*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, 15 Dec 2009 06:38:55 -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/15/t1260884406l1w62gwovlugtct.htm/, Retrieved Tue, 15 Dec 2009 14:40:18 +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/15/t1260884406l1w62gwovlugtct.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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2350.44 0 2440.25 0 2408.64 0 2472.81 0 2407.6 0 2454.62 0 2448.05 0 2497.84 0 2645.64 0 2756.76 0 2849.27 0 2921.44 0 2981.85 0 3080.58 0 3106.22 0 3119.31 0 3061.26 0 3097.31 0 3161.69 0 3257.16 0 3277.01 0 3295.32 0 3363.99 0 3494.17 0 3667.03 0 3813.06 0 3917.96 0 3895.51 0 3801.06 0 3570.12 0 3701.61 0 3862.27 0 3970.1 0 4138.52 0 4199.75 0 4290.89 0 4443.91 0 4502.64 0 4356.98 0 4591.27 0 4696.96 0 4621.4 0 4562.84 0 4202.52 0 4296.49 0 4435.23 0 4105.18 0 4116.68 0 3844.49 0 3720.98 0 3674.4 0 3857.62 0 3801.06 0 3504.37 0 3032.6 0 3047.03 0 2962.34 1 2197.82 1 2014.45 1 1862.83 1 1905.41 1 1810.99 1 1670.07 1 1864.44 1 2052.02 1 2029.6 1 2070.83 1 2293.41 1 2443.27 1 2513.17 1 2466.92 1

