*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: Wed, 18 Nov 2009 08:05:53 -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/18/t1258556839ftp5ey3v9r2zh43.htm/, Retrieved Wed, 18 Nov 2009 16:07:31 +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/18/t1258556839ftp5ey3v9r2zh43.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:

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No (this computation is public)

User-defined keywords:

Dataseries X:

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900.1 33 880.4 849.4 819.3 785.8 937.2 31.3 900.1 880.4 849.4 819.3 948.9 29 937.2 900.1 880.4 849.4 952.6 28.7 948.9 937.2 900.1 880.4 947.3 28 952.6 948.9 937.2 900.1 974.2 29.7 947.3 952.6 948.9 937.2 1000.8 30.7 974.2 947.3 952.6 948.9 1032.8 24 1000.8 974.2 947.3 952.6 1050.7 29 1032.8 1000.8 974.2 947.3 1057.3 33 1050.7 1032.8 1000.8 974.2 1075.4 28 1057.3 1050.7 1032.8 1000.8 1118.4 28.7 1075.4 1057.3 1050.7 1032.8 1179.8 31.7 1118.4 1075.4 1057.3 1050.7 1227 34 1179.8 1118.4 1075.4 1057.3 1257.8 35.3 1227 1179.8 1118.4 1075.4 1251.5 27 1257.8 1227 1179.8 1118.4 1236.3 31.3 1251.5 1257.8 1227 1179.8 1170.6 38.7 1236.3 1251.5 1257.8 1227 1213.1 37.3 1170.6 1236.3 1251.5 1257.8 1265.5 37.3 1213.1 1170.6 1236.3 1251.5 1300.8 37.7 1265.5 1213.1 1170.6 1236.3 1348.4 34.7 1300.8 1265.5 1213.1 1170.6 1371.9 34.7 1348.4 1300.8 1265.5 1213.1 1403.3 33.7 1371.9 1348.4 1300.8 1265.5 1451.8 38.3 1403.3 1371.9 1348.4 1300.8 1474.2 38 1451.8 1403.3 1371.9 1348.4 1438.2 38.3 1474.2 1451.8 14 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 89.7311676142938 + 6.0227316227574X[t] + 1.03930927110254Y1[t] -0.149311449946610Y2[t] -0.0396953938960989Y3[t] -0.094152509972795Y4[t] -9.91198743304521M1[t] -18.0063243920921M2[t] -37.757433211208M3[t] + 20.7736076777111M4[t] -4.61820414700779M5[t] -57.0772202441823M6[t] -54.6300927530562M7[t] -4.68641875812672M8[t] + 9.28292783233514M9[t] -50.7958567312177M10[t] -29.8191819838307M11[t] + 1.37269517473825t + 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) 89.7311676142938 47.458048 1.8907 0.066104 0.033052 X 6.0227316227574 2.514388 2.3953 0.021505 0.010753 Y1 1.03930927110254 0.166109 6.2568 0 0 Y2 -0.149311449946610 0.234821 -0.6359 0.528587 0.264293 Y3 -0.0396953938960989 0.242334 -0.1638 0.870731 0.435365 Y4 -0.094152509972795 0.156609 -0.6012 0.551189 0.275595 M1 -9.91198743304521 38.075056 -0.2603 0.795981 0.39799 M2 -18.0063243920921 38.214605 -0.4712 0.640131 0.320065 M3 -37.757433211208 37.886248 -0.9966 0.325102 0.162551 M4 20.7736076777111 37.450927 0.5547 0.582275 0.291137 M5 -4.61820414700779 38.048225 -0.1214 0.904015 0.452008 M6 -57.0772202441823 39.347995 -1.4506 0.154894 0.077447 M7 -54.6300927530562 39.056067 -1.3988 0.169787 0.084894 M8 -4.68641875812672 37.051435 -0.1265 0.899999 0.449999 M9 9.28292783233514 37.80429 0.2456 0.807317 0.403658 M10 -50.7958567312177 39.986767 -1.2703 0.211498 0.105749 M11 -29.8191819838307 39.950948 -0.7464 0.459905 0.229952 t 1.37269517473825 0.923816 1.4859 0.145346 0.072673 Multiple Linear Regression - Regression Statistics Multiple R 0.985107834520052 R-squared 0.970437445632787 Adjusted R-squared 0.95755120398554 F-TEST (value) 75.3080279105372 F-TEST (DF numerator) 17 F-TEST (DF denominator) 39 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 55.0451702605518 Sum Squared Residuals 118168.859991512 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 900.1 961.617277045312 -61.517277045312 2 937.2 956.153788754332 -18.9537887543321 3 948.9 955.475483010608 -6.57548301060806 4 952.6 1016.49213619741 -63.8921361974092 5 947.3 987.028104190193 -39.7281041901932 6 974.2 936.162141296223 38.0378587037767 7 1000.8 973.505008338124 27.2949916618764 8 1032.8 1007.96024554383 24.8397544561695 9 1050.7 1082.13335973657 -31.4333597365699 10 1057.3 1057.75526639732 -0.455266397324585 11 1075.4 1050.40303507094 24.9969649290564 12 1118.4 1099.91343873288 18.4865612671182 13 1179.8 1149.48278322800 30.3172167720044 14 1227 1212.66672787868 14.333272121319 15 1257.8 1238.59447754517 19.2055224548336 16 1251.5 1266.98691113837 -15.4869111383651 17 1236.3 1250.06451269571 -13.7645126957143 18 1170.6 1223.02295039287 -52.4229503928718 19 1213.1 1149.74804738901 63.3519526109895 20 1265.5 1256.24135364208 9.25864635792217 21 1300.8 1326.14566276998 -25.3456627699837 22 1348.4 1282.73384147024 65.6661585297575 23 1371.9 1343.20211819973 28.6978818002656 24 1403.3 1379.35296766189 23.9470323381095 25 1451.8 1422.43064855565 29.369351444352 26 1474.2 1454.20580617340 19.9941938266014 27 1438.2 1450.21411501344 -12.0141150134365 28 1513.6 1490.97654456162 22.6234554383757 29 1562.2 1539.21825397114 22.9817460288574 30 1546.2 1514.65883500967 31.5411649903301 31 1527.5 1492.58053788064 34.9194621193598 32 1418.7 1517.82251148454 -99.1225114845368 33 1448.5 1409.30277238903 39.1972276109702 34 1492.1 1403.67556803234 88.4244319676616 35 1395.4 1469.35521278486 -73.9552127848577 36 1403.7 1398.38086242114 5.31913757885518 37 1316.6 1392.10841497032 -75.508414970322 38 1274.5 1265.65258032868 8.8474196713204 39 1264.4 1255.41300772005 8.98699227994683 40 1323.9 1297.52035978246 26.3796402175418 41 1332.1 1336.48140635051 -4.38140635050854 42 1250.2 1277.35267108198 -27.1526710819800 43 1096.7 1211.48596983716 -114.785969837164 44 1080.8 1027.66024700039 53.1397529996119 45 1039.2 1039.83011785466 -0.63011785466105 46 792 945.635324100095 -153.635324100095 47 746.6 726.339633944464 20.2603660555358 48 688.8 736.552731184083 -47.7527311840829 49 715.8 638.460876200722 77.3391237992777 50 672.9 697.121096864909 -24.2210968649087 51 629.5 639.102916710736 -9.60291671073588 52 681.2 650.824048320143 30.3759516798568 53 755.4 720.507722792441 34.8922772075587 54 760.6 750.603402219255 9.99659778074505 55 765.9 776.680436555062 -10.7804365550621 56 836.8 824.915642329167 11.8843576708332 57 904.9 886.688087249756 18.2119127502445 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.0625703745636458 0.125140749127292 0.937429625436354 22 0.0262296946282870 0.0524593892565741 0.973770305371713 23 0.00730444770864296 0.0146088954172859 0.992695552291357 24 0.00183221945397437 0.00366443890794873 0.998167780546026 25 0.000449616018408385 0.00089923203681677 0.999550383981592 26 9.67248575731702e-05 0.000193449715146340 0.999903275142427 27 0.000237135012076653 0.000474270024153306 0.999762864987923 28 0.000269587142836578 0.000539174285673157 0.999730412857163 29 0.000442866123657741 0.000885732247315483 0.999557133876342 30 0.000220452256841372 0.000440904513682743 0.999779547743159 31 0.000191329832277194 0.000382659664554387 0.999808670167723 32 0.00157648753621753 0.00315297507243507 0.998423512463782 33 0.0195604479590753 0.0391208959181506 0.980439552040925 34 0.186744083396987 0.373488166793974 0.813255916603013 35 0.468221124568351 0.936442249136702 0.531778875431649 36 0.413090053753477 0.826180107506953 0.586909946246523 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 9 0.5625 NOK 5% type I error level 11 0.6875 NOK 10% type I error level 12 0.75 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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