multiple regression

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

Title produced by software: Multiple Regression

Date of computation: Fri, 09 May 2008 13:45:35 -0600

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/May/09/t1210362429zv5oy7u6jj4c4s1.htm/, Retrieved Fri, 09 May 2008 21:47:21 +0200

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Dataseries X:

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99.2 96.7 101.0 56421 53152 53536 52408 41454 38271 35306 26414 31917 38030 27534 18387 50556 43901 48572 43899 37532 40357 35489 29027 34485 42598 30306 26451 47460 50104 61465 53726 39477 43895 31481 29896 33842 39120 33702 25094 51442 45594 52518 48564 41745 49585 32747 33379 35645 37034 35681 20972 58552 54955 65540 51570 51145 46641 35704 33253 35193 41668 34865 21210 56126 49231 59723 48103 47472 50497 40059 34149 36860 46356 36577

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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 Cons[t] = + 24299.5343115768 + 0.738824388900419Inc[t] -0.0898517536329588Price[t] -6536.71567957236M1[t] -531.990415969853M2[t] -14141.2818594424M3[t] -3650.53062245462M4[t] -3802.69363891778M5[t] -889.32270429815M6[t] -13992.8694521565M7[t] + 366.865286563432M8[t] -1985.93931189468M9[t] -2692.52241802025M10[t] -10634.5167019668M11[t] -150.070877000498t + 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) 24299.5343115768 8122.312827 2.9917 0.004094 0.002047 Inc 0.738824388900419 0.147048 5.0244 5e-06 3e-06 Price -0.0898517536329588 0.142958 -0.6285 0.532176 0.266088 M1 -6536.71567957236 7042.134671 -0.9282 0.357203 0.178601 M2 -531.990415969853 7098.415877 -0.0749 0.940521 0.47026 M3 -14141.2818594424 6818.992721 -2.0738 0.042626 0.021313 M4 -3650.53062245462 6668.210782 -0.5475 0.586205 0.293102 M5 -3802.69363891778 6820.323413 -0.5576 0.579333 0.289666 M6 -889.32270429815 6957.058183 -0.1278 0.898733 0.449367 M7 -13992.8694521565 6809.353296 -2.0549 0.044477 0.022239 M8 366.865286563432 6656.549033 0.0551 0.956241 0.47812 M9 -1985.93931189468 6757.357127 -0.2939 0.769907 0.384954 M10 -2692.52241802025 6890.945601 -0.3907 0.697451 0.348725 M11 -10634.5167019668 6835.609448 -1.5558 0.125302 0.062651 t -150.070877000498 68.709114 -2.1841 0.03308 0.01654 Multiple Linear Regression - Regression Statistics Multiple R 0.623675279460502 R-squared 0.388970854210135 Adjusted R-squared 0.238893520156484 F-TEST (value) 2.59180279729038 F-TEST (DF numerator) 14 F-TEST (DF denominator) 57 p-value 0.00581633877315157 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 11507.7072810966 Sum Squared Residuals 7548357631.442 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 99.2 17675.1170462937 -17575.9170462937 2 56421 57927.0925779469 -1506.09257794688 3 52408 36896.5495753239 15511.4504246761 4 35306 36696.2291688327 -1390.22916883267 5 38030 38437.1728175914 -407.17281759142 6 50556 50580.6364649329 -24.63646493287 7 43899 33359.578463262 10539.4215367380 8 35489 41813.1503947161 -6324.15039471609 9 42598 40977.1003013483 1620.89969865170 10 47460 51601.6222679683 -4141.62226796833 11 53726 37236.7656375077 16489.2343624923 12 31481 41545.8146716911 -10064.8146716911 13 39120 38457.0168800544 662.983119945612 14 51442 50633.6764078299 808.323592170087 15 48564 34294.1142078846 14269.8857921154 16 32747 39706.3231759744 -6959.32317597444 17 37034 42423.2578068159 -5389.25780681595 18 58552 55422.1461801881 3129.85381981194 19 51570 41051.7159255279 10518.2840744721 20 35704 43070.9566966311 -7366.95669663114 21 41668 43015.4632071297 -1347.46320712967 22 56126 49312.2998072809 6813.70019271911 23 48103 40749.6148252757 7353.38517472428 24 40059 42616.0116812144 -2557.01168121435 25 46356 41026.1130858422 5329.88691415782 26 96.7 14870.7965651477 -14774.0965651477 27 53152 40951.0905528977 12200.9094471023 28 41454 41550.2613069509 -96.2613069508825 29 26414 36308.7810695178 -9894.78106951778 30 27534 27950.3040793078 -416.304079307817 31 43901 37596.2437573427 6304.7562426573 32 37532 46492.1185122984 -8960.11851229838 33 29027 39012.1101086398 -9985.11010863982 34 30306 31782.8817589243 -1476.88175892434 35 50104 48997.0026626725 1106.99733732752 36 39477 48499.0562342235 -9022.05623422354 37 29896 33698.4905500326 -3802.4905500326 38 33702 31982.7558742684 1719.24412573157 39 45594 38743.5069419561 6850.49305804386 40 41745 48338.400556511 -6593.40055651098 41 33379 37351.760213951 -3972.76021395099 42 35681 27340.8599785603 8340.14002143973 43 54955 47642.5126620806 7312.4873379194 44 51145 49314.7223211115 1830.27767888845 45 33253 37817.909382854 -4564.90938285398 46 34865 25331.1973157080 9533.80268429195 47 49231 46414.3564638801 2816.64353611991 48 47472 50805.1759830746 -3333.17598307460 49 34149 33477.24474244 671.755257560005 50 36577 16328.6027603846 20248.3972396154 51 101 39414.0481621605 -39313.0481621605 52 53536 47840.9120634887 5695.08793651129 53 38271 36254.6738456898 2016.32615431021 54 31917 40929.8975746047 -9012.89757460474 55 18387 35460.1905934019 -17073.1905934019 56 48572 45323.7663170996 3248.23368290043 57 40357 37371.5668956368 2985.43310436325 58 34485 41652.2951003072 -7167.29510030721 59 26451 35373.5090997687 -8922.50909976873 60 61465 51442.2831314425 10022.7168685575 61 43895 29181.2176953372 14713.7823046628 62 33842 40337.7758144224 -6495.77581442242 63 25094 34613.6905597772 -9519.69055977722 64 52518 43173.8737282423 9344.12627175767 65 49585 31937.3542464341 17647.6457535659 66 35645 37661.1557224062 -2016.15572240624 67 20972 38573.7585983849 -17601.7585983849 68 65540 47967.2857581432 17572.7142418568 69 46641 35349.8501043915 11291.1498956085 70 35193 38754.7037498112 -3561.70374981118 71 21210 40053.7513108953 -18843.7513108953 72 59723 44768.6582983539 14954.3417016461 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.0734548733840805 0.146909746768161 0.92654512661592 19 0.024570606326046 0.049141212652092 0.975429393673954 20 0.00786481872837341 0.0157296374567468 0.992135181271627 21 0.00228542299153647 0.00457084598307294 0.997714577008464 22 0.00888398416530918 0.0177679683306184 0.99111601583469 23 0.00958403954244728 0.0191680790848946 0.990415960457553 24 0.00584626431091812 0.0116925286218362 0.994153735689082 25 0.00401524628049521 0.00803049256099043 0.995984753719505 26 0.00260475718426234 0.00520951436852467 0.997395242815738 27 0.0032012780129711 0.0064025560259422 0.996798721987029 28 0.00135899180092118 0.00271798360184236 0.99864100819908 29 0.00078310205880556 0.00156620411761112 0.999216897941194 30 0.00045406077358185 0.0009081215471637 0.999545939226418 31 0.000360292711119294 0.000720585422238589 0.99963970728888 32 0.000236432829687751 0.000472865659375503 0.999763567170312 33 0.000169012636286791 0.000338025272573583 0.999830987363713 34 6.58128602248873e-05 0.000131625720449775 0.999934187139775 35 0.000227594891618054 0.000455189783236109 0.999772405108382 36 0.000176673889465311 0.000353347778930622 0.999823326110535 37 0.000111222073812905 0.000222444147625809 0.999888777926187 38 0.000107089580418926 0.000214179160837853 0.99989291041958 39 0.000663676952134049 0.00132735390426810 0.999336323047866 40 0.000496400368522685 0.000992800737045369 0.999503599631477 41 0.000337056213309491 0.000674112426618982 0.99966294378669 42 0.000502887572900963 0.00100577514580193 0.999497112427099 43 0.0118424860693377 0.0236849721386754 0.988157513930662 44 0.00939035314105762 0.0187807062821152 0.990609646858942 45 0.00568570546492124 0.0113714109298425 0.994314294535079 46 0.00784797186954401 0.0156959437390880 0.992152028130456 47 0.330114182462494 0.660228364924989 0.669885817537506 48 0.256355553992289 0.512711107984577 0.743644446007711 49 0.191466177692932 0.382932355385864 0.808533822307068 50 0.192217462686041 0.384434925372082 0.807782537313959 51 0.981122836153604 0.0377543276927926 0.0188771638463963 52 0.955912789734142 0.088174420531716 0.044087210265858 53 0.947901668220097 0.104196663559805 0.0520983317799027 54 0.876025991878788 0.247948016242424 0.123974008121212 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 19 0.513513513513513 NOK 5% type I error level 29 0.783783783783784 NOK 10% type I error level 30 0.810810810810811 NOK

Charts produced by software:

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Parameters (Session):

par1 = 71 ; par2 = Do not include Seasonal Dummies ; par3 = No 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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