Chapter 12 Testing for Autocorrelation (EC220).Ppt

8/11/2019 Chapter 12 Testing for Autocorrelation (EC220).Ppt

1/81

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

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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


3/81


4/81


3


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


5/81

4


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.


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


6/81

0

5

-0.5 0 0.5 1

T= 25

T= 50

T= 100

T= 200

0.7

true value

5



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


7/81

0

5

-0.5 0 0.5 1

T= 25

T= 50

T= 100

T= 200

0.7

true value

6



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


8/81

0

5

-0.5 0 0.5 1

T= 25

T= 50

T= 100

T= 200

0.7

true value

7



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.


9/818


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


10/819


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


11/8110


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


12/8111


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


13/8112


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


14/8113


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



15/8114


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



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15


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



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16


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



18/81

17


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


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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19


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)



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20


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



22/81

21


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


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.


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.


DurbinWatson test

In large samples

No autocorrelation



2d

22d

T

t

t

T

t

tt

e

ee

d

1

2

2

2

1 )(



26/81

25

If there is severe positive autocorrelation, will be near 1 and dwill be near 0.


DurbinWatson test

In large samples

No 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.


DurbinWatson test

In large samples

No 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.


DurbinWatson test

In large samples

No autocorrelation



2d

0d

4d

22d

2

4

0

positive

autocorrelation

negative

autocorrelation

no

autocorrelation



29/81

28

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.


DurbinWatson test

In large samples

No 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.


DurbinWatson test

In large samples

No autocorrelation



2d

0d

4d

22d

2

4

0 dcrit

positive

autocorrelation

negative

autocorrelation

no

autocorrelation

dcrit



31/81

30

Unfortunately, they also depend on the actual data for the explanatory variables in the

sample, and thus vary from sample to sample.


DurbinWatson test

In large samples

No autocorrelation



2d

0d

4d

22d

2

4

0 dcrit

positive

autocorrelation

negative

autocorrelation

no

autocorrelation

dcrit



32/81

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


DurbinWatson test

In large samples

No autocorrelation



22d



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32

2d

0d

4d

2

4

0dL d

Ud

crit

positive

autocorrelation

negative

autocorrelation

no

autocorrelation

dcrit


DurbinWatson test

In large samples

No 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



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.)



35/81

34

2d

0d

4d

2

4

0dL d

Ud

crit

positive

autocorrelation

negative

autocorrelation

no

autocorrelation

dcrit

DurbinWatson test

In large samples

No 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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35

2d

0d

4d

DurbinWatson test

In large samples

No 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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36

2d

0d

4d

DurbinWatson test

In large samples

No 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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37

2d

0d

4d

DurbinWatson test

In large samples

No 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.



39/81

38

2d

0d

4d

DurbinWatson test

In large samples

No 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.



40/81

39

2d

0d

4d

DurbinWatson test

In large samples

No 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.



41/81

40

2d

0d

4d

DurbinWatson test

In large samples

No 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.



42/81

41

2d

0d

4d

DurbinWatson test

In large samples

No 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.



43/81

42

2d

0d

4d

DurbinWatson test

In large samples

No 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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43

2d

0d

4d

DurbinWatson test

In large samples

No 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.



45/81

44

2d

0d

4d

DurbinWatson test

In large samples

No 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



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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46

2d

0d4d

DurbinWatson test

In large samples

No 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.



48/81

47

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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48

The other is the Durbin htest, appropriate for the detection of AR(1) autocorrelation.

Durbins htest

2

)1(1

Yb

ns

n

h



50/81

49

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



51/81

50

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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51

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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52

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



54/81

53

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



55/81

54

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



56/81

55

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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56

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



58/81

57

============================================================

Dependent Variable: ELGFOOD


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



59/81

58

============================================================





============================================================


============================================================

ELGFOOD(-1) 0.790169 0.106603 7.412228 0.0000

============================================================

R-squared 0.560960 Mean dependent var 3.28E-05




============================================================

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



60/81

59

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.

============================================================





============================================================


============================================================

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




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



61/81

60

============================================================





============================================================


============================================================

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







============================================================

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.



62/81

61

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


Presample missing value lagged residuals set to zero.

============================================================


============================================================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






============================================================



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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).

============================================================


============================================================



============================================================

Test Equation:




============================================================


============================================================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

============================================================







============================================================



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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.

============================================================


============================================================



============================================================

Test Equation:




============================================================


============================================================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

============================================================







============================================================



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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.

============================================================


============================================================



============================================================

Test Equation:




============================================================


============================================================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

============================================================







============================================================



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65

============================================================





============================================================


============================================================

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







============================================================

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.

============================================================


============================================================



============================================================

Test Equation:




============================================================


============================================================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

============================================================







============================================================



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The DurbinWatson statistic is 0.48. dLis 1.24 for a 1 percent significance test (2

explanatory variables, 45 observations).

============================================================



Sample: 1959 2003


============================================================


============================================================

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






============================================================

dL= 1.24 (1% level, 2 explanatory variables, 45 observations)



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68

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

============================================================





============================================================


============================================================

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






============================================================

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.

============================================================





============================================================


============================================================

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============================================================







============================================================



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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.

============================================================



Sample: 1961 2003


============================================================


============================================================

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





============================================================



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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.

============================================================



Sample: 1961 2003


============================================================


============================================================

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





============================================================

72.2838/006935.0

2/006935.0017419.038,2 F

47.835,2crit,0.1% F



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73

Here is the output using the auto(2)command in EViews. The conclusions for the two

tests are the same.

============================================================


============================================================



============================================================

Test Equation:




============================================================


============================================================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

============================================================







============================================================



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74

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.

============================================================



Sample (adjusted): 1960 2003

Included observations: 44 after adjustments

============================================================


============================================================

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







============================================================


0 04


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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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76

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.

============================================================





============================================================


============================================================

ELGFOOD(-1) 0.431010 0.143277 3.008226 0.0044

============================================================





============================================================

143.0 tt ee



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77

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.)

============================================================





============================================================


============================================================

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============================================================







============================================================

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.

============================================================





============================================================


============================================================

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








============================================================

33.41102.0441

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Ybns

nh



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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.

============================================================





============================================================


============================================================

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








============================================================

33.41102.0441

44445.0

1

22)1(

Ybns

nh


81/81

Copyright Christopher Dougherty 2011.

These slideshows may be downloaded by anyone, anywhere for personal use.

Subject to respect for copyright and, where appropriate, attribution, they may beused as a resource for teaching an econometrics course. There is no need to

refer to the author.

The content of this slideshow comes from Section 12.2 of C. Dougherty,

In troduct ion to Econom etr ics, fourth edition 2011, Oxford University Press.

Additional (free) resources for both students and instructors may be

downloaded from the OUP Online Resource Centrehttp://www.oup.com/uk/orc/bin/9780199567089/.

Individuals studying econometrics on their own and who feel that they might

benefit from participation in a formal course should consider the London School

of Economics summer school course

EC212 Introduction to Econometrics

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.
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