Multiple Linear Regression (M4)

*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: Sun, 20 Dec 2009 05:11:15 -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/20/t126131122325kd4edllsr56zo.htm/, Retrieved Sun, 20 Dec 2009 13:13:55 +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/Dec/20/t126131122325kd4edllsr56zo.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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921365 0 0 987921 0 0 1132614 0 0 1332224 0 0 1418133 0 0 1411549 0 0 1695920 0 0 1636173 0 0 1539653 0 0 1395314 0 0 1127575 0 0 1036076 0 0 989236 0 0 1008380 0 0 1207763 0 0 1368839 0 0 1469798 0 0 1498721 0 0 1761769 0 0 1653214 0 0 1599104 0 0 1421179 0 0 1163995 0 0 1037735 0 0 1015407 0 0 1039210 0 0 1258049 0 0 1469445 0 0 1552346 0 0 1549144 0 0 1785895 0 0 1662335 0 0 1629440 0 0 1467430 0 0 1202209 0 0 1076982 0 0 1039367 1 0 1063449 1 0 1335135 1 0 1491602 1 0 1591972 1 0 1641248 1 0 1898849 1 0 1798580 1 0 1762444 1 0 1622044 1 0 1368955 1 0 1262973 1 0 1195650 1 0 1269530 1 0 1479279 1 0 1607819 1 0 1712466 1 0 1721766 1 0 1949843 1 0 1821326 1 0 1757802 1 1 1590367 1 1 1260647 1 1 1149235 1 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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 952333.91111111 + 52432.972222222X[t] -87550.5749999999Crisis[t] -49993.2215277782M1[t] -12855.8663888886M2[t] + 191658.48875M3[t] + 358720.64388889M4[t] + 449322.199027778M5[t] + 460509.154166666M6[t] + 710123.109305555M7[t] + 601637.864444445M8[t] + 558155.334583333M9[t] + 395377.889722222M10[t] + 116431.644861111M11[t] + 4355.64486111111t + 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) 952333.91111111 20235.112255 47.0634 0 0 X 52432.972222222 18171.667241 2.8854 0.005982 0.002991 Crisis -87550.5749999999 21113.569379 -4.1466 0.000147 7.4e-05 M1 -49993.2215277782 22746.362239 -2.1979 0.033146 0.016573 M2 -12855.8663888886 22634.487496 -0.568 0.572874 0.286437 M3 191658.48875 22534.882292 8.505 0 0 M4 358720.64388889 22447.709954 15.9803 0 0 M5 449322.199027778 22373.115811 20.0831 0 0 M6 460509.154166666 22311.226023 20.6402 0 0 M7 710123.109305555 22262.146544 31.8982 0 0 M8 601637.864444445 22225.96224 27.0691 0 0 M9 558155.334583333 21865.545174 25.5267 0 0 M10 395377.889722222 21832.484954 18.1096 0 0 M11 116431.644861111 21812.62477 5.3378 3e-06 1e-06 t 4355.64486111111 537.525166 8.1031 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.993887953915085 R-squared 0.987813264937514 Adjusted R-squared 0.984021836251407 F-TEST (value) 260.53853223119 F-TEST (DF numerator) 14 F-TEST (DF denominator) 45 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 34478.3144177152 Sum Squared Residuals 53493937428.9072 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 921365 906696.334444445 14668.6655555549 2 987921 948189.334444444 39731.6655555558 3 1132614 1157059.33444444 -24445.3344444442 4 1332224 1328477.13444444 3746.86555555624 5 1418133 1423434.33444444 -5301.3344444448 6 1411549 1438976.93444444 -27427.9344444444 7 1695920 1692946.53444445 2973.46555555489 8 1636173 1588816.93444444 47356.0655555555 9 1539653 1549690.04944444 -10037.049444444 10 1395314 1391268.24944444 4045.75055555558 11 1127575 1116677.64944445 10897.3505555551 12 1036076 1004601.64944444 31474.3505555556 13 989236 958964.072777777 30271.9272222226 14 1008380 1000457.07277778 7922.92722222221 15 1207763 1209327.07277778 -1564.07277777785 16 1368839 1380744.87277778 -11905.8727777779 17 1469798 1475702.07277778 -5904.07277777768 18 1498721 1491244.67277778 