Model 4

*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: Sat, 19 Dec 2009 09:23:35 -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/19/t1261239934uxbxqz74eig97oi.htm/, Retrieved Sat, 19 Dec 2009 17:25:46 +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/19/t1261239934uxbxqz74eig97oi.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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2172 2155 3016 0 2150 2172 2155 0 2533 2150 2172 0 2058 2533 2150 0 2160 2058 2533 0 2260 2160 2058 0 2498 2260 2160 0 2695 2498 2260 0 2799 2695 2498 0 2946 2799 2695 0 2930 2946 2799 0 2318 2930 2946 0 2540 2318 2930 0 2570 2540 2318 0 2669 2570 2540 0 2450 2669 2570 0 2842 2450 2669 0 3440 2842 2450 0 2678 3440 2842 0 2981 2678 3440 0 2260 2981 2678 0 2844 2260 2981 0 2546 2844 2260 0 2456 2546 2844 0 2295 2456 2546 0 2379 2295 2456 0 2479 2379 2295 0 2057 2479 2379 0 2280 2057 2479 0 2351 2280 2057 0 2276 2351 2280 0 2548 2276 2351 0 2311 2548 2276 0 2201 2311 2548 0 2725 2201 2311 1 2408 2725 2201 1 2139 2408 2725 1 1898 2139 2408 1 2537 1898 2139 1 2068 2537 1898 1 2063 2068 2537 1 2520 2063 2068 1 2434 2520 2063 1 2190 2434 2520 1 2794 2190 2434 1 2070 2794 2190 1 2615 2070 2794 1 2265 2615 2070 1 2139 2265 2615 1 2428 2139 2265 1 2137 2428 2139 1 1823 2137 2428 1 2063 1823 2137 1 1806 2063 1823 1 1758 1806 2063 1 2243 1758 1806 1 1993 2243 1758 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation y[t] = + 1112.02652796567 + 0.223233492472836`y(t-1)`[t] + 0.319262189799833`y(t-2)`[t] + 35.0304516301858x[t] -124.87050947952M1[t] + 65.4880752713608M2[t] + 271.293404992809M3[t] -153.152721244156M4[t] + 68.4632365716367M5[t] + 346.830003968164M6[t] + 101.994476866418M7[t] + 281.262139157106M8[t] + 188.653515284143M9[t] + 119.308989555359M10[t] + 414.965821439792M11[t] -5.81464467737814t + 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) 1112.02652796567 496.606678 2.2393 0.030366 0.015183 `y(t-1)` 0.223233492472836 0.140731 1.5862 0.120012 0.060006 `y(t-2)` 0.319262189799833 0.143454 2.2255 0.031341 0.01567 x 35.0304516301858 138.184168 0.2535 0.801086 0.400543 M1 -124.87050947952 187.035294 -0.6676 0.507937 0.253968 M2 65.4880752713608 183.43175 0.357 0.722826 0.361413 M3 271.293404992809 182.782316 1.4842 0.145038 0.072519 M4 -153.152721244156 176.53059 -0.8676 0.390446 0.195223 M5 68.4632365716367 191.92322 0.3567 0.723045 0.361522 M6 346.830003968164 186.683923 1.8578 0.070047 0.035023 M7 101.994476866418 176.609715 0.5775 0.566605 0.283302 M8 281.262139157106 179.772029 1.5645 0.125019 0.06251 M9 188.653515284143 175.446437 1.0753 0.288247 0.144124 M10 119.308989555359 177.412922 0.6725 0.504868 0.252434 M11 414.965821439792 175.003944 2.3712 0.022282 0.011141 t -5.81464467737814 4.046632 -1.4369 0.157981 0.07899 Multiple Linear Regression - Regression Statistics Multiple R 0.744050670630097 R-squared 0.553611400465098 Adjusted R-squared 0.397894447138969 F-TEST (value) 3.55524166534155 F-TEST (DF numerator) 15 F-TEST (DF denominator) 43 p-value 0.000553619578918085 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 254.921020964398 Sum Squared Residuals 2794343.25796983 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2172 2425.30431452403 -253.30431452403 2 2150 2338.75847855192 -188.758478551918 3 2533 2539.26548398818 -6.26548398818243 4 2058 2187.47937251534 -129.479372515339 5 2160 2419.52219542249 -259.522195422492 6 2260 2563.19459421895 -303.194594218949 7 2498 2367.43251504669 130.567484953307 8 2695 2625.94132284852 69.0586771514796 9 2799 2647.47945348769 151.520546512311 10 2946 2658.43121768927 287.568782310732 