Seatbelt law & tutorial Q3

*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: Sun, 23 Nov 2008 09:25:09 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Nov/23/t12274575861dx30tw0w0agee0.htm/, Retrieved Sun, 23 Nov 2008 16:26:36 +0000

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/2008/Nov/23/t12274575861dx30tw0w0agee0.htm/}, year = {2008}, } @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 = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

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User-defined keywords:

Dataseries X:

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377,2 0 332,2 0 364,8 0 352,4 0 341,6 0 298,2 0 355,3 0 330,9 0 314,5 0 418,9 0 433,2 0 367 0 422,9 0 352,1 0 419,8 0 432,7 0 414,2 0 387,7 0 297,2 0 357,4 0 384,2 0 425,2 0 385,3 0 355,4 0 409,8 1 421,2 1 421,8 1 464,2 1 494 1 404,2 1 411,4 1 403,4 1 403,3 1 520,9 1 439,8 1 434,8 1 476,5 1 454,3 1 522 1 498,4 1 439,9 1 450,7 1 447,1 1 451,3 1 466,8 1 498 1 533,6 1 451,9 1 477,1 1 410,4 1 469,5 1 485,4 1 406,7 1 439,7 1 412,2 1 440,2 1 411,1 1 477,7 1 463,2 1 320,5 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation invoer[t] = + 332.101666666667 + 63.2222222222222`1euro>125yen`[t] + 51.6337499999998M1[t] + 12.5325000000001M2[t] + 57.63125M3[t] + 64.2299999999999M4[t] + 36.4487499999999M5[t] + 12.8275M6[t] + 0.926249999999977M7[t] + 12.485M8[t] + 11.3837500000000M9[t] + 83.1025M10[t] + 65.54125M11[t] + 0.44125t + 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) 332.101666666667 19.110014 17.3784 0 0 `1euro>125yen` 63.2222222222222 18.388619 3.4381 0.001254 0.000627 M1 51.6337499999998 22.82585 2.2621 0.028462 0.014231 M2 12.5325000000001 22.695857 0.5522 0.583489 0.291745 M3 57.63125 22.5776 2.5526 0.014077 0.007038 M4 64.2299999999999 22.471264 2.8583 0.006378 0.003189 M5 36.4487499999999 22.377018 1.6288 0.110176 0.055088 M6 12.8275 22.295016 0.5754 0.567857 0.283928 M7 0.926249999999977 22.225393 0.0417 0.966938 0.483469 M8 12.485 22.168267 0.5632 0.576039 0.288019 M9 11.3837500000000 22.123733 0.5145 0.609332 0.304666 M10 83.1025 22.091868 3.7617 0.000476 0.000238 M11 65.54125 22.072727 2.9693 0.004728 0.002364 t 0.44125 0.530834 0.8312 0.410132 0.205066 Multiple Linear Regression - Regression Statistics Multiple R 0.837209151077354 R-squared 0.700919162647664 Adjusted R-squared 0.61639631730896 F-TEST (value) 8.29265933770815 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 3.018004712807e-08 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 34.8899518015571 Sum Squared Residuals 55996.2018888889 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 377.2 384.176666666667 -6.97666666666701 2 332.2 345.516666666666 -13.3166666666664 3 364.8 391.056666666667 -26.2566666666665 4 352.4 398.096666666667 -45.6966666666668 5 341.6 370.756666666667 -29.1566666666667 6 298.2 347.576666666667 -49.3766666666666 7 355.3 336.116666666667 19.1833333333334 8 330.9 348.116666666667 -17.2166666666667 9 314.5 347.456666666667 -32.9566666666667 10 418.9 419.616666666667 -0.716666666666741 11 433.2 402.496666666667 30.7033333333333 12 367 337.396666666667 29.6033333333334 13 422.9 389.471666666667 33.4283333333334 14 352.1 350.811666666667 1.28833333333335 15 419.8 396.351666666667 23.4483333333333 16 432.7 403.391666666667 29.3083333333334 17 414.2 376.051666666667 38.1483333333334 18 387.7 352.871666666667 34.8283333333333 19 297.2 341.411666666667 -44.2116666666667 20 357.4 353.411666666667 3.98833333333334 