*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 08:01:29 -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/t12587294079cucmk81rlh4eh5.htm/, Retrieved Fri, 20 Nov 2009 16:03:40 +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/t12587294079cucmk81rlh4eh5.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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107.1 32.5 115.6 100.6 33.5 111.9 99.2 31.5 107 108.4 31.2 107.1 103 27 100.6 99.8 26.7 99.2 115 26.5 108.4 90.8 26 103 95.9 27.2 99.8 114.4 30.5 115 108.2 33.7 90.8 112.6 34.2 95.9 109.1 36.7 114.4 105 36.2 108.2 105 38.5 112.6 118.5 40 109.1 103.7 42.5 105 112.5 43.5 105 116.6 43.3 118.5 96.6 45.5 103.7 101.9 44.3 112.5 116.5 43 116.6 119.3 43.5 96.6 115.4 41.5 101.9 108.5 42.5 116.5 111.5 41.3 119.3 108.8 39.5 115.4 121.8 38.5 108.5 109.6 41 111.5 112.2 44.5 108.8 119.6 46 121.8 104.1 44 109.6 105.3 41.5 112.2 115 41.3 119.6 124.1 38 104.1 116.8 38 105.3 107.5 36.2 115 115.6 38.7 124.1 116.2 38.7 116.8 116.3 39.2 107.5 119 35.7 115.6 111.9 36.5 116.2 118.6 36.7 116.3 106.9 34.7 119 103.2 35 111.9 118.6 28.2 118.6 118.7 23.7 106.9 102.8 15 103.2 100.6 8.7 118.6 94.9 11 118.7 94.5 7.5 102.8 102.9 5.7 100.6 95.3 9.3 94.9 92.5 10.2 94.5 102.7 15.7 102.9 91.5 18.1 95.3 89.5 20.8 92.5

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 Ipzb[t] = + 51.0021787849945 + 0.358521246109701Cvn[t] + 0.469136445952469Y3[t] -11.5251626923923M1[t] -13.1131829422958M2[t] -11.0020240353219M3[t] -0.0947099776184548M4[t] -7.1883374189211M5[t] -7.64246158228107M6[t] -3.61421218000356M7[t] -16.6986202389138M8[t] -15.4519614926944M9[t] -4.3902634927507M10[t] + 5.74431082396985M11[t] + 0.0569955207934599t + 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) 51.0021787849945 8.636948 5.9051 1e-06 0 Cvn 0.358521246109701 0.060597 5.9165 1e-06 0 Y3 0.469136445952469 0.097855 4.7942 2.1e-05 1e-05 M1 -11.5251626923923 2.71509 -4.2449 0.000118 5.9e-05 M2 -13.1131829422958 2.708507 -4.8415 1.8e-05 9e-06 M3 -11.0020240353219 2.456632 -4.4785 5.7e-05 2.8e-05 M4 -0.0947099776184548 2.313072 -0.0409 0.967533 0.483767 M5 -7.1883374189211 2.284485 -3.1466 0.003034 0.001517 M6 -7.64246158228107 2.260646 -3.3807 0.001573 0.000786 M7 -3.61421218000356 2.49769 -1.447 0.155314 0.077657 M8 -16.6986202389138 2.270694 -7.354 0 0 M9 -15.4519614926944 2.264783 -6.8227 0 0 M10 -4.3902634927507 2.771358 -1.5842 0.12066 0.06033 M11 5.74431082396985 2.379751 2.4138 0.020224 0.010112 t 0.0569955207934599 0.034523 1.6509 0.106214 0.053107 Multiple Linear Regression - Regression Statistics Multiple R 0.945402534098933 R-squared 0.893785951480684 Adjusted R-squared 0.858381268640912 F-TEST (value) 25.2448512397531 F-TEST (DF numerator) 14 F-TEST (DF denominator) 42 p-value 5.55111512312578e-16 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.33903353153817 Sum Squared Residuals 468.264086838924 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 107.1 105.418125264066 1.68187473593373 2 100.6 102.509816931042 -1.90981693104196 3 99.2 101.662160281423 -2.46216028142278 4 108.4 112.565827130682 -4.16582713068201 5 103 100.974019077821 2.02598092217896 6 99.8 99.8125430370881 -0.0125430370881479 7 115 108.1421390137 6.85786098630009 8 90.8 92.402129044385 -1.60212904438496 9 95.9 92.6347721796816 3.26522782031844 10 114.4 112.067459791058 2.33254020894176 11 108.2 112.053195624074 -3.85319562407355 12 112.6 108.937736818310 