*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: Fri, 27 Nov 2009 07:22:07 -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/27/t1259331763r1117jrnrf6p0cg.htm/, Retrieved Fri, 27 Nov 2009 15:22:55 +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/27/t1259331763r1117jrnrf6p0cg.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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416.25 1111.92 398.35 1131.13 400.00 1144.94 427.25 1113.89 391.25 1107.30 397.20 1120.68 394.80 1140.84 391.50 1101.72 407.65 1104.24 418.10 1114.58 429.10 1130.20 452.85 1173.78 427.75 1211.92 420.90 1181.27 433.45 1203.60 427.15 1180.59 427.90 1156.85 415.35 1191.50 432.60 1191.33 431.65 1234.18 439.60 1220.33 466.10 1228.81 459.50 1207.01 499.75 1249.48 530.00 1248.29 568.25 1280.08 564.25 1280.66 587.00 1302.88 661.00 1310.61 625.00 1270.05 622.95 1270.06 637.25 1278.53 621.05 1303.80 600.60 1335.83 614.10 1377.76 648.75 1400.63 639.75 1418.03 660.20 1437.90 670.40 1406.80 658.25 1420.83 673.60 1482.37 666.50 1530.63 654.75 1504.66 665.75 1455.18 672.00 1473.96 742.50 1527.29 790.25 1545.79 784.25 1479.63 846.75 1467.97 914.75 1378.60 988.50 1330.45 887.75 1326.41 853.00 1385.97 888.25 1399.62 937.50 1276.69 912.50 1269.42 822.25 1287.83 880.00 1164.17 729.50 968.67 778.00 888.61

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 S&P500[t] = + 1050.30650339918 + 0.0886434819321215Gold[t] + 98.9169823147264M1[t] + 83.611958587221M2[t] + 69.76921869153M3[t] + 62.9584613505614M4[t] + 78.6478279445748M5[t] + 89.1124244764496M6[t] + 58.7730879173032M7[t] + 46.2655331371204M8[t] + 54.173103801218M9[t] + 44.0432918683740M10[t] + 13.6299886258529M11[t] + 3.66758313090918t + 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) 1050.30650339918 116.953732 8.9805 0 0 Gold 0.0886434819321215 0.330092 0.2685 0.789483 0.394741 M1 98.9169823147264 87.450659 1.1311 0.263868 0.131934 M2 83.611958587221 88.010283 0.95 0.347065 0.173532 M3 69.76921869153 88.589149 0.7876 0.434994 0.217497 M4 62.9584613505614 86.892063 0.7246 0.472392 0.236196 M5 78.6478279445748 86.507707 0.9091 0.368014 0.184007 M6 89.1124244764496 85.941572 1.0369 0.305205 0.152602 M7 58.7730879173032 85.883988 0.6843 0.4972 0.2486 M8 46.2655331371204 85.571244 0.5407 0.591346 0.295673 M9 54.173103801218 85.480751 0.6337 0.529385 0.264693 M10 44.0432918683740 85.415983 0.5156 0.60858 0.30429 M11 13.6299886258529 85.572301 0.1593 0.874145 0.437073 t 3.66758313090918 3.374967 1.0867 0.28283 0.141415 Multiple Linear Regression - Regression Statistics Multiple R 0.555452976967711 R-squared 0.308528009622292 Adjusted R-squared 0.113112012341636 F-TEST (value) 1.57882677936128 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0.126606643347300 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 134.947330579272 Sum Squared Residuals 837695.973401681 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1111.92 1189.78891819906 -77.8689181990577 2 1131.13 1176.56475927588 -45.4347592758774 3 1144.94 1166.53586425628 -21.5958642562836 4 1113.89 1165.80822492887 -51.9182249288745 5 1107.3 1181.97400930424 -74.674009304241 6 1120.68 1196.63361768452 -75.9536176845209 7 1140.84 1169.74911989965 -28.9091198996467 8 1101.72 1160.61662476000 -58.8966247599971 9 1104.24 1173.62337078821 -69.3833707882077 10 1114.58 1168.08746637246 -53.5074663724635 11 1130.2 1142.31682456210 -12.1168245621048 12 1173.78 1134.45970176305 39.320298236951 13 1211.92 1234.81931581219 -22.8993158121882 14 1181.27 1222.57466736436 -41.3046673643571 15 1203.6 1213.51198629782 -9.91198629782346 16 1180.59 1209.81035815159 -29.2203581515917 17 1156.85 1229.23379048796 -72.3837904879634 18 1191.5 1242.2534944525 -50.7534944524991 19 1191.33 1217.11084108759 -25.7808410875911 20 1234.18 1208.18665813048 25.9933418695182 