*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, 20 Dec 2009 05:56:21 -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/20/t1261317869sjn2k1j62v0qlr1.htm/, Retrieved Sun, 20 Dec 2009 15:04:41 +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/20/t1261317869sjn2k1j62v0qlr1.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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2849,27 10872 2921,44 10625 2981,85 10407 3080,58 10463 3106,22 10556 3119,31 10646 3061,26 10702 3097,31 11353 3161,69 11346 3257,16 11451 3277,01 11964 3295,32 12574 3363,99 13031 3494,17 13812 3667,03 14544 3813,06 14931 3917,96 14886 3895,51 16005 3801,06 17064 3570,12 15168 3701,61 16050 3862,27 15839 3970,1 15137 4138,52 14954 4199,75 15648 4290,89 15305 4443,91 15579 4502,64 16348 4356,98 15928 4591,27 16171 4696,96 15937 4621,4 15713 4562,84 15594 4202,52 15683 4296,49 16438 4435,23 17032 4105,18 17696 4116,68 17745 3844,49 19394 3720,98 20148 3674,4 20108 3857,62 18584 3801,06 18441 3504,37 18391 3032,6 19178 3047,03 18079 2962,34 18483 2197,82 19644 2014,45 19195 1862,83 19650 1905,41 20830 1810,99 23595 1670,07 22937 1864,44 21814 2052,02 21928 2029,6 21777 2070,83 21383 2293,41 21467 2443,27 22052 2513,17 22680

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 Y[t] = + 4631.31499317083 -0.0601286484348822X[t] -217.697070655946M1[t] -171.152424772174M2[t] -88.8065967430589M3[t] -7.2881058426518M4[t] -53.1668728563952M5[t] + 60.4791439189899M6[t] + 95.079829363615M7[t] -35.4023754623143M8[t] -72.7180483006568M9[t] -51.051837586295M10[t] + 32.5249358282749M11[t] -7.51276375132147t + 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) 4631.31499317083 1380.136138 3.3557 0.001595 0.000797 X -0.0601286484348822 0.130735 -0.4599 0.647736 0.323868 M1 -217.697070655946 562.582197 -0.387 0.70057 0.350285 M2 -171.152424772174 561.439817 -0.3048 0.761859 0.380929 M3 -88.8065967430589 566.524246 -0.1568 0.876122 0.438061 M4 -7.2881058426518 587.798332 -0.0124 0.990161 0.49508 M5 -53.1668728563952 572.844379 -0.0928 0.926456 0.463228 M6 60.4791439189899 562.567538 0.1075 0.914855 0.457428 M7 95.079829363615 561.594902 0.1693 0.8663 0.43315 M8 -35.4023754623143 557.946298 -0.0635 0.949682 0.474841 M9 -72.7180483006568 557.621901 -0.1304 0.896813 0.448406 M10 -51.051837586295 560.624488 -0.0911 0.927838 0.463919 M11 32.5249358282749 559.201149 0.0582 0.953871 0.476935 t -7.51276375132147 27.799471 -0.2702 0.788178 0.394089 Multiple Linear Regression - Regression Statistics Multiple R 0.409740584844696 R-squared 0.167887346868874 Adjusted R-squared -0.0672749246681839 F-TEST (value) 0.713921267095847 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0.740144560883919 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 881.049552609896 Sum Squared Residuals 35707422.4510885 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2849.27 3752.38649297954 -903.116492979536 2 2921.44 3806.27015127539 -884.830151275392 3 2981.85 3894.21126091199 -912.361260911992 4 3080.58 3964.84978374872 -884.269783748724 5 3106.22 3905.86628867921 -799.646288679215 6 3119.31 4006.58796334414 -887.27796334414 7 3061.26 4030.30868072509 -969.048680725088 8 3097.31 3853.16996201673 -755.85996201673 9 3161.69 3808.76242596611 -647.07242596611 10 3257.16 3816.60236484349 -559.442364843488 11 3277.01 3861.82037785964 -584.810377859641 12 3295.32 3785.10420273477 -489.784202734767 13 3363.99 3532.41557599276 -168.425575992759 14 3494.17 3524.48698369757 -30.3169836975661 15 3667.03 3555.30587732103 111.724122678974 16 3813.06 3606.04181752581 207.018182474188 17 3917.96 3555.35607594032 362.603924059683 18 3895.51 3594.20537136575 301.304628634253 19 3801.06 3557.61705436651 243.442945633489 20 3570.12 3533.6260032218 36.4939967782032 21 3701.61 