Model 4 - WZM & WZM<25j

*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: Thu, 19 Nov 2009 14:57:32 -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/Nov/19/t12586679315bezu9vmz4kccag.htm/, Retrieved Thu, 19 Nov 2009 22:59:03 +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/19/t12586679315bezu9vmz4kccag.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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6.5 15.8 6.8 7.5 8 8.2 6.6 15.8 6.5 6.8 7.5 8 7.6 23.2 6.6 6.5 6.8 7.5 8 23.2 7.6 6.6 6.5 6.8 8.1 23.2 8 7.6 6.6 6.5 7.7 20.9 8.1 8 7.6 6.6 7.5 20.9 7.7 8.1 8 7.6 7.6 20.9 7.5 7.7 8.1 8 7.8 19.8 7.6 7.5 7.7 8.1 7.8 19.8 7.8 7.6 7.5 7.7 7.8 19.8 7.8 7.8 7.6 7.5 7.5 20.6 7.8 7.8 7.8 7.6 7.5 20.6 7.5 7.8 7.8 7.8 7.1 20.6 7.5 7.5 7.8 7.8 7.5 21.1 7.1 7.5 7.5 7.8 7.5 21.1 7.5 7.1 7.5 7.5 7.6 21.1 7.5 7.5 7.1 7.5 7.7 22.4 7.6 7.5 7.5 7.1 7.7 22.4 7.7 7.6 7.5 7.5 7.9 22.4 7.7 7.7 7.6 7.5 8.1 20.5 7.9 7.7 7.7 7.6 8.2 20.5 8.1 7.9 7.7 7.7 8.2 20.5 8.2 8.1 7.9 7.7 8.2 18.4 8.2 8.2 8.1 7.9 7.9 18.4 8.2 8.2 8.2 8.1 7.3 18.4 7.9 8.2 8.2 8.2 6.9 17.6 7.3 7.9 8.2 8.2 6.6 17.6 6.9 7.3 7.9 8.2 6.7 17.6 6.6 6.9 7.3 7.9 6.9 18.5 6.7 6.6 6.9 7.3 7 18.5 6.9 6.7 6.6 6.9 7.1 18.5 7 6.9 6.7 6.6 7.2 17.3 7.1 7 6.9 6.7 7.1 17.3 7.2 7.1 7 6.9 6.9 17.3 7.1 7.2 7.1 7 7 16.2 6.9 7.1 7.2 7.1 6.8 16.2 7 6.9 7.1 7.2 6.4 16.2 6.8 7 6.9 7.1 6.7 18.5 6.4 6.8 7 6.9 6.6 18.5 6.7 6.4 6.8 7 6.4 18.5 6.6 6.7 6 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] = + 0.0438440963714834 + 0.091191104880777X[t] + 1.10242811026445`Y(t-1)`[t] -0.486419144668766`Y(t-2)`[t] -0.188441799569958`Y(t-3)`[t] + 0.342364461886313`Y(t-4)`[t] -0.189339107617867M1[t] -0.248582058103744M2[t] + 0.00334296687708956M3[t] -0.53509934530751M4[t] -0.353680066285393M5[t] -0.188467403632904M6[t] -0.259620844371123M7[t] -0.0467034678873066M8[t] -0.0132662230148949M9[t] -0.113338664654363M10[t] -0.0864416950016453M11[t] + 0.00188780839695149t + 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) 0.0438440963714834 0.401943 0.1091 0.913603 0.456802 X 0.091191104880777 0.0215 4.2415 0.000103 5.2e-05 `Y(t-1)` 1.10242811026445 0.156443 7.0468 0 0 `Y(t-2)` -0.486419144668766 0.229403 -2.1204 0.039281 0.019641 `Y(t-3)` -0.188441799569958 0.221128 -0.8522 0.398434 0.199217 `Y(t-4)` 0.342364461886313 0.118904 2.8793 0.005982 0.002991 M1 -0.189339107617867 0.106143 -1.7838 0.080911 0.040456 M2 -0.248582058103744 0.11166 -2.2262 0.030829 0.015415 M3 0.00334296687708956 0.14087 0.0237 0.981168 0.490584 M4 -0.53509934530751 0.126357 -4.2348 0.000106 5.3e-05 M5 -0.353680066285393 0.143484 -2.4649 0.017413 0.008707 M6 -0.188467403632904 0.119133 -1.582 0.120358 0.060179 M7 -0.259620844371123 0.121117 -2.1436 0.037274 0.018637 M8 -0.0467034678873066 0.121031 -0.3859 0.701326 0.350663 M9 -0.0132662230148949 0.117847 -0.1126 0.91085 0.455425 M10 -0.113338664654363 0.117581 -0.9639 0.340021 0.170011 M11 -0.0864416950016453 0.111035 -0.7785 0.440173 0.220086 t 0.00188780839695149 0.001441 1.3099 0.19659 0.098295 Multiple Linear Regression - Regression Statistics Multiple R 0.97556268486673 R-squared 0.951722552104384 Adjusted R-squared 0.934260496482565 F-TEST (value) 54.5023205008703 F-TEST (DF numerator) 17 F-TEST (DF denominator) 47 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.169643885658411 Sum Squared Residuals 1.35261525324035 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 6.5 6.44543400995743 0.054565990042572 2 6.6 6.42359184346504 0.176408156534963 3 7.6 7.56911443614346 0.03088556385654 4 8 7.90322354470395 0.0967764552960502 5 8.1 7.91952921303714 0.180470786962857 6 7.7 7.63835994263841 0.061640057361593 7 7.5 7.34646889378281 0.153531106217185 8 7.6 7.65345771927573 -0.0534577192757292 9 7.8 7.90561236315305 -0.105612363153049 10 7.8 7.88001401265601 -0.0800140126560126 11 7.8 7.72419788943767 0.0758021105623299 12 7.5 7.88202836301553 -0.382028363015528 13 7.5 7.43232152309254 0.0676784769074585 