Multiple Regression

*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 11:17:03 -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/t1261333432iyg4ggrcvwbddve.htm/, Retrieved Sun, 20 Dec 2009 19:24:05 +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/t1261333432iyg4ggrcvwbddve.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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149,8 137,7 151,7 156,8 180 180,4 170,4 191,6 199,5 218,2 217,5 205 194 199,3 219,3 211,1 215,2 240,2 242,2 240,7 255,4 253 218,2 203,7 205,6 215,6 188,5 202,9 214 230,3 230 241 259,6 247,8 270,3 289,7 322,7 315 320,2 329,5 360,6 382,2 435,4 464 468,8 403 351,6 252 188 146,5 152,9 148,1 165,1 177 206,1 244,9 228,6 253,4 241,1 261,4 273,7

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 Indexprijs[t] = + 186.456470588235 -12.2956209150327M1[t] -24.0112418300654M2[t] -21.8641176470589M3[t] -20.2569934640523M4[t] -4.50986928104575M5[t] + 8.97725490196074M6[t] + 22.2243790849673M7[t] + 40.2915032679738M8[t] + 44.6786274509804M9[t] + 35.8257516339870M10[t] + 18.9328758169935M11[t] + 1.55287581699346t + 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) 186.456470588235 38.450053 4.8493 1.3e-05 7e-06 M1 -12.2956209150327 44.841807 -0.2742 0.785108 0.392554 M2 -24.0112418300654 47.066221 -0.5102 0.612278 0.306139 M3 -21.8641176470589 47.006117 -0.4651 0.643939 0.32197 M4 -20.2569934640523 46.952274 -0.4314 0.668082 0.334041 M5 -4.50986928104575 46.904715 -0.0961 0.923802 0.461901 M6 8.97725490196074 46.863458 0.1916 0.848893 0.424447 M7 22.2243790849673 46.828519 0.4746 0.637231 0.318615 M8 40.2915032679738 46.799914 0.8609 0.393556 0.196778 M9 44.6786274509804 46.777653 0.9551 0.344299 0.172149 M10 35.8257516339870 46.761746 0.7661 0.447348 0.223674 M11 18.9328758169935 46.752199 0.405 0.687304 0.343652 t 1.55287581699346 0.545514 2.8466 0.006482 0.003241 Multiple Linear Regression - Regression Statistics Multiple R 0.503660876987238 R-squared 0.253674279007554 Adjusted R-squared 0.0670928487594425 F-TEST (value) 1.35959017288175 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 0.218275722619624 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 73.9166856665979 Sum Squared Residuals 262256.468156863 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 149.8 175.713725490196 -25.9137254901962 2 137.7 165.550980392157 -27.8509803921568 3 151.7 169.250980392157 -17.5509803921568 4 156.8 172.410980392157 -15.6109803921568 5 180 189.710980392157 -9.71098039215681 6 180.4 204.750980392157 -24.3509803921569 7 170.4 219.550980392157 -49.1509803921568 8 191.6 239.170980392157 -47.5709803921569 9 199.5 245.110980392157 -45.6109803921567 10 218.2 237.810980392157 -19.6109803921568 11 217.5 222.470980392157 -4.97098039215689 12 205 205.090980392157 -0.0909803921568542 13 194 194.348235294118 -0.348235294117580 14 199.3 184.185490196078 15.1145098039215 15 219.3 187.885490196078 31.4145098039216 16 211.1 191.045490196078 20.0545098039215 17 215.2 208.345490196078 6.85450980392158 18 240.2 223.385490196078 16.8145098039216 19 242.2 238.185490196078 4.01450980392158 20 240.7 257.805490196078 -17.1054901960784 21 255.4 263.745490196078 -8.34549019607845 22 253 256.445490196078 -3.44549019607845 23 218.2 241.105490196078 -22.9054901960784 24 203.7 223.725490196078 -20.0254901960784 25 205.6 212.982745098039 -7.3827450980392 26 215.6 202.82 12.7800000000000 27 188.5 206.52 -18.02 28 202.9 209.68 -6.78000000000001 29 