3435.76409871257 265.845901287433 22 3862.27 3462.60469049537 399.665309504633 23 3970.1 3580.8790113599 389.220988640097 24 4138.52 3551.84485444389 586.67514555611 25 4199.75 3284.90573802281 914.844261977185 26 4290.89 3344.56174656843 946.32825343157 27 4443.91 3402.91956117507 1040.99043882493 28 4502.64 3430.68635767773 1071.95364232227 29 4356.98 3402.54885925531 954.431140744687 30 4591.27 3494.0708507097 1097.1991492903 31 4696.96 3535.22887613677 1161.73112386323 32 4621.4 3410.70272480893 1210.69727519107 33 4562.84 3373.02959738302 1189.81040261698 34 4202.52 3381.83159463535 820.68840536465 35 4296.49 3412.49847473026 883.991525269736 36 4435.23 3336.74435798035 1098.48564201965 37 4105.18 3071.60910101232 1033.57089898768 38 4116.68 3107.69467937146 1008.98532062854 39 3844.49 3083.37560238013 761.114397619867 40 3720.98 3112.04432860932 608.935671390683 41 3674.4 3061.05794378165 613.342056218353 42 3857.62 3258.82725702047 598.792742979528 43 3801.06 3294.51357543996 506.546424560037 44 3504.37 3159.52503928446 344.844960715543 45 3032.6 3067.37535637654 -34.7753563765403 46 3047.03 3147.61018796952 -100.580187969516 47 2962.34 3199.38222366507 -237.042223665072 48 2197.82 3089.53516325258 -891.715163252577 49 2014.45 2891.32309199257 -876.873091992573 50 1862.83 2902.99643908715 -1040.16643908715 51 1905.41 2906.87769821178 -1001.46769821178 52 1810.99 2814.62771243842 -1003.63771243842 53 1670.07 2800.80083234351 -1130.73083234351 54 1864.44 2974.45855755994 -1110.01855755994 55 2052.02 2994.69181333167 -942.671813331672 56 2029.6 2865.77627066809 -836.176270668088 57 2070.83 2844.63852156177 -773.808521561767 58 2293.41 2853.74116205628 -560.331162056278 59 2443.27 2894.62991238512 -451.35991238512 60 2513.17 2816.83142158842 -303.661421588418 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.000488768515828888 0.000977537031657775 0.999511231484171 18 5.3221383395391e-05 0.000106442766790782 0.999946778616605 19 2.19248023966544e-05 4.38496047933089e-05 0.999978075197603 20 2.32176408237918e-05 4.64352816475837e-05 0.999976782359176 21 9.07265228488631e-06 1.81453045697726e-05 0.999990927347715 22 1.53370269669279e-06 3.06740539338559e-06 0.999998466297303 23 7.78308941575227e-07 1.55661788315045e-06 0.999999221691058 24 3.91318480141808e-06 7.82636960283616e-06 0.999996086815199 25 2.23679678373037e-06 4.47359356746073e-06 0.999997763203216 26 9.04963288439733e-07 1.80992657687947e-06 0.999999095036712 27 2.36807659732578e-07 4.73615319465157e-07 0.99999976319234 28 4.35901401632557e-08 8.71802803265114e-08 0.99999995640986 29 3.95505502647736e-08 7.91011005295471e-08 0.99999996044945 30 9.07559482674129e-09 1.81511896534826e-08 0.999999990924405 31 9.0771130791594e-09 1.81542261583188e-08 0.999999990922887 32 6.16052438348609e-09 1.23210487669722e-08 0.999999993839476 33 1.20528537405724e-09 2.41057074811448e-09 0.999999998794715 34 4.95503983617461e-08 9.91007967234921e-08 0.999999950449602 35 2.14635864996637e-07 4.29271729993275e-07 0.999999785364135 36 1.75790866556913e-07 3.51581733113827e-07 0.999999824209133 37 5.90826699221808e-06 1.18165339844362e-05 0.999994091733008 38 6.02460258359048e-05 0.000120492051671810 0.999939753974164 39 0.00125789819497254 0.00251579638994507 0.998742101805028 40 0.00862799624636901 0.0172559924927380 0.991372003753631 41 0.0328857795216561 0.0657715590433122 0.967114220478344 42 0.0923351507943564 0.184670301588713 0.907664849205644 43 0.17571125481271 0.35142250962542 0.82428874518729 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 23 0.851851851851852 NOK 5% type I error level 24 0.888888888888889 NOK 10% type I error level 25 0.925925925925926 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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