7476.3272222222 19 1761769 1745214.27277778 16554.7272222225 20 1653214 1641084.67277778 12129.3272222223 21 1599104 1601957.78777778 -2853.78777777782 22 1421179 1443535.98777778 -22356.9877777778 23 1163995 1168945.38777778 -4950.3877777777 24 1037735 1056869.38777778 -19134.3877777778 25 1015407 1011231.81111111 4175.18888888898 26 1039210 1052724.81111111 -13514.8111111112 27 1258049 1261594.81111111 -3545.81111111128 28 1469445 1433012.61111111 36432.3888888887 29 1552346 1527969.81111111 24376.188888889 30 1549144 1543512.41111111 5631.58888888884 31 1785895 1797482.01111111 -11587.0111111109 32 1662335 1693352.41111111 -31017.4111111111 33 1629440 1654225.52611111 -24785.5261111112 34 1467430 1495803.72611111 -28373.7261111111 35 1202209 1221213.12611111 -19004.1261111110 36 1076982 1109137.12611111 -32155.1261111112 37 1039367 1115932.52166667 -76565.5216666665 38 1063449 1157425.52166667 -93976.5216666667 39 1335135 1366295.52166667 -31160.5216666667 40 1491602 1537713.32166667 -46111.3216666668 41 1591972 1632670.52166667 -40698.5216666666 42 1641248 1648213.12166667 -6965.12166666665 43 1898849 1902182.72166667 -3333.72166666657 44 1798580 1798053.12166667 526.878333333297 45 1762444 1758926.23666667 3517.7633333331 46 1622044 1600504.43666667 21539.5633333333 47 1368955 1325913.83666667 43041.1633333335 48 1262973 1213837.83666667 49135.1633333333 49 1195650 1168200.26 27449.7400000001 50 1269530 1209693.26 59836.7399999999 51 1479279 1418563.26 60715.74 52 1607819 1589981.06 17837.9399999998 53 1712466 1684938.26 27527.7400000000 54 1721766 1700480.86 21285.1400000000 55 1949843 1954450.46 -4607.45999999989 56 1821326 1850320.86 -28994.8600000001 57 1757802 1723643.4 34158.5999999999 58 1590367 1565221.6 25145.4 59 1260647 1290631 -29983.9999999999 60 1149235 1178555 -29320 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.219946285219054 0.439892570438108 0.780053714780946 19 0.113648087708745 0.227296175417491 0.886351912291255 20 0.122583449611831 0.245166899223662 0.877416550388169 21 0.0620405012208582 0.124081002441716 0.937959498779142 22 0.0401598164000995 0.080319632800199 0.9598401835999 23 0.0211115947503885 0.0422231895007770 0.978888405249611 24 0.0273107983854356 0.0546215967708711 0.972689201614564 25 0.0152690465467333 0.0305380930934666 0.984730953453267 26 0.0101534396727774 0.0203068793455548 0.989846560327223 27 0.00638239210757405 0.0127647842151481 0.993617607892426 28 0.0200065939812660 0.0400131879625319 0.979993406018734 29 0.0283628415086551 0.0567256830173102 0.971637158491345 30 0.0221569912430821 0.0443139824861641 0.977843008756918 31 0.0200941507504292 0.0401883015008583 0.97990584924957 32 0.0518074839583102 0.103614967916620 0.94819251604169 33 0.0301699661915055 0.060339932383011 0.969830033808494 34 0.0172051141287687 0.0344102282575375 0.982794885871231 35 0.00928334314555144 0.0185666862911029 0.990716656854449 36 0.00568916141770414 0.0113783228354083 0.994310838582296 37 0.00367916438445057 0.00735832876890115 0.99632083561555 38 0.0185469183373142 0.0370938366746285 0.981453081662686 39 0.0594084928536995 0.118816985707399 0.9405915071463 40 0.0490966776683569 0.0981933553367137 0.950903322331643 41 0.0682286238102211 0.136457247620442 0.931771376189779 42 0.0735784926526052 0.147156985305210 0.926421507347395 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.04 NOK 5% type I error level 12 0.48 NOK 10% type I error level 17 0.68 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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