christopher dougherty ec220 - introduction to econometrics (chapter 12) slideshow: testing for...

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 12)Slideshow: testing for autocorrelation

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Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 12). [Teaching Resource]

© 2012 The Author

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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


2


t1 tt uu

After all, provided that the conditions for the consistency of the OLS estimators are satisfied, as the sample size becomes large, the regression parameters will approach their true values, the location of the regression line will converge on the true relationship, and the residuals will coincide with the values of the disturbance term.

1tt ee error


3


t1 tt uu

Hence a regression of et on et–1 is sufficient, at least in large samples. Of course, there is the issue that, in this regression, et–1 is a lagged dependent variable, but that does not matter in large samples.

1tt ee error

4


This is illustrated with the simulation shown in the figure. The true model is as shown, with ut being 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

1ˆ tt ee

tt utY 0.210

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 Yt make no difference to the distributions of the estimator of .

t17.0 tt uu

1ˆ tt ee

tt utY 0.210

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 et is regressed on et–1, 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

1ˆ tt ee

tt utY 0.210

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

1ˆ 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.

8


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.

Breusch–Godfrey test

t

k

jjtjt uXY

21

9


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.


t

k

jjtjt uXY

21

12

1

t

k

jjtjt eXe

10


As a repercussion, a simple regression of et on et–1 will produce an inconsistent estimate of . The solution is to include all of the explanatory variables in the original model in the residuals autoregression.


12

1

t

k

jjtjt eXe

t

k

jjtjt uXY

21

11


If the original model is the first equation where, say, one of the X variables is Yt–1, then the residuals regression would be the second equation.


12

1

t

k

jjtjt eXe

t

k

jjtjt uXY

21

12


The idea is that, by including the X variables, one is controlling for the effects of any endogeneity on the residuals.


12

1

t

k

jjtjt eXe

t

k

jjtjt uXY

21

13


The underlying theory is complex and relates to maximum-likelihood estimation, as does the test statistic. The test is known as the Breusch–Godfrey test.


12

1

t

k

jjtjt eXe

t

k

jjtjt uXY

21

14


Several asymptotically-equivalent versions of the test have been proposed. The most popular involves the computation of the lagrange multiplier statistic nR2 when the residuals regression is fitted, n being the actual number of observations in the regression.


12

1

t

k

jjtjt eXe

t

k

jjtjt uXY

21

Test statistic: nR2, distributed as 2(1) when testing for first-order autocorrelation

15


Asymptotically, under the null hypothesis of no autocorrelation, nR2 is distributed as a chi-squared statistic with one degree of freedom.


12

1

t

k

jjtjt eXe

t

k

jjtjt uXY

21

Test statistic: nR2, distributed as 2(1) when testing for first-order autocorrelation

16


A simple t test on the coefficient of et–1 has also been proposed, again with asymptotic validity.


12

1

t

k

jjtjt eXe

t

k

jjtjt uXY

21

Alternatively, simple t test on coefficient of et–1

17


The procedure can be extended to test for higher order autocorrelation. If AR(q) autocorrelation is suspected, the residuals regression includes q lagged residuals.


12

1

t

k

jjtjt eXe

q

ssts

k

jjtjt eXe

121

t

k

jjtjt uXY

21

18


For the lagrange multiplier version of the test, the test statistic remains nR2 (with n smaller than before, the inclusion of the additional lagged residuals leading to a further loss of initial observations).


12

1

t

k

jjtjt eXe

q

ssts

k

jjtjt eXe

121

t

k

jjtjt uXY

21

Test statistic: nR2, distributed as 2(q)

19


Under the null hypothesis of no autocorrelation, nR2 has a chi-squared distribution with q degrees of freedom.


12

1

t

k

jjtjt eXe

q

ssts

k

jjtjt eXe

121

t

k

jjtjt uXY

21

Test statistic: nR2, distributed as 2(q)

20


The t test version becomes an F test comparing RSS for the residuals regression with RSS for the same specification without the residual terms. Again, the test is valid only asymptotically.


12

1

t

k

jjtjt eXe

q

ssts

k

jjtjt eXe

121

t

k

jjtjt uXY

21

Alternatively, F test on the lagged residuals H0: 1 = ... = q = 0, H1: not H0

21


The lagrange multiplier version of the test has been shown to be asymptotically valid for the case of MA(q) moving average autocorrelation.


12

1

t

k

jjtjt eXe

q

ssts

k

jjtjt eXe

121

t

k

jjtjt uXY

21

Test statistic: nR2, distributed as 2(q),valid also for MA(q) autocorrelation

22


The first major test to be developed and popularised for the detection of autocorrelation was the Durbin–Watson test for AR(1) autocorrelation based on the Durbin–Watson d statistic calculated from the residuals using the expression shown.

Durbin–Watson test

T

tt

T

ttt

e

eed

1

2

2

21 )(

23

It can be shown that in large samples d tends to 2 – 2, where is the parameter in the AR(1) relationship ut = ut–1 + t.



In large samples

No autocorrelation

Severe positive autocorrelation

Severe negative autocorrelation

22 d

T

tt

T

ttt

e

eed

1

2

2

21 )(

24

If there is no autocorrelation, is 0 and d should be distributed randomly around 2.



