Paper - multiple linear regression3

*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: Sun, 21 Dec 2008 07:32:44 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/21/t1229870083jisxl3caax5q57o.htm/, Retrieved Sun, 21 Dec 2008 15:34:43 +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/2008/Dec/21/t1229870083jisxl3caax5q57o.htm/}, year = {2008}, } @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 = {2008}, 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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2490 0 3266 0 3475 0 3127 0 2955 0 3870 0 2852 0 3142 0 3029 0 3180 0 2560 0 2733 0 2452 0 2553 0 2777 0 2520 0 2318 0 2873 0 2311 0 2395 0 2099 0 2268 0 2316 0 2181 0 2175 0 2627 0 2578 0 3090 0 2634 0 3225 0 2938 0 3174 0 3350 0 2588 0 2061 0 2691 0 2061 0 2918 0 2223 0 2651 0 2379 0 3146 0 2883 0 2768 0 3258 0 2839 0 2470 0 5072 1 1463 1 1600 1 2203 1 2013 1 2169 1 2640 1 2411 1 2528 1 2292 1 1988 1 1774 1 2279 1

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 6 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 3379.88904347826 -165.048695652174X[t] -994.603217391305M1[t] -521.040173913043M2[t] -453.677130434783M3[t] -415.714086956522M4[t] -595.951043478261M5[t] + 72.812M6[t] -390.024956521739M7[t] -258.661913043478M8[t] -245.498869565217M9[t] -469.535826086956M10[t] -796.972782608696M11[t] -8.96304347826086t + 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) 3379.88904347826 272.265907 12.4139 0 0 X -165.048695652174 232.754084 -0.7091 0.481834 0.240917 M1 -994.603217391305 325.048468 -3.0599 0.003688 0.001844 M2 -521.040173913043 324.614152 -1.6051 0.115315 0.057657 M3 -453.677130434783 324.275949 -1.399 0.168506 0.084253 M4 -415.714086956522 324.034159 -1.2829 0.205942 0.102971 M5 -595.951043478261 323.888998 -1.84 0.072228 0.036114 M6 72.812 323.840597 0.2248 0.8231 0.41155 M7 -390.024956521739 323.888998 -1.2042 0.234675 0.117337 M8 -258.661913043478 324.034159 -0.7983 0.428824 0.214412 M9 -245.498869565217 324.275949 -0.7571 0.45287 0.226435 M10 -469.535826086956 324.614152 -1.4464 0.154833 0.077417 M11 -796.972782608696 325.048468 -2.4519 0.018068 0.009034 t -8.96304347826086 5.599193 -1.6008 0.116273 0.058137 Multiple Linear Regression - Regression Statistics Multiple R 0.608437199909348 R-squared 0.370195826233528 Adjusted R-squared 0.192207690169091 F-TEST (value) 2.0798904602243 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0.0344943095704632 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 509.495800539598 Sum Squared Residuals 11940954.6553044 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2490 2376.3227826087 113.677217391302 2 3266 2840.92278260870 425.077217391305 3 3475 2899.32278260870 575.677217391304 4 3127 2928.32278260870 198.677217391304 5 2955 2739.12278260870 215.877217391305 6 3870 3398.92278260870 471.077217391305 7 2852 2927.12278260870 -75.1227826086955 8 3142 3049.52278260870 92.4772173913045 9 3029 3053.72278260870 -24.7227826086954 10 3180 2820.72278260870 359.277217391305 11 2560 2484.32278260870 75.6772173913045 12 2733 3272.33252173913 -539.33252173913 13 2452 2268.76626086956 183.233739130436 14 2553 2733.36626086957 -180.366260869565 15 2777 2791.76626086957 -14.7662608695651 16 2520 2820.76626086957 -300.766260869565 17 2318 2631.56626086957 -313.566260869565 18 2873 3291.36626086956 -418.366260869565 19 2311 2819.56626086957 -508.566260869565 20 2395 2941.96626086957 -546.966260869565 