Model 2

*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: Wed, 18 Nov 2009 11:51:44 -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/18/t12585703828dzdu3wum05ws0h.htm/, Retrieved Wed, 18 Nov 2009 19:53:14 +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/18/t12585703828dzdu3wum05ws0h.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:

» Textbox « » Textfile « » CSV «

562 0 561 0 555 0 544 0 537 0 543 0 594 0 611 0 613 0 611 0 594 0 595 0 591 0 589 0 584 0 573 0 567 0 569 0 621 0 629 0 628 0 612 0 595 0 597 0 593 0 590 0 580 0 574 0 573 0 573 0 620 0 626 0 620 0 588 0 566 0 557 0 561 0 549 0 532 0 526 0 511 0 499 0 555 0 565 0 542 0 527 1 510 1 514 1 517 1 508 1 493 1 490 1 469 1 478 1 528 1 534 1 518 1 506 1 502 1 516 1 528 1

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] = + 584.371764705883 -71.4294117647059X[t] -1.89529411764702M1[t] -10.6858823529412M2[t] -21.2858823529412M3[t] -28.6858823529412M4[t] -38.6858823529411M5[t] -37.6858823529412M6[t] + 13.5141176470588M7[t] + 22.9141176470588M8[t] + 14.1141176470588M9[t] + 13M10[t] -2.40000000000001M11[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) 584.371764705883 10.562659 55.3243 0 0 X -71.4294117647059 6.780601 -10.5344 0 0 M1 -1.89529411764702 13.829767 -0.137 0.891569 0.445784 M2 -10.6858823529412 14.500555 -0.7369 0.464752 0.232376 M3 -21.2858823529412 14.500555 -1.4679 0.148644 0.074322 M4 -28.6858823529412 14.500555 -1.9783 0.053654 0.026827 M5 -38.6858823529411 14.500555 -2.6679 0.010377 0.005189 M6 -37.6858823529412 14.500555 -2.5989 0.012386 0.006193 M7 13.5141176470588 14.500555 0.932 0.356015 0.178007 M8 22.9141176470588 14.500555 1.5802 0.120624 0.060312 M9 14.1141176470588 14.500555 0.9734 0.335259 0.16763 M10 13 14.437002 0.9005 0.372369 0.186185 M11 -2.40000000000001 14.437002 -0.1662 0.868667 0.434333 Multiple Linear Regression - Regression Statistics Multiple R 0.872704588342877 R-squared 0.761613298514711 Adjusted R-squared 0.702016623143389 F-TEST (value) 12.7794594877888 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 3.67905705900284e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 22.8269045980705 Sum Squared Residuals 25011.2435294118 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 562 582.476470588235 -20.4764705882351 2 561 573.685882352941 -12.6858823529411 3 555 563.085882352941 -8.08588235294119 4 544 555.685882352941 -11.6858823529412 5 537 545.685882352941 -8.6858823529411 6 543 546.685882352941 -3.68588235294122 7 594 597.885882352941 -3.88588235294121 8 611 607.285882352941 3.71411764705878 9 613 598.485882352941 14.5141176470588 10 611 597.371764705882 13.6282352941177 11 594 581.971764705882 12.0282352941176 12 595 584.371764705882 10.6282352941176 13 591 582.476470588235 8.52352941176466 14 589 573.685882352941 15.3141176470588 15 584 563.085882352941 20.9141176470588 16 573 555.685882352941 17.3141176470588 17 567 545.685882352941 21.3141176470588 18 569 546.685882352941 22.3141176470588 19 621 597.885882352941 23.1141176470588 20 629 607.285882352941 21.7141176470588 21 628 598.485882352941 29.5141176470588 22 612 597.371764705882 14.6282352941177 23 595 581.971764705882 13.0282352941176 24 597 584.371764705882 12.6282352941176 25 593 582.476470588235 10.5235294117647 26 590 573.685882352941 16.3141176470588 27 580 563.085882352941 16.9141176470588 28 574 555.685882352941 18.3141176470588 29 573 545.685882352941 27.3141176470588 30 573 546.685882352941 26.3141176470588 31 620 597.885882352941 22.1141176470588 32 626 607.285882352941 18.7141176470588 33 620 598.485882352941 21.5141176470588 34 588 597.371764705882 -9.37176470588236 35 566 581.971764705882 -15.9717647058824 36 557 584.371764705882 -27.3717647058824 37 561 582.476470588235 -21.4764705882353 38 549 573.685882352941 -24.6858823529412 39 532 563.085882352941 -31.0858823529412 40 526 555.685882352941 -29.6858823529412 41 511 545.685882352941 -34.6858823529412 42 499 546.685882352941 -47.6858823529412 43 555 597.885882352941 -42.8858823529412 44 565 607.285882352941 -42.2858823529412 45 542 598.485882352941 -56.4858823529412 46 527 525.942352941177 