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: Thu, 19 Nov 2009 12:31:56 -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/19/t1258659292ovn42x6atb29k99.htm/, Retrieved Thu, 19 Nov 2009 20:35:04 +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/19/t1258659292ovn42x6atb29k99.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:

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Dataseries X:

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22 591 19 19 18 19 23 589 22 19 19 18 20 584 23 22 19 19 14 573 20 23 22 19 14 567 14 20 23 22 14 569 14 14 20 23 15 621 14 14 14 20 11 629 15 14 14 14 17 628 11 15 14 14 16 612 17 11 15 14 20 595 16 17 11 15 24 597 20 16 17 11 23 593 24 20 16 17 20 590 23 24 20 16 21 580 20 23 24 20 19 574 21 20 23 24 23 573 19 21 20 23 23 573 23 19 21 20 23 620 23 23 19 21 23 626 23 23 23 19 27 620 23 23 23 23 26 588 27 23 23 23 17 566 26 27 23 23 24 557 17 26 27 23 26 561 24 17 26 27 24 549 26 24 17 26 27 532 24 26 24 17 27 526 27 24 26 24 26 511 27 27 24 26 24 499 26 27 27 24 23 555 24 26 27 27 23 565 23 24 26 27 24 542 23 23 24 26 17 527 24 23 23 24 21 510 17 24 23 23 19 514 21 17 24 23 22 517 19 21 17 24 22 508 22 19 21 17 18 493 22 22 19 21 16 490 18 22 22 19 14 469 16 18 22 22 12 478 14 16 18 22 14 528 12 14 16 18 16 534 14 12 14 16 8 518 16 14 12 14 3 506 8 16 14 12 0 502 3 8 16 14 5 516 0 3 8 16 1 528 5 0 3 8 1 533 1 5 0 3 3 536 1 1 5 0 6 537 3 1 1 5 7 524 6 3 1 1 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation y[t] = -37.9811272503563 + 0.0670198079699449x[t] + 0.682921562538848y1[t] + 0.119179732248385y2[t] + 0.150514391120164y3[t] + 0.0814757034058975y4[t] -1.58003334572513M1[t] -2.61700254763576M2[t] -2.23468565236653M3[t] -3.49583107087434M4[t] -1.40348894920972M5[t] -2.27169803811806M6[t] -3.62374782576128M7[t] -5.63694467436919M8[t] -4.37543160126942M9[t] -6.46479144004884M10[t] -4.23386625939827M11[t] + 0.109731952455411t + 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) -37.9811272503563 17.518128 -2.1681 0.036314 0.018157 x 0.0670198079699449 0.026158 2.5621 0.01438 0.00719 y1 0.682921562538848 0.154769 4.4125 7.8e-05 3.9e-05 y2 0.119179732248385 0.188236 0.6331 0.530338 0.265169 y3 0.150514391120164 0.191686 0.7852 0.437074 0.218537 y4 0.0814757034058975 0.16312 0.4995 0.620246 0.310123 M1 -1.58003334572513 2.338083 -0.6758 0.503167 0.251584 M2 -2.61700254763576 2.407656 -1.087 0.283731 0.141866 M3 -2.23468565236653 2.333619 -0.9576 0.344161 0.17208 M4 -3.49583107087434 2.23898 -1.5614 0.12652 0.06326 M5 -1.40348894920972 2.285549 -0.6141 0.542734 0.271367 M6 -2.27169803811806 2.263027 -1.0038 0.321648 0.160824 M7 -3.62374782576128 2.392632 -1.5145 0.137951 0.068976 M8 -5.63694467436919 2.422322 -2.3271 0.025245 0.012623 M9 -4.37543160126942 2.395629 -1.8264 0.075449 0.037724 M10 -6.46479144004884 2.339435 -2.7634 0.008685 0.004343 M11 -4.23386625939827 2.375336 -1.7824 0.082466 0.041233 t 0.109731952455411 0.065306 1.6803 0.100897 0.050449 Multiple Linear Regression - Regression Statistics Multiple R 0.932188746251565 R-squared 0.868975858638064 Adjusted R-squared 0.811862771377733 F-TEST (value) 15.2150041316647 F-TEST (DF numerator) 17 F-TEST (DF denominator) 39 p-value 3.25228732833693e-12 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.21930628543235 Sum Squared Residuals 404.193385417546 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 22 19.6544998724439 