21 1220.33 1220.46652760685 -0.136527606849205 22 1228.81 1216.35335107612 12.4566489238846 23 1207.01 1189.02258398375 17.9874160162484 24 1249.48 1182.62807863658 66.8519213634244 25 1248.29 1287.89410941066 -39.604109410658 26 1280.08 1279.64728199797 0.432718002034516 27 1280.66 1269.11755130545 11.5424486945450 28 1302.88 1267.99101630935 34.8889836906487 29 1310.61 1293.90758369725 16.7024163027489 30 1270.05 1304.84859801048 -34.7985980104785 31 1270.06 1277.99512544428 -7.93512544428057 32 1278.53 1270.42275558664 8.10724441336372 33 1303.8 1280.56188497434 23.2381150256572 34 1335.83 1272.28689696690 63.543103033104 35 1377.76 1246.73786386137 131.022136138632 36 1400.63 1239.84695501537 160.783044984628 37 1418.03 1341.63372912362 76.3962708763815 38 1437.9 1331.80904773253 106.090952267466 39 1406.8 1322.53805448346 84.26194551654 40 1420.83 1318.31786196793 102.512138032075 41 1482.37 1339.03548914051 143.334510859494 42 1530.63 1352.53830008157 178.091699918428 43 1504.66 1324.82498574063 179.835014259368 44 1455.18 1316.96009239261 138.219907607388 45 1473.96 1329.08926794969 144.870732050306 46 1527.29 1328.87640462397 198.413595376026 47 1545.79 1306.36341077462 239.426589225379 48 1479.63 1295.86914438808 183.760855611915 49 1467.97 1403.99392745448 63.9760725455222 50 1378.6 1398.38424362927 -19.784243629266 51 1330.45 1394.74654365698 -64.296543656978 52 1326.41 1382.67253864226 -56.2625386422572 53 1385.97 1398.94912737004 -12.9791273700387 54 1399.62 1416.20598977093 -16.5859897709299 55 1276.69 1393.89992782785 -117.209927827850 56 1269.42 1382.84386913027 -113.423869130273 57 1287.83 1386.41894868091 -98.588948680906 58 1164.17 1385.07588096055 -220.905880960551 59 968.67 1344.98931681815 -376.319316818155 60 888.61 1339.32612019692 -450.716120196919 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.00109057808360239 0.00218115616720479 0.998909421916398 18 8.58923399562958e-05 0.000171784679912592 0.999914107660044 19 5.97652863613641e-06 1.19530572722728e-05 0.999994023471364 20 4.56392914525523e-05 9.12785829051046e-05 0.999954360708547 21 1.37742626097482e-05 2.75485252194964e-05 0.99998622573739 22 2.71535982390933e-06 5.43071964781867e-06 0.999997284640176 23 3.79780465551594e-07 7.59560931103189e-07 0.999999620219534 24 5.12840642033831e-08 1.02568128406766e-07 0.999999948715936 25 1.79996093388865e-08 3.59992186777730e-08 0.99999998200039 26 3.22284136241513e-09 6.44568272483025e-09 0.999999996777159 27 4.33797053489750e-10 8.67594106979499e-10 0.999999999566203 28 1.69046912483306e-10 3.38093824966611e-10 0.999999999830953 29 6.98148417753035e-11 1.39629683550607e-10 0.999999999930185 30 6.16275344994378e-11 1.23255068998876e-10 0.999999999938372 31 4.95074615942361e-11 9.90149231884722e-11 0.999999999950492 32 4.30476165160437e-11 8.60952330320873e-11 0.999999999956952 33 1.76344787162719e-10 3.52689574325437e-10 0.999999999823655 34 9.5845089065071e-10 1.91690178130142e-09 0.99999999904155 35 1.11894447430857e-07 2.23788894861714e-07 0.999999888105553 36 7.8688147410268e-06 1.57376294820536e-05 0.99999213118526 37 0.000353611435955858 0.000707222871911715 0.999646388564044 38 0.000457079719638151 0.000914159439276301 0.999542920280362 39 0.000194709091825589 0.000389418183651179 0.999805290908174 40 0.000157233481145535 0.00031446696229107 0.999842766518855 41 0.00859291468352392 0.0171858293670478 0.991407085316476 42 0.0525098795499005 0.105019759099801 0.9474901204501 43 0.112209664759354 0.224419329518708 0.887790335240646 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 24 0.888888888888889 NOK 5% type I error level 25 0.925925925925926 NOK 10% type I error level 25 0.925925925925926 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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