W6Q3 (1)

*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: Thu, 27 Nov 2008 06:09:34 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Nov/27/t1227791443iv8mccu59jqy6g8.htm/, Retrieved Thu, 27 Nov 2008 13:10:52 +0000

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/2008/Nov/27/t1227791443iv8mccu59jqy6g8.htm/}, year = {2008}, } @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 = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

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Dataseries X:

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149.7 0 163.6 0 173.9 0 164.5 0 154.2 0 147.9 0 159.3 0 170.3 0 170 0 174.2 0 190.8 0 179.9 0 240.8 0 241.9 0 241.1 0 239.6 0 220.8 0 209.3 0 209.9 0 228.3 0 242.1 0 226.4 0 231.5 0 229.7 0 257.6 0 260 0 264.4 0 268.8 0 271.4 0 273.8 0 277.4 0 268.2 0 264.6 0 266.6 0 266 0 267.4 0 289.8 0 294 0 310.3 0 311.7 0 302.1 0 298.2 0 299.2 0 296.2 0 299 0 300 0 299.4 0 300.2 0 470.2 0 472.1 0 484.8 0 513.4 1 547.2 1 548.1 1 544.7 1 521.1 1 459 1 413.2 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation x[t] = + 109.775000000000 + 132.360714285714y[t] + 59.7408333333334M1[t] + 59.9566666666667M2[t] + 64.0525M3[t] + 37.7961904761904M4[t] + 32.8520238095238M5[t] + 24.6878571428571M6[t] + 22.8436904761905M7[t] + 17.0795238095238M8[t] + 2.71535714285712M9[t] -12.6288095238096M10[t] + 7.10916666666663M11[t] + 4.48416666666666t + 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) 109.775000000000 19.616496 5.5961 1e-06 1e-06 y 132.360714285714 17.388166 7.6121 0 0 M1 59.7408333333334 22.850856 2.6144 0.012194 0.006097 M2 59.9566666666667 22.829797 2.6262 0.011833 0.005917 M3 64.0525 22.813403 2.8077 0.007411 0.003706 M4 37.7961904761904 23.120905 1.6347 0.109243 0.054622 M5 32.8520238095238 23.086205 1.423 0.161784 0.080892 M6 24.6878571428571 23.05609 1.0708 0.29011 0.145055 M7 22.8436904761905 23.030577 0.9919 0.326677 0.163339 M8 17.0795238095238 23.009682 0.7423 0.461865 0.230932 M9 2.71535714285712 22.993417 0.1181 0.906532 0.453266 M10 -12.6288095238096 22.981792 -0.5495 0.585431 0.292716 M11 7.10916666666663 24.027428 0.2959 0.768716 0.384358 t 4.48416666666666 0.326942 13.7155 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.962175862244345 R-squared 0.925782389885649 Adjusted R-squared 0.90385445962459 F-TEST (value) 42.2193238880252 F-TEST (DF numerator) 13 F-TEST (DF denominator) 44 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 33.9767684191389 Sum Squared Residuals 50794.5148571429 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 149.7 174 -24.3000000000000 2 163.6 178.700000000000 -15.0999999999998 3 173.9 187.28 -13.3800000000001 4 164.5 165.507857142857 -1.00785714285706 5 154.2 165.047857142857 -10.8478571428573 6 147.9 161.367857142857 -13.4678571428571 7 159.3 164.007857142857 -4.7078571428571 8 170.3 162.727857142857 7.57214285714293 9 170 152.847857142857 17.1521428571428 10 174.2 141.987857142857 32.2121428571428 11 190.8 166.21 24.59 12 179.9 163.585 16.315 13 240.8 227.81 12.9899999999999 14 241.9 232.51 9.39 15 241.1 241.09 0.0100000000000149 16 239.6 219.317857142857 20.2821428571428 17 220.8 218.857857142857 1.94214285714288 18 209.3 215.177857142857 -5.87785714285714 19 209.9 217.817857142857 -7.91785714285714 