11 2930 3014.29199602901 -84.2919960290127 12 2318 2636.87133593285 -318.871335932853 13 2540 2364.45908934578 175.540910654218 14 2570 2403.17240459076 166.827595409244 15 2669 2680.73630054457 -11.7363005445745 16 2450 2282.15351107904 167.846488920963 17 2842 2480.67364615608 361.326353843916 18 3440 2770.81487835842 669.185121641579 19 2678 2778.80911347959 -100.809113479587 20 2981 2973.07699932890 7.92300067110365 21 2260 2699.01569037035 -439.015690370352 22 2844 2559.64161540062 284.358384599376 23 2546 2749.66412336614 -203.664123366136 24 2456 2448.80919533516 7.19080466483707 25 2295 2202.89289429536 92.1071057046404 26 2379 2322.76264499875 56.2373550012491 27 2479 2490.10373085277 -11.1037308527658 28 2057 2108.98433312889 -51.9843331288926 29 2280 2262.50733142375 17.4926685762466 30 2351 2450.11187886881 -99.1118788688148 31 2276 2286.50675338063 -10.5067533806252 32 2548 2465.88487453426 82.1151254657393 33 2311 2404.23645170154 -93.2364517015436 34 2201 2363.01025920487 -162.010259204873 35 2725 2587.66207488754 137.337925112458 36 2408 2248.73711794816 159.262882051844 37 2139 2214.58033413248 -75.5803341324816 38 1898 2237.86835056424 -339.868350564244 39 2537 2298.17823486621 238.821765133794 40 2068 1933.62147790025 134.378522099754 41 2063 2248.73482235099 -185.734822350993 42 2520 2370.43681059166 149.563189408344 43 2434 2220.20803392362 213.791966076381 44 2190 2520.36579192279 -330.365791922789 45 2794 2340.01700288629 453.98299711371 46 2070 2321.79088762256 -251.790887622561 47 2615 2642.84638891838 -27.8463889183824 48 2265 2112.58235078383 152.417649216172 49 2139 2077.76336770235 61.2366322976532 50 2428 2122.43812129433 305.561878705669 51 2137 2346.71624974827 -209.716249748271 52 1823 1943.76130537649 -120.761305376485 53 2063 1996.56200464668 66.4379953533226 54 1806 2222.44183796216 -416.441837962159 55 1758 1991.04358416948 -233.043584169476 56 2243 2071.73101136553 171.268988634467 57 1993 2066.25140155413 -73.2514015541256 58 1932 2090.12602008267 -158.126020082673 59 2465 2286.53541679893 178.464583201073 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.9228131509526 0.1543736980948 0.0771868490474 20 0.871305105500001 0.257389788999997 0.128694894499999 21 0.984940747942303 0.0301185041153943 0.0150592520576971 22 0.979330728410618 0.0413385431787632 0.0206692715893816 23 0.976941752670021 0.0461164946599578 0.0230582473299789 24 0.959478231910724 0.081043536178551 0.0405217680892755 25 0.931896616627819 0.136206766744363 0.0681033833721815 26 0.889014984708678 0.221970030582644 0.110985015291322 27 0.835480112328612 0.329039775342776 0.164519887671388 28 0.770777444203218 0.458445111593564 0.229222555796782 29 0.68438173009147 0.63123653981706 0.31561826990853 30 0.62644898695351 0.74710202609298 0.37355101304649 31 0.516734304998641 0.966531390002718 0.483265695001359 32 0.434614962934235 0.86922992586847 0.565385037065765 33 0.340806889675309 0.681613779350618 0.659193110324691 34 0.31104403520434 0.62208807040868 0.68895596479566 35 0.222429617765780 0.444859235531559 0.77757038223422 36 0.145634303519472 0.291268607038943 0.854365696480529 37 0.122837224132581 0.245674448265161 0.87716277586742 38 0.417357875312944 0.834715750625889 0.582642124687056 39 0.357913879119111 0.715827758238222 0.642086120880889 40 0.342421016676329 0.684842033352657 0.657578983323671 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 3 0.136363636363636 NOK 10% type I error level 4 0.181818181818182 NOK

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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