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Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 12)
Slideshow: testing for autocorrelation
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 12). [Teaching Resource]
2012 The Author
This version available at: http://learningresources.lse.ac.uk/138/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
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Simple autoregression of the residuals
1
TESTS FOR AUTOCORRELATION
We will initially confine the discussion of the tests for autocorrelation to its most common
form, the AR(1) process. If the disturbance term follows the AR(1) process, it is reasonable
to hypothesize that, as an approximation, the residuals will conform to a similar process.
1tt ee
t1 tt uu
error
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Simple autoregression of the residuals
3
TESTS FOR AUTOCORRELATION
t1 tt uu
Hence a regression of eton et1is sufficient, at least in large samples. Of course, there is
the issue that, in this regression, et1is a lagged dependent variable, but that does not
matter in large samples.
1tt ee error
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4
TESTS FOR AUTOCORRELATION
This is illustrated with the simulation shown in the figure. The true model is as shown, with
utbeing generated as an AR (1) process with = 0.7.
Simple autoregression of the residuals
0
5
-0.5 0 0.5 1
T= 25
T= 50
T= 100
T= 200
0.7
true value
t17.0 tt
uu
1 tt ee
tt utY 0.210
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0
5
-0.5 0 0.5 1
T= 25
T= 50
T= 100
T= 200
0.7
true value
5
TESTS FOR AUTOCORRELATION
Simple autoregression of the residuals
The values of the parameters in the model for Ytmake no difference to the distributions of
the estimator of .
t17.0 tt
uu
1 tt ee
tt utY 0.210
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0
5
-0.5 0 0.5 1
T= 25
T= 50
T= 100
T= 200
0.7
true value
6
TESTS FOR AUTOCORRELATION
Simple autoregression of the residuals
As can be seen, when etis regressed on et1, the distribution of the estimator of is left
skewed and heavily biased downwards for T= 25. The mean of the distribution is 0.47.
T mean
25 0.47
50 0.59
100 0.65
200 0.68
t17.0 tt
uu
1 tt ee
tt utY 0.210
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0
5
-0.5 0 0.5 1
T= 25
T= 50
T= 100
T= 200
0.7
true value
7
TESTS FOR AUTOCORRELATION
Simple autoregression of the residuals
T mean
25 0.47
50 0.59
100 0.65
200 0.68
t17.0 tt
uu
1 tt ee
tt utY 0.210
However, as the sample size increases, the downwards bias diminishes and it is clear that it
is converging on 0.7 as the sample becomes large. Inference in finite samples will be
approximate, given the autoregressive nature of the regression.
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TESTS FOR AUTOCORRELATION
The simple estimator of the autocorrelation coefficient depends on Assumption C.7 part (2)
being satisfied when the original model (the model for Yt) is fitted. Generally, one might
expect this not to be the case.
BreuschGodfrey test
t
k
j
jtjt uXY
2
1
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TESTS FOR AUTOCORRELATION
If the original model contains a lagged dependent variable as a regressor, or violates
Assumption C.7 part (2) in any other way, the estimates of the parameters will be
inconsistent if the disturbance term is subject to autocorrelation.
BreuschGodfrey test
t
k
j
jtjt uXY
2
1
1
2
1
t
k
j
jtjt eXe
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TESTS FOR AUTOCORRELATION
As a repercussion, a simple regression of eton et1will produce an inconsistent estimate of
. The solution is to include all of the explanatory variables in the original model in the
residuals autoregression.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
t
k
j
jtjt uXY
2
1
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TESTS FOR AUTOCORRELATION
If the original model is the first equation where, say, one of the Xvariables is Yt1, then the
residuals regression would be the second equation.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
t
k
j
jtjt uXY
2
1
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TESTS FOR AUTOCORRELATION
The idea is that, by including the Xvariables, one is controlling for the effects of any
endogeneity on the residuals.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
t
k
j
jtjt uXY
2
1
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TESTS FOR AUTOCORRELATION
The underlying theory is complex and relates to maximum-likelihood estimation, as does
the test statistic. The test is known as the BreuschGodfrey test.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
t
k
j
jtjt uXY
2
1
TESTS FOR AUTOCORRELATION
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TESTS FOR AUTOCORRELATION
Several asymptotically-equivalent versions of the test have been proposed. The most
popular involves the computation of the lagrange multiplier statistic nR2when the residuals
regression is fitted, nbeing the actual number of observations in the regression.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
t
k
j
jtjt uXY
2
1
Test statistic: nR2, distributed as 2(1) when
testing for first-order autocorrelation
TESTS FOR AUTOCORRELATION
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15
TESTS FOR AUTOCORRELATION
Asymptotically, under the null hypothesis of no autocorrelation, nR2is distributed as a chi-
squared statistic with one degree of freedom.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
t
k
j
jtjt uXY
2
1
Test statistic: nR2, distributed as 2(1) when
testing for first-order autocorrelation
TESTS FOR AUTOCORRELATION
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TESTS FOR AUTOCORRELATION
A simple ttest on the coefficient of et1has also been proposed, again with asymptotic
validity.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
t
k
j
jtjt uXY
2
1
Alternatively, simple ttest on coefficient of et1
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TESTS FOR AUTOCORRELATION
The procedure can be extended to test for higher order autocorrelation. If AR(q)
autocorrelation is suspected, the residuals regression includes qlagged residuals.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
q
s
sts
k
j
jtjt eXe
12
1
t
k
j
jtjt uXY
2
1
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18
TESTS FOR AUTOCORRELATION
For the lagrange multiplier version of the test, the test statistic remains nR2(with nsmaller
than before, the inclusion of the additional lagged residuals leading to a further loss of
initial observations).