3.66226318169039 13 109.1 107.044897012106 2.05510298789429 14 105 102.425965695036 2.57403430496448 15 105 107.482919351046 -2.48291935104603 16 118.5 117.343033237874 1.15696676212616 17 103.7 109.279245004234 -5.57924500423379 18 112.5 109.240637607777 3.25936239222302 19 116.6 119.587520301984 -2.98752030198435 20 96.6 100.405635105212 -3.80563510521238 21 101.9 105.407464601275 -3.50746460127532 22 116.5 117.983539930475 -1.48353993047498 23 119.3 118.971641471994 0.328358528005538 24 115.4 115.053706840147 0.346293159853243 25 108.5 110.793453025564 -2.29345302556367 26 111.5 110.145784849789 1.35421515021108 27 108.8 109.838968895344 -1.03896889534417 28 121.8 117.207715750659 4.59228424934067 29 109.6 112.474796283282 -2.87479628328181 30 112.2 112.065823598028 0.134176401972424 31 119.6 122.787624187645 -3.18762418764521 32 104.1 103.319704516689 0.780295483311092 33 105.3 104.946810427904 0.353189572096055 34 115 119.465409399467 -4.46540939946741 35 124.1 121.202244212556 2.89775578744385 36 116.8 116.077892644523 0.722107355477284 37 107.5 108.515010755665 -1.01501075566537 38 115.6 112.149430799997 3.45056920000292 39 116.2 110.892889172311 5.30711082768863 40 116.3 117.673490426505 -1.37349042650517 41 119 113.182039356827 5.81796064317297 42 111.9 113.353209578720 -1.45320957871975 43 118.6 117.557072395608 1.04292760439208 44 106.9 105.079285769343 1.82071423065660 45 103.2 103.159627643927 0.0403723560733419 46 118.6 114.983590878999 3.61640912100062 47 118.7 118.072918691376 0.62708130862416 48 102.8 107.530663697021 -4.73066369702092 49 100.6 101.028513942599 -0.428513942598980 50 94.9 100.369001724137 -5.46900172413651 51 94.5 93.8230622998756 0.676937700124361 52 102.9 103.109933454280 -0.209933454279638 53 95.3 94.6899002778363 0.610099722163675 54 92.5 94.4277861783875 -1.92778617838755 55 102.7 104.425644101063 -1.72564410106262 56 91.5 88.6932455643704 2.80675443562964 57 89.5 89.6513251472125 -0.151325147212517 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.873847963918947 0.252304072162105 0.126152036081052 19 0.874428091373619 0.251143817252762 0.125571908626381 20 0.804084304750084 0.391831390499832 0.195915695249916 21 0.698659613108289 0.602680773783423 0.301340386891711 22 0.627872981234583 0.744254037530835 0.372127018765417 23 0.639654982627519 0.720690034744963 0.360345017372481 24 0.585742407374648 0.828515185250703 0.414257592625352 25 0.624379061851658 0.751241876296683 0.375620938148342 26 0.572057531329474 0.855884937341051 0.427942468670525 27 0.488569532841829 0.977139065683659 0.511430467158171 28 0.590148867815914 0.819702264368172 0.409851132184086 29 0.594170352428188 0.811659295143624 0.405829647571812 30 0.5128899053316 0.9742201893368 0.4871100946684 31 0.470528397910771 0.941056795821542 0.529471602089229 32 0.408832667395718 0.817665334791437 0.591167332604282 33 0.330428305808291 0.660856611616582 0.669571694191709 34 0.660317441055427 0.679365117889147 0.339682558944573 35 0.567947831169296 0.864104337661407 0.432052168830704 36 0.525323698111024 0.949352603777951 0.474676301888976 37 0.569887932728465 0.86022413454307 0.430112067271535 38 0.732149409188331 0.535701181623338 0.267850590811669 39 0.683462583639897 0.633074832720206 0.316537416360103 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):

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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