workshop 10

*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, 24 Dec 2010 17:47:56 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/24/t1293212745bvxlckccqu046cw.htm/, Retrieved Fri, 24 Dec 2010 18:45:55 +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/2010/Dec/24/t1293212745bvxlckccqu046cw.htm/}, year = {2010}, } @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 = {2010}, 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 «

235.1 9700 280.7 9081 264.6 9084 240.7 9743 201.4 8587 240.8 9731 241.1 9563 223.8 9998 206.1 9437 174.7 10038 203.3 9918 220.5 9252 299.5 9737 347.4 9035 338.3 9133 327.7 9487 351.6 8700 396.6 9627 438.8 8947 395.6 9283 363.5 8829 378.8 9947 357 9628 369 9318 464.8 9605 479.1 8640 431.3 9214 366.5 9567 326.3 8547 355.1 9185 331.6 9470 261.3 9123 249 9278 205.5 10170 235.6 9434 240.9 9655 264.9 9429 253.8 8739 232.3 9552 193.8 9687 177 9019 213.2 9672 207.2 9206 180.6 9069 188.6 9788 175.4 10312 199 10105 179.6 9863 225.8 9656 234 9295 200.2 9946 183.6 9701 178.2 9049 203.2 10190 208.5 9706 191.8 9765 172.8 9893 148 9994 159.4 10433 154.5 10073 213.2 10112 196.4 9266 182.8 9820 176.4 10097 153.6 9115 173.2 10411 171 9678 151.2 10408 161.9 10153 157.2 10368 201.7 10581 236.4 10597 356.1 10680 398.3 9738 403.7 9556

