Multiple Regression

*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: Tue, 14 Dec 2010 19:12:52 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/14/t1292353915c7gjp5p255g7oeu.htm/, Retrieved Tue, 14 Dec 2010 20:14:05 +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/2010/Dec/14/t1292353915c7gjp5p255g7oeu.htm/}, year = {2010}, } @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 = {2010}, 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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6000 10800 10100 16100 17700 13900 17700 6000 10900 10000 15800 17700 13500 19800 6000 11000 10000 16900 17700 13900 19400 6000 11000 10000 17800 17700 13700 18500 6000 11100 10600 17600 17400 13800 18400 6000 11000 12200 18300 17800 15100 18200 6000 11000 12400 18000 17800 15100 18300 6000 11100 13400 15700 17800 14500 19100 6100 11100 13000 14500 17800 13000 18500 6100 11100 10500 14000 18100 12900 18100 6100 11100 10000 15500 18400 14400 18300 6100 11100 10000 15800 18000 14600 17900 6100 11200 10100 15800 17800 15000 18000 6100 11100 10200 15900 17600 13900 18200 6200 11100 10600 18000 17400 14800 18800 6200 11200 10900 19900 17200 15200 20100 6200 11200 10900 20600 17300 16800 19700 6300 11100 11500 20600 17700 17400 19200 6300 11200 12500 20800 18100 17200 19800 6300 11100 13700 20000 18300 17400 20200 6300 11100 15100 18500 18700 18300 19000 6300 11000 13500 17700 18900 19900 19400 6300 11000 13200 17000 18200 18500 19600 6400 11000 13000 16600 17900 16800 18400 6300 11100 13900 etc...

