Multiple regression - Model 6

*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 09:28:13 -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/t1258561844i4cnlkdwzybvjze.htm/, Retrieved Wed, 18 Nov 2009 17:30:56 +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/t1258561844i4cnlkdwzybvjze.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?

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Dataseries X:

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2.97 101.1 2.98 3.01 3.06 3.12 3.58 3.02 100.93 2.97 2.98 3.01 3.06 3.52 3.07 100.85 3.02 2.97 2.98 3.01 3.45 3.18 100.93 3.07 3.02 2.97 2.98 3.36 3.29 99.6 3.18 3.07 3.02 2.97 3.27 3.43 101.88 3.29 3.18 3.07 3.02 3.21 3.61 101.81 3.43 3.29 3.18 3.07 3.19 3.74 102.38 3.61 3.43 3.29 3.18 3.16 3.87 102.74 3.74 3.61 3.43 3.29 3.12 3.88 102.82 3.87 3.74 3.61 3.43 3.06 4.09 101.72 3.88 3.87 3.74 3.61 3.01 4.19 103.47 4.09 3.88 3.87 3.74 2.98 4.2 102.98 4.19 4.09 3.88 3.87 2.97 4.29 102.68 4.2 4.19 4.09 3.88 3.02 4.37 102.9 4.29 4.2 4.19 4.09 3.07 4.47 103.03 4.37 4.29 4.2 4.19 3.18 4.61 101.29 4.47 4.37 4.29 4.2 3.29 4.65 103.69 4.61 4.47 4.37 4.29 3.43 4.69 103.68 4.65 4.61 4.47 4.37 3.61 4.82 104.2 4.69 4.65 4.61 4.47 3.74 4.86 104.08 4.82 4.69 4.65 4.61 3.87 4.87 104.16 4.86 4.82 4.69 4.65 3.88 5.01 103.05 4.87 4.86 4.82 4.69 4.09 5.03 104.66 5.01 4.87 4.86 4.82 4.19 5.13 104.46 5.03 5.01 4.87 4.86 4.2 5.18 104.95 5.13 5.03 5.01 4.87 4.29 5.21 105.85 5.18 5.13 5.03 5.01 4.37 5.26 106. 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] = -1.39732048460966 + 0.0161718963298520X[t] + 2.00157312691411y1[t] -1.56424107271801y2[t] + 0.790011335492396y3[t] -0.262256734663169y4[t] -0.0424262947310977y12[t] + 0.113985026448485M1[t] + 0.0815647206202224M2[t] -0.0294406678719151M3[t] + 0.0370825333025232M4[t] + 0.0878443689176442M5[t] + 0.0948251430698515M6[t] + 0.0467429337123727M7[t] + 0.104716315344380M8[t] + 0.124092289711188M9[t] + 0.0282583110835392M10[t] + 0.220714286663328M11[t] -0.00206216097793336t + 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) -1.39732048460966 2.096764 -0.6664 0.510411 0.255206 X 0.0161718963298520 0.02228 0.7258 0.473752 0.236876 y1 2.00157312691411 0.184845 10.8284 0 0 y2 -1.56424107271801 0.396628 -3.9439 0.000466 0.000233 y3 0.790011335492396 0.402015 1.9651 0.059043 0.029522 y4 -0.262256734663169 0.201902 -1.2989 0.204201 0.102101 y12 -0.0424262947310977 0.067502 -0.6285 0.534581 0.267291 M1 0.113985026448485 0.097171 1.173 0.250325 0.125162 M2 0.0815647206202224 0.088295 0.9238 0.363229 0.181614 M3 -0.0294406678719151 0.090153 -0.3266 0.746342 0.373171 M4 0.0370825333025232 0.098302 0.3772 0.708748 0.354374 M5 0.0878443689176442 0.105279 0.8344 0.41088 0.20544 M6 0.0948251430698515 0.092153 1.029 0.311986 0.155993 M7 0.0467429337123727 0.091547 0.5106 0.613502 0.306751 M8 0.104716315344380 0.093043 1.1255 0.269622 0.134811 M9 0.124092289711188 0.085637 1.4491 0.15805 0.079025 M10 0.0282583110835392 0.083982 0.3365 0.738931 0.369466 M11 0.220714286663328 0.093076 2.3713 0.024582 0.012291 t -0.00206216097793336 