -307.770833333332 55 16879 17370.0565476190 -491.056547619048 56 17473 17765.4851190476 -292.485119047618 57 16998 17051.4851190476 -53.4851190476179 58 17307 17881.7708333333 -574.770833333332 59 17418 17624.7708333333 -206.770833333332 60 20169 20181.1994047619 -12.1994047619047 61 17871 18126.1130952381 -255.113095238096 62 17226 17477.6845238095 -251.684523809522 63 19062 18906.6845238095 155.315476190475 64 17804 18240.8273809524 -436.82738095238 65 19100 18882.5416666667 217.458333333333 66 18522 18289.8273809524 232.172619047621 67 18060 18303.1130952381 -243.113095238096 68 18869 18698.5416666667 170.458333333333 69 18127 17984.5416666667 142.458333333333 70 18871 18814.8273809524 56.1726190476184 71 18890 18557.8273809524 332.172619047620 72 21263 21114.2559523810 148.744047619049 73 19547 19059.1696428571 487.830357142856 74 18450 18410.7410714286 39.2589285714303 75 20254 19839.7410714286 414.258928571429 76 19240 19173.8839285714 66.1160714285725 77 20216 19815.5982142857 400.401785714284 78 19420 19222.8839285714 197.116071428572 79 19415 19236.1696428571 178.830357142856 80 20018 19631.5982142857 386.401785714287 81 18652 18917.5982142857 -265.598214285715 82 19978 19747.8839285714 230.116071428572 83 19509 19490.8839285714 18.1160714285717 84 21971 22047.3125 -76.3124999999989 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0480796323752217 0.0961592647504433 0.951920367624778 17 0.0132432466795157 0.0264864933590315 0.986756753320484 18 0.0432107824099886 0.0864215648199773 0.956789217590011 19 0.0315084969136549 0.0630169938273099 0.968491503086345 20 0.0152933905266358 0.0305867810532717 0.984706609473364 21 0.0158812131725257 0.0317624263450514 0.984118786827474 22 0.0096668261431016 0.0193336522862032 0.990333173856898 23 0.00630198435563797 0.0126039687112759 0.993698015644362 24 0.00319094541650994 0.00638189083301988 0.99680905458349 25 0.00143352618919697 0.00286705237839393 0.998566473810803 26 0.00212110096484131 0.00424220192968263 0.997878899035159 27 0.00238615697093017 0.00477231394186035 0.99761384302907 28 0.00131420023077466 0.00262840046154933 0.998685799769225 29 0.00104463247716715 0.00208926495433430 0.998955367522833 30 0.000514169460455889 0.00102833892091178 0.999485830539544 31 0.000776122767264067 0.00155224553452813 0.999223877232736 32 0.000452014186878448 0.000904028373756897 0.999547985813122 33 0.0023910490053108 0.0047820980106216 0.99760895099469 34 0.00481588271422685 0.0096317654284537 0.995184117285773 35 0.00686617241902962 0.0137323448380592 0.99313382758097 36 0.102272467142474 0.204544934284947 0.897727532857526 37 0.113338559214991 0.226677118429981 0.88666144078501 38 0.159719499815923 0.319438999631846 0.840280500184077 39 0.139186574074145 0.278373148148289 0.860813425925855 40 0.291585373792782 0.583170747585564 0.708414626207218 41 0.584474474166785 0.83105105166643 0.415525525833215 42 0.564639796808961 0.870720406382078 0.435360203191039 43 0.702621746976863 0.594756506046275 0.297378253023137 44 0.782353228856718 0.435293542286564 0.217646771143282 45 0.797662920721426 0.404674158557148 0.202337079278574 46 0.898354404500992 0.203291190998015 0.101645595499008 47 0.876890356415658 0.246219287168684 0.123109643584342 48 0.858653848981129 0.282692302037743 0.141346151018871 49 0.857155395926084 0.285689208147832 0.142844604073916 50 0.83488474687285 0.330230506254299 0.165115253127149 51 0.785695729980796 0.428608540038408 0.214304270019204 52 0.77295482680727 0.454090346385459 0.227045173192730 53 0.809578338257665 0.38084332348467 0.190421661742335 54 0.770831147074807 0.458337705850385 0.229168852925192 55 0.753727922744181 0.492544154511638 0.246272077255819 56 0.730345718803357 0.539308562393286 0.269654281196643 57 0.701912347381598 0.596175305236803 0.298087652618402 58 0.7867422343356 0.4265155313288 0.2132577656644 59 0.731444268566858 0.537111462866284 0.268555731433142 60 0.669796407678478 0.660407184643043 0.330203592321522 61 0.800781167812374 0.398437664375251 0.199218832187626 62 0.744179576169956 0.511640847660089 0.255820423830044 63 0.704109506968415 0.59178098606317 0.295890493031585 64 0.776403232124108 0.447193535751784 0.223596767875892 65 0.730615425479785 0.538769149040429 0.269384574520215 66 0.628788517161013 0.742422965677975 0.371211482838987 67 0.724631049343983 0.550737901312033 0.275368950656017 68 0.75285278661892 0.494294426762161 0.247147213381080 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 11 0.207547169811321 NOK 5% type I error level 17 0.320754716981132 NOK 10% type I error level 20 0.377358490566038 NOK

Charts produced by software:

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Parameters (Session):

par1 = 12 ;

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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