Paper - multiple regression analysis (2)

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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Fri, 04 Dec 2009 04:01:55 -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/04/t1259924589rj1o4jg78w6glz8.htm/, Retrieved Fri, 04 Dec 2009 12:03:24 +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/04/t1259924589rj1o4jg78w6glz8.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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User-defined keywords:

Dataseries X:

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216234 213587 209465 204045 200237 203666 241476 260307 243324 244460 233575 237217 235243 230354 227184 221678 217142 219452 256446 265845 248624 241114 229245 231805 219277 219313 212610 214771 211142 211457 240048 240636 230580 208795 197922 194596 194581 185686 178106 172608 167302 168053 202300 202388 182516 173476 166444 171297 169701 164182 161914 159612 151001 158114 186530 187069 174330 169362 166827 178037 186412 189226 191563 188906 186005 195309 223532 226899 214126

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Werkl[t] = + 237091.616 -3807.56888888884M1[t] -6032.53511111112M2[t] -8658.50133333337M3[t] -10903.8008888889M4[t] -14743.9337777778M5[t] -9915.23333333334M6[t] + 23423.3004444444M7[t] + 29850.3342222222M8[t] + 15868.0346666666M9[t] + 2934.26577777775M10[t] -4746.16711111114M11[t] -958.367111111112t + 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) 237091.616 9352.044003 25.3518 0 0 M1 -3807.56888888884 11385.813789 -0.3344 0.739317 0.369658 M2 -6032.53511111112 11380.650284 -0.5301 0.598159 0.29908 M3 -8658.50133333337 11376.632605 -0.7611 0.449804 0.224902 M4 -10903.8008888889 11373.761965 -0.9587 0.341841 0.17092 M5 -14743.9337777778 11372.039233 -1.2965 0.200118 0.100059 M6 -9915.23333333334 11371.464932 -0.8719 0.386964 0.193482 M7 23423.3004444444 11372.039233 2.0597 0.044084 0.022042 M8 29850.3342222222 11373.761965 2.6245 0.011164 0.005582 M9 15868.0346666666 11376.632605 1.3948 0.168587 0.084293 M10 2934.26577777775 11879.307021 0.247 0.805807 0.402903 M11 -4746.16711111114 11877.657614 -0.3996 0.69098 0.34549 t -958.367111111112 114.287523 -8.3856 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.802972043958226 R-squared 0.644764103378451 Adjusted R-squared 0.568642125530976 F-TEST (value) 8.47014386134791 F-TEST (DF numerator) 12 F-TEST (DF denominator) 56 p-value 7.57113860494485e-09 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 18779.3562689802 Sum Squared Residuals 19749196425.128 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 216234 232325.680000000 -16091.6799999996 2 213587 229142.346666667 -15555.3466666667 3 209465 225558.013333333 -16093.0133333334 4 204045 222354.346666667 -18309.3466666667 5 200237 217555.846666667 -17318.8466666666 6 203666 221426.18 -17760.1800000000 7 241476 253806.346666667 -12330.3466666667 8 260307 259275.013333333 1031.98666666669 9 243324 244334.346666667 -1010.34666666669 10 244460 230442.210666667 14017.7893333333 11 233575 221803.410666667 11771.5893333333 12 237217 225591.210666667 11625.7893333333 13 235243 220825.274666667 14417.7253333332 14 230354 217641.941333333 12712.0586666667 15 227184 214057.608 13126.392 16 221678 210853.941333333 10824.0586666667 17 217142 206055.441333333 11086.5586666667 18 219452 209925.774666667 9526.22533333333 19 256446 242305.941333333 14140.0586666667 20 265845 247774.608 18070.392 21 248624 232833.941333333 15790.0586666667 22 241114 218941.805333333 22172.1946666667 23 229245 210303.005333333 18941.9946666666 24 231805 214090.805333333 17714.1946666666 25 219277 209324.869333333 9952.13066666658 26 219313 206141.536 13171.464 27 212610 202557.202666667 10052.7973333333 28 214771 199353.536 15417.464 29 211142 194555.036 16586.964 30 211457 198425.369333333 13031.6306666667 31 240048 230805.536 9242.464 32 240636 236274.202666667 4361.79733333332 33 230580 221333.536 9246.464 34 208795 207441.4 1353.60000000000 35 197922 198802.6 -880.600000000006 36 