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 8 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 3521.28142857143 -1377.44342857143X[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) 3521.28142857143 84.82554 41.512 0 0 X -1377.44342857143 184.548355 -7.4639 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.668365905570427 R-squared 0.446712983728976 Adjusted R-squared 0.438694331319252 F-TEST (value) 55.7092340337892 F-TEST (DF numerator) 1 F-TEST (DF denominator) 69 p-value 1.90738980165861e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 634.77621662073 Sum Squared Residuals 27802918.3179257 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2350.44 3521.28142857142 -1170.84142857142 2 2440.25 3521.28142857143 -1081.03142857143 3 2408.64 3521.28142857143 -1112.64142857143 4 2472.81 3521.28142857143 -1048.47142857143 5 2407.6 3521.28142857143 -1113.68142857143 6 2454.62 3521.28142857143 -1066.66142857143 7 2448.05 3521.28142857143 -1073.23142857143 8 2497.84 3521.28142857143 -1023.44142857143 9 2645.64 3521.28142857143 -875.64142857143 10 2756.76 3521.28142857143 -764.521428571428 11 2849.27 3521.28142857143 -672.011428571429 12 2921.44 3521.28142857143 -599.841428571429 13 2981.85 3521.28142857143 -539.431428571429 14 3080.58 3521.28142857143 -440.701428571429 15 3106.22 3521.28142857143 -415.061428571429 16 3119.31 3521.28142857143 -401.971428571429 17 3061.26 3521.28142857143 -460.021428571428 18 3097.31 3521.28142857143 -423.971428571429 19 3161.69 3521.28142857143 -359.591428571429 20 3257.16 3521.28142857143 -264.121428571429 21 3277.01 3521.28142857143 -244.271428571428 22 3295.32 3521.28142857143 -225.961428571429 23 3363.99 3521.28142857143 -157.291428571429 24 3494.17 3521.28142857143 -27.1114285714286 25 3667.03 3521.28142857143 145.748571428572 26 3813.06 3521.28142857143 291.778571428571 27 3917.96 3521.28142857143 396.678571428571 28 3895.51 3521.28142857143 374.228571428572 29 3801.06 3521.28142857143 279.778571428571 30 3570.12 3521.28142857143 48.8385714285712 31 3701.61 3521.28142857143 180.328571428571 32 3862.27 3521.28142857143 340.988571428571 33 3970.1 3521.28142857143 448.818571428571 34 4138.52 3521.28142857143 617.238571428572 35 4199.75 3521.28142857143 678.468571428571 36 4290.89 3521.28142857143 769.608571428572 37 4443.91 3521.28142857143 922.62857142857 38 4502.64 3521.28142857143 981.358571428572 39 4356.98 3521.28142857143 835.69857142857 40 4591.27 3521.28142857143 1069.98857142857 41 4696.96 3521.28142857143 1175.67857142857 42 4621.4 3521.28142857143 1100.11857142857 43 4562.84 3521.28142857143 1041.55857142857 44 4202.52 3521.28142857143 681.238571428572 45 4296.49 3521.28142857143 775.208571428571 46 4435.23 3521.28142857143 913.94857142857 47 4105.18 3521.28142857143 583.898571428572 48 4116.68 3521.28142857143 595.398571428572 49 3844.49 3521.28142857143 323.208571428571 50 3720.98 3521.28142857143 199.698571428571 51 3674.4 3521.28142857143 153.118571428571 52 3857.62 3521.28142857143 336.338571428571 53 3801.06 3521.28142857143 279.778571428571 54 3504.37 3521.28142857143 -16.9114285714288 55 3032.6 3521.28142857143 -488.681428571429 56 3047.03 3521.28142857143 -474.251428571429 57 2962.34 2143.838 818.502 58 2197.82 2143.838 53.9820000000003 59 2014.45 2143.838 -129.388000000000 60 1862.83 2143.838 -281.008 61 1905.41 2143.838 -238.428 62 1810.99 2143.838 -332.848 63 1670.07 2143.838 -473.768 64 1864.44 2143.838 -279.398 65 2052.02 2143.838 -91.8179999999999 66 2029.6 2143.838 -114.238 67 2070.83 2143.838 -73.008 68 2293.41 2143.838 149.572 69 2443.27 2143.838 299.432 70 2513.17 2143.838 369.332 71 2466.92 2143.838 323.082 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.00105548270290344 0.00211096540580687 0.998944517297097 6 0.000117634388455769 0.000235268776911538 0.999882365611544 7 1.20811842465913e-05 2.41623684931825e-05 0.999987918815753 8 3.32728342187998e-06 6.65456684375997e-06 0.999996672716578 9 3.23788117540966e-05 6.47576235081932e-05 0.999967621188246 10 0.000209700086659408 0.000419400173318815 0.99979029991334 11 0.0008602189128698 0.0017204378257396 0.99913978108713 12 0.00249445280695747 0.00498890561391494 0.997505547193043 13 0.00579419663923052 0.0115883932784610 0.99420580336077 14 0.0141298176263541 0.0282596352527083 0.985870182373646 15 0.025821686045509 0.051643372091018 0.97417831395449 16 0.0395056067774849 0.0790112135549698 0.960494393222515 17 0.0486464924561493 0.0972929849122986 0.95135350754385 18 0.0619535225692082 0.123907045138416 0.938046477430792 19 0.083965891322891 0.167931782645782 0.91603410867711 20 0.122582990553412 0.245165981106823 0.877417009446589 21 0.168687824241940 0.337375648483881 0.83131217575806 22 0.222341803861625 0.44468360772325 0.777658196138375 23 0.292567060981050 0.585134121962101 0.70743293901895 24 0.391941486960955 0.78388297392191 0.608058513039045 25 0.52806512100223 0.94386975799554 0.47193487899777 26 0.671554175002168 0.656891649995664 0.328445824997832 27 0.786867562352446 0.426264875295108 0.213132437647554 28 0.847328130139871 0.305343739720257 0.152671869860129 29 0.872348258910868 0.255303482178264 0.127651741089132 30 0.879779933188242 0.240440133623516 0.120220066811758 31 0.888163805562956 0.223672388874087 0.111836194437044 32 0.900349441497694 0.199301117004611 0.0996505585023055 33 0.913583390728151 0.172833218543698 0.0864166092718488 34 0.932360297426268 0.135279405147464 0.0676397025737322 35 0.946749044914818 0.106501910170364 0.0532509550851821 36 0.959910461293132 0.0801790774137367 0.0400895387068684 37 0.974787942547103 0.0504241149057943 0.0252120574528971 38 0.984912985233729 0.0301740295325423 0.0150870147662711 39 0.987288035667174 0.0254239286656513 0.0127119643328257 40 0.993284415703355 0.0134311685932891 0.00671558429664453 41 0.997545419581455 0.00490916083708889 0.00245458041854445 42 0.99900094582106 0.00199810835787876 0.000999054178939379 43 0.999583802826376 0.000832394347248086 0.000416197173624043 44 0.999524632198955 0.000950735602089178 0.000475367801044589 45 0.999605074160017 0.000789851679966631 0.000394925839983316 46 0.999844930471777 0.000310139056445294 0.000155069528222647 47 0.999834937718634 0.000330124562731582 0.000165062281365791 48 0.999858761327876 0.000282477344248680 0.000141238672124340 49 0.999768654063253 0.000462691873494559 0.000231345936747280 50 0.999558675211142 0.000882649577715929 0.000441324788857964 51 0.999156636658403 0.00168672668319359 0.000843363341596795 52 0.999052979296748 0.00189404140650487 0.000947020703252437 53 0.99916340954819 0.0016731809036219 0.00083659045181095 54 0.99885855420682 0.00228289158636207 0.00114144579318103 55 0.99758863820111 0.0048227235977782 0.0024113617988891 56 0.99503756000676 0.00992487998648171 0.00496243999324085 57 0.999366383562465 0.00126723287507063 0.000633616437535316 58 0.99852935838535 0.00294128322929868 0.00147064161464934 59 0.996473650335183 0.00705269932963472 0.00352634966481736 60 0.993388556355917 0.0132228872881668 0.0066114436440834 61 0.986979545829928 0.0260409083401448 0.0130204541700724 62 0.98057062363484 0.0388587527303206 0.0194293763651603 63 0.987411251209115 0.0251774975817698 0.0125887487908849 64 0.986495885167811 0.0270082296643773 0.0135041148321887 65 0.972343342698034 0.0553133146039319 0.0276566573019660 66 0.958952566239626 0.0820948675207484 0.0410474337603742 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 27 0.435483870967742 NOK 5% type I error level 37 0.596774193548387 NOK 10% type I error level 44 0.709677419354839 NOK

Charts produced by software:

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Parameters (Session):

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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