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
q
s
sts
k
j
jtjt eXe
12
1
t
k
j
jtjt uXY
2
1
Test statistic: nR2, distributed as 2(q)
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TESTS FOR AUTOCORRELATION
Under the null hypothesis of no autocorrelation, nR2has a chi-squared distribution with q
degrees of freedom.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
q
s
sts
k
j
jtjt eXe
12
1
t
k
j
jtjt uXY
2
1
Test statistic: nR2, distributed as 2(q)
TESTS FOR AUTOCORRELATION
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20
TESTS FOR AUTOCORRELATION
The ttest version becomes an Ftest comparing RSSfor the residuals regression with RSS
for the same specification without the residual terms. Again, the test is valid only
asymptotically.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
q
s
sts
k
j
jtjt eXe
12
1
t
k
j
jtjt uXY
2
1
Alternatively, Ftest on the lagged residuals
H0: 1= ... = q= 0, H1: not H0
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21
TESTS FOR AUTOCORRELATION
The lagrange multiplier version of the test has been shown to be asymptotically valid for the
case of MA(q) moving average autocorrelation.
BreuschGodfrey test
1
2
1
t
k
j
jtjt eXe
q
s
sts
k
j
jtjt eXe
12
1
t
k
j
jtjt uXY
2
1
Test statistic: nR2, distributed as 2(q),
valid also for MA(q) autocorrelation
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22
TESTS FOR AUTOCORRELATION
The first major test to be developed and popularised for the detection of autocorrelation
was the DurbinWatson test for AR(1) autocorrelation based on the DurbinWatson d
statistic calculated from the residuals using the expression shown.
DurbinWatson test
T
t
t
T
t
tt
e
ee
d
1
2
2
21 )(
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23
It can be shown that in large samples dtends to 22, where is the parameter in the
AR(1) relationship ut= ut1+ t.
TESTS FOR AUTOCORRELATION
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
T
t
t
T
t
tt
e
ee
d
1
2
2
21 )(
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24
If there is no autocorrelation, is 0 and dshould be distributed randomly around 2.
TESTS FOR AUTOCORRELATION
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
2d
22d
T
t
t
T
t
tt
e
ee
d
1
2
2
2
1 )(
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25
If there is severe positive autocorrelation, will be near 1 and dwill be near 0.
TESTS FOR AUTOCORRELATION
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
2d
0d
22d
T
t
t
T
t
tt
e
ee
d
1
2
2
2
1 )(
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26
Likewise, if there is severe positive autocorrelation, will be near1 and dwill be near 4.
TESTS FOR AUTOCORRELATION
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
2d
0d
4d
22d
T
t
t
T
t
tt
e
ee
d
1
2
2
2
1 )(
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27
Thus dbehaves as illustrated graphically above.
TESTS FOR AUTOCORRELATION
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
2d
0d
4d
22d
2
4
0
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
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To perform the DurbinWatson test, we define critical values of d. The null hypothesis is H0:
= 0 (no autocorrelation). If dlies between these values, we do not reject the null
hypothesis.
TESTS FOR AUTOCORRELATION
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
2d
0d
4d
22d
2
4
0 dcrit
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dcrit
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29
The critical values, at any significance level, depend on the number of observations in the
sample and the number of explanatory variables.
TESTS FOR AUTOCORRELATION
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
2d
0d
4d
22d
2
4
0 dcrit
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dcrit
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30
Unfortunately, they also depend on the actual data for the explanatory variables in the
sample, and thus vary from sample to sample.
TESTS FOR AUTOCORRELATION
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
2d
0d
4d
22d
2
4
0 dcrit
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dcrit
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31
However Durbin and Watson determined upper and lower bounds, dUand dL, for the critical
values, and these are presented in standard tables.
2d
0d
4d
2
4
0 dL
dU
dcrit
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dcrit
TESTS FOR AUTOCORRELATION
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
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2d
0d
4d
2
4
0dL d
Ud
crit
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dcrit
TESTS FOR AUTOCORRELATION
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
If dis less than dL, it must also be less than the critical value of dfor positive
autocorrelation, and so we would reject the null hypothesis and conclude that there is
positive autocorrelation.