21 384.2 352.751666666667 31.4483333333333 22 425.2 424.911666666667 0.288333333333357 23 385.3 407.791666666667 -22.4916666666666 24 355.4 342.691666666667 12.7083333333333 25 409.8 457.988888888889 -48.1888888888888 26 421.2 419.328888888889 1.87111111111107 27 421.8 464.868888888889 -43.0688888888889 28 464.2 471.908888888889 -7.70888888888884 29 494 444.568888888889 49.4311111111111 30 404.2 421.388888888889 -17.1888888888889 31 411.4 409.928888888889 1.47111111111108 32 403.4 421.928888888889 -18.5288888888889 33 403.3 421.268888888889 -17.9688888888889 34 520.9 493.428888888889 27.4711111111111 35 439.8 476.308888888889 -36.5088888888889 36 434.8 411.208888888889 23.5911111111111 37 476.5 463.283888888889 13.2161111111112 38 454.3 424.623888888889 29.6761111111111 39 522 470.163888888889 51.8361111111111 40 498.4 477.203888888889 21.1961111111111 41 439.9 449.863888888889 -9.9638888888889 42 450.7 426.683888888889 24.0161111111111 43 447.1 415.223888888889 31.8761111111111 44 451.3 427.223888888889 24.0761111111111 45 466.8 426.563888888889 40.2361111111111 46 498 498.723888888889 -0.72388888888886 47 533.6 481.603888888889 51.9961111111111 48 451.9 416.503888888889 35.396111111111 49 477.1 468.578888888889 8.52111111111119 50 410.4 429.918888888889 -19.5188888888889 51 469.5 475.458888888889 -5.95888888888893 52 485.4 482.498888888889 2.90111111111114 53 406.7 455.158888888889 -48.4588888888889 54 439.7 431.978888888889 7.72111111111109 55 412.2 420.518888888889 -8.3188888888889 56 440.2 432.518888888889 7.68111111111113 57 411.1 431.858888888889 -20.7588888888889 58 477.7 504.018888888889 -26.3188888888889 59 463.2 486.898888888889 -23.6988888888889 60 320.5 421.798888888889 -101.298888888889 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.159446743002159 0.318893486004318 0.840553256997841 18 0.122160167729612 0.244320335459223 0.877839832270388 19 0.634385002131098 0.731229995737803 0.365614997868902 20 0.508673025640443 0.982653948719113 0.491326974359557 21 0.412727447823504 0.825454895647009 0.587272552176496 22 0.340285075274316 0.680570150548633 0.659714924725684 23 0.459829330173488 0.919658660346975 0.540170669826512 24 0.396166346325722 0.792332692651444 0.603833653674278 25 0.357679666793394 0.715359333586787 0.642320333206607 26 0.339021956862654 0.678043913725308 0.660978043137346 27 0.356407507415939 0.712815014831877 0.643592492584061 28 0.316139943825119 0.632279887650239 0.683860056174881 29 0.39124701447653 0.78249402895306 0.60875298552347 30 0.359705199624421 0.719410399248843 0.640294800375579 31 0.309469920741597 0.618939841483194 0.690530079258403 32 0.326208559253357 0.652417118506714 0.673791440746643 33 0.372046221448044 0.744092442896088 0.627953778551956 34 0.30645264855814 0.61290529711628 0.69354735144186 35 0.73748482095513 0.525030358089741 0.262515179044870 36 0.64708159650183 0.705836806996339 0.352918403498169 37 0.62596827680369 0.748063446392621 0.374031723196310 38 0.513828303700258 0.972343392599484 0.486171696299742 39 0.441143809046744 0.882287618093487 0.558856190953256 40 0.367052072694038 0.734104145388075 0.632947927305962 41 0.304082173570664 0.608164347141329 0.695917826429336 42 0.256427283020717 0.512854566041434 0.743572716979283 43 0.161547045905818 0.323094091811637 0.838452954094182 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):

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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