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*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, 09 Dec 2009 12:43:51 -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/09/t1260388003nnbn2ryllj7649m.htm/, Retrieved Sun, 19 May 2013 22:07:12 +0000

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

hypotheseMD

System-generated keywords (parent):

t1258034663fagi4z9rmty7nh6 (pk = 55996)

Estimated Impact

Dataseries X:

» Textfile « » CSV « » Correlation Matrix « » Notched Boxplots «

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

Y[t] = + 661.39268292683 + 75.0182926829268X[t] + 7.60121951219552M1[t] -8.06544715447153M2[t] + 98.4345528455284M3[t] -9.06544715447148M4[t] -2.89878048780486M5[t] + 134.101219512195M6[t] -253.735162601626M7[t] -342.568495934959M8[t] + 146.098170731707M9[t] + 76.4315040650406M10[t] -29.2M11[t] + 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)	661.39268292683	28.884941	22.8975	0	0
X	75.0182926829268	15.573716	4.817	1.1e-05	6e-06
M1	7.60121951219552	38.204132	0.199	0.842999	0.4215
M2	-8.06544715447153	38.204132	-0.2111	0.833551	0.416775
M3	98.4345528455284	38.204132	2.5765	0.012594	0.006297
M4	-9.06544715447148	38.204132	-0.2373	0.813283	0.406641
M5	-2.89878048780486	38.204132	-0.0759	0.939783	0.469892
M6	134.101219512195	38.204132	3.5101	0.000883	0.000441
M7	-253.735162601626	38.221763	-6.6385	0	0
M8	-342.568495934959	38.221763	-8.9627	0	0
M9	146.098170731707	38.221763	3.8224	0.000329	0.000165
M10	76.4315040650406	38.221763	1.9997	0.050308	0.025154
M11	-29.2	39.888177	-0.732	0.46714	0.23357

Multiple Linear Regression - Regression Statistics
Multiple R	0.930216186377923
R-squared	0.865302153399486
Adjusted R-squared	0.836944712009904
F-TEST (value)	30.5141123810057
F-TEST (DF numerator)	12
F-TEST (DF denominator)	57
p-value	0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation	63.0687447979535
Sum Squared Residuals	226726.994512195

Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or Index	Actuals	Interpolation Forecast	Residuals Prediction Error
1	627	668.993902439022	-41.9939024390224
2	696	653.327235772358	42.6727642276422
3	825	759.827235772358	65.1727642276424
4	677	652.327235772358	24.6727642276422
5	656	658.493902439024	-2.49390243902429
6	785	795.493902439025	-10.4939024390246
7	412	407.657520325203	4.34247967479673
8	352	318.82418699187	33.1758130081301
9	839	807.490853658536	31.5091463414637
10	729	737.82418699187	-8.82418699187037
11	696	632.192682926829	63.8073170731707
12	641	661.392682926829	-20.3926829268292
13	695	668.993902439025	26.0060975609752
14	638	653.327235772358	-15.3272357723577
15	762	759.827235772358	2.17276422764226
16	635	652.327235772358	-17.3272357723577
17	721	658.493902439024	62.5060975609756
18	854	795.493902439024	58.5060975609756
19	418	407.657520325203	10.3424796747967
20	367	318.82418699187	48.1758130081300
21	824	807.490853658537	16.5091463414633
22	687	737.82418699187	-50.8241869918699
23	601	632.192682926829	-31.1926829268293
24	676	661.392682926829	14.6073170731707
25	740	668.993902439025	71.0060975609752
26	691	653.327235772358	37.6727642276423
27	683	759.827235772358	-76.8272357723577
28	594	652.327235772358	-58.3272357723577
29	729	658.493902439024	70.5060975609756
30	731	795.493902439024	-64.4939024390244
31	386	407.657520325203	-21.6575203252033
32	331	318.82418699187	12.1758130081300
33	707	807.490853658537	-100.490853658537
34	715	737.82418699187	-22.8241869918699
35	657	632.192682926829	24.8073170731707
36	653	661.392682926829	-8.39268292682928
37	642	668.993902439025	-26.9939024390248
38	643	653.327235772358	-10.3272357723577
39	718	759.827235772358	-41.8272357723578
40	654	652.327235772358	1.67276422764227
41	632	658.493902439024	-26.4939024390245
42	731	795.493902439024	-64.4939024390244
43	392	482.67581300813	-90.67581300813
44	344	393.842479674797	-49.8424796747967
45	792	882.509146341463	-90.5091463414634
46	852	812.842479674797	39.1575203252033
47	649	707.210975609756	-58.2109756097561
48	629	736.410975609756	-107.410975609756
49	685	744.012195121952	-59.0121951219516
50	617	728.345528455284	-111.345528455284
51	715	834.845528455285	-119.845528455285
52	715	727.345528455284	-12.3455284552845
53	629	733.512195121951	-104.512195121951
54	916	870.512195121951	45.4878048780488
55	531	482.67581300813	48.3241869918699
56	357	393.842479674797	-36.8424796747967
57	917	882.509146341464	34.4908536585365
58	828	812.842479674797	15.1575203252034
59	708	707.210975609756	0.789024390243946
60	858	736.410975609756	121.589024390244
61	775	744.012195121952	30.9878048780484
62	785	728.345528455284	56.6544715447155
63	1006	834.845528455285	171.154471544715
64	789	727.345528455284	61.6544715447155
65	734	733.512195121951	0.487804878048803
66	906	870.512195121951	35.4878048780488
67	532	482.67581300813	49.3241869918699
68	387	393.842479674797	-6.84247967479674
69	991	882.509146341463	108.490853658537
70	841	812.842479674797	28.1575203252033

