2. Remedial Measures
(Generalized Least Squares GLS)
If we know ρ
If we don’t
know ρ
Assume ρ=1
First Difference
Method
Estimate ρ from
DW d-statistic
Estimate ρ from
OLS residuals (et)
3. If the regression holds at time t, it should hold at
time t-1, i.e.
(2)
Multiplying the second equation by
gives
(3)
Subtracting (3) from (1) gives
t
11211 ttt uXY
11211
ttt uXY
)()1( 1211 tttt XXpYY
ttt uXY 21 11
AR(1) scheme
Follows all CLRM
assumptions. So
the TRICK is to
REPLACE ut with
this term in the
regression
(1)
4. The equation can be re-written as:
The error term satisfies all the OLS assumptions
When we apply OLS to transformed models, the
estimators thus obtained are called generalized least
squares (GLS) estimators
The estimators thus obtained will have the desirable
BLUE property
5. Note that when we use GLS, we loose the first
observation. Eg. If n=50, now the GLS regression
will be run for 49 observations.
t
To avoid this loss of one observation, the first
observation of Y and X is transformed as follows:
Not generally used in large samples
In small samples, results may be sensitive if we
exclude first observation.
)()1( 1211 tttt XXpYY
6. Method 1: Assume that = +1 (First difference Method)
The generalized Least squares equation reduces to the
first difference equation
t
Putting = 1 , we get
OR (First difference)
This assumption is appropriate if the coefficient of
autocorrelation is very high, say in excess of 0.8, or the
Durbin-Watson d is quite low.
Feature :There is no intercept. Thus, we have to use
the regression through the origin.
)()( 112 tttttt uuXXYY
ttt XY 2
)()1( 1211 tttt XXpYY
7. Method 2: Estimate from Durbin-Watson d
statistic
We know that , i.e.
Obtain rho from this equation and use it to
transform the data as shown in the GLS.
t
Good for reasonably large samples
d-statistic is reported along with regression
results by the software.
Eg. If computed d=1.755, we can calculate ρ-hat.
2
1
d
ˆ12 d
)()1( 1211 tttt XXpYY
8. Method 3: Estimate from OLS residuals
The sample counterpart of the following
regression is given by
Estimate the above equation using OLS
residuals and obtain estimated-ρ. Use it to run
GLS. t
In small samples, this procedure gives biased
estimate of ρ.
ttt uu 11
ttt vuu 1
ˆ.ˆˆ
)()1( 1211 tttt XXpYY
9. Describe how can the problem of
autocorrelation be remedied (Assuming
disturbance term follows AR(1) scheme) if
P=0.5.
The GLS equation is:
t
Put ρ=0.5, we get
i.e.
ttt uXY 21
)()1( 1211 tttt XXpYY