21 2099 2946.16626086957 -847.166260869565 22 2268 2713.16626086957 -445.166260869565 23 2316 2376.76626086957 -60.766260869565 24 2181 3164.776 -983.776 25 2175 2161.20973913043 13.7902608695659 26 2627 2625.80973913043 1.19026086956512 27 2578 2684.20973913043 -106.209739130435 28 3090 2713.20973913043 376.790260869565 29 2634 2524.00973913043 109.990260869565 30 3225 3183.80973913043 41.1902608695651 31 2938 2712.00973913043 225.990260869565 32 3174 2834.40973913043 339.590260869565 33 3350 2838.60973913043 511.390260869565 34 2588 2605.60973913043 -17.6097391304349 35 2061 2269.20973913043 -208.209739130435 36 2691 3057.21947826087 -366.219478260870 37 2061 2053.65321739130 7.34678260869619 38 2918 2518.25321739130 399.746782608696 39 2223 2576.65321739130 -353.653217391305 40 2651 2605.65321739130 45.3467826086955 41 2379 2416.45321739130 -37.4532173913046 42 3146 3076.25321739130 69.7467826086955 43 2883 2604.45321739130 278.546782608696 44 2768 2726.85321739130 41.1467826086955 45 3258 2731.05321739130 526.946782608695 46 2839 2498.05321739130 340.946782608695 47 2470 2161.65321739130 308.346782608696 48 5072 2784.61426086956 2287.38573913044 49 1463 1781.048 -318.047999999999 50 1600 2245.648 -645.648 51 2203 2304.048 -101.048000000000 52 2013 2333.048 -320.048 53 2169 2143.848 25.1519999999998 54 2640 2803.648 -163.648000000000 55 2411 2331.848 79.152 56 2528 2454.248 73.7519999999999 57 2292 2458.448 -166.448 58 1988 2225.448 -237.448 59 1774 1889.048 -115.048000000000 60 2279 2677.05773913044 -398.057739130435 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0833554552114319 0.166710910422864 0.916644544788568 18 0.057795949060125 0.11559189812025 0.942204050939875 19 0.0213856948345541 0.0427713896691082 0.978614305165446 20 0.00792956894218911 0.0158591378843782 0.99207043105781 21 0.00501726630165043 0.0100345326033009 0.99498273369835 22 0.00233414802896307 0.00466829605792613 0.997665851971037 23 0.00180661333762281 0.00361322667524563 0.998193386662377 24 0.00215633029390282 0.00431266058780564 0.997843669706097 25 0.0051565329187928 0.0103130658375856 0.994843467081207 26 0.00572082065396532 0.0114416413079306 0.994279179346035 27 0.0027348306299748 0.0054696612599496 0.997265169370025 28 0.0137186226112566 0.0274372452225133 0.986281377388743 29 0.0127107000905672 0.0254214001811345 0.987289299909433 30 0.0081317661098247 0.0162635322196494 0.991868233890175 31 0.0127090396159899 0.0254180792319797 0.98729096038401 32 0.0152579275936099 0.0305158551872197 0.98474207240639 33 0.0287236365499042 0.0574472730998083 0.971276363450096 34 0.0178416174444520 0.0356832348889040 0.982158382555548 35 0.0223491705779931 0.0446983411559863 0.977650829422007 36 0.911939598166415 0.176120803667171 0.0880604018335855 37 0.85384296035681 0.292314079286380 0.146157039643190 38 0.946040262095966 0.107919475808067 0.0539597379040337 39 0.968842279919971 0.0623154401600579 0.0311577200800289 40 0.93356434905516 0.132871301889681 0.0664356509448407 41 0.929968027621635 0.140063944756731 0.0700319723783653 42 0.859959419514091 0.280081160971817 0.140040580485909 43 0.766838337973583 0.466323324052834 0.233161662026417 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.148148148148148 NOK 5% type I error level 16 0.592592592592593 NOK 10% type I error level 18 0.666666666666667 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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