Multiple Regression Lineair Trend WS7

*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: Sun, 22 Nov 2009 09:29:30 -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/22/t1258907447afqy3rzrcvez7s5.htm/, Retrieved Sun, 22 Nov 2009 17:30:59 +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/22/t1258907447afqy3rzrcvez7s5.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:

KVN WS7

Dataseries X:

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9487 1169 8700 2154 9627 2249 8947 2687 9283 4359 8829 5382 9947 4459 9628 6398 9318 4596 9605 3024 8640 1887 9214 2070 9567 1351 8547 2218 9185 2461 9470 3028 9123 4784 9278 4975 10170 4607 9434 6249 9655 4809 9429 3157 8739 1910 9552 2228 9687 1594 9019 2467 9672 2222 9206 3607 9069 4685 9788 4962 10312 5770 10105 5480 9863 5000 9656 3228 9295 1993 9946 2288 9701 1580 9049 2111 10190 2192 9706 3601 9765 4665 9893 4876 9994 5813 10433 5589 10073 5331 10112 3075 9266 2002 9820 2306 10097 1507 9115 1992 10411 2487 9678 3490 10408 4647 10153 5594 10368 5611 10581 5788 10597 6204 10680 3013 9738 1931 9556 2549

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] = + 9387.4473706367 -0.208349443842813X[t] + 129.516229799319M1[t] -556.032767060951M2[t] + 383.208337781761M3[t] + 148.071092904942M4[t] + 536.948383907607M5[t] + 686.295868512073M6[t] + 1256.28633537861M7[t] + 1249.82740380037M8[t] + 946.679869485758M9[t] + 487.085170332202M10[t] -533.952818160934M11[t] + 19.6360507434562t + 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) 9387.4473706367 248.828986 37.7265 0 0 X -0.208349443842813 0.106474 -1.9568 0.056456 0.028228 M1 129.516229799319 169.928878 0.7622 0.449844 0.224922 M2 -556.032767060951 148.61417 -3.7415 0.000506 0.000253 M3 383.208337781761 148.740468 2.5764 0.01326 0.00663 M4 148.071092904942 185.679248 0.7975 0.429283 0.214642 M5 536.948383907607 294.309868 1.8244 0.074586 0.037293 M6 686.295868512073 343.500351 1.9979 0.051655 0.025827 M7 1256.28633537861 351.850852 3.5705 0.000848 0.000424 M8 1249.82740380037 414.822958 3.0129 0.004197 0.002099 M9 946.679869485758 344.252864 2.75 0.008492 0.004246 M10 487.085170332202 171.74094 2.8362 0.006766 0.003383 M11 -533.952818160934 151.818432 -3.517 0.000994 0.000497 t 19.6360507434562 1.925746 10.1966 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.917773465845367 R-squared 0.842308134609817 Adjusted R-squared 0.797743042216939 F-TEST (value) 18.9006257898936 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 3.16413562018170e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 233.245923115510 Sum Squared Residuals 2502568.3899003 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 9487 9293.0391513272 193.960848672802 2 8700 8421.90200302524 278.097996974755 3 9627 9360.98596144635 266.014038553654 4 8947 9054.22771090983 -107.227710909832 5 9283 9114.38078255077 168.619217449232 6 8829 9070.2228368475 -241.222836847493 7 9947 9852.1558911244 94.8441088755972 8 9628 9461.3434386784 166.656561321594 9 9318 9553.277652912 -235.277652911996 10 9605 9440.8443302228 164.1556697772 11 8640 8676.3357101224 -36.3357101223976 12 9214 9191.79663080355 22.2033691964461 13 9567 9490.75216146931 76.2478385306889 14 8547 8644.20024754078 -97.2002475407793 15 9185 9552.44848827314 -367.448488273143 16 9470 9218.8131594809 251.186840519094 17 9123 9261.46487783905 -138.464877839047 18 9278 9390.653669413 -112.653669412992 19 10170 10056.9527823571 