WS 7: Seasonal effects

*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: Wed, 25 Nov 2009 13:38:31 -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/25/t1259181624vh0180y7cqmodrj.htm/, Retrieved Wed, 25 Nov 2009 21:40:36 +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/25/t1259181624vh0180y7cqmodrj.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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8.1 10.9 7.7 10.0 7.5 9.2 7.6 9.2 7.8 9.5 7.8 9.6 7.8 9.5 7.5 9.1 7.5 8.9 7.1 9.0 7.5 10.1 7.5 10.3 7.6 10.2 7.7 9.6 7.7 9.2 7.9 9.3 8.1 9.4 8.2 9.4 8.2 9.2 8.2 9.0 7.9 9.0 7.3 9.0 6.9 9.8 6.6 10.0 6.7 9.8 6.9 9.3 7.0 9.0 7.1 9.0 7.2 9.1 7.1 9.1 6.9 9.1 7.0 9.2 6.8 8.8 6.4 8.3 6.7 8.4 6.6 8.1 6.4 7.7 6.3 7.9 6.2 7.9 6.5 8.0 6.8 7.9 6.8 7.6 6.4 7.1 6.1 6.8 5.8 6.5 6.1 6.9 7.2 8.2 7.3 8.7 6.9 8.3 6.1 7.9 5.8 7.5 6.2 7.8 7.1 8.3 7.7 8.4 7.9 8.2 7.7 7.7 7.4 7.2 7.5 7.3 8.0 8.1 8.1 8.5

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] = + 3.1325199786894 + 0.448188598827917X[t] -0.196529035695261M1[t] -0.199326052210974M2[t] -0.129014384656366M3[t] + 0.0461667554608419M4[t] + 0.305492807671817M5[t] + 0.434456579648375M6[t] + 0.444094299413959M7[t] + 0.420623335109216M8[t] + 0.326116142781033M9[t] + 0.117152370804475M10[t] + 0.129637719765584M11[t] + 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) 3.1325199786894 0.778925 4.0216 0.000208 0.000104 X 0.448188598827917 0.081127 5.5245 1e-06 1e-06 M1 -0.196529035695261 0.345024 -0.5696 0.571655 0.285827 M2 -0.199326052210974 0.344688 -0.5783 0.565836 0.282918 M3 -0.129014384656366 0.347362 -0.3714 0.712 0.356 M4 0.0461667554608419 0.346394 0.1333 0.894543 0.447271 M5 0.305492807671817 0.345127 0.8852 0.380576 0.190288 M6 0.434456579648375 0.345237 1.2584 0.214452 0.107226 M7 0.444094299413959 0.346759 1.2807 0.206583 0.103291 M8 0.420623335109216 0.349854 1.2023 0.235277 0.117638 M9 0.326116142781033 0.354563 0.9198 0.36239 0.181195 M10 0.117152370804475 0.354181 0.3308 0.742288 0.371144 M11 0.129637719765584 0.34476 0.376 0.708592 0.354296 Multiple Linear Regression - Regression Statistics Multiple R 0.6765523075421 R-squared 0.45772302484054 Adjusted R-squared 0.319269329055146 F-TEST (value) 3.30596465658829 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0.00159533698611725 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.544509777370832 Sum Squared Residuals 13.9350721896644 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.1 7.82124667021845 0.278753329781554 2 7.7 7.41507991475759 0.284920085242410 3 7.5 7.12684070324987 0.373159296750134 4 7.6 7.30202184336707 0.297978156632925 5 7.8 7.69580447522642 0.104195524773575 6 7.8 7.86958710708577 -0.0695871070857745 7 7.8 7.83440596696857 -0.0344059669685666 8 7.5 7.63165956313266 -0.131659563132658 9 7.5 7.44751465103889 0.0524853489611086 10 7.1 7.28336973894512 -0.183369738945125 11 7.5 7.78886254661694 -0.288862546616941 12 7.5 7.74886254661694 -0.248862546616941 13 7.6 7.50751465103889 0.0924853489611116 14 7.7 7.23580447522643 0.464195524773575 15 7.7 7.12684070324987 0.573159296750134 16 7.9 7.34684070324987 0.553159296750134 17 8.1 7.65098561534363 0.449014384656367 18 8.2 7.77994938732019 0.420050612679808 19 8.2 7.69994938732019 0.500050612679808 20 8.2 7.58684070324987 0.613159296750133 21 7.9 7.49233351092168 0.407666489078317 22 7.3 7.28336973894513 0.0166302610548752 23 6.9 7.65440596696857 -0.754405966968567 24 6.6 7.61440596696857 -1.01440596696857 25 6.7 7.32823921150772 -0.628239211507722 26 6.9 7.10134789557805 -0.20134789557805 27 7 7.03720298348428 -0.0372029834842833 28 7.1 7.2123841236015 -0.112384123601492 29 7.2 7.51652903569526 -0.316529035695258 30 7.1 7.64549280767182 -0.545492807671816 31 6.9 7.6551305274374 -0.7551305274374 32 7 7.67647842301545 -0.676478423015449 33 6.8 7.4026957911561 -0.6026957911561 34 6.4 6.96963771976558 -0.569637719765583 35 6.7 7.02694192860948 -0.326941928609484 36 6.6 6.76284762919552 -0.162847629195525 37 6.4 6.3870431539691 0.0129568460309029 38 6.3 6.47388385721897 -0.173883857218968 39 6.2 6.54419552477358 -0.344195524773575 40 6.5 6.76419552477358 -0.264195524773575 41 6.8 6.97870271710176 -0.178702717101759 42 6.8 6.97320990942994 -0.173209909429942 43 6.4 6.75875332978157 -0.358753329781567 44 6.1 6.60082578582845 -0.500825785828451 45 5.8 6.37186201385189 -0.571862013851893 46 6.1 6.3421736814065 -0.242173681406501 47 7.2 6.9373042088439 0.2626957911561 48 7.3 7.03176078849227 0.268239211507725 49 6.9 6.65595631326585 0.244043686734153 50 6.1 6.47388385721897 -0.373883857218968 51 5.8 6.36492008524241 -0.564920085242409 52 6.2 6.67455780500799 -0.474557805007992 53 7.1 7.15797815663293 -0.0579781566329257 54 7.7 7.33176078849228 0.368239211507725 55 7.9 7.25176078849228 0.648239211507725 56 7.7 7.00419552477357 0.695804475226425 57 7.4 6.68559403303143 0.714405966968567 58 7.5 6.52144912093767 0.978550879062333 59 8 6.89248534896111 1.10751465103889 60 8.1 6.94212306872669 1.15787693127331 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0260356253292302 0.0520712506584604 0.97396437467077 17 0.0177052556515251 0.0354105113030503 0.982294744348475 18 0.0195832449421587 0.0391664898843175 0.980416755057841 19 0.0185891664020576 0.0371783328041152 0.981410833597942 20 0.0357056445972754 0.0714112891945508 0.964294355402725 21 0.0269332443137380 0.0538664886274761 0.973066755686262 22 0.0135731993444336 0.0271463986888672 0.986426800655566 23 0.0133401153186102 0.0266802306372205 0.98665988468139 24 0.0300301079921946 0.0600602159843892 0.969969892007805 25 0.0313305910906898 0.0626611821813795 0.96866940890931 26 0.0194156002054765 0.0388312004109531 0.980584399794523 27 0.0150835011996666 0.0301670023993332 0.984916498800333 28 0.0106636471950221 0.0213272943900441 0.989336352804978 29 0.00644859414466569 0.0128971882893314 0.993551405855334 30 0.00465363711349979 0.00930727422699958 0.9953463628865 31 0.0086922297835737 0.0173844595671474 0.991307770216426 32 0.0279623899324669 0.0559247798649337 0.972037610067533 33 0.0517542429340556 0.103508485868111 0.948245757065944 34 0.211861924464144 0.423723848928288 0.788138075535856 35 0.618881821674431 0.762236356651138 0.381118178325569 36 0.649555474025257 0.700889051949486 0.350444525974743 37 0.583746010559743 0.832507978880514 0.416253989440257 38 0.487899793404532 0.975799586809064 0.512100206595468 39 0.412647576214355 0.825295152428709 0.587352423785645 40 0.313542094258642 0.627084188517284 0.686457905741358 41 0.227899764598937 0.455799529197874 0.772100235401063 42 0.160248327951017 0.320496655902034 0.839751672048983 43 0.107322358884908 0.214644717769816 0.892677641115092 44 0.0540735552047762 0.108147110409552 0.945926444795224 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.0344827586206897 NOK 5% type I error level 11 0.379310344827586 NOK 10% type I error level 17 0.586206896551724 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No 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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