WS 7 Multiple Regression - autoregression met 4 lags

*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 06:27:42 -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/t1258723756pbdmg6ot5avlopf.htm/, Retrieved Fri, 20 Nov 2009 14:29:29 +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/2009/Nov/20/t1258723756pbdmg6ot5avlopf.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:

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No (this computation is public)

User-defined keywords:

SHW WS 7 Multiple Regression - autoregression met 4 lags

Dataseries X:

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15.6 0 14.6 11.9 13.5 14.2 14.1 -0.2 15.6 14.6 11.9 13.5 14.9 1 14.1 15.6 14.6 11.9 14.2 0.4 14.9 14.1 15.6 14.6 14.6 1 14.2 14.9 14.1 15.6 17.2 1.7 14.6 14.2 14.9 14.1 15.4 3.1 17.2 14.6 14.2 14.9 14.3 3.3 15.4 17.2 14.6 14.2 17.5 3.1 14.3 15.4 17.2 14.6 14.5 3.5 17.5 14.3 15.4 17.2 14.4 6 14.5 17.5 14.3 15.4 16.6 5.7 14.4 14.5 17.5 14.3 16.7 4.7 16.6 14.4 14.5 17.5 16.6 4.2 16.7 16.6 14.4 14.5 16.9 3.6 16.6 16.7 16.6 14.4 15.7 4.4 16.9 16.6 16.7 16.6 16.4 2.5 15.7 16.9 16.6 16.7 18.4 -0.6 16.4 15.7 16.9 16.6 16.9 -1.9 18.4 16.4 15.7 16.9 16.5 -1.9 16.9 18.4 16.4 15.7 18.3 0.7 16.5 16.9 18.4 16.4 15.1 -0.9 18.3 16.5 16.9 18.4 15.7 -1.7 15.1 18.3 16.5 16.9 18.1 -3.1 15.7 15.1 18.3 16.5 16.8 -2.1 18.1 15.7 15.1 18.3 18.9 0.2 16.8 18.1 15.7 15.1 19 1.2 18.9 16.8 18.1 15.7 18.1 3.8 19 18.9 16.8 18.1 17.8 4 18.1 19 18.9 16.8 21.5 6.6 17.8 18.1 19 18.9 17.1 5.3 21.5 17.8 18.1 19 18.7 7.6 17.1 21.5 17.8 18.1 19 4.7 18.7 17.1 21.5 17.8 16.4 6.6 19 18.7 17.1 21.5 16.9 4.4 16.4 19 18. 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 6.08577545459943 + 0.0937233198980662X[t] + 0.353117280750058Y1[t] + 0.297388312914167Y2[t] + 0.390002938543844Y3[t] -0.412813823279897Y4[t] + 1.32708847505859M1[t] -0.690848167073465M2[t] -2.44667437247958M3[t] -1.21771370872316M4[t] -0.853217792826271M5[t] + 1.39824803727689M6[t] -1.35047284337896M7[t] -1.46577081148973M8[t] -0.108516967765969M9[t] -1.34277021223409M10[t] -2.37859652074837M11[t] + 0.0403527438512034t + 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) 6.08577545459943 1.845808 3.2971 0.001784 0.000892 X 0.0937233198980662 0.038234 2.4513 0.017699 0.00885 Y1 0.353117280750058 0.133995 2.6353 0.011107 0.005554 Y2 0.297388312914167 0.131808 2.2562 0.028373 0.014186 Y3 0.390002938543844 0.124509 3.1323 0.002872 0.001436 Y4 -0.412813823279897 0.137136 -3.0102 0.004052 0.002026 M1 1.32708847505859 0.762706 1.74 0.087895 0.043948 M2 -0.690848167073465 0.889778 -0.7764 0.441083 0.220541 M3 -2.44667437247958 0.790693 -3.0943 0.003199 0.001599 M4 -1.21771370872316 0.641573 -1.898 0.063361 0.03168 M5 -0.853217792826271 0.656872 -1.2989 0.199817 0.099909 M6 1.39824803727689 0.673195 2.077 0.042854 0.021427 M7 -1.35047284337896 0.809549 -1.6682 0.101409 0.050705 M8 -1.46577081148973 0.797244 -1.8385 0.071808 0.035904 M9 -0.108516967765969 0.639952 -0.1696 0.866019 0.433009 M10 -1.34277021223409 0.78138 -1.7185 0.091779 0.045889 M11 -2.37859652074837 0.742783 -3.2023 0.00235 0.001175 t 0.0403527438512034 0.017783 2.2692 0.027519 0.013759 Multiple Linear Regression - Regression Statistics Multiple R 0.940286933379321 R-squared 0.884139517083888 Adjusted R-squared 0.84551935611185 F-TEST (value) 22.8932115980574 F-TEST (DF numerator) 17 F-TEST (DF denominator) 51 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.983511283994648 Sum Squared Residuals 49.3320167329849 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 15.6 15.550733275906 0.049266724093985 2 14.1 14.3754354138896 -0.275435413889626 3 14.9 14.2536523793177 0.646347620682323 4 14.2 14.5785487659034 -0.378548765903390 5 14.6 14.0325417403010 0.567458259699044 6 17.2 17.254264817199 -0.0542648171989887 7 15.4 15.1109164677628 0.289083532237153 8 14.3 15.6372852674231 -1.33728526742308 9 17.5 16.9413013298499 0.558698670150091 10 14.5 14.8124170814802 -0.312417081480165 11 14.4 14.257604225143 0.142395774856993 12 16.6 17.4230644359038 -0.82306443590384 13 16.7 16.9528884711471 -0.252888471147079 14 16.6 16.8174501053887 -0.217450105388667 15 16.9 15.9394576022358 0.960542397764226 16 15.7 16.4907559013341 -0.790755901334058 