Goldfeld-Quandt test for Heteroskedasticity
p-values	Alternative Hypothesis
breakpoint index	greater	2-sided	less
16	0.283994948779162	0.567989897558324	0.716005051220838
17	0.225272970620628	0.450545941241257	0.774727029379372
18	0.189595830558681	0.379191661117362	0.810404169441319
19	0.104801763121337	0.209603526242674	0.895198236878663
20	0.0587301108500135	0.117460221700027	0.941269889149986
21	0.0299888351723092	0.0599776703446183	0.97001116482769
22	0.0184325135810061	0.0368650271620122	0.981567486418994
23	0.0261332836250375	0.052266567250075	0.973866716374962
24	0.0149217914654033	0.0298435829308066	0.985078208534597
25	0.0208588884519247	0.0417177769038494	0.979141111548075
26	0.0127749201709348	0.0255498403418695	0.987225079829065
27	0.0286287560252879	0.0572575120505758	0.971371243974712
28	0.0246289237725529	0.0492578475451058	0.975371076227447
29	0.0229282514103949	0.0458565028207898	0.977071748589605
30	0.0273216716961565	0.054643343392313	0.972678328303843
31	0.0167567237978379	0.0335134475956759	0.983243276202162
32	0.0113592598925707	0.0227185197851414	0.98864074010743
33	0.0267879254099278	0.0535758508198556	0.973212074590072
34	0.0161453400743632	0.0322906801487264	0.983854659925637
35	0.0105986107544418	0.0211972215088836	0.989401389245558
36	0.0058082255841414	0.0116164511682828	0.994191774415859
37	0.00372967643800005	0.00745935287600011	0.996270323562
38	0.00230395755264915	0.00460791510529829	0.997696042447351
39	0.00136074582422031	0.00272149164844063	0.99863925417578
40	0.000675784261014619	0.00135156852202924	0.999324215738985
41	0.000728127876403002	0.00145625575280600	0.999271872123597
42	0.000475030468647181	0.000950060937294363	0.999524969531353
43	0.000499514600915729	0.000999029201831458	0.999500485399084
44	0.000229270405653065	0.00045854081130613	0.999770729594347
45	0.000388730782884489	0.000777461565768979	0.999611269217116
46	0.000769685901222576	0.00153937180244515	0.999230314098777
47	0.000418259657201288	0.000836519314402575	0.999581740342799
48	0.00232952952782195	0.00465905905564389	0.997670470472178
49	0.00159435587518208	0.00318871175036416	0.998405644124818
50	0.00439131111480057	0.00878262222960114	0.9956086888852
51	0.488826292470925	0.97765258494185	0.511173707529075
52	0.537228702366755	0.92554259526649	0.462771297633245
53	0.817798065158885	0.364403869682230	0.182201934841115
54	0.720668173247186	0.558663653505629	0.279331826752814

Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description	# significant tests	% significant tests	OK/NOK
1% type I error level	14	0.358974358974359	NOK
5% type I error level	25	0.641025641025641	NOK
10% type I error level	30	0.769230769230769	NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No 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')
}