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33
2d
0d
4d
2
4
0dL d
Ud
crit
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dcrit
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
If dis above than dU, it must also be above the critical value of d, and so we would not reject
the null hypothesis. (Of course, if it were above 2, we should consider testing for negative
autocorrelation instead.)
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2d
0d
4d
2
4
0dL d
Ud
crit
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dcrit
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
If dlies between dLand dU, we cannot tell whether it is above or below the critical value and
so the test is indeterminate.
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2d
0d
4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
Here are dLand dUfor 45 observations and two explanatory variables, at the 5% significance
level.
1.43 1.62(n = 45, k = 3, 5% level)
2
40 d
L d
U
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
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2d
0d
4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
1.43 1.62(n = 45, k = 3, 5% level)
2
40 d
L d
U
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
There are similar bounds for the critical value in the case of negative autocorrelation. They
are not given in the standard tables because negative autocorrelation is uncommon, but it
is easy to calculate them because are they are located symmetrically to the right of 2.
2.38 2.57
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2d
0d
4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
1.43 1.62(n = 45, k = 3, 5% level)
2
40 d
L d
U
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
2.38 2.57
So if d< 1.43, we reject the null hypothesis and conclude that there is positive
autocorrelation.
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2d
0d
4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
1.43 1.62(n = 45, k = 3, 5% level)
2
40 d
L d
U
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
2.38 2.57
If 1.43 < d< 1.62, the test is indeterminate and we do not come to any conclusion.
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39
2d
0d
4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
1.43 1.62(n = 45, k = 3, 5% level)
2
40 d
L d
U
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
2.38 2.57
If 1.62 < d< 2.38, we do not reject the null hypothesis of no autocorrelation.
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40
2d
0d
4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
1.43 1.62(n = 45, k = 3, 5% level)
2
40 d
L d
U
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
2.38 2.57
If 2.38 < d< 2.57, we do not come to any conclusion.
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2d
0d
4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
1.43 1.62(n = 45, k = 3, 5% level)
2
40 d
L
dU
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
2.38 2.57
If d> 2.57, we conclude that there is significant negative autocorrelation.
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2d
0d
4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
2
40
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
Here are the bounds for the critical values for the 1% test, again with 45 observations and
two explanatory variables.
dL
dU
1.24 1.42 2.58 2.76(n = 45, k = 3, 1% level)
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2d
0d
4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
2
40
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dL
dU
1.24 1.42 2.58 2.76(n = 45, k = 3, 1% level)
The Durbin-Watson test is valid only when all the explanatory variables are deterministic.
This is in practice a serious limitation since usually interactions and dynamics in a system
of equations cause Assumption C.7 part (2) to be violated.
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2d
0d
4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
2
40
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dL
dU
1.24 1.42 2.58 2.76(n = 45, k = 3, 1% level)
In particular, if the lagged dependent variable is used as a regressor, the statistic is biased
towards 2 and therefore will tend to under-reject the null hypothesis. It is also restricted to
testing for AR(1) autocorrelation.
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45
2d
0d4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
2
40
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dL
dU
1.24 1.42 2.58 2.76(n = 45, k = 3, 1% level)
Despite these shortcomings, it remains a popular test and some major applications produce
the dstatistic automatically as part of the standard regression output.
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2d
0d4d
DurbinWatson test
In large samples
No autocorrelation
Severe positive autocorrelation
Severe negative autocorrelation
22d
2
40
positive
autocorrelation
negative
autocorrelation
no
autocorrelation
dL
dU
1.24 1.42 2.58 2.76(n = 45, k = 3, 1% level)
It does have the appeal of the test statistic being part of standard regression output.
Further, it is appropriate for finite samples, subject to the zone of indeterminacy and the
deterministic regressor requirement.
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Durbin proposed two tests for the case where the use of the lagged dependent variable as a
regressor made the original DurbinWatson test inapplicable. One was a precursor to the
Breusch
Godrey test.
Durbins htest
2
)1(1
Yb
ns
n
h
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The other is the Durbin htest, appropriate for the detection of AR(1) autocorrelation.
Durbins htest
2
)1(1
Yb
ns
n
h
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The Durbin hstatistic is defined as shown, where is an estimate of in the AR(1)
process, is an estimate of the variance of the coefficient of the lagged dependent
variable, and nis the number of observations in the regression.
Durbins htest
2
)1(1
Yb
ns
n
h
2
)1(Ybs
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There are various ways in which one might estimate but, since this test is valid only for
large samples, it does not matter which is used. The most convenient is to take advantage
of the fact that dtends to 22in large samples. The estimator is then 10.5d.
