WS7 Multiple Regression5

*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 10:52:33 -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/t1258739613gjonqgxg195ln40.htm/, Retrieved Fri, 20 Nov 2009 18:53:45 +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/t1258739613gjonqgxg195ln40.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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No (this computation is public)

User-defined keywords:

Dataseries X:

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2.08 1.00 2.05 2.09 2.06 1.00 2.08 2.05 2.06 1.00 2.06 2.08 2.08 1.00 2.06 2.06 2.07 1.00 2.08 2.06 2.06 1.00 2.07 2.08 2.07 1.00 2.06 2.07 2.06 1.00 2.07 2.06 2.09 1.00 2.06 2.07 2.07 1.00 2.09 2.06 2.09 1.00 2.07 2.09 2.28 1.25 2.09 2.07 2.33 1.25 2.28 2.09 2.35 1.25 2.33 2.28 2.52 1.50 2.35 2.33 2.63 1.50 2.52 2.35 2.58 1.50 2.63 2.52 2.70 1.75 2.58 2.63 2.81 1.75 2.70 2.58 2.97 2.00 2.81 2.70 3.04 2.00 2.97 2.81 3.28 2.25 3.04 2.97 3.33 2.25 3.28 3.04 3.50 2.50 3.33 3.28 3.56 2.50 3.50 3.33 3.57 2.50 3.56 3.50 3.69 2.75 3.57 3.56 3.82 2.75 3.69 3.57 3.79 2.75 3.82 3.69 3.96 3.00 3.79 3.82 4.06 3.00 3.96 3.79 4.05 3.00 4.06 3.96 4.03 3.00 4.05 4.06 3.94 3.00 4.03 4.05 4.02 3.00 3.94 4.03 3.88 3.00 4.02 3.94 4.02 3.00 3.88 4.02 4.03 3.00 4.02 3.88 4.09 3.00 4.03 4.02 3.99 3.00 4.09 4.03 4.01 3.00 3.99 4.09 4.01 3.00 4.01 3.99 4.19 3.25 4.01 4.01 4.30 3.25 4.19 4.01 4.27 3.25 4.30 4.19 3.82 3.25 4.27 4.30 3.15 2.75 3.82 4.27 2.49 2.00 3.15 3.82 1.81 1.00 2.49 3 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] = + 0.599180932557525 + 0.690170760852934X[t] + 0.799964668425404`Y-1`[t] -0.415248318667088`Y-2`[t] + 0.0905745590290102M1[t] + 0.0163192906230318M2[t] + 0.105507076694788M3[t] + 0.0676117496179909M4[t] + 0.0712801719111382M5[t] + 0.0656905940929301M6[t] + 0.00799840973995277M7[t] + 0.0361116251349147M8[t] + 0.0549776843366118M9[t] -0.0445931299432913M10[t] -0.0115226106672737M11[t] -0.00761139671202985t + 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) 0.599180932557525 0.116311 5.1515 7e-06 4e-06 X 0.690170760852934 0.117726 5.8625 1e-06 0 `Y-1` 0.799964668425404 0.179596 4.4543 6.6e-05 3.3e-05 `Y-2` -0.415248318667088 0.110474 -3.7588 0.000546 0.000273 M1 0.0905745590290102 0.073231 1.2368 0.223363 0.111682 M2 0.0163192906230318 0.072225 0.226 0.82239 0.411195 M3 0.105507076694788 0.071539 1.4748 0.148091 0.074046 M4 0.0676117496179909 0.075284 0.8981 0.374511 0.187255 M5 0.0712801719111382 0.074097 0.962 0.341838 0.170919 M6 0.0656905940929301 0.072807 0.9023 0.372322 0.186161 M7 0.00799840973995277 0.07443 0.1075 0.914959 0.45748 M8 0.0361116251349147 0.073909 0.4886 0.627798 0.313899 M9 0.0549776843366118 0.077543 0.709 0.482437 0.241218 M10 -0.0445931299432913 0.07592 -0.5874 0.560257 0.280128 M11 -0.0115226106672737 0.075387 -0.1528 0.879288 0.439644 t -0.00761139671202985 0.001516 -5.0195 1.1e-05 6e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.996738388731846 R-squared 0.993487415571757 Adjusted R-squared 0.991045196411166 F-TEST (value) 406.796994963937 F-TEST (DF numerator) 15 F-TEST (DF denominator) 40 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.105929688689683 Sum Squared Residuals 0.44884395783573 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2.08 2.14437343998530 -0.0643734399853037 2 2.06 2.10311564766674 -0.0431156476667425 3 2.06 2.15623529409795 -0.0962352940979458 4 2.08 2.11903353668246 -0.0390335366824608 5 2.07 2.13108985563209 -0.0610898556320865 6 2.06 2.10158426804425 -0.0415842680442524 7 2.07 2.03243352348166 0.0375664765183375 8 2.06 2.06508747203552 -0.00508747203551918 9 2.09 