Single equation estimators and system
estimators;Testing over-identifying restrictions.
Ragnar Nymoen
Department of Economics, UiO
13March 2009
ECON 4610: Lecture 9
GIVE and 2SLS
With reference to the notation in lecture 9 we have that the
Generalized IV estimator (GIVE !), for the …rst equation in the
model:
bIV
1
D .W10 Z1 / 1 W10 y1
h
b1 : X1 0
D
Y
Y1 : X1
i
1
0
b1 : X1
Y
y1
b1 is obtained from the reduced form equations
where W1 D Y
for the included endogenous variables, is identical to the 2SLS
estimator:
bSLS D . Y
b1 : X1
1
0
b1 : X1 /
Y
1
b1 : X1
Y
where it is understood that (??) is the second stage of a
b is from the …rst stage.
2-step procedure, where Y
For the exactly identi…ed case we also have
bIV D b2SLS D bILS .
1
1
1
ECON 4610: Lecture 9
0
y1
Single equation and system methods of estimation
IV and 2SLS are single equation estimators: They can be used
to estimate any identi…ed equation in a model.
2SLS
The rest of the model still plays a role for b1 , via the
reduced form that gives the relevant and valid instruments,
but 1 is not estimated jointly with the other j of the
structural model.
A system estimator estimates all identi…ed parameters of a
model jointly.
The system version of 2SLS is three stage least squares
(3SLS), which allows for contemporaneous correlation
between the structural disturbances.
3SLS is an application of the SURE procedure to a
simultaneous equations model: From the 2 stage residuals the
correlation matrix of the disturbances are estimated
3SLS
(consistently) and then bj
are obtained for j D 1, 2, ...M
for all the equations of the model.
ECON 4610: Lecture 9
Maximum likelihood estimation
Maximum likelihood is an estimation principle that “chooses”
the values of the parameters that makes it most likely that the
model has generated the data.
Full information maximum likelihood (FIML) estimation
requires that we specify the distribution of the disturbances
(and therefore also the distribution of the endogenous
variables).
In the single equation case with exogenous regressors and
independently normally distributed disturbances, FIML yields
the same estimator as OLS.
For simultaneous equations models, 3SLS, if it is iterated
rather than stopped at the third stage, converges to FIML.
2SLS can be shown to have the same large sample distribution
as the limited information maximum likelihood estimator
(LIML), which is the single equation version of FIML.
ECON 4610: Lecture 9
Examples of the use of the estimation methods
We can estimate the following cases of our single market model,
using PcGive
Qcase3,t
Qcase3,t
D 1.2
D
0.8Pcase3,t C "dt
0.2 C 0.5Pcase3,t C 2Xst C "st
and
Qcase4,t
Qcase4,t
D 1.2
D
0.8Pcase4,t C 1.5Xdt C "dt
0.2 C 0.5Pcase4,t C 2Xst C "st
In Case 3 it makes sense to estimate the demand equation, in case
4 we can estimate both. We can check OLS, 2SLS, 3SLS and
FIML.
ECON 4610: Lecture 9
Equivalence of method for exactly identi…ed equations and
models
The results shows that in both cases (case 3 and case 4) all
the valid estimators ( 2SLS, 3SLS and FIML) give identical
results.
This is because in both cases, the identi…ed equations are
exactly identi…ed.
There is in fact a theorem saying that 3SLS = 2SLS when all
the equations of the system are exactly identi…ed, or when the
disturbances are independent.
PcGive “delivers” according to this theorem.
ECON 4610: Lecture 9
Testing over-identi…cation
Before we give examples of di¤erent estimators in an
over-identi…ed equation, we can consider testing any such
over-identifying restrictions
This is relevant because identifying restrictions should be
theoretically motivated, so tests of over-identifying restrictions
are tests of economic hypotheses.
The principle for testing is simple:
Obtain the unrestricted reduced form (URF) of an exactly
identi…ed version of the system and estimate that URF by OLS
on each equation.
Obtain a system counterpart to the unrestricted residual sum of
squares (RRSU / that we considered in the single equation case.
If we impose 1 or more over-identifying restrictions, and derive
the restricted reduced form, the RRSR RRSU .
If the di¤erence in signi…cant the restrictions are rejected. In
practice, in PcGive, the log-likelihood value of the reduced
form takes the place of RRSU and RSSR .
ECON 4610: Lecture 9
The simple Keynes model revisited
Let Yt denote GDP. Ct is “consumption”, and let Xt and Zt
denote “exogenous expenditure”.
Assume that Ct depends on GDP, then our example model is
Yt
Ct
Zt
D Ct C Xt C Zt
(1)
D 100 C 0.8Yt C "ct ,
D 80 C 0.5 Zt
For simplicity we assume normality "t
1
(2)
C " zt
N.0,
2 /.
"
The parameter of interest is the marginal propensity to
consume b2 , which we set equal to 0.8.
We consider estimation of (2) and the testing of any
over-identifying instruments.
ECON 4610: Lecture 9