multiple regression, include lineaire trend

*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, 19 Nov 2009 11:17:28 -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/19/t1258654743pgj9wiv51i4jd82.htm/, Retrieved Thu, 19 Nov 2009 19:19:15 +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/19/t1258654743pgj9wiv51i4jd82.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:

y = aantal bouwvergunningen x= rente

Dataseries X:

» Textbox « » Textfile « » CSV «

2360 2 2214 2 2825 2 2355 2 2333 2 3016 2 2155 2 2172 2 2150 2 2533 2 2058 2 2160 2 2260 2 2498 2 2695 2 2799 2 2947 2 2930 2 2318 2 2540 2 2570 2 2669 2 2450 2 2842 2 3440 2 2678 2 2981 2 2260 2,21 2844 2,25 2546 2,25 2456 2,45 2295 2,5 2379 2,5 2479 2,64 2057 2,75 2280 2,93 2351 3 2276 3,17 2548 3,25 2311 3,39 2201 3,5 2725 3,5 2408 3,65 2139 3,75 1898 3,75 2537 3,9 2069 4 2063 4 2524 4 2437 4 2189 4 2793 4 2074 4 2622 4 2278 4 2144 4 2427 4 2139 4 1828 4,18 2072 4,25 1800 4,25

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] = + 2996.21858257135 -397.434462871071X[t] + 177.028669075707M1[t] + 114.595914455059M2[t] + 334.238409008073M3[t] + 204.342364556124M4[t] + 178.748941589334M5[t] + 453.032484736411M6[t] + 22.3364402844630M7[t] -44.4569826823279M8[t] -31.3734395352507M9[t] + 164.561302458348M10[t] -197.155266290631M11[t] + 13.7164568529229t + 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) 2996.21858257135 185.868852 16.1201 0 0 X -397.434462871071 100.09187 -3.9707 0.000244 0.000122 M1 177.028669075707 149.576482 1.1835 0.24255 0.121275 M2 114.595914455059 156.950654 0.7301 0.468929 0.234464 M3 334.238409008073 156.647487 2.1337 0.038116 0.019058 M4 204.342364556124 156.539162 1.3054 0.198119 0.09906 M5 178.748941589334 156.325218 1.1434 0.258646 0.129323 M6 453.032484736411 156.129905 2.9016 0.005633 0.002817 M7 22.3364402844630 156.024256 0.1432 0.886776 0.443388 M8 -44.4569826823279 155.91773 -0.2851 0.776796 0.388398 M9 -31.3734395352507 155.908904 -0.2012 0.841388 0.420694 M10 164.561302458348 155.827815 1.056 0.296346 0.148173 M11 -197.155266290631 155.758956 -1.2658 0.211833 0.105917 t 13.7164568529229 5.039211 2.7219 0.009075 0.004537 Multiple Linear Regression - Regression Statistics Multiple R 0.729466690504987 R-squared 0.532121652556298 Adjusted R-squared 0.402708492625061 F-TEST (value) 4.11180480284261 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 0.000165474103349661 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 246.259297428538 Sum Squared Residuals 2850251.15378987 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2360 2392.09478275783 -32.0947827578258 2 2214 2343.37848499011 -129.378484990108 3 2825 2576.73743639605 248.262563603954 4 2355 2460.55784879702 -105.557848797021 5 2333 2448.68088268315 -115.680882683153 6 3016 2736.68088268315 279.319117316848 7 2155 2319.70129508413 -164.701295084128 8 2172 2266.62432897026 -94.62432897026 9 2150 2293.42432897026 -143.42432897026 10 2533 2503.07552781678 29.9244721832179 11 2058 2155.07541592073 -97.0754159207257 12 2160 2365.94713906428 -205.947139064279 13 2260 2556.69226499291 -296.692264992910 14 2498 2507.97596722518 -9.97596722518373 15 2695 2741.33491863112 -46.334918631121 16 2799 2625.15533103210 173.844668967904 17 2947 2613.27836491823 333.721635081772 18 2930 2901.27836491823 28.7216350817717 19 2318 2484.29877731920 -166.298777319203 20 2540 2431.22181120534 108.778188794665 21 2570 2458.02181120534 111.978188794665 22 2669 