Recursive systems
Ragnar Nymoen
Department of Economics, UiO
27 February 2009
ECON 4610: Lecture 7
Overview
If a system of equations has a special causal structure, called
recursive structure, or diagonal structure,the parameters can
be estimated consistently by OLS,
and the parameters are identi…ed despite violation of the order
condition.
Identi…cation hinges on an extra restriction on the covariance
matrix of the structural disturbances.
Reference is G Ch 13.2.3;. B Ch 9.7.d
ECON 4610: Lecture 7
Notation for the simultaneous equation model
M endogenous variables and M linear equations, M structural
disturbances ("1 , ...,"M ). K exogenous variables
Let yt , xt denote vectors with observations .t D 1,....,T /, and
let et contain the disturbances. The structural form of the
model is :
yt0 0 C x0t B D e0t
(1)
where 0 is a M M coe¢ cient matrix and B is a K M
coe¢ cient matrix. (1) is called the structural form of the
model.
0 1 exists (0 is non-singular). The reduced form:
yt0
D
1
x0t B0
0
0
D xt 5 C vt
C e0t 0
1
(2)
5 is the matrix of reduced form coe¢ cients. vt is the reduced
form disturbances.
ECON 4610: Lecture 7
Correlation matrix of the disturbances
Assumptions about structural disturbances
E [et j xt ] D 0
E [et e0t j xt ] D 6
E [et e0s j xt , Xs ] D 0
The …rst assumption is standard. The second allows for
contemporaneous correlation among the disturbances. The
last implies no autocorrelation or heteroscedasticity.
Properties of the reduced form disturbances:
E [vt j xt ]
D
0
E [vt vt0 j xt ]
D
E [.0
6 D0
0
•0
1
0
et /e0t 0
1
j xt ] D 0
ECON 4610: Lecture 7
1
0
60
1
D •.
Recursive structure
If the following two conditions hold:
1
0 is upper triangular. For the case of M D 2 this means
0D
1
0
12
1
in Greene’s general notation.
2
6 is a diagonal matrix with each disturbance’s variance along
the main diagonal
the system is said to have a recursive structure.
The structure is causal. Ssince y1 is determined by x1t (only),
we can …rst solve for y1t . Then we can solve for y2 , given the
solution for y1 , and so on recursively.
The equations form a causal chain, there are no feed-backs
between current endogenous variables.
ECON 4610: Lecture 7
The cobweb model
A partial market equilibrium model for the case of completely
inelastic short-run supply:
Qt
Qt
D
21 Pt
D
12
11
22 x1t
21 x1t
32 Pt 1
31 x2t
C "1t , demand(3)
C "2t , supply
(4)
where 32 < 0 (and 21 > 0 as before).
Qt is determined “…rst”, from the supply equation, then Pt ,
so we have a causal structure.
It is now only a matter of re-arranging, so that the (rede…ned)
0 becomes upper triangular: We de…ne Qt as the …rst
endogenous variable, and Pt as the second, and regard the
supply equation as the …rst equation of the system, and write
the demand equation as
Pt D J 12 Qt C J 12 C J 22 x1t C J 32 x2t C "J 1t
where the suitably re-de…ned parameters and disturbance are
denoted J 12 and so on.
ECON 4610: Lecture 7
Identi…cation and estimation
Using the cobweb model as an example, it seems like
identi…cation can be problematic:
Assume that x2t D Pt
the order condition.
1.
Then the demand equation fails on
However it turns out that the restrictions on the disturbances
in assumption 2, (a restriction of a kind not considered in
Lecture 6) is su¢ cient to identity the demand curve.
It is easy to see why: Because "2t is uncorrelated with "1t , Qt
is an exogenous variable in the demand equation.
Another way of expressing this is by noting that assumption 2
secures that the parameters of interest, speci…cally the slope
coe¢ cient of the demand curve, are parameters in the
regression function (the conditional mean of Pt .
But then we also now that OLS on each equation of the
system gives consistent estimates.
ECON 4610: Lecture 7