Workshop 7: Multiple Regression 4 (4lags)

*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, 25 Dec 2009 12:23:32 -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/Dec/25/t1261769088yrk1xlqpmanmd4n.htm/, Retrieved Fri, 25 Dec 2009 20:25:00 +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/Dec/25/t1261769088yrk1xlqpmanmd4n.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:

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Dataseries X:

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20.7 7.8 21.3 22 23.7 25.6 20.4 7.8 20.7 21.3 22 23.7 20.3 7.8 20.4 20.7 21.3 22 20.4 7.5 20.3 20.4 20.7 21.3 19.8 7.5 20.4 20.3 20.4 20.7 19.5 7.1 19.8 20.4 20.3 20.4 23.1 7.5 19.5 19.8 20.4 20.3 23.5 7.5 23.1 19.5 19.8 20.4 23.5 7.6 23.5 23.1 19.5 19.8 22.9 7.7 23.5 23.5 23.1 19.5 21.9 7.7 22.9 23.5 23.5 23.1 21.5 7.9 21.9 22.9 23.5 23.5 20.5 8.1 21.5 21.9 22.9 23.5 20.2 8.2 20.5 21.5 21.9 22.9 19.4 8.2 20.2 20.5 21.5 21.9 19.2 8.2 19.4 20.2 20.5 21.5 18.8 7.9 19.2 19.4 20.2 20.5 18.8 7.3 18.8 19.2 19.4 20.2 22.6 6.9 18.8 18.8 19.2 19.4 23.3 6.6 22.6 18.8 18.8 19.2 23 6.7 23.3 22.6 18.8 18.8 21.4 6.9 23 23.3 22.6 18.8 19.9 7 21.4 23 23.3 22.6 18.8 7.1 19.9 21.4 23 23.3 18.6 7.2 18.8 19.9 21.4 23 18.4 7.1 18.6 18.8 19.9 21.4 18.6 6.9 18.4 18.6 18.8 19.9 19.9 7 18.6 18.4 18.6 18.8 19.2 6.8 19.9 18.6 18.4 18.6 18.4 6.4 19.2 19.9 18.6 18.4 21.1 6.7 18.4 19.2 19.9 18.6 20.5 6.6 21.1 18.4 19.2 19.9 19.1 6.4 20.5 21.1 18.4 19.2 18.1 6.3 19.1 20.5 21.1 18.4 17 6.2 18.1 19.1 20.5 21 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] = + 1.21788686486356 + 0.484134343472141X[t] + 1.47743594518220Y1[t] -1.00247950308246Y2[t] + 0.142867662547923Y3[t] + 0.160203862531591Y4[t] -0.889739069238092M1[t] -1.03979564822641M2[t] -0.877831720066912M3[t] -0.372343703812319M4[t] -1.29197609960059M5[t] + 0.190290825479914M6[t] + 3.04103217720423M7[t] -2.16551128616970M8[t] + 0.636778689366172M9[t] + 0.306992689379753M10[t] -0.656329660215681M11[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) 1.21788686486356 1.070239 1.138 0.262081 0.13104 X 0.484134343472141 0.249159 1.9431 0.059251 0.029625 Y1 1.47743594518220 0.178676 8.2688 0 0 Y2 -1.00247950308246 0.298226 -3.3615 0.001746 0.000873 Y3 0.142867662547923 0.294579 0.485 0.630397 0.315198 Y4 0.160203862531591 0.151234 1.0593 0.295975 0.147988 M1 -0.889739069238092 0.416499 -2.1362 0.038992 0.019496 M2 -1.03979564822641 0.43316 -2.4005 0.021243 0.010621 M3 -0.877831720066912 0.435959 -2.0136 0.050993 0.025496 M4 -0.372343703812319 0.439712 -0.8468 0.402279 0.201139 M5 -1.29197609960059 0.430019 -3.0045 0.004631 0.002316 M6 0.190290825479914 0.42282 0.4501 0.655165 0.327582 M7 3.04103217720423 0.481829 6.3114 0 0 M8 -2.16551128616970 0.791093 -2.7374 0.009281 0.00464 M9 0.636778689366172 0.790135 0.8059 0.425181 0.212591 M10 0.306992689379753 0.689957 0.4449 0.658819 0.32941 M11 -0.656329660215681 0.439887 -1.492 0.143734 0.071867 Multiple Linear Regression - Regression Statistics Multiple R 0.980441846539918 R-squared 0.961266214446603 Adjusted R-squared 0.945375430629825 F-TEST (value) 60.4920578827372 F-TEST (DF numerator) 16 F-TEST (DF denominator) 39 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.5710872110191 Sum Squared Residuals 12.7194835009934 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 20.7 21.0064147224696 -0.306414722469565 2 20.4 20.1243698633882 0.275630136611821 3 20.3 20.0722367797552 0.227763220244780 4 20.4 20.3876214480738 0.0123785519261583 5 19.8 