Autocorrelation- Detection- part 2- Breusch-Godfrey Test and Durbin's h test

This ppt explains the two tests with examples

1. Autocorrelation
How to detect Autocorrelation?
Part 2
ShilpaChaudhary
JDMC

2. Detection of
Autocorrelation
Graphical
Examination
Formal tests
Durbin-Watson
d test
Durbin’s h test
Breusch
Godfrey (BG)
Test

3. Breusch-Godfrey
Test

4. Features of BG Test
The BG test is a more general test for autocorrelation as
compared to DW d test:
it allows for nonstochastic regressors such as the
lagged values of the regressand (Yt-1)
It allows for higher-order autoregressive schemes
such as AR(1), AR (2)...
Note that Durbin-Watson d test could not be used in
the above two cases.

5. Steps
Runtheoriginalregression (1) (1)
ObtaintheOLSresiduals(et)
Regressresiduals(et)onet-1andallexplanatoryvariablesandobtainthe
R-square.TheregressorsmaycontainlaggedvaluesofY.
(1)(2)(2) (2)
Ifthesamplesizeislarge,BreuschandGodfreyhaveshownthatnR
2
followsachi-squaredstatisticwith1d.f.(becauseitisAR(1)).
IngeneralnR
2
̴(wherek:numberoflagsinresidualterm)
Noten:no.ofobservationsineqn(2)=no.ofobservationsineqn(1)–1:
asfirstobservationlost)
IfnR
2
exceedsthecriticalvalueatthechosenlevelofsignificance,we
rejectthenullhypothesis,inwhichwedofindevidenceofautocorrelation
(ρisstatisticallydifferentfromzero).
Ho:ρ=0 (no autocorrelation),
Ha: ρ≠0 (Evidence of autocorrelation)

6. Let’s try a
question…….

7. Ques.In the regression model Y
i= β
1+ β
2X
2i+ β
3X
3i+ u
i,
an auxiliary regression results are as follows:
R
2
=
0.567 , n=34
Use Breusch-Godfrey test to check for the presence of
autocorrelation of AR(1) scheme. Use 1% level of
significance.
Solution
H
0: No Autocorrelation, H
a: Presence of autocorrelation
nR
2
̴
nR
2
=34×0.567= 19.278 (Under Ho)
Critical value at 5% level of significance ( )= 6.64
As computed chi-square statistic exceeds critical value, H
o
is Rejected. So, there is evidence of Autocorrelation.

8. Durbin’s h test

9. Step 1: Run the original regression and obtain results:
Step 2: Use estimated-rho and compute h-statistic:
(d= 2(1-p), p=1-d/2)
Where n: number of observations in the original regression
(total observations -1 as first observation is lost due to Yt-
1 term)
: estimated variance of the coefficient of lagged
dependent variable. (square of this s.e)
Under the Ho of no autocorrelation, h is distributed as a
normal variable with mean 0 and unit variance. h ̴ N(0,1)
Original model
if denominator<0, cannot compute h
If denominator is approx 0, h approaches infinity.
Ho:ρ=0 (no autocorrelation),
Ha: ρ≠0 (Evidence of autocorrelation)
h test can be used only in large samples.
Refer to z-table for critical
values
Y-hat
t= …. + ….. Y
t-1 + …..X
t
Se = (….) (…) (...)
Y
t= A + B Y
t-1 + C X
t+ u
t

10. Let’s try a
question…….

11. Ques. A researcher estimated the following demand function for
money for 100 40 years for a country XYZ
ln_ M
t = 2.6027 -0.4024 ln_R
t+ 0.59 ln_Y
t + 0.524 ln_M
t-1
(se) = (1.24) (0.36) (0.34) (.03)
R
2
=0.92 Durbin-Watson d-statistic=1.32
M
t: real cash balances R
t:long term interest rate Y
t: aggregate real
national income
Use Durbin’s h-statistic to check if there is an evidence of
autocorrelation.
Solution
Ho:ρ=0 (no autocorrelation), Ha: ρ≠0 (Evidence of autocorrelation)
= = =2.69
Critical value: z
.05
=1.96
As computed value of h exceeds critical value, we reject Ho.
There is an evidence of autocorrelation.

12. Question (cond.)
a. Can we use Durbin-Watson d-statistic? Give reasons.
b. Can we use Breusch-Godfrey test to check for first-
order autocorrelation in the above regression?
Solution:
a.No. DW d-test cannot be used as there is a lagged
dependent term as one of the explanatory variables
b.Yes

13. Always remember
Model Misspecification vs. Pure Autocorrelation
When we find evidence of autocorrelation, t
could be due to
pure autocorrelation OR
result of mis-specification of the model.

14. Thank You!
Will start with Remedial
Measures in next
session…..
Take care!