1.05764705882353 47 510 510.542352941176 -0.54235294117646 48 514 512.942352941177 1.05764705882353 49 517 511.047058823529 5.95294117647056 50 508 502.256470588235 5.74352941176472 51 493 491.656470588235 1.34352941176471 52 490 484.256470588235 5.74352941176471 53 469 474.256470588235 -5.2564705882353 54 478 475.256470588235 2.74352941176472 55 528 526.456470588235 1.54352941176472 56 534 535.856470588235 -1.85647058823527 57 518 527.056470588235 -9.05647058823529 58 506 525.942352941177 -19.9423529411765 59 502 510.542352941176 -8.54235294117646 60 516 512.942352941177 3.05764705882353 61 528 511.047058823529 16.9529411764705 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.496360041911489 0.992720083822977 0.503639958088511 17 0.438345957071831 0.876691914143662 0.561654042928169 18 0.372862096241454 0.745724192482907 0.627137903758546 19 0.326787701931623 0.653575403863245 0.673212298068377 20 0.257947274556336 0.515894549112673 0.742052725443664 21 0.216983732445957 0.433967464891914 0.783016267554043 22 0.151140184286404 0.302280368572808 0.848859815713596 23 0.101406234794569 0.202812469589138 0.898593765205431 24 0.0664126530748274 0.132825306149655 0.933587346925173 25 0.0484350605965526 0.0968701211931052 0.951564939403447 26 0.0377880887476351 0.0755761774952701 0.962211911252365 27 0.0299065820858898 0.0598131641717796 0.97009341791411 28 0.0267808897780102 0.0535617795560205 0.97321911022199 29 0.0436792661944382 0.0873585323888764 0.956320733805562 30 0.080296168937857 0.160592337875714 0.919703831062143 31 0.139466396831954 0.278932793663908 0.860533603168046 32 0.264766013255535 0.52953202651107 0.735233986744465 33 0.894615445856426 0.210769108287147 0.105384554143574 34 0.98367190694881 0.0326561861023782 0.0163280930511891 35 0.997920139129348 0.00415972174130465 0.00207986087065232 36 0.998463600857489 0.00307279828502234 0.00153639914251117 37 0.99735368128066 0.00529263743868113 0.00264631871934056 38 0.99682766003038 0.0063446799392398 0.0031723399696199 39 0.996544028577424 0.00691194284515112 0.00345597142257556 40 0.99467482570977 0.0106503485804616 0.0053251742902308 41 0.996165745558624 0.00766850888275199 0.00383425444137599 42 0.994412860357124 0.0111742792857522 0.0055871396428761 43 0.987414502452798 0.0251709950944048 0.0125854975472024 44 0.971157847490238 0.057684305019524 0.028842152509762 45 0.934123433561852 0.131753132876296 0.0658765664381478 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 6 0.2 NOK 5% type I error level 10 0.333333333333333 NOK 10% type I error level 16 0.533333333333333 NOK

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/10khzt1258570299.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/10khzt1258570299.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/1uj071258570299.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/1uj071258570299.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/2g9h51258570299.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/2g9h51258570299.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/3ot8i1258570299.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/3ot8i1258570299.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/4ot6g1258570299.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/4ot6g1258570299.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/5w3dt1258570299.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/5w3dt1258570299.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/6t6qo1258570299.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/6t6qo1258570299.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/7msq01258570299.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/7msq01258570299.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/8frqj1258570299.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/8frqj1258570299.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/9anhb1258570299.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t12585703828dzdu3wum05ws0h/9anhb1258570299.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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