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*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, 20 Nov 2009 10:17:20 -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/20/t1258737662uu5nmqy2mvdqgsa.htm/, Retrieved Fri, 20 Nov 2009 18:21:14 +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/20/t1258737662uu5nmqy2mvdqgsa.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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3956.2 3977.7 3142.7 3983.4 3884.3 4152.9 3892.2 4286.1 3613 4348.1 3730.5 3949.3 3481.3 4166.7 3649.5 4217.9 4215.2 4528.2 4066.6 4232.2 4196.8 4470.9 4536.6 5121.2 4441.6 4170.8 3548.3 4398.6 4735.9 4491.4 4130.6 4251.8 4356.2 4901.9 4159.6 4745.2 3988 4666.9 4167.8 4210.4 4902.2 5273.6 3909.4 4095.3 4697.6 4610.1 4308.9 4718.1 4420.4 4185.5 3544.2 4314.7 4433 4422.6 4479.7 5059.2 4533.2 5043.6 4237.5 4436.6 4207.4 4922.6 4394 4454.8 5148.4 5058.7 4202.2 4768.9 4682.5 5171.8 4884.3 4989.3 5288.9 5202.1 4505.2 4838.4 4611.5 4876.5 5104 5875.5 4586.6 5717.9 4529.3 4778.8 4504.1 6195.9 4604.9 4625.4 4795.4 5549.8 5391.1 6397.6 5213.9 5856.7 5415 6343.8 5990.3 6615.5 4241.8 5904.6 5677.6 6861 5164.2 6553.5 3962.3 5481 4011 5435.3 3310.3 5278 3837.3 4671.8 4145.3 4891.5 3796.7 4241.6 3849.6 4152.1 4285 4484.4 4189.6 4124.7