194596 202590.4 -7994.40000000002 37 194581 197824.464 -3243.46400000007 38 185686 194641.130666667 -8955.13066666666 39 178106 191056.797333333 -12950.7973333333 40 172608 187853.130666667 -15245.1306666667 41 167302 183054.630666667 -15752.6306666667 42 168053 186924.964 -18871.9640000000 43 202300 219305.130666667 -17005.1306666667 44 202388 224773.797333333 -22385.7973333333 45 182516 209833.130666667 -27317.1306666667 46 173476 195940.994666667 -22464.9946666666 47 166444 187302.194666667 -20858.1946666667 48 171297 191089.994666667 -19792.9946666667 49 169701 186324.058666667 -16623.0586666667 50 164182 183140.725333333 -18958.7253333333 51 161914 179556.392 -17642.392 52 159612 176352.725333333 -16740.7253333333 53 151001 171554.225333333 -20553.2253333333 54 158114 175424.558666667 -17310.5586666666 55 186530 207804.725333333 -21274.7253333333 56 187069 213273.392 -26204.392 57 174330 198332.725333333 -24002.7253333333 58 169362 184440.589333333 -15078.5893333333 59 166827 175801.789333333 -8974.78933333332 60 178037 179589.589333333 -1552.58933333334 61 186412 174823.653333333 11588.3466666666 62 189226 171640.32 17585.6800000000 63 191563 168055.986666667 23507.0133333334 64 188906 164852.32 24053.68 65 186005 160053.82 25951.18 66 195309 163924.153333333 31384.8466666667 67 223532 196304.32 27227.6800000000 68 226899 201772.986666667 25126.0133333333 69 214126 186832.32 27293.6800000000 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 4.81719072650606e-05 9.63438145301211e-05 0.999951828092735 17 1.92620033663612e-06 3.85240067327224e-06 0.999998073799663 18 2.37716964781009e-07 4.75433929562018e-07 0.999999762283035 19 4.40947225369467e-08 8.81894450738934e-08 0.999999955905277 20 9.89883520438864e-06 1.97976704087773e-05 0.999990101164796 21 1.18608325443965e-05 2.37216650887929e-05 0.999988139167456 22 8.7311053019765e-05 0.00017462210603953 0.99991268894698 23 0.000177701788075052 0.000355403576150104 0.999822298211925 24 0.000258133523568103 0.000516267047136207 0.999741866476432 25 0.000629513490914568 0.00125902698182914 0.999370486509085 26 0.000506711584784105 0.00101342316956821 0.999493288415216 27 0.000434074466372467 0.000868148932744935 0.999565925533628 28 0.000234649942516538 0.000469299885033075 0.999765350057483 29 0.000142048857538833 0.000284097715077666 0.999857951142461 30 9.03053700711777e-05 0.000180610740142355 0.999909694629929 31 0.000120142945931869 0.000240285891863739 0.999879857054068 32 0.000833522401571682 0.00166704480314336 0.999166477598428 33 0.00248584900830683 0.00497169801661367 0.997514150991693 34 0.0296810740746778 0.0593621481493557 0.970318925925322 35 0.119797891419957 0.239595782839915 0.880202108580043 36 0.290196836751097 0.580393673502195 0.709803163248903 37 0.389580988325621 0.779161976651242 0.610419011674379 38 0.485535941829383 0.971071883658767 0.514464058170617 39 0.539775139171115 0.92044972165777 0.460224860828885 40 0.581476991333973 0.837046017332055 0.418523008666027 41 0.63204368381623 0.735912632367539 0.367956316183769 42 0.637765801914518 0.724468396170963 0.362234198085482 43 0.702900932275321 0.594198135449358 0.297099067724679 44 0.812865213054016 0.374269573891968 0.187134786945984 45 0.887791572604205 0.224416854791591 0.112208427395795 46 0.957989501562597 0.0840209968748064 0.0420104984374032 47 0.989215642287367 0.0215687154252660 0.0107843577126330 48 0.99848968052852 0.00302063894295833 0.00151031947147916 49 0.99967371579495 0.000652568410098678 0.000326284205049339 50 0.999744859315168 0.000510281369664209 0.000255140684832104 51 0.99948575637418 0.00102848725163892 0.000514243625819461 52 0.999842097940794 0.000315804118411808 0.000157902059205904 53 0.999477173030987 0.0010456539380258 0.0005228269690129 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 24 0.631578947368421 NOK 5% type I error level 25 0.657894736842105 NOK 10% type I error level 27 0.710526315789474 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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