Paper hypothese testing

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

Title produced by software: Multiple Regression

Date of computation: Sun, 19 Dec 2010 12:54:13 +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/19/t1292763337kfoagztah54rbg9.htm/, Retrieved Sun, 19 Dec 2010 13:55:37 +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/19/t1292763337kfoagztah54rbg9.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:

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Dataseries X:

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104,17 89.00 103.88 103.77 104,18 86.40 103.91 103.88 104,2 84.50 103.91 103.91 104,5 82.70 103.92 103.91 104,78 80.80 104.05 103.92 104,88 81.80 104.23 104.05 104,89 81.80 104.30 104.23 104,9 82.90 104.31 104.30 104,95 83.80 104.31 104.31 105,24 86.20 104.34 104.31 105,35 86.10 104.55 104.34 105,44 86.20 104.65 104.55 105,46 88.80 104.73 104.65 105,47 89.60 104.75 104.73 105,48 87.80 104.75 104.75 105,75 88.30 104.76 104.75 106,1 88.60 104.94 104.76 106,19 91.00 105.29 104.94 106,23 91.50 105.38 105.29 106,24 95.40 105.43 105.38 106,25 98.70 105.43 105.43 106,35 99.90 105.42 105.43 106,48 98.60 105.52 105.42 106,52 100.30 105.69 105.52 106,55 100.20 105.72 105.69 106,55 100.40 105.74 105.72 106,56 101.40 105.74 105.74 106,89 103.00 105.74 105.74 107,09 109.10 105.95 105.74 107,24 111.40 106.17 105.95 107,28 114.10 106.34 106.17 107,3 121.80 106.37 106.34 107,31 127.60 106.37 106.37 107,47 129.90 106.36 106.37 107,35 128.00 106.44 106.36 107,31 123.50 106.29 106.44 107,32 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 5 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 25.7267919840376 -0.00823946509526631X[t] + 1.23002986579691Y1[t] -0.467759327195804Y2[t] + 0.0460425932424228t + 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) 25.7267919840376 4.255304 6.0458 0 0 X -0.00823946509526631 0.002317 -3.5568 0.000832 0.000416 Y1 1.23002986579691 0.157414 7.814 0 0 Y2 -0.467759327195804 0.163279 -2.8648 0.006087 0.003043 t 0.0460425932424228 0.004376 10.5206 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.996714490043566 R-squared 0.993439774662806 Adjusted R-squared 0.99291495663583 F-TEST (value) 1892.9223532734 F-TEST (DF numerator) 4 F-TEST (DF denominator) 50 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0992785185599964 Sum Squared Residuals 0.492811212373377 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 104.17 104.275639259676 -0.105639259676094 2 104.18 104.328551832149 -0.148551832148691 3 104.2 104.376216629256 -0.176216629256249 4 104.5 104.449390558328 0.0506094416718703 5 104.78 104.666314424533 0.113685575466810 6 104.88 104.864714215988 0.0152857840116470 7 104.89 104.912662220941 -0.0226622209412984 8 104.9 104.929198548333 -0.0291985483331953 9 104.95 104.963148029718 -0.0131480297179209 10 105.24 105.026316802706 0.213683197294379 11 105.35 105.317456834459 0.0325431655409587 12 105.44 105.387449009061 0.0525509909394801 13 105.46 105.463695449599 -0.00369544959942092 14 105.47 105.490326321906 -0.0203263219058955 15 105.48 105.541844765776 -0.0618447657758782 16 105.75 105.596067925129 0.153932074871353 17 106.1 105.856366461414 0.243633538586029 18 106.19 106.228948112561 -0.0389481125614405 19 106.23 106.217857896659 0.0121421033406026 20 106.24 106.251169729873 -0.0111697298725330 21 106.25 106.246634121941 0.0033658780592237 22 106.35 106.270489058411 0.0795109415890903 23 106.48 106.454923536129 0.0250764638711859 24 106.52 106.649288183175 -0.129288183175192 25 106.55 106.653536533278 -0.103536533277762 26 106.55 106.708499051001 -0.158499051001190 27 106.56 106.736946992604 -0.176946992604427 28 106.89 106.769806441694 0.120193558305575 29 107.09 107.023894569673 0.0661054303269173 30 