Durbins htest
2
)1(1
Yb
ns
n
h
22d
d5.01
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The estimate of the variance of the coefficient of the lagged dependent variable is obtained
by squaring its standard error.
Durbins htest
2
)1(1
Yb
ns
n
h
22d
d5.01
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Thus hcan be calculated from the usual regression results. In large samples, under the
null hypothesis of no autocorrelation, his distributed as a normal variable with zero mean
and unit variance.
Durbins htest
2
)1(1
Yb
ns
n
h
22d
d5.01
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An occasional problem with this test is that the hstatistic cannot be computed if n
is greater than 1, which can happen if the sample size is not very large.
Durbins htest
2
)1(1
Yb
ns
n
h
22d
d5.01
2
)1(Ybs
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An even worse problem occurs when n is near to, but less than, 1. In such a situation
the hstatistic could be enormous, without there being any problem of autocorrelation.
Durbins htest
2
)1(1
Yb
ns
n
h
22d
d5.01
2
)1(Ybs
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The output shown in the table gives the result of a logarithmic regression of expenditure on
food on disposable personal income and the relative price of food.
============================================================
Dependent Variable: LGFOOD
Method: Least Squares
Sample: 1959 2003
Included observations: 45
============================================================
Variable Coefficient Std. Error t-Statistic Prob.
============================================================
C 2.236158 0.388193 5.760428 0.0000
LGDPI 0.500184 0.008793 56.88557 0.0000
LGPRFOOD -0.074681 0.072864 -1.024941 0.3113
============================================================
R-squared 0.992009 Mean dependent var 6.021331Adjusted R-squared 0.991628 S.D. dependent var 0.222787
S.E. of regression 0.020384 Akaike info criter-4.883747
Sum squared resid 0.017452 Schwarz criterion -4.763303
Log likelihood 112.8843 Hannan-Quinn crite-4.838846
F-statistic 2606.860 Durbin-Watson stat 0.478540
Prob(F-statistic) 0.000000
============================================================
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The plot of the residuals is shown. All the tests indicate highly significant autocorrelation.
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003
Residuals, static logarithmic regression for FOOD
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============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
ELGFOOD(-1) 0.790169 0.106603 7.412228 0.0000
============================================================
R-squared 0.560960 Mean dependent var 3.28E-05
Adjusted R-squared 0.560960 S.D. dependent var 0.020145
S.E. of regression 0.013348 Akaike info criter-5.772439Sum squared resid 0.007661 Schwarz criterion -5.731889
Log likelihood 127.9936 Durbin-Watson stat 1.477337
============================================================
ELGFOODin the regression above is the residual from the LGFOODregression. A simple
regression of ELGFOODon ELGFOOD(1)yields a coefficient of 0.79 with standard error
0.11.
179.0 tt ee
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============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
ELGFOOD(-1) 0.790169 0.106603 7.412228 0.0000
============================================================
R-squared 0.560960 Mean dependent var 3.28E-05
Adjusted R-squared 0.560960 S.D. dependent var 0.020145
S.E. of regression 0.013348 Akaike info criter-5.772439Sum squared resid 0.007661 Schwarz criterion -5.731889
Log likelihood 127.9936 Durbin-Watson stat 1.477337
============================================================
179.0 tt ee
Technical note for EViews users: EViews places the residuals from the most recentregression in a pseudo-variable called resid. residcannot be used directly. So the
residuals were saved as ELGFOODusing the genrcommand:
genr ELGFOOD = resid
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Adding an intercept, LGDPIand LGPRFOODto the specification, the coefficient of the
lagged residuals becomes 0.81 with standard error 0.11. R2is 0.5720, so nR2 is 25.17.
============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
C 0.175732 0.265081 0.662936 0.5112
LGDPI -7.36E-05 0.006180 -0.011917 0.9906
LGPRFOOD -0.037373 0.049496 -0.755058 0.4546
ELGFOOD(-1) 0.805744 0.110202 7.311504 0.0000
============================================================R-squared 0.572006 Mean dependent var 3.28E-05
Adjusted R-squared 0.539907 S.D. dependent var 0.020145
S.E. of regression 0.013664 Akaike info criter-5.661558
Sum squared resid 0.007468 Schwarz criterion -5.499359
Log likelihood 128.5543 F-statistic 17.81977
Durbin-Watson stat 1.513911 Prob(F-statistic) 0.000000
============================================================
181.0... tt ee 5720.02R
17.255720.0442
nR 83.101%1.0
2
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============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
C 0.175732 0.265081 0.662936 0.5112
LGDPI -7.36E-05 0.006180 -0.011917 0.9906
LGPRFOOD -0.037373 0.049496 -0.755058 0.4546
ELGFOOD(-1) 0.805744 0.110202 7.311504 0.0000
============================================================R-squared 0.572006 Mean dependent var 3.28E-05
Adjusted R-squared 0.539907 S.D. dependent var 0.020145
S.E. of regression 0.013664 Akaike info criter-5.661558
Sum squared resid 0.007468 Schwarz criterion -5.499359
Log likelihood 128.5543 F-statistic 17.81977
Durbin-Watson stat 1.513911 Prob(F-statistic) 0.000000
============================================================
181.0... tt ee 5720.02R
17.255720.0442
nR 83.101%1.0
2
(Note that here n= 44. There are 45 observations in the regression in Table 12.1, and one
fewer in the residuals regression.) The critical value of chi-squared with one degree of
freedom at the 0.1 percent level is 10.83.