2.34550012755607 2 23 20.7110263823796 2.28897361762041 3 20 21.9899126529444 -1.98991265294441 4 14 18.6232395172149 -4.62323951721494 5 14 16.3630676728749 -2.36306767287492 6 14 14.6534842889170 -0.65348428891698 7 15 15.7486830112276 -0.748683011227628 8 11 14.5754439209382 -3.57544392093815 9 17 13.2671626206164 3.73283737938362 10 16 13.9865426441330 2.01345735586703 11 20 14.6994380116266 5.30056198837341 12 24 22.3667658904246 1.63323410957544 13 23 24.1751302737392 -1.17513027373922 14 20 23.3612128279036 -3.36121282790361 15 21 21.9430795541681 -0.943079554168122 16 19 20.8903180285932 -1.89031802859317 17 23 21.2456900251476 1.75430997485245 18 23 22.8866269552557 0.113373044744275 19 23 25.0514059448144 -2.05140594481443 20 23 23.9891660541505 -0.989166054150464 21 27 25.2841950455096 1.71580495449044 22 26 23.8916195543027 2.10838044569729 23 17 24.5516382785246 -7.5516382785246 24 24 22.6286419880314 1.37135801196859 25 26 25.3096415966814 0.690358403318623 26 24 24.3421626789158 -0.342162678915839 27 27 22.8877105377586 4.11228946224144 28 27 24.0159421530879 2.98405784691213 29 26 25.4321809289754 0.567819071024645 30 24 23.4751363008929 0.524863699107063 31 23 24.7453319649136 -1.74533196491365 32 23 22.4402697303048 0.559730269695181 33 24 21.7683749546567 2.23162504534334 34 17 19.1529057133904 -2.15290571339037 35 21 15.6114792020778 5.38852079792216 36 19 22.2710991613482 -3.27109916134816 37 22 19.1406079614689 2.85939203853113 38 22 19.4523253040433 2.5476746959567 39 18 19.3214902603472 -1.32149026034718 40 16 15.5259228867782 0.474077113221756 41 14 14.7224460496759 -0.722446049675899 42 12 12.3608870308974 -0.360887030897355 43 14 12.2384254087684 1.76157459123160 44 16 11.4005828319644 4.59941716803562 45 8 12.8397333305228 -4.83973333052278 46 3 4.96893208817396 -1.96893208817396 47 0 3.13744450777096 -3.13744450777096 48 5 4.73349296019588 0.266507039804116 49 1 5.7201202956666 -4.7201202956666 50 1 2.13327280675766 -1.13327280675766 51 3 2.85780699478172 0.142193005218279 52 6 2.94457741432579 3.05542258567421 53 7 6.23661532332627 0.763384676673734 54 8 7.623865424037 0.376134575962994 55 14 11.2161536702759 2.7838463297241 56 14 14.5945374626422 -0.594537462642183 57 13 15.8405340486946 -2.84053404869461 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.152681842250322 0.305363684500643 0.847318157749678 22 0.389023526356137 0.778047052712274 0.610976473643863 23 0.575435573959904 0.849128852080193 0.424564426040096 24 0.470033147153822 0.940066294307644 0.529966852846178 25 0.540925438542755 0.918149122914491 0.459074561457245 26 0.439256771975076 0.878513543950151 0.560743228024924 27 0.656521126852212 0.686957746295576 0.343478873147788 28 0.593939835848916 0.812120328302168 0.406060164151084 29 0.486900792385123 0.973801584770246 0.513099207614877 30 0.363962108957986 0.727924217915972 0.636037891042014 31 0.405954710759082 0.811909421518164 0.594045289240918 32 0.588352971148111 0.823294057703777 0.411647028851888 33 0.474235138563298 0.948470277126595 0.525764861436702 34 0.533988409766375 0.932023180467251 0.466011590233625 35 0.408220643014274 0.816441286028547 0.591779356985726 36 0.719930271047051 0.560139457905898 0.280069728952949 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 0 0 OK

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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