14 7.1 7.52089212440425 -0.420892124404246 15 7.5 7.43586180598763 0.0641381940123728 16 7.5 7.43213686560737 0.0678631343926295 17 7.6 7.49625301498692 0.103746985013084 18 7.7 7.6798222288253 0.0201777711746977 19 7.7 7.80910327779813 -0.109103277798129 20 7.9 7.95642236825502 -0.0564223682550237 21 8.1 8.05436221053544 0.0456377894645638 22 8.2 8.11361581660069 0.0863841833993132 23 8.2 8.11767121682906 0.0823287831709441 24 8.2 7.99664201797441 0.203357982025585 25 7.9 7.85881943117377 0.0411805688262343 26 7.3 7.50497230219414 -0.204972302194139 27 6.9 7.17030112890926 -0.270301128909262 28 6.6 6.54115940768808 0.0588405923119164 29 6.7 6.5986614610714 0.101338538928596 30 6.9 6.97396052363681 -0.0739605236368146 31 7 6.99612535399802 0.00387464600197746 32 7.1 7.10233600244859 -0.00233600244859139 33 7.2 7.08638071269523 0.113619287304771 34 7.1 7.09942568843255 0.000574311567450727 35 6.9 6.98471800722053 -0.0847180072205312 36 7 6.8162858538959 0.183714146104103 37 6.8 6.8894418207808 -0.089441820780806 38 6.4 6.56641105589747 -0.166411055897474 39 6.7 6.59895894299476 0.101041057005237 40 6.6 6.65962533625658 -0.0596253362565784 41 6.4 6.59366769669929 -0.193667696699291 42 6.3 6.1948257047995 0.105174295200500 43 6.2 6.23415460888843 -0.0341546088884289 44 6.5 6.39081081093499 0.109189189065011 45 6.8 6.80147305177069 -0.00147305177068493 46 6.8 6.87269884197524 -0.0726988419752369 47 6.4 6.66478889056466 -0.264788890564657 48 6.1 6.17594173879083 -0.075941738790829 49 5.8 5.95503900292398 -0.155039002923979 50 6.1 5.78825789098433 0.311742109015668 51 7.2 7.04017294817167 0.159827051828328 52 7.3 7.52418682357938 -0.224186823579379 53 6.9 7.12343378445237 -0.223433784452367 54 6.1 6.21303160009998 -0.113031600099976 55 5.8 5.81414786553260 -0.0141478655326054 56 6.2 6.19697309908567 0.00302690091433393 57 7.1 7.1521716618456 -0.0521716618456001 58 7.7 7.63424564033551 0.0657543596644856 59 7.9 7.70862399594809 0.191376004051914 60 7.7 7.62910202632333 0.0708979736766684 61 7.4 7.31894421207148 0.0810557879285197 62 7.5 7.19587478305477 0.304125216945229 63 8 8.08559073779321 -0.085590737793215 64 8.1 8.03966802216464 0.0603319778353614 65 8 7.96845482975288 0.0315451702471214 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.490944935436138 0.981889870872276 0.509055064563862 22 0.509505400477714 0.980989199044572 0.490494599522286 23 0.493335967556963 0.986671935113927 0.506664032443037 24 0.692589578058237 0.614820843883527 0.307410421941763 25 0.625794627741329 0.748410744517342 0.374205372258671 26 0.624404583975842 0.751190832048316 0.375595416024158 27 0.847031689645292 0.305936620709416 0.152968310354708 28 0.78859442162503 0.422811156749941 0.211405578374970 29 0.747594051165028 0.504811897669944 0.252405948834972 30 0.733265991823236 0.533468016353527 0.266734008176764 31 0.650679780070438 0.698640439859124 0.349320219929562 32 0.558044015797843 0.883911968404315 0.441955984202157 33 0.499548354871492 0.999096709742983 0.500451645128508 34 0.3936543173521 0.7873086347042 0.6063456826479 35 0.324127149796449 0.648254299592897 0.675872850203551 36 0.342152412759102 0.684304825518204 0.657847587240898 37 0.258559134787522 0.517118269575043 0.741440865212478 38 0.456274835979265 0.91254967195853 0.543725164020735 39 0.410085051087323 0.820170102174646 0.589914948912677 40 0.397733682675406 0.795467365350813 0.602266317324593 41 0.340030359454569 0.680060718909138 0.659969640545431 42 0.301927583419318 0.603855166838636 0.698072416580682 43 0.188143087253542 0.376286174507084 0.811856912746458 44 0.268237520017897 0.536475040035794 0.731762479982103 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 0 0 OK 10% type I error level 0 0 OK

Charts produced by software:

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Parameters (Session):

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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