0.005175 -0.3985 0.693167 0.346583 Multiple Linear Regression - Regression Statistics Multiple R 0.995022541782589 R-squared 0.990069858655483 Adjusted R-squared 0.983906322648542 F-TEST (value) 160.6334184696 F-TEST (DF numerator) 18 F-TEST (DF denominator) 29 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.114958626496740 Sum Squared Residuals 0.38324908837449 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2.97 3.05321092845207 -0.0832109284520703 2 3.02 3.02167115517124 -0.00167115517124363 3 3.07 3.01541325836723 0.0545867416327652 4 3.18 3.1068206081907 0.0731793918092982 5 3.29 3.32191415168089 -0.0319141516808896 6 3.43 3.44074452217412 -0.0107445221741233 7 3.61 3.57225877493024 0.0377412250697556 8 3.74 3.8380031840895 -0.0980031840894992 9 3.87 3.92323039151194 -0.0532303915119448 10 3.88 3.9915128458779 -0.111512845877897 11 4.09 4.03839854244391 0.0516014575560893 12 4.19 4.31849174625441 -0.128491746254406 13 4.2 4.26838807073998 -0.0683880707399832 14 4.29 4.25380415740239 0.0361958425976128 15 4.37 4.33059950035339 0.0394004996466066 16 4.47 4.39351458814947 0.0764854118505262 17 4.61 4.55290475047395 0.0570952495260492 18 4.65 4.7540901647934 -0.104090164793398 19 4.69 4.61523711211532 0.0747628878846814 20 4.82 4.77591149621634 0.0440885037836637 21 4.86 4.97828863788755 -0.118288637887551 22 4.87 4.77908375669744 0.090916243302561 23 5.01 4.99227453706753 0.0177254629324665 24 5.03 5.0533771179985 -0.0233771179985028 25 5.13 4.98008889758193 0.149911102418068 26 5.18 5.22656380431092 -0.0465638043109186 27 5.21 5.04739569089014 0.162604309109864 28 5.26 5.16935056124637 0.0906494387536255 29 5.25 5.25638137412178 -0.00638137412177796 30 5.2 5.2136861464644 -0.0136861464643993 31 5.16 5.12181714155653 0.0381828584434746 32 5.19 5.15290693955938 0.0370930604406221 33 5.39 5.26946412099723 0.120535879002774 34 5.58 5.51833413616976 0.0616658638302374 35 5.76 5.7879342242742 -0.0279342242741952 36 5.89 5.79611885560434 0.0938811443956576 37 5.98 5.97831210322602 0.00168789677398566 38 6.02 6.00796088311545 0.0120391168845495 39 5.62 5.87659155038924 -0.256591550389236 40 4.87 5.11031424241345 -0.24031424241345 41 4.24 4.25879972372338 -0.0187997237233816 42 4.02 3.89147916656808 0.128520833431921 43 3.74 3.89068697139791 -0.150686971397912 44 3.45 3.43317838013479 0.0168216198652134 45 3.34 3.28901684960328 0.0509831503967224 46 3.21 3.2510692612549 -0.0410692612549018 47 3.12 3.16139269621436 -0.0413926962143606 48 3.04 2.98201228014275 0.057987719857251 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 22 0.0972645976134416 0.194529195226883 0.902735402386558 23 0.0703816052642715 0.140763210528543 0.929618394735728 24 0.0246872633953323 0.0493745267906646 0.975312736604668 25 0.0334800415267999 0.0669600830535999 0.9665199584732 26 0.188825354837066 0.377650709674132 0.811174645162934 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 0.2 NOK 10% type I error level 2 0.4 NOK

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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