workshop 7-model 4/part 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: Sun, 22 Nov 2009 12:52:12 -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/22/t12589195954nfe2o0epg2pyu6.htm/, Retrieved Sun, 22 Nov 2009 20:53:27 +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/22/t12589195954nfe2o0epg2pyu6.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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267366 0 267413 262813 258113 267037 269645 264777 0 267366 267413 262813 258113 267037 258863 0 264777 267366 267413 262813 258113 254844 0 258863 264777 267366 267413 262813 254868 0 254844 258863 264777 267366 267413 277267 0 254868 254844 258863 264777 267366 285351 0 277267 254868 254844 258863 264777 286602 0 285351 277267 254868 254844 258863 283042 0 286602 285351 277267 254868 254844 276687 0 283042 286602 285351 277267 254868 277915 0 276687 283042 286602 285351 277267 277128 0 277915 276687 283042 286602 285351 277103 0 277128 277915 276687 283042 286602 275037 0 277103 277128 277915 276687 283042 270150 0 275037 277103 277128 277915 276687 267140 0 270150 275037 277103 277128 277915 264993 0 267140 270150 275037 277103 277128 287259 0 264993 267140 270150 275037 277103 291186 0 287259 264993 267140 270150 275037 292300 0 291186 287259 264993 267140 270150 288186 0 292300 291186 287259 264993 267140 281477 0 288186 292300 291186 287259 264993 282656 0 281477 288186 292300 2911 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] = + 21386.0345240392 + 4570.5834593585x[t] + 0.937414189322683y1[t] + 0.134008529853577y2[t] + 0.126857011464927y3[t] -0.259803480419487y4[t] + 0.00246372856114373y5[t] -3682.13734938526M1[t] -8617.31854672714M2[t] -7066.4047264724M3[t] -9287.79908724293M4[t] -5147.02671990541M5[t] + 17983.3445346842M6[t] + 782.48771753359M7[t] -6645.70370737072M8[t] -15236.7700945788M9[t] -10308.5541016025M10[t] -1107.26843820283M11[t] -68.7912791567189t + 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) 21386.0345240392 10753.816064 1.9887 0.052452 0.026226 x 4570.5834593585 2098.741718 2.1778 0.034369 0.017184 y1 0.937414189322683 0.142928 6.5586 0 0 y2 0.134008529853577 0.194882 0.6876 0.494989 0.247494 y3 0.126857011464927 0.198556 0.6389 0.525927 0.262963 y4 -0.259803480419487 0.198715 -1.3074 0.197298 0.098649 y5 0.00246372856114373 0.149128 0.0165 0.986887 0.493444 M1 -3682.13734938526 2465.403429 -1.4935 0.141845 0.070923 M2 -8617.31854672714 2763.500072 -3.1183 0.003072 0.001536 M3 -7066.4047264724 2705.339604 -2.612 0.011979 0.00599 M4 -9287.79908724293 2398.918476 -3.8717 0.000326 0.000163 M5 -5147.02671990541 2443.087351 -2.1068 0.040389 0.020195 M6 17983.3445346842 2229.790985 8.065 0 0 M7 782.48771753359 3408.61334 0.2296 0.819408 0.409704 M8 -6645.70370737072 4500.77405 -1.4766 0.146322 0.073161 M9 -15236.7700945788 4515.96289 -3.374 0.001474 0.000737 M10 -10308.5541016025 4463.807728 -2.3094 0.025271 0.012636 M11 -1107.26843820283 2654.333191 -0.4172 0.678425 0.339212 t -68.7912791567189 36.037059 -1.9089 0.062263 0.031132 Multiple Linear Regression - Regression Statistics Multiple R 0.986093368796538 R-squared 0.972380131984505 Adjusted R-squared 0.962022681478694 F-TEST (value) 93.8821895831396 F-TEST (DF numerator) 18 F-TEST (DF denominator) 48 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3381.62366621292 Sum Squared Residuals 548898173.754784 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 267366 267561.66414859 -195.664148589884 2 264777 266038.361251580 -1261.36125158039 3 258863 264427.699636706 -5564.69963670626 4 254844 255057.219632103 -213.219632102641 5 254868 254264.31776012 603.682239880169 6 277267 276732.098444375 534.901555624953 7 285351 281483.067840301 3867.93215969854 8 286602 285598.422768286 1003.57723171426 9 283042 282019.928399285 1022.07160071451 10 276687 278916.036322215 -2229.03632221504 11 277915 279727.825008706 -1812.82500870615 12 277128 280309.114251889 -3181.11425188940 13 277103 276106.809337874 996.19066212559 14 275037 272771.997448115 2265.00255188492 15 270150 271879.539923842 -1729.53992384200 16 267140 264935.668890494 2204.33110950609 17 264993 265273.603130365 -280.603130365346 18 287259 285835.931348767 1423.06865123332 19 291186 290033.761219416 1152.23878058413 20 292300 289699.444193514 2600.55580648610 21 288186 285985.097897559 2200.90210244081 22 281477 281845.379702544 -368.37970254389 23 282656 283313.379022110 -657.379022110357 24 280190 283756.368523286 -3566.36852328586 25 280408 278072.264982716 2335.73501728379 26 276836 274824.633952222 2011.36604777802 27 275216 272351.860019964 2864.13998003634 28 274352 268735.619871884 5616.38012811592 29 271311 271264.735323811 46.2646761886604 30 289802 292082.902146032 -2280.90214603200 31 290726 292041.936627159 -1315.93662715915 32 292300 287723.783154503 4576.21684549669 33 278506 283796.886025194 -5290.88602519388 34 269826 271242.246361088 -1416.24636108839 35 265861 270394.643246172 -4533.64324617248 36 269034 264396.559296288 4637.44070371187 37 264176 265575.190327744 -1399.19032774368 38 255198 258160.590272551 -2962.59027255119 39 253353 251986.847717212 1366.15228278776 40 246057 245313.607828147 743.392171853363 41 235372 242429.989723120 -7057.98972311957 42 258556 256584.068519531 1971.93148046879 43 260993 259147.219157699 1845.78084230060 44 254663 257577.082035586 -2914.0820355856 45 250643 249009.049116564 1633.95088343566 46 243422 243511.336502552 -89.3365025524604 47 247105 243957.021854291 3147.9781457088 48 248541 248620.91483001 -79.9148300099232 49 245039 246822.446502492 -1783.44650249211 50 237080 241061.436900374 -3981.43690037385 51 237085 233820.901903244 3264.09809675594 52 225554 229660.572305417 -4106.57230541694 53 226839 222827.615167561 4011.38483243935 54 247934 247608.302226826 325.69777317352 55 248333 253372.494842279 -5039.49484227919 56 246969 252235.267848111 -5266.26784811145 57 245098 244664.038561397 433.961438602899 58 246263 242160.0011116 4102.99888839978 59 255765 251909.13086872 3855.86913128019 60 264319 262129.043098527 2189.95690147333 61 268347 268300.624700584 46.3752994162896 62 273046 269116.980175157 3929.01982484249 63 273963 274163.150799032 -200.150799031771 64 267430 271674.311471956 -4244.31147195579 65 271993 269315.738895023 2677.26110497673 66 292710 294684.697314469 -1974.69731446858 67 295881 296391.520313145 -510.520313144927 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 22 0.0355083895027452 0.0710167790054903 0.964491610497255 23 0.00871887752105566 0.0174377550421113 0.991281122478944 24 0.00239835073951609 0.00479670147903218 0.997601649260484 25 0.000774647317205566 0.00154929463441113 0.999225352682794 26 0.000195390925715357 0.000390781851430714 0.999804609074285 27 0.00042512727670525 0.0008502545534105 0.999574872723295 28 0.00162283619066268 0.00324567238132536 0.998377163809337 29 0.000566561545307381 0.00113312309061476 0.999433438454693 30 0.000281215838677075 0.00056243167735415 0.999718784161323 31 0.000101699601654305 0.000203399203308610 0.999898300398346 32 0.000128346068149485 0.00025669213629897 0.99987165393185 33 0.0541125428396283 0.108225085679257 0.945887457160372 34 0.0349889020746427 0.0699778041492854 0.965011097925357 35 0.0518788321108202 0.103757664221640 0.94812116788918 36 0.0758682111566794 0.151736422313359 0.92413178884332 37 0.0466920185122089 0.0933840370244178 0.953307981487791 38 0.0414070404918526 0.0828140809837051 0.958592959508147 39 0.0722440452516329 0.144488090503266 0.927755954748367 40 0.121866685817795 0.243733371635591 0.878133314182205 41 0.277673246387021 0.555346492774041 0.72232675361298 42 0.299025501317978 0.598051002635956 0.700974498682022 43 0.519371598375563 0.961256803248873 0.480628401624437 44 0.461766495812319 0.923532991624639 0.538233504187681 45 0.70105208786846 0.59789582426308 0.29894791213154 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 9 0.375 NOK 5% type I error level 10 0.416666666666667 NOK 10% type I error level 14 0.583333333333333 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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