*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 09:21:47 -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/t1258561612z55p2yj3p5mudoq.htm/, Retrieved Wed, 18 Nov 2009 17:27:05 +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/t1258561612z55p2yj3p5mudoq.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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318672 441977 326225 327532 338653 344744 317756 439148 318672 326225 327532 338653 337302 488180 317756 318672 326225 327532 349420 520564 337302 317756 318672 326225 336923 501492 349420 337302 317756 318672 330758 485025 336923 349420 337302 317756 321002 464196 330758 336923 349420 337302 320820 460170 321002 330758 336923 349420 327032 467037 320820 321002 330758 336923 324047 460070 327032 320820 321002 330758 316735 447988 324047 327032 320820 321002 315710 442867 316735 324047 327032 320820 313427 436087 315710 316735 324047 327032 310527 431328 313427 315710 316735 324047 330962 484015 310527 313427 315710 316735 339015 509673 330962 310527 313427 315710 341332 512927 339015 330962 310527 313427 339092 502831 341332 339015 330962 310527 323308 470984 339092 341332 339015 330962 325849 471067 323308 339092 341332 339015 330675 476049 325849 323308 339092 341332 332225 474605 330675 325849 323308 339092 331735 470439 332225 330675 325849 323308 328047 461251 331735 332225 330675 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 8 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 45157.3216699624 + 0.492045256054949X[t] + 0.469069737125908`yt-1`[t] -0.0789174470844457`yt-2`[t] + 0.0872661433584767`yt-3`[t] -0.289706583131103`yt-4`[t] + 3347.42238425707M1[t] + 4680.4007143094M2[t] -5912.44143343331M3[t] -20106.8179521148M4[t] -20373.1414736510M5[t] -17006.4450401302M6[t] -10613.8820516124M7[t] -2691.84006111352M8[t] -2069.86719893052M9[t] -1154.20523955606M10[t] -2543.26157898661M11[t] -311.187481241182t + 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) 45157.3216699624 14651.865569 3.082 0.003341 0.00167 X 0.492045256054949 0.069907 7.0386 0 0 `yt-1` 0.469069737125908 0.123888 3.7862 0.000411 0.000205 `yt-2` -0.0789174470844457 0.137762 -0.5729 0.569312 0.284656 `yt-3` 0.0872661433584767 0.138214 0.6314 0.530666 0.265333 `yt-4` -0.289706583131103 0.10223 -2.8339 0.006616 0.003308 M1 3347.42238425707 1934.151162 1.7307 0.089674 0.044837 M2 4680.4007143094 1973.884906 2.3712 0.021626 0.010813 M3 -5912.44143343331 4091.049188 -1.4452 0.154636 0.077318 M4 -20106.8179521148 4438.569686 -4.53 3.7e-05 1.8e-05 M5 -20373.1414736510 4236.880926 -4.8085 1.4e-05 7e-06 M6 -17006.4450401302 3546.159356 -4.7957 1.5e-05 7e-06 M7 -10613.8820516124 1997.729216 -5.313 3e-06 1e-06 M8 -2691.84006111352 2418.919532 -1.1128 0.271104 0.135552 M9 -2069.86719893052 2472.791929 -0.8371 0.406544 0.203272 M10 -1154.20523955606 2413.719748 -0.4782 0.634603 0.317301 M11 -2543.26157898661 1988.365583 -1.2791 0.206776 0.103388 t -311.187481241182 84.895304 -3.6655 0.000597 0.000299 Multiple Linear Regression - Regression Statistics Multiple R 0.99481067895626 R-squared 0.989648286965417 Adjusted R-squared 0.98612870453366 F-TEST (value) 281.183437567921 F-TEST (DF numerator) 17 F-TEST (DF denominator) 50 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 2933.95709978877 Sum Squared Residuals 430405213.170048 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 318672 322518.863375641 -3846.86337564090 2 317756 319503.035591463 -1747.03559146340 3 337302 335999.134617619 1302.86538238138 4 349420 346388.214976538 3031.78502346251 5 336923 342676.301539132 -5753.30153913222 6 330758 332782.090399556 -2024.09039955607 7 321002 322103.957926635 -1101.95792663474 8 320820 319042.890572957 1777.10942704312 9 327032 326499.566044411 532.43395558866 10 324047 327538.857996403 -3491.85799640324 11 316735 320813.810032402 -4078.81003240201 12 315710 317926.674916245 -2216.6749162452 13 313427 315662.944143400 -2235.94414340046 14 310527 313579.779902458 -3052.77990245839 15 330962 329448.891712182 1513.10828781758 