In large samples

No autocorrelation



2d22 d

T

tt

T

ttt

e

eed

1

2

2

21 )(

25

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



In large samples

No autocorrelation



2d0d

22 d

T

tt

T

ttt

e

eed

1

2

2

21 )(

26

Likewise, if there is severe positive autocorrelation, will be near –1 and d will be near 4.



In large samples

No autocorrelation



2d0d4d

22 d

T

tt

T

ttt

e

eed

1

2

2

21 )(

27

Thus d behaves as illustrated graphically above.



In large samples

No autocorrelation



2d0d4d

22 d

2 40

positiveautocorrelation

negativeautocorrelation

noautocorrelation

28

To perform the Durbin–Watson test, we define critical values of d. The null hypothesis is H0: = 0 (no autocorrelation). If d lies between these values, we do not reject the null hypothesis.



In large samples

No autocorrelation



2d0d4d

22 d

2 40 dcrit



noautocorrelation

dcrit

29

The critical values, at any significance level, depend on the number of observations in the sample and the number of explanatory variables.



In large samples

No autocorrelation



2d0d4d

22 d

2 40 dcrit



noautocorrelation

dcrit

30

Unfortunately, they also depend on the actual data for the explanatory variables in the sample, and thus vary from sample to sample.



In large samples

No autocorrelation



2d0d4d

22 d

2 40 dcrit



noautocorrelation

dcrit

31

However Durbin and Watson determined upper and lower bounds, dU and dL, for the critical values, and these are presented in standard tables.

2d0d4d

2 40 dL dUdcrit



noautocorrelation

dcrit



In large samples

No autocorrelation



22 d

32

2d0d4d

2 40 dL dUdcrit



noautocorrelation

dcrit



In large samples

No autocorrelation



22 d

If d is less than dL, it must also be less than the critical value of d for positive autocorrelation, and so we would reject the null hypothesis and conclude that there is positive autocorrelation.

33

2d0d4d

2 40 dL dUdcrit



noautocorrelation

dcrit



In large samples

No autocorrelation



22 d

If d is 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.)

34

2d0d4d

2 40 dL dUdcrit



noautocorrelation

dcrit



In large samples

No autocorrelation



22 d

If d lies between dL and dU, we cannot tell whether it is above or below the critical value and so the test is indeterminate.

35

2d0d4d



In large samples

No autocorrelation



22 d

Here are dL and dU for 45 observations and two explanatory variables, at the 5% significance level.

1.43 1.62(n = 45, k = 3, 5% level)

2 40 dL dU



noautocorrelation

36

2d0d4d



In large samples

No autocorrelation



22 d

1.43 1.62(n = 45, k = 3, 5% level)

2 40 dL dU



noautocorrelation

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

37

2d0d4d



In large samples

No autocorrelation



22 d

1.43 1.62(n = 45, k = 3, 5% level)

2 40 dL dU



noautocorrelation

2.38 2.57

So if d < 1.43, we reject the null hypothesis and conclude that there is positive autocorrelation.

38

2d0d4d



In large samples

No autocorrelation



22 d

1.43 1.62(n = 45, k = 3, 5% level)

2 40 dL dU



noautocorrelation

2.38 2.57

If 1.43 < d < 1.62, the test is indeterminate and we do not come to any conclusion.

39

2d0d4d



In large samples

No autocorrelation



22 d

1.43 1.62(n = 45, k = 3, 5% level)

2 40 dL dU



noautocorrelation

2.38 2.57

If 1.62 < d < 2.38, we do not reject the null hypothesis of no autocorrelation.

40

2d0d4d



In large samples

No autocorrelation



22 d

1.43 1.62(n = 45, k = 3, 5% level)

2 40 dL dU



noautocorrelation

2.38 2.57

If 2.38 < d < 2.57, we do not come to any conclusion.

41

2d0d4d



In large samples

No autocorrelation



22 d

1.43 1.62(n = 45, k = 3, 5% level)

2 40 dL dU



noautocorrelation

2.38 2.57

If d > 2.57, we conclude that there is significant negative autocorrelation.

42

2d0d4d



In large samples

No autocorrelation



22 d

2 40



noautocorrelation

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)

43

2d0d4d



In large samples

No autocorrelation



22 d

2 40



noautocorrelation

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.

44

2d0d4d



In large samples

No autocorrelation



22 d

2 40



noautocorrelation

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.

45

2d0d4d



In large samples

No autocorrelation



22 d

2 40



noautocorrelation

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 d statistic automatically as part of the standard regression output.

46

2d0d4d



In large samples

No autocorrelation



22 d

2 40



noautocorrelation

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.

47

Durbin proposed two tests for the case where the use of the lagged dependent variable as a regressor made the original Durbin–Watson test inapplicable. One was a precursor to the Breusch–Godrey test.

Durbin’s h test

2

)1(1

ˆ

Ybnsn

h


48

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

Durbin’s h test

2

)1(1

ˆ

Ybnsn

h


49

The Durbin h statistic 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 n is the number of observations in the regression.