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Technical note for EViews users: one can perform the test simply by following the LGFOODregression with the command auto(1). EViews allows itself to use residdirectly.
============================================================
Breusch-Godfrey Serial Correlation LM Test:
============================================================
F-statistic 54.78773 Probability 0.000000
Obs*R-squared 25.73866 Probability 0.000000
============================================================
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Presample missing value lagged residuals set to zero.
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================C 0.171665 0.258094 0.665124 0.5097
LGDPI 9.50E-05 0.005822 0.016324 0.9871
LGPRFOOD -0.036806 0.048504 -0.758819 0.4523
RESID(-1) 0.805773 0.108861 7.401873 0.0000
============================================================
R-squared 0.571970 Mean dependent var-1.85E-18
Adjusted R-squared 0.540651 S.D. dependent var 0.019916
S.E. of regression 0.013498 Akaike info criter-5.687865
Sum squared resid 0.007470 Schwarz criterion -5.527273
Log likelihood 131.9770 F-statistic 18.26258
Durbin-Watson stat 1.514975 Prob(F-statistic) 0.000000
============================================================
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The argument in the autocommand relates to the order of autocorrelation being tested. At
the moment we are concerned only with first-order autocorrelation. This is why thecommand is auto(1).
============================================================
Breusch-Godfrey Serial Correlation LM Test:
============================================================
F-statistic 54.78773 Probability 0.000000
Obs*R-squared 25.73866 Probability 0.000000
============================================================
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Presample missing value lagged residuals set to zero.
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================C 0.171665 0.258094 0.665124 0.5097
LGDPI 9.50E-05 0.005822 0.016324 0.9871
LGPRFOOD -0.036806 0.048504 -0.758819 0.4523
RESID(-1) 0.805773 0.108861 7.401873 0.0000
============================================================
R-squared 0.571970 Mean dependent var-1.85E-18
Adjusted R-squared 0.540651 S.D. dependent var 0.019916
S.E. of regression 0.013498 Akaike info criter-5.687865
Sum squared resid 0.007470 Schwarz criterion -5.527273
Log likelihood 131.9770 F-statistic 18.26258
Durbin-Watson stat 1.514975 Prob(F-statistic) 0.000000
============================================================
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When we performed the test, resid(1), and hence ELGFOOD(1), were not defined for the
first observation in the sample, so we had 44 observations from 1960 to 2003.
============================================================
Breusch-Godfrey Serial Correlation LM Test:
============================================================
F-statistic 54.78773 Probability 0.000000
Obs*R-squared 25.73866 Probability 0.000000
============================================================
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Presample missing value lagged residuals set to zero.
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================C 0.171665 0.258094 0.665124 0.5097
LGDPI 9.50E-05 0.005822 0.016324 0.9871
LGPRFOOD -0.036806 0.048504 -0.758819 0.4523
RESID(-1) 0.805773 0.108861 7.401873 0.0000
============================================================
R-squared 0.571970 Mean dependent var-1.85E-18
Adjusted R-squared 0.540651 S.D. dependent var 0.019916
S.E. of regression 0.013498 Akaike info criter-5.687865
Sum squared resid 0.007470 Schwarz criterion -5.527273
Log likelihood 131.9770 F-statistic 18.26258
Durbin-Watson stat 1.514975 Prob(F-statistic) 0.000000
============================================================
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EViews uses the first observation by assigning a value of zero to the first observation forresid(1). Hence the test results are very slightly different.
============================================================
Breusch-Godfrey Serial Correlation LM Test:
============================================================
F-statistic 54.78773 Probability 0.000000
Obs*R-squared 25.73866 Probability 0.000000
============================================================
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Presample missing value lagged residuals set to zero.
