*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 06:11:30 -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/t1258723001c0tkxox6b295540.htm/, Retrieved Fri, 20 Nov 2009 14:16:59 +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/t1258723001c0tkxox6b295540.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?

No (this computation is public)

User-defined keywords:

Dataseries X:

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280 1258 557 1199 831 1158 1081 1427 1318 934 1578 709 1859 1186 2141 986 2428 1033 2715 1257 3004 1105 3309 1179 269 1092 537 1092 813 1087 1068 2028 1411 2039 1675 2010 1958 754 2242 760 2524 715 2836 855 3143 971 3522 815 285 915 574 843 865 761 1147 1858 1516 2968 1789 4061 2087 3661 2372 3269 2669 2857 2966 2568 3270 2274 3652 1987 329 683 658 381 988 71 1303 1772 1603 3485 1929 5181 2235 4479 2544 3782 2872 3067 3198 2489 3544 1903 3903 1330 332 736 665 483 1001 242 1329 1334 1639 2423 1975 3523 2304 2986 2640 2462 2992 1908 3330 1575 3690 1237 4063 904

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] = + 3580.46697505064 + 0.0879589903051959X[t] -3363.86695716855M1[t] -3052.59898369868M2[t] -2739.25415281523M3[t] -2542.97232292653M4[t] -2291.51219027590M5[t] -2063.65837622777M6[t] -1721.72140851618M7[t] -1390.73302941988M8[t] -1051.99640047540M9[t] -725.289657296368M10[t] -382.029542527825M11[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) 3580.46697505064 84.024248 42.6123 0 0 X 0.0879589903051959 0.026002 3.3828 0.001454 0.000727 M1 -3363.86695716855 109.974273 -30.5878 0 0 M2 -3052.59898369868 110.289953 -27.6779 0 0 M3 -2739.25415281523 110.714779 -24.7415 0 0 M4 -2542.97232292653 110.282907 -23.0586 0 0 M5 -2291.51219027590 113.531376 -20.184 0 0 M6 -2063.65837622777 119.809846 -17.2244 0 0 M7 -1721.72140851618 115.326853 -14.9291 0 0 M8 -1390.73302941988 112.778531 -12.3315 0 0 M9 -1051.99640047540 111.072832 -9.4712 0 0 M10 -725.289657296368 110.471349 -6.5654 0 0 M11 -382.029542527825 109.885915 -3.4766 0.001104 0.000552 Multiple Linear Regression - Regression Statistics Multiple R 0.989585570274023 R-squared 0.979279600894564 Adjusted R-squared 0.973989286229346 F-TEST (value) 185.108006397625 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 173.428308377894 Sum Squared Residuals 1413636.77290044 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 280 327.252427686036 -47.2524276860357 2 557 633.330820727895 -76.3308207278947 3 831 943.069333008828 -112.069333008828 4 1081 1163.01213128962 -82.0121312896247 5 1318 1371.1084817198 -53.1084817198011 6 1578 1579.17152294925 -1.17152294925396 7 1859 1963.06492903642 -104.064929036423 8 2141 2276.46151007168 -135.461510071683 9 2428 2619.33221156051 -191.332211560512 10 2715 2965.74176856790 -250.741768567903 11 3004 3295.63211681006 -291.632116810059 12 3309 3684.17062462047 -375.170624620467 13 269 312.651235295365 -43.6512352953646 14 537 623.919208765239 -86.9192087652394 15 813 936.82424469716 -123.824244697159 16 1068 1215.87548446305 -147.875484463049 17 1411 1468.30316600704 -57.3031660070411 18 1675 1693.60616933631 -18.6061693363131 19 1958 1925.06664522458 32.9333547754197 20 2242 2256.58277826271 -14.5827782627088 21 2524 2591.36125264346 -67.3612526434598 22 2836 2930.38225446522 -94.3822544652162 23 3143 3283.84561210916 -140.845612109162 24 3522 3652.15355214938 -130.153552149377 25 285 297.082494011345 -12.0824940113451 26 574 602.017420179246 -28.0174201792459 