2667.67301005186 1.32698994814249 23 2450 2319.6728981558 130.327101844199 24 2842 2530.54462129935 311.455378700645 25 3440 2721.28974722798 718.710252772015 26 2678 2672.57344946026 5.42655053974068 27 2981 2905.9324008662 75.0675991338036 28 2260 2706.29157606425 -446.291576064246 29 2844 2678.51723143554 165.482768564464 30 2546 2966.51723143554 -420.517231435536 31 2456 2470.05075126230 -14.0507512622963 32 2295 2397.10206200487 -102.102062004875 33 2379 2423.90206200488 -44.9020620048750 34 2479 2577.91243604945 -98.912436049447 35 2057 2186.19453323757 -129.194533237573 36 2280 2325.52805306433 -45.5280530643336 37 2351 2488.45276659199 -137.452766591988 38 2276 2372.17261013618 -96.172610136181 39 2548 2573.73680451243 -25.7368045124324 40 2311 2401.91639211146 -90.9163921114575 41 2201 2346.32163508177 -145.321635081772 42 2725 2634.32163508177 90.6783649182282 43 2408 2157.72687805209 250.273121947914 44 2139 2064.90646565111 74.093534348889 45 1898 2091.70646565111 -193.706465651111 46 2537 2241.74249506697 295.257504933028 47 2069 1853.99893688381 215.001063116191 48 2063 2064.87066002736 -1.87066002736259 49 2524 2255.61578595599 268.384214044008 50 2437 2206.89948818827 230.100511811733 51 2189 2440.25843959420 -251.258439594204 52 2793 2324.07885199518 468.921148004821 53 2074 2312.20188588131 -238.201885881311 54 2622 2600.20188588131 21.7981141186885 55 2278 2183.22229828229 94.7777017177136 56 2144 2130.14533216842 13.8546678315815 57 2427 2156.94533216842 270.054667831581 58 2139 2366.59653101494 -227.596531014941 59 1828 1947.05821580209 -119.058215802092 60 2072 2130.10952654467 -58.10952654467 61 1800 2320.8546524733 -520.8546524733 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.613502739050481 0.772994521899038 0.386497260949519 18 0.521476708883348 0.957046582233305 0.478523291116652 19 0.3950987812497 0.7901975624994 0.6049012187503 20 0.293785755094465 0.587571510188929 0.706214244905535 21 0.220018748901669 0.440037497803339 0.77998125109833 22 0.140409218817058 0.280818437634115 0.859590781182942 23 0.0942655454993997 0.188531090998799 0.9057344545006 24 0.129717823218669 0.259435646437338 0.87028217678133 25 0.605875059336117 0.788249881327765 0.394124940663883 26 0.538583705024501 0.922832589950997 0.461416294975499 27 0.582057131553342 0.835885736893316 0.417942868446658 28 0.575651762317718 0.848696475364563 0.424348237682282 29 0.779126471480211 0.441747057039578 0.220873528519789 30 0.765456711305048 0.469086577389904 0.234543288694952 31 0.830746708353578 0.338506583292844 0.169253291646422 32 0.766864673498232 0.466270653003537 0.233135326501768 33 0.699038363742738 0.601923272514525 0.300961636257263 34 0.610263975214537 0.779472049570926 0.389736024785463 35 0.511815861349589 0.976368277300823 0.488184138650411 36 0.434811043222344 0.869622086444689 0.565188956777656 37 0.361029466058470 0.722058932116941 0.63897053394153 38 0.283712620104940 0.567425240209879 0.71628737989506 39 0.334815597174628 0.669631194349256 0.665184402825372 40 0.334059736432856 0.668119472865712 0.665940263567144 41 0.235209337936339 0.470418675872677 0.764790662063661 42 0.181864112472554 0.363728224945108 0.818135887527446 43 0.159252960820254 0.318505921640508 0.840747039179746 44 0.0860757802891491 0.172151560578298 0.91392421971085 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 0 0 OK

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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