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Case: the Seatbelt Law & Tutorial Q3 met seasonal dummies and linear 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: Mon, 24 Nov 2008 11:17:29 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Nov/24/t1227550704oc6quq0ra8x2qpp.htm/, Retrieved Mon, 24 Nov 2008 18:18:36 +0000

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/2008/Nov/24/t1227550704oc6quq0ra8x2qpp.htm/},
    year = {2008},
}
@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 = {2008},
    note = {{ISBN} 3-900051-07-0},
    url = {http://www.R-project.org},
}

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)

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Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

7.4 0 7.2 0 7.1 0 6.9 0 6.8 0 6.8 0 6.8 0 6.9 0 6.7 0 6.6 0 6.5 0 6.4 0 6.3 0 6.3 0 6.3 0 6.5 0 6.6 0 6.5 0 6.4 0 6.5 0 6.7 0 7.1 0 7.1 0 7.2 0 7.2 0 7.3 0 7.3 0 7.3 0 7.3 0 7.4 0 7.6 0 7.6 0 7.6 0 7.7 0 7.8 0 7.9 0 8.1 0 8.1 0 8.1 0 8.2 0 8.2 0 8.2 0 8.2 0 8.2 0 8.2 0 8.3 0 8.3 0 8.4 0 8.4 0 8.4 0 8.3 1 8 1 8 1 8.2 1 8.6 1 8.7 1 8.7 1 8.5 1 8.4 1 8.4 1 8.4 1 8.5 1 8.5 1 8.5 1 8.5 1 8.5 1 8.4 1 8.4 1 8.4 1 8.5 1 8.6 1 8.6 1 8.6 1 8.6 1 8.5 1 8.4 1 8.4 1 8.3 1 8.2 1 8.1 1 8.2 1 8.1 1 8 1 7.9 1 7.8 1 7.7 1 7.7 1 7.9 1 7.8 1 7.6 1 7.4 1 7.3 1 7.1 1 7.1 1 7 1 7 1 7 1 6.9 1 6.8 1 6.7 1 6.6 1 6.6 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	2 seconds
R Server	'George Udny Yule' @ 72.249.76.132

Multiple Linear Regression - Estimated Regression Equation

y[t] = + 7.39270833333334 + 0.638958333333334x[t] + 0.000572916666670294M1[t] -0.0218865740740734M2[t] -0.137563657407408M3[t] -0.160023148148148M4[t] -0.182482638888889M5[t] -0.193831018518519M6[t] -0.0238136574074078M7[t] -0.0115509259259264M8[t] -0.0242881944444446M9[t] + 0.0129745370370371M10[t] -0.0122627314814816M11[t] + 0.000237268518518472t + 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)	7.39270833333334	0.291181	25.3887	0	0
x	0.638958333333334	0.280392	2.2788	0.025099	0.01255
M1	0.000572916666670294	0.341525	0.0017	0.998665	0.499333
M2	-0.0218865740740734	0.341415	-0.0641	0.949032	0.474516
M3	-0.137563657407408	0.342499	-0.4016	0.688918	0.344459
M4	-0.160023148148148	0.342146	-0.4677	0.641152	0.320576
M5	-0.182482638888889	0.341859	-0.5338	0.594829	0.297415
M6	-0.193831018518519	0.341638	-0.5674	0.571916	0.285958
M7	-0.0238136574074078	0.351981	-0.0677	0.946213	0.473106
M8	-0.0115509259259264	0.351692	-0.0328	0.973873	0.486937
M9	-0.0242881944444446	0.351467	-0.0691	0.945063	0.472531
M10	0.0129745370370371	0.351306	0.0369	0.970623	0.485311
M11	-0.0122627314814816	0.351209	-0.0349	0.972226	0.486113
t	0.000237268518518472	0.004754	0.0499	0.960307	0.480154

Multiple Linear Regression - Regression Statistics
Multiple R	0.45096899656541
R-squared	0.203373035863213
Adjusted R-squared	0.0856895070702781
F-TEST (value)	1.72813509204886
F-TEST (DF numerator)	13
F-TEST (DF denominator)	88
p-value	0.0687166433524351
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation	0.702354370996517
Sum Squared Residuals	43.4105462962963

Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or Index	Actuals	Interpolation Forecast	Residuals Prediction Error
1	7.4	7.39351851851849	0.00648148148151031
2	7.2	7.3712962962963	-0.171296296296294
3	7.1	7.25585648148148	-0.155856481481483
4	6.9	7.23363425925926	-0.333634259259261
5	6.8	7.21141203703704	-0.411412037037037
6	6.8	7.20030092592593	-0.400300925925928
7	6.8	7.37055555555556	-0.570555555555557
8	6.9	7.38305555555556	-0.483055555555556
9	6.7	7.37055555555556	-0.670555555555556
10	6.6	7.40805555555556	-0.808055555555557
11	6.5	7.38305555555556	-0.883055555555558
12	6.4	7.39555555555556	-0.995555555555557
13	6.3	7.39636574074074	-1.09636574074075
14	6.3	7.37414351851852	-1.07414351851852
15	6.3	7.2587037037037	-0.958703703703705
16	6.5	7.23648148148148	-0.736481481481482
17	6.6	7.21425925925926	-0.61425925925926
18	6.5	7.20314814814815	-0.703148148148149
19	6.4	7.37340277777778	-0.973402777777778
20	6.5	7.38590277777778	-0.885902777777778
21	6.7	7.37340277777778	-0.673402777777778
22	7.1	7.41090277777778	-0.310902777777779
23	7.1	7.38590277777778	-0.285902777777778
24	7.2	7.39840277777778	-0.198402777777778
25	7.2	7.39921296296297	-0.199212962962967
26	7.3	7.37699074074074	-0.0769907407407419
27	7.3	7.26155092592593	0.0384490740740738
28	7.3	7.2393287037037	0.0606712962962960
29	7.3	7.21710648148148	0.0828935185185185
30	7.4	7.20599537037037	0.19400462962963
31	7.6	7.37625	0.223750000000000
32	7.6	7.38875	0.21125
33	7.6	7.37625	0.22375
34	7.7	7.41375	0.28625
35	7.8	7.38875	0.41125
36	7.9	7.40125	0.49875
37	8.1	7.4020601851852	0.697939814814811
38	8.1	7.37983796296296	0.720162037037036
39	8.1	7.26439814814815	0.835601851851852
40	8.2	7.24217592592593	0.957824074074074
41	8.2	7.2199537037037	0.980046296296296
42	8.2	7.20884259259259	0.991157407407407
43	8.2	7.37909722222222	0.820902777777778
44	8.2	7.39159722222222	0.808402777777778
45	8.2	7.37909722222222	0.820902777777778
46	8.3	7.41659722222222	0.88340277777778
47	8.3	7.39159722222222	0.90840277777778
48	8.4	7.40409722222222	0.995902777777779
49	8.4	7.40490740740741	0.99509259259259
50	8.4	7.38268518518518	1.01731481481482
51	8.3	7.9062037037037	0.393796296296296
52	8	7.88398148148148	0.116018518518518
53	8	7.86175925925926	0.138240740740740
54	8.2	7.85064814814815	0.349351851851850
55	8.6	8.02090277777778	0.579097222222221
56	8.7	8.03340277777778	0.666597222222221
57	8.7	8.02090277777778	0.679097222222221
58	8.5	8.05840277777778	0.441597222222222
59	8.4	8.03340277777778	0.366597222222222
60	8.4	8.04590277777778	0.354097222222222
61	8.4	8.04671296296297	0.353287037037033
62	8.5	8.02449074074074	0.475509259259258
63	8.5	7.90905092592593	0.590949074074074
64	8.5	7.8868287037037	0.613171296296296
65	8.5	7.86460648148148	0.635393518518519
66	8.5	7.85349537037037	0.64650462962963
67	8.4	8.02375	0.376250000000000
68	8.4	8.03625	0.363750000000001
69	8.4	8.02375	0.376250000000001
70	8.5	8.06125	0.43875
71	8.6	8.03625	0.56375
72	8.6	8.04875	0.551249999999999
73	8.6	8.0495601851852	0.550439814814811
74	8.6	8.02733796296296	0.572662037037036
75	8.5	7.91189814814815	0.588101851851852
76	8.4	7.88967592592593	0.510324074074075
77	8.4	7.8674537037037	0.532546296296297
78	8.3	7.8563425925926	0.443657407407409
79	8.2	8.02659722222222	0.173402777777778
80	8.1	8.03909722222222	0.0609027777777783
81	8.2	8.02659722222222	0.173402777777778
82	8.1	8.06409722222222	0.0359027777777779
83	8	8.03909722222222	-0.0390972222222215
84	7.9	8.05159722222222	-0.151597222222221
85	7.8	8.0524074074074	-0.252407407407410
86	7.7	8.03018518518519	-0.330185185185185
87	7.7	7.91474537037037	-0.214745370370369
88	7.9	7.89252314814815	0.00747685185185319
89	7.8	7.87030092592592	-0.0703009259259249
90	7.6	7.85918981481481	-0.259189814814814
91	7.4	8.02944444444444	-0.629444444444443
92	7.3	8.04194444444444	-0.741944444444443
93	7.1	8.02944444444444	-0.929444444444443
94	7.1	8.06694444444444	-0.966944444444444
95	7	8.04194444444444	-1.04194444444444
96	7	8.05444444444444	-1.05444444444444
97	7	8.05525462962963	-1.05525462962963
98	6.9	8.0330324074074	-1.13303240740741
99	6.8	7.9175925925926	-1.11759259259259
100	6.7	7.89537037037037	-1.19537037037037
101	6.6	7.87314814814815	-1.27314814814815
102	6.6	7.86203703703704	-1.26203703703704

Charts produced by software:

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/24/t1227550704oc6quq0ra8x2qpp/1a4l71227550645.png (open in new window)

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http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/24/t1227550704oc6quq0ra8x2qpp/6303c1227550645.ps (open in new window)

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http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/24/t1227550704oc6quq0ra8x2qpp/75hip1227550645.ps (open in new window)

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http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/24/t1227550704oc6quq0ra8x2qpp/9wifz1227550645.ps (open in new window)

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)

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))

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')

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()

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')

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Software written by Ed van Stee & Patrick Wessa

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