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================C 0.171665 0.258094 0.665124 0.5097
LGDPI 9.50E-05 0.005822 0.016324 0.9871
LGPRFOOD -0.036806 0.048504 -0.758819 0.4523
RESID(-1) 0.805773 0.108861 7.401873 0.0000
============================================================
R-squared 0.571970 Mean dependent var-1.85E-18
Adjusted R-squared 0.540651 S.D. dependent var 0.019916
S.E. of regression 0.013498 Akaike info criter-5.687865
Sum squared resid 0.007470 Schwarz criterion -5.527273
Log likelihood 131.9770 F-statistic 18.26258
Durbin-Watson stat 1.514975 Prob(F-statistic) 0.000000
============================================================
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============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
C 0.175732 0.265081 0.662936 0.5112
LGDPI -7.36E-05 0.006180 -0.011917 0.9906
LGPRFOOD -0.037373 0.049496 -0.755058 0.4546
ELGFOOD(-1) 0.805744 0.110202 7.311504 0.0000
============================================================R-squared 0.572006 Mean dependent var 3.28E-05
Adjusted R-squared 0.539907 S.D. dependent var 0.020145
S.E. of regression 0.013664 Akaike info criter-5.661558
Sum squared resid 0.007468 Schwarz criterion -5.499359
Log likelihood 128.5543 F-statistic 17.81977
Durbin-Watson stat 1.513911 Prob(F-statistic) 0.000000
============================================================
We can also perform the test with a ttest on the coefficient of the lagged variable.
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Here is the corresponding output using the autocommand built into EViews. The test is
presented as an Fstatistic. Of course, when there is only one lagged residual, the F
statistic is the square of the tstatistic.
============================================================
Breusch-Godfrey Serial Correlation LM Test:
============================================================
F-statistic 54.78773 Probability 0.000000
Obs*R-squared 25.73866 Probability 0.000000
============================================================
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Presample missing value lagged residuals set to zero.
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================C 0.171665 0.258094 0.665124 0.5097
LGDPI 9.50E-05 0.005822 0.016324 0.9871
LGPRFOOD -0.036806 0.048504 -0.758819 0.4523
RESID(-1) 0.805773 0.108861 7.401873 0.0000
============================================================
R-squared 0.571970 Mean dependent var-1.85E-18
Adjusted R-squared 0.540651 S.D. dependent var 0.019916
S.E. of regression 0.013498 Akaike info criter-5.687865
Sum squared resid 0.007470 Schwarz criterion -5.527273
Log likelihood 131.9770 F-statistic 18.26258
Durbin-Watson stat 1.514975 Prob(F-statistic) 0.000000
============================================================
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The DurbinWatson statistic is 0.48. dLis 1.24 for a 1 percent significance test (2
explanatory variables, 45 observations).
============================================================
Dependent Variable: LGFOOD
Method: Least Squares
Sample: 1959 2003
Included observations: 45
============================================================
Variable Coefficient Std. Error t-Statistic Prob.
============================================================
C 2.236158 0.388193 5.760428 0.0000
LGDPI 0.500184 0.008793 56.88557 0.0000
LGPRFOOD -0.074681 0.072864 -1.024941 0.3113
============================================================
R-squared 0.992009 Mean dependent var 6.021331Adjusted R-squared 0.991628 S.D. dependent var 0.222787
S.E. of regression 0.020384 Akaike info criter-4.883747
Sum squared resid 0.017452 Schwarz criterion -4.763303
Log likelihood 112.8843 Hannan-Quinn crite-4.838846
F-statistic 2606.860 Durbin-Watson stat 0.478540
Prob(F-statistic) 0.000000
============================================================
dL= 1.24 (1% level, 2 explanatory variables, 45 observations)
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The BreuschGodfrey test for higher-order autocorrelation is a straightforward extension of
the first-order test. If we are testing for order q, we add qlagged residuals to the right side
of the residuals regression. We will perform the test for second-order autocorrelation.
tttt uuu 2211
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Here is the regression for ELGFOODwith two lagged residuals. The BreuschGodfrey test
statistic is 25.89. With two lagged residuals, the test statistic has a chi-squared distribution
with two degrees of freedom under the null hypothesis. It is significant at the 0.1 percent
level
============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample(adjusted): 1961 2003
Included observations: 43 after adjusting endpoints
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
C 0.071220 0.277253 0.256879 0.7987
LGDPI 0.000251 0.006491 0.038704 0.9693
LGPRFOOD -0.015572 0.051617 -0.301695 0.7645
ELGFOOD(-1) 1.009693 0.163240 6.185318 0.0000
ELGFOOD(-2) -0.289159 0.171960 -1.681548 0.1009============================================================
R-squared 0.602010 Mean dependent var 0.000149
Adjusted R-squared 0.560117 S.D. dependent var 0.020368
S.E. of regression 0.013509 Akaike info criter-5.661981
Sum squared resid 0.006935 Schwarz criterion -5.457191
Log likelihood 126.7326 F-statistic 14.36996
Durbin-Watson stat 1.892212 Prob(F-statistic) 0.000000
============================================================
89.256020.0432
nR 82.132 %1.02
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We will also perform an Ftest, comparing the RSSwith the RSSfor the same regression
without the lagged residuals. We know the result, because one of the tstatistics is very
high.