Durbin’s h test

2

)1(1

ˆ

Ybnsn

h

2

)1( Ybs


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 d tends to 2 – 2 in large samples. The estimator is then 1 – 0.5d.

Durbin’s h test

2

)1(1

ˆ

Ybnsn

h

22 d

d5.01ˆ


51

The estimate of the variance of the coefficient of the lagged dependent variable is obtained by squaring its standard error.

Durbin’s h test

2

)1(1

ˆ

Ybnsn

h


22 d

d5.01ˆ

52

Thus h can be calculated from the usual regression results. In large samples, under the null hypothesis of no autocorrelation, h is distributed as a normal variable with zero mean and unit variance.

Durbin’s h test

2

)1(1

ˆ

Ybnsn

h


22 d

d5.01ˆ

53

An occasional problem with this test is that the h statistic cannot be computed if n is greater than 1, which can happen if the sample size is not very large.

Durbin’s h test

2

)1(1

ˆ

Ybnsn

h


22 d

d5.01ˆ

2

)1( Ybs

54

An even worse problem occurs when n is near to, but less than, 1. In such a situation the h statistic could be enormous, without there being any problem of autocorrelation.

Durbin’s h test

2

)1(1

ˆ

Ybnsn

h


22 d

d5.01ˆ

2

)1( Ybs

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 6.021331 Adjusted 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 ============================================================


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

57

============================================================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-5.772439 Sum squared resid 0.007661 Schwarz criterion -5.731889 Log likelihood 127.9936 Durbin-Watson stat 1.477337 ============================================================


ELGFOOD in the regression above is the residual from the LGFOOD regression. A simple regression of ELGFOOD on ELGFOOD(–1) yields a coefficient of 0.79 with standard error 0.11.

179.0ˆ tt ee

58

============================================================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-5.772439 Sum squared resid 0.007661 Schwarz criterion -5.731889 Log likelihood 127.9936 Durbin-Watson stat 1.477337 ============================================================


179.0ˆ tt ee

Technical note for EViews users: EViews places the residuals from the most recent regression in a pseudo-variable called resid. resid cannot be used directly. So the residuals were saved as ELGFOOD using the genr command:genr ELGFOOD = resid

59

Adding an intercept, LGDPI and LGPRFOOD to the specification, the coefficient of the lagged residuals becomes 0.81 with standard error 0.11. R2 is 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...ˆ tt ee


5720.02 R17.255720.0442 nR 83.101 %1.0

2

60


181.0...ˆ tt ee


5720.02 R17.255720.0442 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.

61

Technical note for EViews users: one can perform the test simply by following the LGFOOD regression with the command auto(1). EViews allows itself to use resid directly.

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


62

The argument in the auto command relates to the order of autocorrelation being tested. At the moment we are concerned only with first-order autocorrelation. This is why the command is auto(1).



63

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.



64

EViews uses the first observation by assigning a value of zero to the first observation for resid(–1). Hence the test results are very slightly different.



65



We can also perform the test with a t test on the coefficient of the lagged variable.

66

Here is the corresponding output using the auto command built into EViews. The test is presented as an F statistic. Of course, when there is only one lagged residual, the F statistic is the square of the t statistic.



67

The Durbin–Watson statistic is 0.48. dL is 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 6.021331 Adjusted 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)

68

The Breusch–Godfrey test for higher-order autocorrelation is a straightforward extension of the first-order test. If we are testing for order q, we add q lagged residuals to the right side of the residuals regression. We will perform the test for second-order autocorrelation.


tttt uuu 2211

69

Here is the regression for ELGFOOD with two lagged residuals. The Breusch–Godfrey 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

70

We will also perform an F test, comparing the RSS with the RSS for the same regression without the lagged residuals. We know the result, because one of the t statistics 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 ============================================================


71

Here is the regression for ELGFOOD without the lagged residuals. Note that the sample period has been adjusted to 1961 to 2003, to make RSS comparable 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 0.000149 Adjusted 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

The F statistic 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 0.000149 Adjusted 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 ============================================================


72.28

38/006935.02/006935.0017419.0

38,2

F 47.835,2crit,0.1% F

73

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


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.

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


75

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

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.

============================================================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-5.882426 Sum squared resid 0.006700 Schwarz criterion -5.841468 Log likelihood 127.4722 Durbin-Watson stat 1.801390 ============================================================


143.0ˆ tt ee

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. R2 is 0.2469, so nR2 is 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

78

The Durbin–Watson statistic is 1.11. From this one obtains an estimate of as 1 – 0.5d = 0.445. The standard error of the coefficient of the lagged dependent variable is 0.1102. Hence the h statistic is as shown.



33.4

1102.0441

44445.0

1ˆ 22

)1(

Yb

nsn

h

79

Under the null hypothesis of no autocorrelation, the h statistic asymptotically has a standardized normal distribution, so this value is above the critical value at the 0.1 percent level, 3.29.



33.4

1102.0441

44445.0

1ˆ 22

)1(

Yb

nsn

h

Copyright Christopher Dougherty 2011.

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Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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2012.02.23

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christopher dougherty ec220 - introduction to econometrics (chapter 12) slideshow: testing for...

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error slide