214 226.98 -12.9800000000000 30 230.3 242.02 -11.7200000000000 31 230 256.82 -26.82 32 241 276.44 -35.44 33 259.6 282.38 -22.78 34 247.8 275.08 -27.28 35 270.3 259.74 10.56 36 289.7 242.36 47.34 37 322.7 231.617254901961 91.0827450980393 38 315 221.454509803922 93.5454901960784 39 320.2 225.154509803922 95.0454901960784 40 329.5 228.314509803922 101.185490196078 41 360.6 245.614509803922 114.985490196078 42 382.2 260.654509803922 121.545490196078 43 435.4 275.454509803922 159.945490196078 44 464 295.074509803922 168.925490196078 45 468.8 301.014509803922 167.785490196078 46 403 293.714509803922 109.285490196078 47 351.6 278.374509803922 73.2254901960785 48 252 260.994509803922 -8.99450980392156 49 188 250.251764705882 -62.2517647058823 50 146.5 240.089019607843 -93.5890196078431 51 152.9 243.789019607843 -90.8890196078431 52 148.1 246.949019607843 -98.8490196078431 53 165.1 264.249019607843 -99.1490196078432 54 177 279.289019607843 -102.289019607843 55 206.1 294.089019607843 -87.9890196078431 56 244.9 313.709019607843 -68.8090196078431 57 228.6 319.649019607843 -91.0490196078432 58 253.4 312.349019607843 -58.9490196078431 59 241.1 297.009019607843 -55.9090196078431 60 261.4 279.629019607843 -18.2290196078431 61 273.7 268.886274509804 4.8137254901961 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.000780490164477813 0.00156098032895563 0.999219509835522 17 0.000253878093438721 0.000507756186877441 0.999746121906561 18 2.57817939181908e-05 5.15635878363817e-05 0.999974218206082 19 5.58425863657065e-06 1.11685172731413e-05 0.999994415741363 20 6.04536223789105e-07 1.20907244757821e-06 0.999999395463776 21 5.31201894679895e-08 1.06240378935979e-07 0.99999994687981 22 1.74153591642974e-08 3.48307183285949e-08 0.99999998258464 23 2.44390843256174e-07 4.88781686512347e-07 0.999999755609157 24 4.21641227882270e-07 8.43282455764541e-07 0.999999578358772 25 2.08063058986654e-07 4.16126117973307e-07 0.99999979193694 26 4.16984492090240e-08 8.33968984180481e-08 0.99999995830155 27 9.16844287700599e-08 1.83368857540120e-07 0.999999908315571 28 3.74978251650155e-08 7.4995650330031e-08 0.999999962502175 29 1.56858798425946e-08 3.13717596851892e-08 0.99999998431412 30 5.41732713649657e-09 1.08346542729931e-08 0.999999994582673 31 2.43318218959333e-09 4.86636437918667e-09 0.999999997566818 32 1.93194310205314e-09 3.86388620410628e-09 0.999999998068057 33 1.78823086892251e-09 3.57646173784501e-09 0.99999999821177 34 6.11095551587227e-09 1.22219110317445e-08 0.999999993889044 35 1.69095799049995e-08 3.3819159809999e-08 0.99999998309042 36 1.54501463424023e-07 3.09002926848046e-07 0.999999845498537 37 8.86376561206864e-06 1.77275312241373e-05 0.999991136234388 38 1.65209022905853e-05 3.30418045811705e-05 0.99998347909771 39 2.04520744430858e-05 4.09041488861716e-05 0.999979547925557 40 2.31888948031983e-05 4.63777896063967e-05 0.999976811105197 41 4.02750683604052e-05 8.05501367208104e-05 0.99995972493164 42 7.25887105038182e-05 0.000145177421007636 0.999927411289496 43 0.000766028462484934 0.00153205692496987 0.999233971537515 44 0.00508084379672542 0.0101616875934508 0.994919156203275 45 0.0555399266030238 0.111079853206048 0.944460073396976 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 28 0.933333333333333 NOK 5% type I error level 29 0.966666666666667 NOK 10% type I error level 29 0.966666666666667 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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