27 865 908.149613857665 -43.1496138576647 28 1147 1200.92245611117 -53.9224561111655 29 1516 1550.01706800057 -34.0170680005679 30 1789 1874.01005845227 -85.0100584522696 31 2087 2180.76343004178 -93.7634300417844 32 2372 2477.27188493845 -105.271884938445 33 2669 2779.76940987719 -110.769409877189 34 2966 3081.05600485802 -115.056004858017 35 3270 3398.45617647683 -128.456176476832 36 3652 3755.24148878707 -103.241488787066 37 329 276.676008260540 52.3239917394605 38 658 561.380366658245 96.6196333417547 39 988 847.45791054708 140.54208945292 40 1303 1193.35798294492 109.642017055082 41 1603 1595.49186598835 7.50813401164582 42 1929 1972.52412759409 -43.5241275940891 43 2235 2252.71388411143 -17.7138841114348 44 2544 2522.39484696501 21.6051530349895 45 2872 2798.24079784128 73.7592021587195 46 3198 3074.10724462391 123.892755376094 47 3544 3365.82339107360 178.176608926396 48 3903 3697.45243215655 205.547567843448 49 332 281.337834746715 50.662165253285 50 665 570.352183669375 94.6478163306251 51 1001 862.498897889269 138.501102110731 52 1329 1154.83194519124 174.168054808757 53 1639 1502.07941828424 136.920581715764 54 1975 1826.68812166807 148.311878331926 55 2304 2121.39111158578 182.608888414223 56 2640 2406.28897976215 233.711020237847 57 2992 2696.29632807756 295.703671922442 58 3330 2993.71272748496 336.287272515043 59 3690 3307.24270353034 382.757296469656 60 4063 3659.98190228654 403.018097713461 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.000691237451845648 0.00138247490369130 0.999308762548154 17 0.000534159291129249 0.00106831858225850 0.99946584070887 18 6.29723796259271e-05 0.000125944759251854 0.999937027620374 19 0.000835645735451927 0.00167129147090385 0.999164354264548 20 0.00102646015224099 0.00205292030448197 0.99897353984776 21 0.00123935926432910 0.00247871852865820 0.99876064073567 22 0.00290586007059366 0.00581172014118733 0.997094139929406 23 0.0105938190795929 0.0211876381591858 0.989406180920407 24 0.102734680565659 0.205469361131317 0.897265319434341 25 0.061596910629788 0.123193821259576 0.938403089370212 26 0.035982915486402 0.071965830972804 0.964017084513598 27 0.0211380832705190 0.0422761665410379 0.97886191672948 28 0.0167281591123491 0.0334563182246982 0.98327184088765 29 0.0192556506932093 0.0385113013864186 0.98074434930679 30 0.0186140516727357 0.0372281033454714 0.981385948327264 31 0.0179607369253205 0.0359214738506410 0.98203926307468 32 0.0200589111066934 0.0401178222133867 0.979941088893307 33 0.0313340386381979 0.0626680772763958 0.968665961361802 34 0.0744344726366698 0.148868945273340 0.92556552736333 35 0.342573110310403 0.685146220620805 0.657426889689597 36 0.95424628746962 0.0915074250607585 0.0457537125303792 37 0.928900269434735 0.142199461130530 0.0710997305652651 38 0.916509331813383 0.166981336373235 0.0834906681866174 39 0.922567163833026 0.154865672333949 0.0774328361669743 40 0.905770296930453 0.188459406139095 0.0942297030695473 41 0.853847370026087 0.292305259947826 0.146152629973913 42 0.842956609685492 0.314086780629017 0.157043390314508 43 0.831563952040514 0.336872095918972 0.168436047959486 44 0.806960183101005 0.38607963379799 0.193039816898995 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 7 0.241379310344828 NOK 5% type I error level 14 0.482758620689655 NOK 10% type I error level 17 0.586206896551724 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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