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 9 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation unemployment[t] = + 943.403030642086 -0.0716993270501178birth[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) 943.403030642086 177.739119 5.3078 1e-06 1e-06 birth -0.0716993270501178 0.018479 -3.8801 0.000227 0.000113 Multiple Linear Regression - Regression Statistics Multiple R 0.413487517133902 R-squared 0.170971926825559 Adjusted R-squared 0.159615377877964 F-TEST (value) 15.0549192025245 F-TEST (DF numerator) 1 F-TEST (DF denominator) 73 p-value 0.000226712823779618 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 79.9683384170429 Sum Squared Residuals 466830.265890337 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 235.1 247.919558255950 -12.8195582559495 2 280.7 292.301441699967 -11.6014416999669 3 264.6 292.086343718817 -27.4863437188166 4 240.7 244.836487192789 -4.13648719278909 5 201.4 327.720909262725 -126.320909262725 6 240.8 245.696879117390 -4.89687911739048 7 241.1 257.742366061810 -16.6423660618103 8 223.8 226.553158795009 -2.75315879500905 9 206.1 266.776481270125 -60.6764812701251 10 174.7 223.685185713004 -48.9851857130044 11 203.3 232.289104959018 -28.9891049590185 12 220.5 280.040856774397 -59.5408567743969 13 299.5 245.26668315509 54.2333168449102 14 347.4 295.599610744272 51.8003892557276 15 338.3 288.573076693361 49.7269233066391 16 327.7 263.191514917619 64.5084850823808 17 351.6 319.618885306062 31.9811146939382 18 396.6 253.153609130603 143.446390869397 19 438.8 301.909151524683 136.890848475317 20 395.6 277.818177635843 117.781822364157 21 363.5 310.369672116597 53.1303278834033 22 378.8 230.209824474565 148.590175525435 23 357 253.081909803553 103.918090196447 24 369 275.308701189089 93.6912988109109 25 464.8 254.730994325705 210.069005674295 26 479.1 323.920844929069 155.179155070931 27 431.3 282.765431202301 148.534568797699 28 366.5 257.45556875361 109.044431246390 29 326.3 330.58888234473 -4.28888234472987 30 355.1 284.844711686755 70.2552883132453 31 331.6 264.410403477471 67.1895965225288 32 261.3 289.290069963862 -27.9900699638621 33 249 278.176674271094 -29.1766742710938 34 205.5 214.220874542389 -8.72087454238881 35 235.6 266.991579251275 -31.3915792512755 36 240.9 251.146027973199 -10.2460279731994 37 264.9 267.350075886526 -2.45007588652607 38 253.8 316.822611551107 -63.0226115511073 39 232.3 258.531058659362 -26.2310586593616 40 193.8 248.851649507596 -55.0516495075957 41 177 296.746799977074 -119.746799977074 42 213.2 249.927139413347 -36.7271394133474 43 207.2 283.339025818702 -76.1390258187023 44 180.6 293.161833624568 -112.561833624568 45 188.6 241.610017475534 -53.0100174755338 46 175.4 204.039570101272 -28.6395701012721 47 199 218.881330800646 -19.8813308006465 48 179.6 236.232567946775 -56.632567946775 49 225.8 251.074328646149 -25.2743286461493 50 234 276.957785711242 -42.9577857112418 51 200.2 230.281523801615 -30.0815238016152 52 183.6 247.847858928894 -64.247858928894 53 178.2 294.595820165571 -116.395820165571 54 203.2 212.786888001386 -9.58688800138647 55 208.5 247.489362293643 -38.9893622936434 56 191.8 243.259101997686 -51.4591019976865 57 172.8 234.081588135271 -61.2815881352714 58 148 226.839956103210 -78.8399561032095 59 159.4 195.363951528208 -35.9639515282078 60 154.5 221.17570926625 -66.6757092662502 61 213.2 218.379435511296 -5.17943551129565 62 196.4 279.037066195695 -82.6370661956952 63 182.8 239.31563900993 -56.51563900993 64 176.4 219.454925417047 -43.0549254170474 65 153.6 289.863664580263 -136.263664580263 66 173.2 196.941336723310 -23.7413367233105 67 171 249.496943451047 -78.4969434510467 68 151.2 197.156434704461 -45.9564347044608 69 161.9 215.439763102241 -53.5397631022408 70 157.2 200.024407786466 -42.8244077864655 71 201.7 184.752451124790 16.9475488752096 72 236.4 183.605261891989 52.7947381080115 73 356.1 177.654217746829 178.445782253171 74 398.3 245.19498382804 153.105016171960 75 403.7 258.244261351161 145.455738648839 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0882557727469245 0.176511545493849 0.911744227253075 6 0.030489966948479 0.060979933896958 0.969510033051521 7 0.00939999630135206 0.0187999926027041 0.990600003698648 8 0.00361206031751288 0.00722412063502576 0.996387939682487 9 0.00219635121202357 0.00439270242404715 0.997803648787976 10 0.00324219521859506 0.00648439043719011 0.996757804781405 11 0.00130314694262338 0.00260629388524677 0.998696853057377 12 0.000490208900183663 0.000980417800367327 0.999509791099816 13 0.00157455622266459 0.00314911244532917 0.998425443777335 14 0.00800848171111023 0.0160169634222205 0.99199151828889 15 0.0120976776356355 0.0241953552712711 0.987902322364364 16 0.0157584250815353 0.0315168501630706 0.984241574918465 17 0.0130870860993605 0.0261741721987211 0.98691291390064 18 0.0697735837861812 0.139547167572362 0.930226416213819 19 0.168462533404591 0.336925066809183 0.831537466595409 20 0.228350321672309 0.456700643344618 0.771649678327691 21 0.188990826120407 0.377981652240814 0.811009173879593 22 0.314861198149580 0.629722396299159 0.68513880185042 23 0.334200584984632 0.668401169969263 0.665799415015368 24 0.340419936914024 0.680839873828048 0.659580063085976 25 0.6967834226302 0.6064331547396 0.3032165773698 26 0.855386161223472 0.289227677553056 0.144613838776528 27 0.94065970364845 0.118680592703099 0.0593402963515494 28 0.961826076829693 0.0763478463406141 0.0381739231703070 29 0.957959394570633 0.084081210858734 0.042040605429367 30 0.969023911703295 0.0619521765934103 0.0309760882967052 31 0.975547603162948 0.0489047936741031 0.0244523968370516 32 0.972237902470099 0.0555241950598024 0.0277620975299012 33 0.967485053475243 0.0650298930495144 0.0325149465247572 34 0.956950504473898 0.086098991052205 0.0430494955261025 35 0.948384918627454 0.103230162745091 0.0516150813725457 36 0.93488177252712 0.130236454945761 0.0651182274728803 37 0.923959424681928 0.152081150636144 0.076040575318072 38 0.92382010543047 0.152359789139059 0.0761798945695296 39 0.907513131886052 0.184973736227896 0.0924868681139478 40 0.894650550070599 0.210698899858803 0.105349449929401 41 0.911148346500955 0.177703306998090 0.0888516534990448 42 0.889235630475254 0.221528739049492 0.110764369524746 43 0.875539197639113 0.248921604721774 0.124460802360887 44 0.878258777200305 0.24348244559939 0.121741222799695 45 0.853931786312115 0.29213642737577 0.146068213687885 46 0.821041402342017 0.357917195315966 0.178958597657983 47 0.774878771026922 0.450242457946157 0.225121228973078 48 0.738524769945918 0.522950460108163 0.261475230054082 49 0.683641602491639 0.632716795016722 0.316358397508361 50 0.632905199254202 0.734189601491597 0.367094800745798 51 0.567433176934746 0.865133646130507 0.432566823065254 52 0.516694794223851 0.966610411552299 0.483305205776149 53 0.505434923338893 0.989130153322215 0.494565076661107 54 0.430413433567429 0.860826867134858 0.569586566432571 55 0.362251415519623 0.724502831039246 0.637748584480377 56 0.302959132227275 0.60591826445455 0.697040867772725 57 0.257179696072587 0.514359392145173 0.742820303927413 58 0.236427687525765 0.472855375051529 0.763572312474235 59 0.195191714918617 0.390383429837235 0.804808285081383 60 0.17094131097388 0.34188262194776 0.82905868902612 61 0.122107222209302 0.244214444418604 0.877892777790698 62 0.0961039536503366 0.192207907300673 0.903896046349663 63 0.0720174615634622 0.144034923126924 0.927982538436538 64 0.0523531230991878 0.104706246198376 0.947646876900812 65 0.0999267752028374 0.199853550405675 0.900073224797163 66 0.0707903086963969 0.141580617392794 0.929209691303603 67 0.132745911425494 0.265491822850987 0.867254088574506 68 0.129076989932014 0.258153979864028 0.870923010067986 69 0.224857783972131 0.449715567944262 0.775142216027869 70 0.423695321185677 0.847390642371353 0.576304678814323 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 6 0.090909090909091 NOK 5% type I error level 12 0.181818181818182 NOK 10% type I error level 19 0.287878787878788 NOK

Charts produced by software:

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Parameters (Session):

par1 = kendall ;

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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