============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample(adjusted): 1961 2003
Included observations: 43 after adjusting endpoints
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
C 0.071220 0.277253 0.256879 0.7987
LGDPI 0.000251 0.006491 0.038704 0.9693
LGPRFOOD -0.015572 0.051617 -0.301695 0.7645
ELGFOOD(-1) 1.009693 0.163240 6.185318 0.0000
ELGFOOD(-2) -0.289159 0.171960 -1.681548 0.1009============================================================
R-squared 0.602010 Mean dependent var 0.000149
Adjusted R-squared 0.560117 S.D. dependent var 0.020368
S.E. of regression 0.013509 Akaike info criter-5.661981
Sum squared resid 0.006935 Schwarz criterion -5.457191
Log likelihood 126.7326 F-statistic 14.36996
Durbin-Watson stat 1.892212 Prob(F-statistic) 0.000000
============================================================
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Here is the regression for ELGFOODwithout the lagged residuals. Note that the sample
period has been adjusted to 1961 to 2003, to make RSScomparable with that for the
previous regression.
============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample: 1961 2003
Included observations: 43
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
C 0.027475 0.412043 0.066680 0.9472
LGDPI -0.001074 0.009986 -0.107528 0.9149
LGPRFOOD -0.003948 0.076191 -0.051816 0.9589
============================================================
R-squared 0.000298 Mean dependent var 0.000149Adjusted R-squared -0.049687 S.D. dependent var 0.020368
S.E. of regression 0.020868 Akaike info criter-4.833974
Sum squared resid 0.017419 Schwarz criterion -4.711100
Log likelihood 106.9304 F-statistic 0.005965
Durbin-Watson stat 0.476550 Prob(F-statistic) 0.994053
============================================================
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The Fstatistic is 28.72. This is significant at the 1% level. The critical value for F(2,35) is
8.47. That for F(2,38) must be slightly lower.
============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample: 1961 2003
Included observations: 43
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
C 0.027475 0.412043 0.066680 0.9472
LGDPI -0.001074 0.009986 -0.107528 0.9149
LGPRFOOD -0.003948 0.076191 -0.051816 0.9589
============================================================
R-squared 0.000298 Mean dependent var 0.000149Adjusted R-squared -0.049687 S.D. dependent var 0.020368
S.E. of regression 0.020868 Akaike info criter-4.833974
Sum squared resid 0.017419 Schwarz criterion -4.711100
Log likelihood 106.9304 F-statistic 0.005965
Durbin-Watson stat 0.476550 Prob(F-statistic) 0.994053
============================================================
72.2838/006935.0
2/006935.0017419.038,2 F
47.835,2crit,0.1% F
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Here is the output using the auto(2)command in EViews. The conclusions for the two
tests are the same.
============================================================
Breusch-Godfrey Serial Correlation LM Test:
============================================================
F-statistic 30.24142 Probability 0.000000
Obs*R-squared 27.08649 Probability 0.000001
============================================================
Test Equation:
Dependent Variable: RESID
Method: Least Squares
Presample missing value lagged residuals set to zero.
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================C 0.053628 0.261016 0.205460 0.8383
LGDPI 0.000920 0.005705 0.161312 0.8727
LGPRFOOD -0.013011 0.049304 -0.263900 0.7932
RESID(-1) 1.011261 0.159144 6.354360 0.0000
RESID(-2) -0.290831 0.167642 -1.734833 0.0905
============================================================
R-squared 0.601922 Mean dependent var-1.85E-18
Adjusted R-squared 0.562114 S.D. dependent var 0.019916
S.E. of regression 0.013179 Akaike info criter-5.715965
Sum squared resid 0.006947 Schwarz criterion -5.515225
Log likelihood 133.6092 F-statistic 15.12071
Durbin-Watson stat 1.894290 Prob(F-statistic) 0.000000
============================================================
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The table above gives the result of a parallel logarithmic regression with the addition of
lagged expenditure on food as an explanatory variable. Again, there is strong evidence that
the specification is subject to autocorrelation.
============================================================
Dependent Variable: LGFOOD
Method: Least Squares
Sample (adjusted): 1960 2003
Included observations: 44 after adjustments
============================================================
Variable Coefficient Std. Error t-Statistic Prob.
============================================================
C 0.985780 0.336094 2.933054 0.0055
LGDPI 0.126657 0.056496 2.241872 0.0306
LGPRFOOD -0.088073 0.051897 -1.697061 0.0975
LGFOOD(-1) 0.732923 0.110178 6.652153 0.0000
============================================================R-squared 0.995879 Mean dependent var 6.030691
Adjusted R-squared 0.995570 S.D. dependent var 0.216227
S.E. of regression 0.014392 Akaike info criter-5.557847
Sum squared resid 0.008285 Schwarz criterion -5.395648
Log likelihood 126.2726 Hannan-Quinn crite-5.497696
F-statistic 3222.264 Durbin-Watson stat 1.112437
Prob(F-statistic) 0.000000
============================================================
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Here is a plot of the residuals.