20 228.3 216.537857142857 11.7621428571429 21 242.1 206.657857142857 35.4421428571428 22 226.4 195.797857142857 30.6021428571429 23 231.5 220.02 11.48 24 229.7 217.395 12.305 25 257.6 281.62 -24.02 26 260 286.32 -26.3200000000001 27 264.4 294.9 -30.5 28 268.8 273.127857142857 -4.32785714285713 29 271.4 272.667857142857 -1.26785714285715 30 273.8 268.987857142857 4.81214285714285 31 277.4 271.627857142857 5.77214285714282 32 268.2 270.347857142857 -2.14785714285716 33 264.6 260.467857142857 4.13214285714290 34 266.6 249.607857142857 16.9921428571429 35 266 273.83 -7.83000000000002 36 267.4 271.205 -3.80500000000003 37 289.8 335.43 -45.63 38 294 340.13 -46.1300000000001 39 310.3 348.71 -38.41 40 311.7 326.937857142857 -15.2378571428572 41 302.1 326.477857142857 -24.3778571428571 42 298.2 322.797857142857 -24.5978571428571 43 299.2 325.437857142857 -26.2378571428572 44 296.2 324.157857142857 -27.9578571428572 45 299 314.277857142857 -15.2778571428571 46 300 303.417857142857 -3.41785714285712 47 299.4 327.64 -28.24 48 300.2 325.015 -24.815 49 470.2 389.24 80.96 50 472.1 393.94 78.16 51 484.8 402.52 82.2800000000001 52 513.4 513.108571428572 0.291428571428519 53 547.2 512.648571428571 34.5514285714286 54 548.1 508.968571428571 39.1314285714286 55 544.7 511.608571428571 33.0914285714286 56 521.1 510.328571428571 10.7714285714286 57 459 500.448571428571 -41.4485714285714 58 413.2 489.588571428571 -76.3885714285715 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0107356826420908 0.0214713652841816 0.98926431735791 18 0.00335735311087733 0.00671470622175466 0.996642646889123 19 0.00223114765479689 0.00446229530959378 0.997768852345203 20 0.000638471840343323 0.00127694368068665 0.999361528159657 21 0.000158507811097574 0.000317015622195149 0.999841492188903 22 8.64415174600628e-05 0.000172883034920126 0.99991355848254 23 0.000101484237881278 0.000202968475762555 0.999898515762119 24 4.6130350942051e-05 9.2260701884102e-05 0.999953869649058 25 8.54384531814587e-05 0.000170876906362917 0.999914561546819 26 0.000100301529980100 0.000200603059960199 0.99989969847002 27 7.7301677245156e-05 0.000154603354490312 0.999922698322755 28 2.85963128441043e-05 5.71926256882086e-05 0.999971403687156 29 8.30112464130695e-06 1.66022492826139e-05 0.99999169887536 30 3.28319759050227e-06 6.56639518100454e-06 0.99999671680241 31 1.11465872685438e-06 2.22931745370876e-06 0.999998885341273 32 3.99353772658274e-07 7.98707545316547e-07 0.999999600646227 33 4.34380771246993e-07 8.68761542493985e-07 0.999999565619229 34 2.51750859616211e-06 5.03501719232422e-06 0.999997482491404 35 1.10715910120663e-05 2.21431820241326e-05 0.999988928408988 36 0.000628411975448827 0.00125682395089765 0.99937158802455 37 0.000495578234306678 0.000991156468613356 0.999504421765693 38 0.000328249503689366 0.000656499007378732 0.99967175049631 39 0.000119246639988522 0.000238493279977044 0.999880753360011 40 3.535182900283e-05 7.070365800566e-05 0.999964648170997 41 2.23379162456083e-05 4.46758324912165e-05 0.999977662083754 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 24 0.96 NOK 5% type I error level 25 1 NOK 10% type I error level 25 1 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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