16 339015 337480.246209252 1534.7537907476 17 341332 341076.919345131 255.080654869102 18 339092 342239.48450244 -3147.48450244010 19 323308 326199.72703026 -2891.72703025979 20 325849 324493.588186651 1355.41181334932 21 330675 328826.555905838 1848.44409416168 22 332225 330055.252292 2169.74770800022 23 331735 331445.822405934 289.177594066029 24 328047 328489.820456986 -442.820456985822 25 326165 325360.354884297 804.645115702978 26 327081 325742.423239547 1338.57676045265 27 346764 345360.205435152 1403.79456484809 28 344190 345025.373348114 -835.373348114505 29 343333 341221.443975107 2111.55602489277 30 345777 343370.303266961 2406.69673303912 31 344094 340140.144772061 3953.85522793919 32 348609 348915.242056091 -306.242056091246 33 354846 355346.157803148 -500.157803147705 34 356427 356604.638622906 -177.63862290585 35 353467 352553.656523993 913.343476007163 36 355996 352112.180952915 3883.81904708533 37 352487 350830.142410579 1656.85758942143 38 355178 352298.416131387 2879.58386861305 39 374556 374691.807854988 -135.807854988002 40 375021 373137.467780870 1883.53221912957 41 375787 371652.923577312 4134.07642268766 42 372720 369778.143278275 2941.85672172550 43 364431 359871.215686351 4559.78431364924 44 370490 367430.908145509 3059.09185449054 45 376974 374456.918373107 2517.08162689259 46 377632 377066.062671388 565.937328611743 47 378205 375375.815537097 2829.18446290288 48 370861 370862.073411086 -1.0734110864272 49 369167 365173.886163525 3993.11383647512 50 371551 368821.81342917 2729.18657082995 51 382842 383362.683149536 -520.683149536432 52 381903 384424.438554914 -2521.43855491353 53 384502 384124.503335237 377.496664763145 54 392058 388133.64321254 3924.35678745983 55 384359 385759.162811686 -1400.16281168602 56 388884 390294.450617333 -1410.45061733278 57 386586 390983.801873495 -4397.80187349523 58 387495 386561.188417303 933.811582697123 59 385705 385657.895500574 47.1044994259362 60 378670 379893.250262768 -1223.25026276788 61 377367 377738.809022558 -371.809022558154 62 376911 379058.531705974 -2147.53170597386 63 389827 393390.277230523 -3563.27723052262 64 387820 390913.259130312 -3093.25913031164 65 387267 388391.908228080 -1124.90822808045 66 380575 384676.335340228 -4101.33534022829 67 372402 375521.791773008 -3119.79177300787 68 376740 381214.920421459 -4474.92042145895 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.213464633058115 0.42692926611623 0.786535366941885 22 0.177298904143395 0.354597808286789 0.822701095856605 23 0.32655072462697 0.65310144925394 0.67344927537303 24 0.298686649910441 0.597373299820882 0.701313350089559 25 0.364781489580082 0.729562979160164 0.635218510419918 26 0.302013684721280 0.604027369442559 0.697986315278720 27 0.910613330437333 0.178773339125334 0.0893866695626672 28 0.937143548241273 0.125712903517454 0.062856451758727 29 0.905547248302536 0.188905503394929 0.0944527516974643 30 0.862560090896088 0.274879818207825 0.137439909103912 31 0.799505326016185 0.40098934796763 0.200494673983815 32 0.859296701138353 0.281406597723295 0.140703298861647 33 0.822198769766059 0.355602460467881 0.177801230233941 34 0.748843435039151 0.502313129921698 0.251156564960849 35 0.774728957525867 0.450542084948266 0.225271042474133 36 0.759179619929041 0.481640760141918 0.240820380070959 37 0.749345968237198 0.501308063525603 0.250654031762802 38 0.728725819251568 0.542548361496865 0.271274180748432 39 0.839872081578054 0.320255836843891 0.160127918421946 40 0.7865629626518 0.4268740746964 0.2134370373482 41 0.701674537616568 0.596650924766865 0.298325462383432 42 0.592512409329291 0.814975181341419 0.407487590670709 43 0.50405582813052 0.99188834373896 0.49594417186948 44 0.396883899770071 0.793767799540142 0.603116100229929 45 0.298553147691187 0.597106295382374 0.701446852308813 46 0.265654504151218 0.531309008302435 0.734345495848782 47 0.161279252498087 0.322558504996175 0.838720747501913 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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