WS 7: Model 1

*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 09:54:33 -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/t1258736168678730wuoyyyijn.htm/, Retrieved Fri, 20 Nov 2009 17:56:20 +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/t1258736168678730wuoyyyijn.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] = + 315.014586098755 + 1.01436997434097X[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) 315.014586098755 14.797195 21.2888 0 0 X 1.01436997434097 0.121032 8.381 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.740084845814715 R-squared 0.547725579004591 Adjusted R-squared 0.539927744159842 F-TEST (value) 70.2407257796007 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 1.41233691408615e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 17.8483841551159 Sum Squared Residuals 18476.7593830185 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 423 430.652763173627 -7.65276317362673 2 427 432.681503122308 -5.68150312230792 3 441 470.213192172924 -29.2131921729241 4 449 479.342521941993 -30.3425219419928 5 452 478.328151967652 -26.3281519676519 6 462 466.15571227556 -4.15571227556016 7 455 456.01201253215 -1.01201253215042 8 461 451.954532634787 9.04546736521348 9 461 446.882682763082 14.1173172369183 10 463 443.839572840059 19.1604271599413 11 462 438.767722968354 23.2322770316461 12 456 433.695873096649 22.304126903351 13 455 428.624023224944 26.3759767750559 14 456 429.638393199285 26.3616068007149 15 472 466.15571227556 5.84428772443984 16 472 474.270672070288 -2.27067207028795 17 471 474.270672070288 -3.27067207028795 18 465 464.126972326878 0.87302767312179 19 459 453.983272583468 5.01672741653153 20 465 448.911422711764 16.0885772882364 21 468 441.810832891377 26.1891671086232 22 467 439.782092942695 27.2179070573052 23 463 433.695873096649 29.304126903351 24 460 430.652763173626 29.3472368263739 25 462 427.609653250603 34.3903467493969 26 461 428.624023224944 32.3759767750559 27 476 461.083862403855 14.9161375961447 28 476 467.170082249901 8.82991775009886 29 471 466.15571227556 4.84428772443984 30 453 450.940162660446 2.05983733955445 31 443 439.782092942695 3.21790705730516 32 442 432.681503122308 9.31849687769198 33 444 433.695873096649 10.304126903351 34 438 427.609653250603 10.3903467493969 35 427 421.523433404557 5.4765665954427 36 424 418.480323481534 5.51967651846562 37 416 411.379733661148 4.62026633885244 38 406 409.350993712466 -3.35099371246562 39 431 440.796462917036 -9.7964629170358 40 434 446.882682763082 -12.8826827630817 41 418 440.796462917036 -22.7964629170358 42 412 431.667133147967 -19.6671331479670 43 404 422.537803378898 -18.5378033788983 44 409 421.523433404557 -12.5234334045573 45 412 421.523433404557 -9.5234334045573 46 406 417.465953507193 -11.4659535071934 47 398 411.379733661148 -13.3797336611476 48 397 409.350993712466 -12.3509937124656 49 385 400.221663943397 -15.2216639433968 50 390 403.26477386642 -13.2647738664198 51 413 432.681503122308 -19.681503122308 52 413 436.738983019672 -23.7389830196719 53 401 433.695873096649 -32.695873096649 54 397 425.580913301921 -28.5809133019212 55 397 421.523433404557 -24.5234334045573 56 409 423.552173353239 -14.5521733532393 57 419 425.580913301921 -6.5809133019212 58 424 425.580913301921 -1.5809133019212 59 428 424.56654332758 3.43345667241977 60 430 423.552173353239 6.44782664676075 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.00802807410300564 0.0160561482060113 0.991971925896994 6 0.119833317014239 0.239666634028478 0.88016668298576 7 0.125488890239168 0.250977780478337 0.874511109760832 8 0.195345815357037 0.390691630714073 0.804654184642963 9 0.240031784361804 0.480063568723609 0.759968215638196 10 0.280432836375595 0.560865672751191 0.719567163624405 11 0.294261364911667 0.588522729823334 0.705738635088333 12 0.252034447360624 0.504068894721248 0.747965552639376 13 0.220615860370385 0.44123172074077 0.779384139629615 14 0.195854059552947 0.391708119105894 0.804145940447053 15 0.203888787599536 0.407777575199072 0.796111212400464 16 0.187735542369193 0.375471084738387 0.812264457630807 17 0.161421872104140 0.322843744208280 0.83857812789586 18 0.121769347790746 0.243538695581492 0.878230652209254 19 0.0839762847840472 0.167952569568094 0.916023715215953 20 0.0681454147720377 0.136290829544075 0.931854585227962 21 0.0771397943335713 0.154279588667143 0.922860205666429 22 0.0884771366814836 0.176954273362967 0.911522863318516 23 0.1059157297998 0.2118314595996 0.8940842702002 24 0.13265778824083 0.26531557648166 0.86734221175917 25 0.235048300406526 0.470096600813051 0.764951699593474 26 0.412792763345829 0.825585526691657 0.587207236654171 27 0.461978761027825 0.92395752205565 0.538021238972175 28 0.468829811495344 0.937659622990687 0.531170188504656 29 0.440008970447382 0.880017940894764 0.559991029552618 30 0.408237649742318 0.816475299484635 0.591762350257682 31 0.428545584977197 0.857091169954393 0.571454415022803 32 0.500250644538914 0.999498710922172 0.499749355461086 33 0.603552523404407 0.792894953191185 0.396447476595593 34 0.733849607506292 0.532300784987415 0.266150392493708 35 0.827712996689425 0.344574006621149 0.172287003310575 36 0.895890004391894 0.208219991216213 0.104109995608106 37 0.940150382058067 0.119699235883865 0.0598496179419327 38 0.958078170847133 0.0838436583057344 0.0419218291528672 39 0.956410089753458 0.0871798204930839 0.0435899102465419 40 0.952896576375722 0.094206847248557 0.0471034236242785 41 0.958421362851975 0.08315727429605 0.041578637148025 42 0.959464678150373 0.0810706436992538 0.0405353218496269 43 0.959537012830988 0.0809259743380248 0.0404629871690124 44 0.947501093910528 0.104997812178945 0.0524989060894723 45 0.929307561732935 0.141384876534130 0.0706924382670648 46 0.903805849781808 0.192388300436383 0.0961941502181916 47 0.871771024545723 0.256457950908555 0.128228975454277 48 0.825253828239941 0.349492343520118 0.174746171760059 49 0.79191219193627 0.41617561612746 0.20808780806373 50 0.807984005710756 0.384031988578487 0.192015994289244 51 0.73318050975012 0.533638980499759 0.266819490249879 52 0.656010065710008 0.687979868579984 0.343989934289992 53 0.663439316594966 0.673121366810068 0.336560683405034 54 0.869769150122112 0.260461699755775 0.130230849877888 55 0.846435187117987 0.307129625764025 0.153564812882013 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 1 0.0196078431372549 OK 10% type I error level 7 0.137254901960784 NOK

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/1043xf1258736069.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/1043xf1258736069.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/10dmu1258736069.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/10dmu1258736069.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/2zdsg1258736069.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/2zdsg1258736069.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/3m4591258736069.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/3m4591258736069.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/4wptq1258736069.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/4wptq1258736069.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/5ue401258736069.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/5ue401258736069.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/6qtno1258736069.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/6qtno1258736069.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/7wzcf1258736069.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/7wzcf1258736069.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/8fccs1258736069.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/8fccs1258736069.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/9fymq1258736069.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736168678730wuoyyyijn/9fymq1258736069.ps (open in new window)

Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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