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003
Residuals, ADL(1,0) logarithmic regression for FOOD
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A simple regression of the residuals on the lagged residuals yields a coefficient of 0.43 with
standard error 0.14. We expect the estimate to be adversely affected by the presence of the
lagged dependent variable in the regression for LGFOOD.
============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample(adjusted): 1961 2003
Included observations: 43 after adjusting endpoints
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
ELGFOOD(-1) 0.431010 0.143277 3.008226 0.0044
============================================================
R-squared 0.176937 Mean dependent var 0.000276
Adjusted R-squared 0.176937 S.D. dependent var 0.013922
S.E. of regression 0.012630 Akaike info criter-5.882426Sum squared resid 0.006700 Schwarz criterion -5.841468
Log likelihood 127.4722 Durbin-Watson stat 1.801390
============================================================
143.0 tt ee
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With an intercept, LGDPI, LGPRFOOD, and LGFOOD(1) added to the specification, the
coefficient of the lagged residuals becomes 0.60 with standard error 0.17. R2is 0.2469, sonR2is 10.62, not quite significant at the 0.1 percent level. (Note that here n= 43.)
============================================================
Dependent Variable: ELGFOOD
Method: Least Squares
Sample(adjusted): 1961 2003
Included observations: 43 after adjusting endpoints
============================================================
Variable CoefficientStd. Errort-Statistic Prob.
============================================================
C 0.417342 0.317973 1.312507 0.1972
LGDPI 0.108353 0.059784 1.812418 0.0778
LGPRFOOD -0.005585 0.046434 -0.120279 0.9049
LGFOOD(-1) -0.214252 0.116145 -1.844700 0.0729
ELGFOOD(-1) 0.604346 0.172040 3.512826 0.0012============================================================
R-squared 0.246863 Mean dependent var 0.000276
Adjusted R-squared 0.167586 S.D. dependent var 0.013922
S.E. of regression 0.012702 Akaike info criter-5.785165
Sum squared resid 0.006131 Schwarz criterion -5.580375
Log likelihood 129.3811 F-statistic 3.113911
Durbin-Watson stat 1.867467 Prob(F-statistic) 0.026046
============================================================
62.102469.0432
nR 83.101 %1.02
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The DurbinWatson statistic is 1.11. From this one obtains an estimate of as 10.5d=
0.445. The standard error of the coefficient of the lagged dependent variable is 0.1102.
Hence the hstatistic is as shown.
============================================================
Dependent Variable: LGFOOD
Method: Least Squares
Sample (adjusted): 1960 2003
Included observations: 44 after adjustments
============================================================
Variable Coefficient Std. Error t-Statistic Prob.
============================================================
C 0.985780 0.336094 2.933054 0.0055
LGDPI 0.126657 0.056496 2.241872 0.0306
LGPRFOOD -0.088073 0.051897 -1.697061 0.0975
LGFOOD(-1) 0.732923 0.110178 6.652153 0.0000
============================================================R-squared 0.995879 Mean dependent var 6.030691
Adjusted R-squared 0.995570 S.D. dependent var 0.216227
S.E. of regression 0.014392 Akaike info criter-5.557847
Sum squared resid 0.008285 Schwarz criterion -5.395648
Log likelihood 126.2726 Hannan-Quinn crite-5.497696
F-statistic 3222.264 Durbin-Watson stat 1.112437
Prob(F-statistic) 0.000000
============================================================
33.41102.0441
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Under the null hypothesis of no autocorrelation, the hstatistic asymptotically has a
standardized normal distribution, so this value is above the critical value at the 0.1 percent
level, 3.29.
============================================================
Dependent Variable: LGFOOD
Method: Least Squares
Sample (adjusted): 1960 2003
Included observations: 44 after adjustments
============================================================
Variable Coefficient Std. Error t-Statistic Prob.
============================================================
C 0.985780 0.336094 2.933054 0.0055
LGDPI 0.126657 0.056496 2.241872 0.0306
LGPRFOOD -0.088073 0.051897 -1.697061 0.0975
LGFOOD(-1) 0.732923 0.110178 6.652153 0.0000
============================================================R-squared 0.995879 Mean dependent var 6.030691
Adjusted R-squared 0.995570 S.D. dependent var 0.216227
S.E. of regression 0.014392 Akaike info criter-5.557847
Sum squared resid 0.008285 Schwarz criterion -5.395648
Log likelihood 126.2726 Hannan-Quinn crite-5.497696
F-statistic 3222.264 Durbin-Watson stat 1.112437
Prob(F-statistic) 0.000000
============================================================
33.41102.0441
44445.0
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Copyright Christopher Dougherty 2011.
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