| N | Number of observations (i=1,2,...,N) |
| G | Number of equations (endogenous variables) (j=1,2,...,G) |
| K | Number of predetermined variables (k=1,2,...,K) |
| Gj | Number of RHS endogenous variables in the equation j; Gj+1 is the number of endogenous variables in the equation j |
| Kj | Number of predetermined variables in the equation j |
| Gj* | Number of endogenous variables not in the equation j Gj+Gj*+1 = G |
| Kj* | Number of predetermined variables not in the equation j Kj+Kj* = K |
| Y | NxG Data matrix of endogenous variables |
| X | NxK Data matrix of predetermined variables |
| Z | Z=Y~X, Nx(G+K) Data matrix of all variables |
| B | GxG parameter (sparse) matrix associated with Y Note: Bjj = -1 (normalization) |
| G | KxG parameter (sparse) matrix associated with X |
| D | D=B|G, (G+K)xG parameter (sparse) matrix associated with Z |
| U, V | NxG error matrices |
| YB + XG = U (or ZD = U) |
| Yi.B + Xi.G = Ui. (i=1,2,...,N) |
| YB.j + XG.j = U.j (j=1,1,...,G) |
| The equation specification may be expressed as |
| yj = Yjbj + Xjgj + ej |
| Note: ej = 0 if the j-th equation is an identity. |
| Y = XP + V |
| Yi. = Xi.P+Vi. (i=1,2,...,N) |
| Y.j = XP.j+V.j (j=1,2,...,G) |
P.j = (X'X)-1X'yj
Given the parameter estimator of P.j for each equation j, can we derive or solve the corresponding structural paramters B.j and G.j through the non-linear relationship P = -GB-1?
The j-th stochastic equation is identified if the structural paramters B.j and G.j are derivable from the reduced form parameters in P. An identity equation is automatically identified. A linear system model is identified if all the stochastic equations are identified.
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| Where | P | (KxG parameter matrix) |
| I | (KxK identity matrix) | |
| B.j | (Gx1 parameter vector) | |
| G.j | (Kx1 parameter vector) | |
K ³ Gj+Kj or Kj* ³ Gj.
Equivalently, Kj* + Gj* ³ G-1, since G = Gj + Gj* + 1.
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Remember that yj = Yjbj + Xjgj + ej.
| Where | P1 | (Kjx1 scalar) |
| P2 | (KjxGj matrix) | |
| P3 | (KjxGj* matrix) | |
| P1* | (Kj*x1 vector) | |
| P2* | (Kj*xGj matrix) | |
| P3* | (Kj*xGj* matrix) | |
rank([P1* P2*]) = rank(P2*) = Gj.
Once bj is solved, gj is obtained from (1).
In practice, the rank condition as derived is difficult to check because the dense matrix P2* is not known prior estimation. The alternative method is to check the structural parameters in B and G in relation with the zeros restrictions for each equation. That is, for each equation j, there must exist a matrix of rank G-1 obtained from the non-zero coefficients appeared in the other equations but not in the jth equation.
yj = Zjdj + ej
where Zj = [Yj Xj] and dj = [bj gj]. Denote dj as the estimator of dj.
Note: the OLS estimator of dj (that is, dj) is biased and inconsistent due to the random regressors problem (because in general there are RHS endogenous variables in the equation). The method of instrumental variables is recommended instead. The appropriate instrumental variables for the RHS endogenous variables can be constructed from the least squares estimator of P.j (that is, (X'X)-1X'yj), for the reduced form equation: yj = Y.j=XP.j+V.j.
yj = Wjdj + ej
where Wj = [X(X'X)-1X'Yj Xj]. Recall that Zj = [Yj Xj] and thus W'jZj = W'jWj. Then the 2SLS estimator of dj is the following:
dj = (W'jZj)-1W'jyj
= (Z'j[X(X'X)-1X']Zj)-1Z'j[X(X'X)-1X']yj
Var(dj) = s2j(W'jZj)-1
= s2j(Z'j[X(X'X)-1X']Zj)-1
s2j = e'jej/N, and
ej = yj - Zjdj
Note: the 2SLS estimator of dj (that is, dj) does not take account of cross equation correlation although the instrumental variables are obtained from all the predetermined variables in the model.
By assuming the normal distribution for the reduced form error matrix V0j with zero mean and variance-covariance matrix W0j, LIML estimator of d is obtained from maximizing the log-likelihood function:
L(P0j) = -½ N ((Gj+1)log(2p) + log(|W0j|) + trace[(Y0j - XP0j)'W0j-1(Y0j - XP0j])
subject to the identification constraint:
PB.j = -G.j
LIML estimator is the same as the least variance ratio estimator, which is a special case of k-class estimator.
S = e'e/N.