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 17 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Vruchtesappen[t] = + 9131.73309840353 + 0.454083662567809Mineraalwater[t] + 0.00771101131009535Appelen[t] -0.0655831488070655Sinaasappelen[t] + 0.0340529156698439Citroenen[t] + 0.0192571859658415Pompelmoezen[t] -0.0490769779229899Bananen[t] + 57.4062005061049M1[t] + 77.7493115271082M2[t] + 250.489345731327M3[t] + 344.236338901192M4[t] + 356.036377383132M5[t] + 359.805646551251M6[t] + 302.852781869608M7[t] + 192.587979854707M8[t] + 49.6309153787849M9[t] + 39.9297724453515M10[t] + 44.9352240195256M11[t] + 4.55639570066448t + 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) 9131.73309840353 991.904759 9.2063 0 0 Mineraalwater 0.454083662567809 0.11238 4.0406 0.000173 8.7e-05 Appelen 0.00771101131009535 0.013222 0.5832 0.562241 0.28112 Sinaasappelen -0.0655831488070655 0.022302 -2.9407 0.004845 0.002423 Citroenen 0.0340529156698439 0.041115 0.8282 0.411251 0.205626 Pompelmoezen 0.0192571859658415 0.017953 1.0726 0.288291 0.144145 Bananen -0.0490769779229899 0.028234 -1.7382 0.087974 0.043987 M1 57.4062005061049 79.157528 0.7252 0.47151 0.235755 M2 77.7493115271082 81.500209 0.954 0.344426 0.172213 M3 250.489345731327 97.271029 2.5752 0.012846 0.006423 M4 344.236338901192 110.500607 3.1152 0.002964 0.001482 M5 356.036377383132 115.737436 3.0762 0.003313 0.001656 M6 359.805646551251 123.043558 2.9242 0.005071 0.002535 M7 302.852781869608 114.185284 2.6523 0.010524 0.005262 M8 192.587979854707 101.554579 1.8964 0.063364 0.031682 M9 49.6309153787849 92.729641 0.5352 0.594735 0.297368 M10 39.9297724453515 77.753222 0.5135 0.609705 0.304852 M11 44.9352240195256 79.954967 0.562 0.576481 0.28824 t 4.55639570066448 2.876237 1.5842 0.119109 0.059554 Multiple Linear Regression - Regression Statistics Multiple R 0.799200768910404 R-squared 0.638721869026981 Adjusted R-squared 0.516023635866333 F-TEST (value) 5.20563216416252 F-TEST (DF numerator) 18 F-TEST (DF denominator) 53 p-value 1.3085519341427e-06 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 129.986803912926 Sum Squared Residuals 895518.167149366 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 10800 10941.9391714998 -141.939171499841 2 10900 10874.9779937080 25.0220062919670 3 11000 11007.4666254807 -7.46662548067558 4 11000 11087.0630233624 -87.0630233623686 5 11100 11117.7802357804 -17.7802357803734 6 11000 11141.0062181885 -141.006218188493 7 11000 11104.9191983194 -104.919198319354 8 11100 11106.9471516536 -6.94715165356724 9 11100 11090.1306309846 9.869369015438 10 11100 11126.4208771537 -26.4208771536528 11 11100 11063.0387536280 36.9612463720369 12 11100 11012.8460427614 87.1539572385911 13 11200 11071.5643335593 128.435666440744 14 11100 11052.8677432502 47.1322567497655 15 11100 11122.0070289226 -22.0070289225945 16 11200 11035.1079584052 164.892041594792 17 11200 11059.4037687044 140.596231295606 18 11100 11167.4783734250 -67.478373424953 19 11200 11089.9998283136 110.000171686368 20 11100 11037.0423837751 62.9576162248964 21 11100 11097.6568611894 2.34313881063982 22 11000 11150.6323044162 -150.632304416210 23 11000 11143.1765555573 -143.176555557299 24 11000 11188.8364334207 -188.836433420730 25 11100 11179.4948537127 -79.4948537126964 26 11000 11190.7545403590 -190.754540358985 27 11000 11140.2045892504 -140.204589250432 28 10900 11150.7382354840 -250.738235484013 29 11000 11114.1019851576 -114.101985157647 30 11000 11064.4661514704 -64.4661514704467 31 11100 11129.5968179685 -29.596817968493 32 11300 11281.6771453080 18.3228546920307 33 11300 11207.1373751130 92.8626248870277 34 11300 11215.0534259733 84.9465740267101 35 11300 11210.0232730845 89.9767269155478 36 11400 11218.9637798106 181.036220189367 37 11400 11216.0874346639 183.9125653361 38 11400 11220.3207937966 179.679206203378 39 11500 11248.6859495975 251.314050402483 40 11500 11279.5755638071 220.424436192851 41 11500 11362.4399732866 137.560026713448 42 11500 11329.8527219905 170.14727800953 43 11500 11379.1734328689 120.826567131071 44 11500 11455.1930287973 44.8069712027277 45 11400 11397.5580699158 2.44193008420276 46 11400 11285.0284987938 114.971501206220 47 11400 11225.8310319541 174.168968045868 48 11300 11205.4725284409 94.5274715591098 49 11200 11184.2894494505 15.7105505495091 50 11300 11225.6694413627 74.3305586373422 51 11300 11279.9558233053 20.0441766947427 52 11300 11347.1410881156 -47.1410881155754 53 11200 11318.5546458387 -118.554645838654 54 11300 11377.2201233352 -77.2201233352117 55 11200 11295.0689557265 -95.0689557265167 56 11200 11297.5326409155 -97.5326409154885 57 11100 11208.0162445651 -108.016244565063 58 11100 11155.5045876359 -55.5045876359424 59 11100 11157.8341436636 -57.8341436635955 60 11100 11200.3643778261 -100.364377826066 61 11400 11506.6247571138 -106.624757113815 62 11500 11635.4094875235 -135.409487523468 63 11500 11601.6799834435 -101.679983443523 64 11600 11600.3741308257 -0.374130825685482 65 11500 11527.7193912324 -27.7193912323806 66 11600 11419.9764115904 180.023588409574 67 11300 11301.2417668031 -1.24176680307487 68 11300 11321.6076495506 -21.6076495505992 69 11200 11199.5008182322 0.499181767755088 70 11200 11167.3603060271 32.6396939728755 71 11100 11200.0962421126 -100.096242112558 72 11100 11173.5168377403 -73.5168377402719 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 22 0.00829046298927137 0.0165809259785427 0.991709537010729 23 0.0073569799727175 0.014713959945435 0.992643020027282 24 0.00188740801332064 0.00377481602664128 0.99811259198668 25 0.00274985532239729 0.00549971064479457 0.997250144677603 26 0.00126014911385750 0.00252029822771499 0.998739850886142 27 0.0231916748448085 0.046383349689617 0.976808325155192 28 0.151391134668211 0.302782269336422 0.848608865331789 29 0.150820988241316 0.301641976482633 0.849179011758684 30 0.429895435762797 0.859790871525593 0.570104564237203 31 0.652309434288732 0.695381131422535 0.347690565711268 32 0.791665404728275 0.416669190543451 0.208334595271725 33 0.79510144584561 0.409797108308781 0.204898554154390 34 0.893025052719179 0.213949894561643 0.106974947280821 35 0.96024526294742 0.0795094741051582 0.0397547370525791 36 0.985135161974964 0.0297296760500723 0.0148648380250361 37 0.979917958458966 0.0401640830820678 0.0200820415410339 38 0.982945934863936 0.0341081302721285 0.0170540651360643 39 0.98850600818474 0.0229879836305213 0.0114939918152606 40 0.990855212939078 0.0182895741218443 0.00914478706092214 41 0.998558070392299 0.00288385921540223 0.00144192960770111 42 0.998355351182763 0.00328929763447414 0.00164464881723707 43 0.998594831951174 0.00281033609765232 0.00140516804882616 44 0.996576475136715 0.0068470497265708 0.0034235248632854 45 0.993300133126541 0.0133997337469172 0.00669986687345861 46 0.98816127686469 0.0236774462706222 0.0118387231353111 47 0.971068838815022 0.0578623223699566 0.0289311611849783 48 0.968801758519278 0.0623964829614435 0.0311982414807217 49 0.974047416734649 0.0519051665307025 0.0259525832653512 50 0.94479766761859 0.110404664762818 0.0552023323814091 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 17 0.586206896551724 NOK 10% type I error level 21 0.724137931034483 NOK

Charts produced by software:

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Parameters (Session):

par1 = kendall ;

Parameters (R input):

par1 = 2 ; 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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