113.04721764286 20 9434 9728.02011473246 -294.020114732459 21 9655 9744.53183029495 -89.5318302949524 22 9429 9648.76646311318 -219.76646311318 23 8739 8907.17628183549 -168.176281835487 24 9552 9394.51002759786 157.489972402137 25 9687 9675.75585553698 11.2441444630183 26 9019 8827.9538449454 191.046155054608 27 9672 9837.87661427305 -165.87661427305 28 9206 9333.8114404174 -127.811440417392 29 9069 9517.72408170096 -448.72408170096 30 9788 9628.99482110442 159.005178895577 31 10312 10050.2749880894 261.725011910577 32 10105 10123.8734459691 -18.8734459690554 33 9863 9940.36969544245 -77.3696954424489 34 9656 9869.60626152181 -213.606261521814 35 9295 9125.515886918 169.484113081992 36 9946 9617.64166988877 328.358330111231 37 9701 9914.30535667226 -213.305356672255 38 9049 9137.7588558749 -88.7588558749075 39 10190 10079.7597065098 110.240293490192 40 9706 9570.69414600192 135.305853998078 41 9765 9757.5236794993 7.47632050070986 42 9893 9882.54548219638 10.4545178036213 43 9994 10276.9485709257 -282.948570925656 44 10433 10336.7959655117 96.204034488337 45 10073 10107.0386384520 -34.0386384519522 46 10112 10137.1163353512 -25.1163353512387 47 9266 9359.2733508449 -93.2733508448967 48 9820 9849.52398882107 -29.523988821072 49 10097 10165.1474749943 -68.1474749942544 50 9115 9398.18504861368 -283.185048613676 51 10411 10253.9292294977 157.070770502348 52 9678 9829.45354318995 -151.453543189949 53 10408 9996.90657840993 411.093421590065 54 10153 9968.58319043871 184.416809561287 55 10368 10554.6677675034 -186.667767503378 56 10581 10530.9670351084 50.032964891583 57 10597 10160.7821828987 436.217817101349 58 10680 10385.6666097910 294.333390209033 59 9738 9609.69877027921 128.301229720790 60 9556 10034.5276828887 -478.527682888742 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.721197820278868 0.557604359442264 0.278802179721132 18 0.697573611594279 0.604852776811442 0.302426388405721 19 0.605192444732662 0.789615110534676 0.394807555267338 20 0.615244038713369 0.769511922573262 0.384755961286631 21 0.560084757655123 0.879830484689753 0.439915242344877 22 0.497661938619243 0.995323877238487 0.502338061380757 23 0.40044309303713 0.80088618607426 0.59955690696287 24 0.368110104587685 0.73622020917537 0.631889895412315 25 0.278221658199597 0.556443316399195 0.721778341800403 26 0.268122614864691 0.536245229729383 0.731877385135309 27 0.205005895960106 0.410011791920212 0.794994104039894 28 0.143653282783491 0.287306565566981 0.85634671721651 29 0.285061061878549 0.570122123757098 0.714938938121451 30 0.341406363551338 0.682812727102677 0.658593636448662 31 0.42954298276571 0.85908596553142 0.57045701723429 32 0.347116607666212 0.694233215332424 0.652883392333788 33 0.283182617428917 0.566365234857833 0.716817382571084 34 0.317502443722642 0.635004887445284 0.682497556277358 35 0.286240006158942 0.572480012317884 0.713759993841058 36 0.566072996435624 0.867854007128753 0.433927003564376 37 0.496465896149105 0.99293179229821 0.503534103850895 38 0.42957046024808 0.85914092049616 0.57042953975192 39 0.352487042267325 0.70497408453465 0.647512957732675 40 0.371943302531563 0.743886605063125 0.628056697468437 41 0.353778179645111 0.707556359290221 0.64622182035489 42 0.230958296836677 0.461916593673353 0.769041703163323 43 0.153630787491261 0.307261574982522 0.84636921250874 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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