Var(d) = [ smn(W'mZm)-1(W'mZn)(W'nZn)-1, m,n=1,2,...,G ]
Where smn is the (m,n)-th element of S, and smm=s2m, snn=s2n.
By stacking all the stochastic equations yj = Wjdj + ej (j=1,2,...,G) as follows:
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Write the above stacked-equation system as: y = Wd + e, where
| y | NGx1 data vector |
| W | NGx(Sj=1,2,...,GGj+Kj) data matrix |
| d | (Sj=1,2,...,GGj+Kj)x1 parameter vector |
| e | NGx1 error vector |
The error structure e satisfies:
E(e) = 0 and
Var(e) = E(ee')
= S Ä I
= [sijI (i,j=1,2,...,G)]
e is clearly heterogeneous and correlated across equations. Denote d as the Generalized Least Squares (GLS) estimator of d. Then
d = [W'(S-1ÄI)Z]-1W'(S-1ÄI)y
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Var(d) = [W'(S-1ÄI)Z]-1
S = e'e/N is the estimated variance-covariance matrix S, where e = [e1, e2, ..., eG] and the estimated residual is ej = yj - Zjdj (j=1,2,...,G). Furthermore, S-1 denotes the inverse of S with the element sjk (j,k=1,2,...,G).
Note: Since S-1 depends on d, iterations of 3SLS may be performed until convergence.
L*(B,G) = -½ NG(1 + log(2p)) + N log(|B|) - ½ N log(|(YB+XG)'(YB+XG)|/N)
Since log(|B|) = ½ log(|B'Y'YB|) - ½ log(|Y'Y|), we can also write
L*(B,G)
= -½ NG(1 + log(2p)) - ½ N log(|Y'Y|)
+ ½ N log(|B'Y'YB|/N) - ½ N log(|(YB+XG)'(YB+XG)|/N)
Instrumental Variables Method
FIML estimator using IV method is obtained by maximizing
L*1(B,G)
= N log(|B|) - ½ N log(|(YB+XG)'(YB+XG)/N|)
The first derivatives of L*1(B,G) are used to set up the normal equations similar to the iterative 3SLS estimation. Let S = |(YB+XG)'(YB+XG)|/N, the normal equations for maximizing L*1(B,G) are:
¶L*1/¶B =
NB'-1 - Y'(YB+XG)S-1 = 0
¶L*1/¶G =
-X'(YB+XG)S-1 = 0
By subsitituting out N, combining terms, and using the parameter restrictions P = -GB-1 in the first equation, it can be re-written as follows:
¶L*1/¶B =
- P'X'(YB+XG)S-1 = 0
Together with the second equation, the normal equation in matrix form looks like this:
[XGB-1 X]'(YB+XG)S-1 = 0
We need to re-arrange the equations and parameters, and define Wj* = [(-XGB-1)j Xj] and write the typical j-th equation: yj = Wj*dj + ej (j=1,2,...,G). The corresponding stacked-equations system y = W*d + e:
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As in the 3SLS, the FIML estimator for d is:
d = [W*'(S-1ÄI)Z]-1W*'(S-1ÄI)y
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Var(d) = [W*'(S-1ÄI)Z]-1
S = e'e/N and e = [e1, e2, ..., eG] with ej = yj - Zjdj (j=1,2,...,G).
Linearized ML Method
FIML estimator using the linearized ML method is obtained by maximizing
L*2(B,G)
= log(|B'Y'YB|/N) - log(|(YB+XG)'(YB+XG)|/N)
Let Q = |B'Y'YB|/N and S = |(YB+XG)'(YB+XG)|/N, then the normal equations for maximizing L*2(B,G) are:
¶L*2/¶B =
Y'YBQ-1 - Y'(YB+XG)S-1 = 0
¶L*2/¶G =
-X'(YB+XG)S-1 = 0
By re-arranging the equations and parameters, and let Zj = [Yj Xj] and Z0j = [Yj 0].
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Then the FIML estimator of d is derived from the following
d = [Z'(S-1ÄI)Z - Z'0(Q-1ÄI)Z0]-1 [Z'(S-1ÄI)y - Z'0(Q-1ÄI)y]
Where S = e'e/N, e = [e1, e2, ...,eG], and ej = yj - Zjdj (j=1,2,...G). Similarly, Q = e0'e0/N, e0 = [e01, e02, ...,e0G], and e0j = yj - Z0jdj (j=1,2,...,G).
Newton Method
Both the first derivatives (gradient) and second derivatives (hessian) of L*2(B,G)
are used in the iterative estimation.
Y = Y-1P1 + XP2 + V
From now on, X denotes the data matrix of current and lagged exogenous variables and Y-1 includes lagged endogenous variables. Then,
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The stability of the model requires that the characteristic roots of P1 lie inside the unit circle. A plot of the period (dynamic) multipliers against the lag length is call the Impulse Response Function.