19.5769979808287 0.223002019171304 6 19.5 19.8165537260885 -0.316553726088509 7 23.1 23.0174721134981 0.0825278865018796 8 23.5 23.3607416924293 0.139258307570727 9 23.5 23.0545106530051 0.44548934699493 10 22.9 22.8384087125459 0.0615912874540699 11 21.9 21.6225057659741 0.277494234025931 12 21.5 21.5637955965641 -0.0637955965640818 13 20.5 21.0966679235012 -0.59666792350124 14 20.2 19.6795906548440 0.520409345155962 15 19.4 20.1834523749806 -0.783452374980565 16 19.2 19.6007862784536 -0.400786278453571 17 18.8 18.8393458317572 -0.0393458317572162 18 18.8 19.4782983845002 -0.678298384500229 19 22.6 22.3796411775338 0.220358822466189 20 23.3 22.5529261652851 0.74707383471487 21 23 22.5643310800698 0.435668919930205 22 21.4 21.7293026307475 -0.32930263074753 23 19.9 19.4600220955361 0.43997790446389 24 18.8 19.6218608822654 -0.821860882265362 25 18.6 18.3824255434616 0.217574456538409 26 18.4 18.5205681206079 -0.120568120607895 27 18.6 18.0932536690529 0.506746330947098 28 19.9 18.9383404280133 0.961659571986684 29 19.2 19.5814376866351 -0.381437686635085 30 18.4 18.5291551186953 -0.129155118695265 31 21.1 21.2626924032918 -0.162692403291790 32 20.5 20.9070538175361 -0.40705381753609 33 19.1 19.7929238651351 -0.692923865135126 34 18.1 18.20538140825 -0.105381408250001 35 17 17.4665108147471 -0.46651081474713 36 17.1 17.3492436963004 -0.249243696300431 37 17.4 17.4880629079208 -0.0880629079207592 38 16.8 17.3636308708445 -0.563630870844548 39 15.3 15.9827981610512 -0.682798161051174 40 14.3 14.7872603433578 -0.487260343357824 41 13.4 13.7110115152001 -0.311011515200116 42 15.3 14.7008820843999 0.59911791560011 43 22.1 21.4103576062187 0.68964239378135 44 23.7 24.1052961897495 -0.405296189749542 45 22.2 22.38823440179 -0.188234401790009 46 19.5 19.1269072484565 0.373092751543461 47 16.6 16.8509613237427 -0.250961323742691 48 17.3 16.1650998248701 1.13490017512987 49 19.8 19.0264289026468 0.773571097353156 50 21.2 21.3118404903153 -0.111840490315340 51 21.5 20.7682590151601 0.731740984839863 52 20.6 20.6859915021014 -0.0859915021014472 53 19.1 18.5912069855789 0.508793014421113 54 19.6 19.0751106863161 0.524889313683893 55 23.5 24.3298366994576 -0.829836699457628 56 24 24.0739821350000 -0.0739821349999634 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 20 0.0980677490405858 0.196135498081172 0.901932250959414 21 0.077831759910241 0.155663519820482 0.922168240089759 22 0.110664936338311 0.221329872676621 0.88933506366169 23 0.0612881930852984 0.122576386170597 0.938711806914702 24 0.0627886477706587 0.125577295541317 0.937211352229341 25 0.0632106496471335 0.126421299294267 0.936789350352867 26 0.0316840630378131 0.0633681260756263 0.968315936962187 27 0.0169137220273416 0.0338274440546831 0.983086277972658 28 0.125101809083701 0.250203618167403 0.874898190916299 29 0.152959612823328 0.305919225646656 0.847040387176672 30 0.110221160697594 0.220442321395187 0.889778839302406 31 0.0733131176384588 0.146626235276918 0.926686882361541 32 0.0656672326977005 0.131334465395401 0.9343327673023 33 0.0525241960963519 0.105048392192704 0.947475803903648 34 0.0398562634174893 0.0797125268349786 0.96014373658251 35 0.0468459147210362 0.0936918294420724 0.953154085278964 36 0.588280239357918 0.823439521284165 0.411719760642082 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 1 0.0588235294117647 NOK 10% type I error level 4 0.235294117647059 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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