17 16.4 16.3027243340676 0.0972756659323625 18 18.4 18.3525990012572 0.0474009987428246 19 16.9 15.8449492558885 1.0550507441115 20 16.5 16.6034833812487 -0.103483381248741 21 18.3 18.1484774196792 0.151522580320833 22 15.1 15.9106433330342 -0.810643333034218 23 15.7 14.7087343368003 0.991265663199682 24 18.1 17.1238295393582 0.976170460641827 25 16.8 17.6198342544706 -0.81983425447056 26 18.9 17.6674994755962 1.23250052440384 27 19 17.0890095752633 1.91099042473669 28 18.1 17.7640738038219 0.335926196178095 29 17.8 19.2552645473719 -1.45526454737192 30 21.5 20.5892703521801 0.910729647819914 31 17.1 18.5840953173915 -1.48409531739147 32 18.7 18.5258660107684 0.174133989231630 33 19 20.4750090626129 -1.47500906261290 34 16.4 16.7975152789615 -0.397515278961452 35 16.9 17.2073474985484 -0.307347498548432 36 18.6 18.5048892182411 0.0951107817588718 37 19.3 19.6146848315424 -0.314684831542436 38 19.4 19.5456925876627 -0.145692587662684 39 17.6 18.4553221011888 -0.855322101188823 40 18.6 18.5681414762751 0.0318585237248646 41 18.1 18.5689560670557 -0.468956067055667 42 20.4 20.3414132937301 0.0585867062698637 43 18.1 19.4202162346514 -1.32021623465139 44 19.6 18.4874387759500 1.11256122405004 45 19.9 20.8247695032484 -0.924769503248421 46 19.2 18.3832697639822 0.816730236017805 47 17.8 18.8861471138953 -1.08614711389535 48 19.2 19.9691278651909 -0.769127865190936 49 22 20.9240201152083 1.07597988479175 50 21.1 20.0288674795133 1.07113252048670 51 19.5 19.9709638719758 -0.470963871975847 52 22.2 20.9029643601119 1.29703563988808 53 20.9 20.4566013351557 0.443398664844337 54 22.2 22.7086309804277 -0.508630980427662 55 23.5 21.926805532973 1.57319446702701 56 21.5 20.8603528017486 0.639647198251358 57 24.3 23.1132040728401 1.18679592715994 58 22.8 22.0961545425420 0.70384545745803 59 20.3 20.0401668256129 0.259833174387105 60 23.7 23.1790889413059 0.520911058694078 61 23.3 23.0378390517257 0.262160948274342 62 19.6 21.2650549379496 -1.66505493794956 63 18 20.1915944700186 -2.19159447001858 64 17.3 17.7955156925536 -0.49551569255359 65 16.8 15.9839119760482 0.816088023951838 66 18.2 18.6538215552059 -0.453821555205949 67 16.5 16.6130171913328 -0.113017191332803 68 16 16.4855737628612 -0.485573762861198 69 18.4 17.8972386117695 0.502761388230456 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.0205545532715187 0.0411091065430375 0.979445446728481 22 0.0188525825624731 0.0377051651249463 0.981147417437527 23 0.0115296806927879 0.0230593613855759 0.988470319307212 24 0.00397383561800473 0.00794767123600946 0.996026164381995 25 0.00373476083644249 0.00746952167288498 0.996265239163558 26 0.00248530267255816 0.00497060534511631 0.997514697327442 27 0.00402869863864868 0.00805739727729735 0.995971301361351 28 0.0449732337856437 0.0899464675712875 0.955026766214356 29 0.0252428348537729 0.0504856697075458 0.974757165146227 30 0.0346128421232181 0.0692256842464362 0.965387157876782 31 0.0898724820973575 0.179744964194715 0.910127517902642 32 0.0620669644663609 0.124133928932722 0.93793303553364 33 0.100849822652059 0.201699645304119 0.89915017734794 34 0.135540760485540 0.271081520971079 0.86445923951446 35 0.132485614292485 0.264971228584969 0.867514385707515 36 0.132970729020964 0.265941458041928 0.867029270979036 37 0.0913660894561851 0.182732178912370 0.908633910543815 38 0.0653520186624892 0.130704037324978 0.934647981337511 39 0.137286602871377 0.274573205742753 0.862713397128623 40 0.0954483498774054 0.190896699754811 0.904551650122595 41 0.0596748486197279 0.119349697239456 0.940325151380272 42 0.0379651767714782 0.0759303535429564 0.962034823228522 43 0.0419086275906883 0.0838172551813766 0.958091372409312 44 0.0639365680816004 0.127873136163201 0.9360634319184 45 0.0420249522836188 0.0840499045672376 0.957975047716381 46 0.032346635587291 0.064693271174582 0.967653364412709 47 0.0587358336450867 0.117471667290173 0.941264166354913 48 0.198406603688857 0.396813207377714 0.801593396311143 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.142857142857143 NOK 5% type I error level 7 0.25 NOK 10% type I error level 14 0.5 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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