107.24 107.223363504961 0.0166364950394015 31 107.28 107.353357567648 -0.0733575676481975 32 107.3 107.293338090008 0.00666190999230525 33 107.31 107.277559005882 0.0324409941183070 34 107.47 107.292350530747 0.177649469252969 35 107.35 107.457128090206 -0.107128090206176 36 107.31 107.318323050332 -0.00832305033209919 37 107.32 107.356608018158 -0.0366080181584469 38 107.32 107.402701989709 -0.0827019897087135 39 107.34 107.447096689932 -0.107096689932073 40 107.53 107.486547711098 0.0434522889017152 41 107.72 107.643175194293 0.0768248057074291 42 107.75 107.730281227271 0.0197187727293466 43 107.79 107.738611299398 0.051388700601715 44 107.81 107.747340703772 0.0626592962277002 45 107.9 107.776203018976 0.123796981023928 46 107.8 107.869330441937 -0.0693304419374059 47 107.86 107.691651628300 0.168348371699751 48 107.8 107.840460506387 -0.0404605063874901 49 107.74 107.773530958288 -0.0335309582883204 50 107.75 107.819087957577 -0.0690879575766127 51 107.83 107.836052026354 -0.00605202635432782 52 107.8 107.900044745852 -0.100044745851548 53 107.81 107.865609386931 -0.0556093869310889 54 107.86 107.778314052048 0.0816859479520488 55 107.83 107.884579087962 -0.0545790879620511 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.298429858298413 0.596859716596827 0.701570141701587 9 0.219985427784868 0.439970855569737 0.780014572215132 10 0.199185290155083 0.398370580310166 0.800814709844917 11 0.396651714677678 0.793303429355355 0.603348285322322 12 0.302660559761953 0.605321119523907 0.697339440238047 13 0.214264623106670 0.428529246213341 0.78573537689333 14 0.165717216432383 0.331434432864765 0.834282783567617 15 0.310558800216585 0.621117600433169 0.689441199783415 16 0.23488793915442 0.46977587830884 0.76511206084558 17 0.299781612672298 0.599563225344596 0.700218387327702 18 0.259734300348173 0.519468600696345 0.740265699651827 19 0.46514702972975 0.9302940594595 0.53485297027025 20 0.426889965357867 0.853779930715734 0.573110034642133 21 0.356378067002508 0.712756134005016 0.643621932997492 22 0.305305948672394 0.610611897344788 0.694694051327606 23 0.275277823288408 0.550555646576817 0.724722176711592 24 0.374136247882223 0.748272495764445 0.625863752117777 25 0.369049459594721 0.738098919189442 0.630950540405279 26 0.535338425912684 0.929323148174631 0.464661574087316 27 0.824045002821542 0.351909994356917 0.175954997178458 28 0.815684783386347 0.368630433227305 0.184315216613653 29 0.809991759838703 0.380016480322594 0.190008240161297 30 0.829566346781312 0.340867306437376 0.170433653218688 31 0.777720363663317 0.444559272673366 0.222279636336683 32 0.778675961644306 0.442648076711388 0.221324038355694 33 0.73484577236818 0.530308455263642 0.265154227631821 34 0.820549678207061 0.358900643585877 0.179450321792939 35 0.881955821616107 0.236088356767786 0.118044178383893 36 0.856090610574528 0.287818778850945 0.143909389425473 37 0.852749143526494 0.294501712947013 0.147250856473506 38 0.89764607714641 0.204707845707181 0.102353922853590 39 0.992587634848458 0.0148247303030846 0.00741236515154231 40 0.999651433617107 0.000697132765785412 0.000348566382892706 41 0.99919914557771 0.00160170884457977 0.000800854422289883 42 0.99835162010041 0.00329675979918062 0.00164837989959031 43 0.995124887062706 0.00975022587458823 0.00487511293729412 44 0.9925743493609 0.0148513012782007 0.00742565063910036 45 0.992623885338347 0.0147522293233067 0.00737611466165333 46 0.982112848344395 0.0357743033112107 0.0178871516556053 47 0.945021915079676 0.109956169840649 0.0549780849203243 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.1 NOK 5% type I error level 8 0.2 NOK 10% type I error level 8 0.2 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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