autoregression with 4 lags

*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, 30 Nov 2010 16:17:19 +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/Nov/30/t1291133793vf1bf9lzc7hghbd.htm/, Retrieved Tue, 30 Nov 2010 17:16:44 +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/Nov/30/t1291133793vf1bf9lzc7hghbd.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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9743 9084 9081 9700 8587 9743 9084 9081 9731 8587 9743 9084 9563 9731 8587 9743 9998 9563 9731 8587 9437 9998 9563 9731 10038 9437 9998 9563 9918 10038 9437 9998 9252 9918 10038 9437 9737 9252 9918 10038 9035 9737 9252 9918 9133 9035 9737 9252 9487 9133 9035 9737 8700 9487 9133 9035 9627 8700 9487 9133 8947 9627 8700 9487 9283 8947 9627 8700 8829 9283 8947 9627 9947 8829 9283 8947 9628 9947 8829 9283 9318 9628 9947 8829 9605 9318 9628 9947 8640 9605 9318 9628 9214 8640 9605 9318 9567 9214 8640 9605 8547 9567 9214 8640 9185 8547 9567 9214 9470 9185 8547 9567 9123 9470 9185 8547 9278 9123 9470 9185 10170 9278 9123 9470 9434 10170 9278 9123 9655 9434 10170 9278 9429 9655 9434 10170 8739 9429 9655 9434 9552 8739 9429 9655 9687 9552 8739 9429 9019 9687 9552 8739 9672 9019 9687 9552 9206 9672 9019 9687 9069 9206 9672 9019 9788 9069 9206 9672 10312 9788 9069 9206 10105 10312 9788 9069 9863 10105 10312 9788 9656 9863 10105 10312 9295 9656 9863 10105 9946 9295 9656 9863 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 7 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation Yt[t] = + 3540.69197220561 + 0.130932028247944`Yt-1`[t] + 0.201818490077343`Yt-2`[t] + 0.264823448542267`Yt-3`[t] + 385.773255960307M1[t] -436.30006975701M2[t] + 469.50316191611M3[t] + 72.8939494758238M4[t] + 333.401103073757M5[t] + 79.6860766985089M6[t] + 718.477094565068M7[t] + 477.862169357032M8[t] + 160.8783375403M9[t] + 134.048534247597M10[t] -554.848198896826M11[t] + 5.16340774563452t + 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) 3540.69197220561 1487.891151 2.3797 0.020759 0.01038 `Yt-1` 0.130932028247944 0.133903 0.9778 0.332371 0.166186 `Yt-2` 0.201818490077343 0.129049 1.5639 0.123476 0.061738 `Yt-3` 0.264823448542267 0.134251 1.9726 0.053486 0.026743 M1 385.773255960307 200.187808 1.9271 0.059052 0.029526 M2 -436.30006975701 220.764862 -1.9763 0.053053 0.026527 M3 469.50316191611 159.044028 2.952 0.004605 0.002303 M4 72.8939494758238 235.72034 0.3092 0.758287 0.379144 M5 333.401103073757 212.06604 1.5722 0.121548 0.060774 M6 79.6860766985089 183.358501 0.4346 0.665529 0.332764 M7 718.477094565068 182.264961 3.9419 0.000227 0.000113 M8 477.862169357032 222.742583 2.1454 0.036275 0.018138 M9 160.8783375403 205.873838 0.7814 0.437834 0.218917 M10 134.048534247597 177.712911 0.7543 0.45383 0.226915 M11 -554.848198896826 184.793507 -3.0025 0.003996 0.001998 t 5.16340774563452 2.422649 2.1313 0.037467 0.018733 Multiple Linear Regression - Regression Statistics Multiple R 0.881471388071222 R-squared 0.776991807988208 Adjusted R-squared 0.717257470842192 F-TEST (value) 13.0074567679376 F-TEST (DF numerator) 15 F-TEST (DF denominator) 56 p-value 3.57713858534225e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 268.914224668373 Sum Squared Residuals 4049632.17282355 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 9743 9522.5163397682 220.483660231804 2 8587 8628.5703692345 -41.5703692344936 3 9731 9521.97243930522 209.027560694778 4 9563 9221.52935298616 341.470647013836 5 9998 9389.9477797177 608.052220282304 6 9437 9467.4041121753 -30.4041121752984 7 10038 10081.2063737689 -43.2063737689387 8 9918 9926.42303246605 -8.42303246604785 9 9252 9571.61772290947 -319.617722909469 10 9737 9597.6912703139 139.308729686109 11 9035 8811.27005039877 223.729949601227 12 9133 9200.87692416954 -67.8769241695381 13 9487 9591.40771915248 -104.407719152483 14 8700 8654.71989033148 45.2801096685198 15 9627 9560.03946696363 66.9605330363746 16 8947 9224.88400154791 -277.884001547911 17 9283 9380.19046998181 -97.1904699818096 18 8829 9283.88677638959 -454.886776389593 19 9947 9756.12912883447 190.870871165534 20 9628 9664.41470316835 -36.4147031683539 21 9318 9416.23018835444 -98.230188354443 22 9605 9585.6673811861 19.3326188139060 23 8640 8792.4691358855 -152.469135885506 24 9214 9201.9579728728 12.0420271272052 25 9567 9549.29910760005 17.7008923999494 26 8547 8638.897381061 -91.8973810609994 27 9185 9639.56393812742 -454.563938127415 28 9470 9219.28058491148 250.719415088518 29 9123 9380.90705346195 -257.907053461946 30 9278 9313.3976508723 -35.3976508723051 31 10170 9983.09020764064 186.909792359363 32 9434 9803.81818869322 -369.818188693224 33 9655 9616.70151950468 38.2984804953184 34 9429 9711.65520960319 -282.655209603186 35 8739 8848.02307400035 -109.023074000347 36 9552 9330.60658452209 221.393415477913 37 9687 9628.88612966969 58.1138703303121 38 9019 8811.0022884502 207.997711549807 39 9672 9877.05329282463 -205.053292824626 40 9206 9472.04251675742 -266.042516757423 41 9069 9631.58416333172 -562.584163331719 42 9788 9443.9771523542 344.022847645804 43 10312 10031.0148461154 280.985153884632 44 10105 9972.99839337021 132.001606629791 45 9863 9930.2359877542 -67.2359877542051 46 9656 9973.87510096127 -317.875100961272 47 9295 9159.38031726819 135.619682731807 48 9946 9566.2617597199 379.738240280093 49 9701 9914.7602450491 -213.760245049090 50 9049 9101.55455227325 -52.5545522732525 51 10190 10050.1080442064 139.891955793585 52 9706 9611.58828331938 94.4117166806158 53 9765 9871.49775171964 -106.497751719637 54 9893 9835.15452834595 57.845471654055 55 9994 10379.6009953940 -385.600995393981 56 10433 10198.8309629785 234.169037021484 57 10073 9998.77076821949 74.2292317805122 58 10112 10045.3143279499 66.6856720501176 59 9266 9410.29018913498 -144.290189134975 60 9820 9772.06777951748 47.9322204825248 61 10097 10075.1304587605 21.8695412395063 62 9115 9182.25551864958 -67.2555186495814 63 10411 10167.2628185727 243.737181427304 64 9678 9820.67526047764 -142.675260477636 65 10408 9991.87278178719 416.127218212808 66 10153 10034.1797798627 118.820220137337 67 10368 10597.9584482466 -229.958448246609 68 10581 10532.5147193236 48.4852806763516 69 10597 10224.4438132577 372.556186742286 70 10680 10304.7967099857 375.203290014326 71 9738 9691.5672333122 46.4327666877946 72 9556 10149.2289791982 -593.228979198198 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.724299647848351 0.551400704303297 0.275700352151649 20 0.63801872896644 0.723962542067119 0.361981271033559 21 0.576243413776385 0.84751317244723 0.423756586223615 22 0.46824636956869 0.93649273913738 0.53175363043131 23 0.377151798888824 0.754303597777648 0.622848201111176 24 0.456071486919627 0.912142973839254 0.543928513080373 25 0.402916691174363 0.805833382348725 0.597083308825637 26 0.31644303204329 0.63288606408658 0.68355696795671 27 0.282430381833860 0.564860763667721 0.71756961816614 28 0.542854502599966 0.914290994800068 0.457145497400034 29 0.499987254005359 0.999974508010719 0.500012745994641 30 0.445781647396238 0.891563294792477 0.554218352603762 31 0.679376031946832 0.641247936106336 0.320623968053168 32 0.715444183258559 0.569111633482883 0.284555816741441 33 0.738157894403984 0.523684211192032 0.261842105596016 34 0.670567578949318 0.658864842101364 0.329432421050682 35 0.69383164136063 0.61233671727874 0.30616835863937 36 0.714734304364504 0.570531391270992 0.285265695635496 37 0.672545520010682 0.654908959978635 0.327454479989317 38 0.685205951650274 0.629588096699453 0.314794048349726 39 0.608290217583566 0.783419564832868 0.391709782416434 40 0.552676505919095 0.89464698816181 0.447323494080905 41 0.782817392073497 0.434365215853007 0.217182607926503 42 0.826191377730364 0.347617244539272 0.173808622269636 43 0.810940497621126 0.378119004757747 0.189059502378874 44 0.805258967974178 0.389482064051644 0.194741032025822 45 0.740707286300687 0.518585427398626 0.259292713699313 46 0.67569233972825 0.648615320543499 0.324307660271750 47 0.588327678901877 0.823344642196246 0.411672321098123 48 0.780907954874467 0.438184090251065 0.219092045125533 49 0.695057528969951 0.609884942060098 0.304942471030049 50 0.700441050762417 0.599117898475167 0.299558949237583 51 0.721353335955171 0.557293328089658 0.278646664044829 52 0.772065792931168 0.455868414137663 0.227934207068832 53 0.629287079623463 0.741425840753074 0.370712920376537 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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