2.06419000465426 0.0258099953457381 10 2.07 1.98515921690176 0.0848407830982383 11 2.09 1.98216159653723 0.107838403462771 12 2.28 2.18291976044756 0.0970802395524438 13 2.33 2.40957124339202 -0.079571243392021 14 2.35 2.28880563114854 0.0611943688514634 15 2.52 2.53816158815665 -0.0181615881566496 16 2.63 2.6203438916268 0.00965610837320004 17 2.58 2.63380481656131 -0.0538048165613066 18 2.7 2.70747098376965 -0.00747098376965239 19 2.81 2.75892557884905 0.0510744211509518 20 2.97 2.99013640303196 -0.0201364030319573 21 3.04 3.08370809741631 -0.0437080974163098 22 3.28 3.13862637244065 0.141373627559345 23 3.33 3.32700963312004 0.00299036687995733 24 3.5 3.44380217422969 0.0561978257703104 25 3.56 3.64199691424563 -0.081996914245634 26 3.57 3.53753591505974 0.0324640849402547 27 3.69 3.77473974219693 -0.0847397421969333 28 3.82 3.82107629543248 -0.00107629543248452 29 3.79 3.87129892966885 -0.0812989296688536 30 3.96 3.95265942387237 0.0073405761276342 31 4.06 4.03580728599969 0.0241927140003100 32 4.05 4.06571335735176 -0.0157133573517571 33 4.03 4.02744354129046 0.00255645870953857 34 3.94 3.90841452011669 0.0315854798833081 35 4.02 3.87018178889573 0.149818211104265 36 3.88 3.97546252500505 -0.0954625250050485 37 4.02 3.91321076824911 0.106789231750894 38 4.03 4.00147392132405 0.0285260786759544 39 4.09 4.03291519275463 0.0570848072453656 40 3.99 4.03125386588466 -0.0412538658846602 41 4.01 3.92239952550321 0.0876004744967872 42 4.01 3.96672267620819 0.0432773237918087 43 4.19 4.06565681898308 0.124343181016924 44 4.3 4.23015227798258 0.0698477220174186 45 4.27 4.25465835663897 0.0153416433610332 46 3.82 4.07779989054089 -0.257799890540892 47 3.15 3.41064698144699 -0.260646981446994 48 2.49 2.54781554031771 -0.0578155403177057 49 1.81 1.69084763412794 0.119152365872065 50 1.26 1.33906888480093 -0.07906888480093 51 1.06 0.917948182793837 0.142051817206163 52 0.84 0.768292410373594 0.0717075896264055 53 0.78 0.67140687263454 0.108593127365460 54 0.7 0.701562648105538 -0.00156264810553799 55 0.36 0.597176792686524 -0.237176792686524 56 0.35 0.378910489598185 -0.028910489598185 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.0143384346741444 0.0286768693482887 0.985661565325856 20 0.00252929998960038 0.00505859997920076 0.9974707000104 21 0.000468839701643676 0.000937679403287353 0.999531160298356 22 0.000512949299928806 0.00102589859985761 0.999487050700071 23 9.58819342595953e-05 0.000191763868519191 0.99990411806574 24 2.08509321733568e-05 4.17018643467136e-05 0.999979149067827 25 4.62565115711951e-06 9.25130231423902e-06 0.999995374348843 26 7.49288453113313e-07 1.49857690622663e-06 0.999999250711547 27 6.8412628114168e-07 1.36825256228336e-06 0.999999315873719 28 1.29571307738209e-07 2.59142615476419e-07 0.999999870428692 29 6.43024642337561e-08 1.28604928467512e-07 0.999999935697536 30 2.03833527143946e-08 4.07667054287893e-08 0.999999979616647 31 4.24973295087452e-09 8.49946590174904e-09 0.999999995750267 32 2.80244334143799e-09 5.60488668287597e-09 0.999999997197557 33 7.76124823151533e-09 1.55224964630307e-08 0.999999992238752 34 2.44315118144257e-07 4.88630236288515e-07 0.999999755684882 35 5.04340426487219e-07 1.00868085297444e-06 0.999999495659573 36 1.25946645896411e-05 2.51893291792822e-05 0.99998740533541 37 3.73895208304896e-06 7.47790416609791e-06 0.999996261047917 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 18 0.947368421052632 NOK 5% type I error level 19 1 NOK 10% type I error level 19 1 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

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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