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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 1677.68894339794 + 0.586252193687844X[t] + 273.970572539457M1[t] -629.564052274936M2[t] + 82.4442739799037M3[t] -175.120587025784M4[t] -456.435752915417M5[t] -281.343885814236M6[t] -737.70923779068M7[t] -147.638849728842M8[t] -3.03609424816335M9[t] -187.498457097383M10[t] + 5.70781212665605M11[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) 1677.68894339794 309.74801 5.4163 2e-06 1e-06 X 0.586252193687844 0.054589 10.7393 0 0 M1 273.970572539457 180.456973 1.5182 0.135522 0.067761 M2 -629.564052274936 188.524052 -3.3394 0.00163 0.000815 M3 82.4442739799037 187.195071 0.4404 0.661611 0.330805 M4 -175.120587025784 187.007064 -0.9364 0.353735 0.176868 M5 -456.435752915417 186.972198 -2.4412 0.018377 0.009188 M6 -281.343885814236 188.6593 -1.4913 0.142431 0.071216 M7 -737.70923779068 187.021625 -3.9445 0.00026 0.00013 M8 -147.638849728842 190.777478 -0.7739 0.442797 0.221398 M9 -3.03609424816335 187.003763 -0.0162 0.987114 0.493557 M10 -187.498457097383 188.136532 -0.9966 0.323952 0.161976 M11 5.70781212665605 187.583098 0.0304 0.975852 0.487926 Multiple Linear Regression - Regression Statistics Multiple R 0.889456603915496 R-squared 0.791133050248887 Adjusted R-squared 0.738916312811109 F-TEST (value) 15.1509475518574 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 1.84519066692701e-12 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 295.615397185516 Sum Squared Residuals 4194646.22655121 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 3956.2 4283.59486676954 -327.394866769543 2 3142.7 3383.40187945917 -240.701879459166 3 3884.3 4194.77995254410 -310.479952544095 4 3892.2 4015.30388373763 -123.103883737629 5 3613 3770.33635385664 -157.336353856642 6 3730.5 3711.63084611511 18.8691538848887 7 3481.3 3382.71672104640 98.5832789535953 8 3649.5 4002.80322142506 -353.30322142506 9 4215.2 4329.32003260708 -114.120032607077 10 4066.6 3971.32702042626 95.2729795737447 11 4196.8 4304.47168828358 -107.671688283582 12 4536.6 4680.00367771213 -143.403677712131 13 4441.6 4396.80016537066 44.7998346293384 14 3548.3 3626.81379027836 -78.5137902783593 15 4735.9 4393.22632010743 342.673679892569 16 4130.6 3995.19543349414 135.404566505865 17 4356.2 4095.00281872097 261.19718127903 18 4159.6 4178.22896707127 -18.6289670712657 19 3988 3675.96006832906 312.039931670936 20 4167.8 3998.4063299724 169.393670027599 21 4902.2 4766.312417782 135.887582218004 22 3909.4 3891.06909511039 18.3309048896106 23 4697.6 4386.07799364493 311.52200635507 24 4308.9 4443.68541843656 -134.785418436562 25 4420.4 4405.41807261787 14.9819273821264 26 3544.2 3577.62723122795 -33.4272312279492 27 4433 4352.89216918171 80.1078308182923 28 4479.7 4468.5354546777 11.1645453222989 29 4533.2 4178.07475456654 355.125245433462 30 4237.5 3997.3115400992 240.188459900802 31 4207.4 3825.86475425505 381.535245744953 32 4394 4141.68636610971 252.313633890290 33 5148.4 4640.32682135848 508.073178641522 34 4202.2 4285.96857277852 -83.7685727785211 35 4682.5 4715.37585083939 -32.8758508393926 36 4884.3 4602.67701336471 281.622986635295 37 5288.9 5001.40205272094 287.497947279064 38 4505.2 3884.64750506227 620.552494937727 39 4611.5 4618.99203989662 -7.49203989661987 40 5104 4947.09312038509 156.906879614912 41 4586.6 4573.38460877025 13.2153912297498 42 4529.3 4197.92704077918 331.372959220822 43 4504.1 4572.33967247778 -68.2396724777777 44 4604.9 4241.70099035286 363.199009647144 45 4795.4 4928.23527367858 -132.835273678579 46 5391.1 5240.79752063791 150.302479362087 47 5213.9 5116.8999782962 97.000021703803 48 5415 5396.75560971489 18.2443902851101 49 5990.3 5830.01090327933 160.289096720666 50 4241.8 4509.70959397225 -267.909593972253 51 5677.6 5782.40951827015 -104.809518270146 52 5164.2 5344.57210770545 -180.372107705446 53 3962.3 4434.5014640856 -472.2014640856 54 4011 4582.80160593525 -571.801605935248 55 3310.3 4034.21878389171 -723.918783891706 56 3837.3 4268.90309213997 -431.603092139972 57 4145.3 4542.30545457387 -397.00545457387 58 3796.7 3976.83779104692 -180.137791046921 59 3849.6 4117.5744889359 -267.974488935898 60 4285 4306.67828077171 -21.6782807717122 61 4189.6 4369.77393924165 -180.173939241652 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.270722889029574 0.541445778059148 0.729277110970426 17 0.151622012065559 0.303244024131118 0.84837798793444 18 0.247921303582830 0.495842607165660 0.75207869641717 19 0.157151153771992 0.314302307543985 0.842848846228008 20 0.203745614972483 0.407491229944967 0.796254385027517 21 0.127206590666521 0.254413181333042 0.872793409333479 22 0.0733345749852377 0.146669149970475 0.926665425014762 23 0.0686811262079924 0.137362252415985 0.931318873792008 24 0.0441841343275634 0.0883682686551268 0.955815865672437 25 0.0252304322535084 0.0504608645070169 0.974769567746492 26 0.0139594264910380 0.0279188529820759 0.986040573508962 27 0.00684000463085306 0.0136800092617061 0.993159995369147 28 0.00503414700169722 0.0100682940033944 0.994965852998303 29 0.00409440560121735 0.0081888112024347 0.995905594398783 30 0.00289271985870217 0.00578543971740433 0.997107280141298 31 0.00342068496937366 0.00684136993874733 0.996579315030626 32 0.00274492079308503 0.00548984158617007 0.997255079206915 33 0.0109170902748189 0.0218341805496377 0.989082909725181 34 0.00965492488814803 0.0193098497762961 0.990345075111852 35 0.00739186518183625 0.0147837303636725 0.992608134818164 36 0.00820096284288892 0.0164019256857778 0.99179903715711 37 0.00507987820133806 0.0101597564026761 0.994920121798662 38 0.0360454057871903 0.0720908115743805 0.96395459421281 39 0.0270516421175059 0.0541032842350117 0.972948357882494 40 0.0246543706547390 0.0493087413094781 0.97534562934526 41 0.0283715483308167 0.0567430966616334 0.971628451669183 42 0.176889601822002 0.353779203644004 0.823110398177998 43 0.31641573611604 0.63283147223208 0.68358426388396 44 0.937226582306678 0.125546835386644 0.0627734176933219 45 0.913186303531024 0.173627392937952 0.086813696468976 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.133333333333333 NOK 5% type I error level 13 0.433333333333333 NOK 10% type I error level 18 0.6 NOK

Charts produced by software:

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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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