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Chapter 16
Simultaneous equations

1 INTRODUCTION

In Chapter 13, we considered the simple linear regression model

(1)

where y_t and u_t are scalar random variables and x_t and β₀ are k × 1 vectors. In Section 15.8, we generalized (1) to the multivariate linear regression model

(2)

where y_t and u_t are random m × 1 vectors, x_t is a k × 1 vector, and B₀ a k × m matrix.

In this chapter, we consider a further generalization, where the model is specified by

(3)

This model is known as the simultaneous equations model.

2 THE SIMULTANEOUS EQUATIONS MODEL

Thus, let economic theory specify a set of economic relations of the form

(4)

where y_t is an m × 1 vector of observed endogenous variables, x_t is a k × 1 vector of observed exogenous (nonstochastic) variables, and u_0t is an m × 1 vector of unobserved random disturbances. The m × m matrix Γ₀ and the k × m matrix B₀ are unknown parameter matrices. We shall make the following assumption.

Given the nonsingularity of Γ₀, we may postmultiply ( 4 ) with , thus obtaining the reduced form

(6)

where

Combining the observations, we define

and similarly

Then we rewrite the structure ( 4 ) as

and the reduced form (6) as

It is clear that the vectors {v_0t, t = 1, …, n} are independent and identically distributed as N(0, Ω₀), where . The loglikelihood function expressed in terms of the reduced‐form parameters (Π, Ω) follows from Equation (19) in Chapter 15:

(7)

where

Rewriting (7) in terms of (B, Γ, Σ), using Π = −BΓ⁻¹ and Ω = Γ⁻¹′ΣΓ⁻¹, we obtain

(8)

where

The essential feature of (8) is the presence of the Jacobian term of the transformation from u_t to y_t.

There are two problems relating to the simultaneous equations model: the identification problem and the estimation problem. We shall discuss the identification problem first.

Exercise

1. 1. In Equation ( 8 ), we write rather than log |Γ|. Why?

3 THE IDENTIFICATION PROBLEM

It is clear that knowledge of the structural parameters (B₀, Γ₀, Σ₀) implies knowledge of the reduced‐form parameters (Π₀, Ω₀), but that the converse is not true. It is also clear that a nonsingular transformation of ( 4 ), say

leads to the same loglikelihood ( 8 ) and the same reduced‐form parameters (Π₀, Ω₀). We say that (B₀, Γ₀, Σ₀) and (B₀G, Γ₀G, G′Σ₀G) are observationally equivalent, and that therefore (B₀, Γ₀, Σ₀) is not identified. The following definition makes these concepts precise.

Proof.

Since Y = (y₁, …, y_n)′ is normally distributed, its density function depends only on its first two moments,

Now, X has full column rank, so X′X is nonsingular and hence knowledge of these two moments is equivalent to knowledge of (Π₀, Ω₀). Thus, the density of Y depends only on (Π₀, Ω₀). This proves (i). But it also shows that if we know the density of Y, we know the value of (Π₀, Ω₀), thus proving (ii).

As a consequence of Theorem 16.1, a structural parameter of (B₀, Γ₀, Σ₀) is identified if and only if its value can be deduced from the reduced‐form parameters (Π₀, Ω₀). Since without a priori restrictions on (B, Γ, Σ) none of the structural parameters are identified (why not?), we introduce constraints, say

(9)

The identifiability of the structure (B₀, Γ₀, Σ₀) satisfying (9) then depends on the uniqueness of solutions of

and

4 IDENTIFICATION WITH LINEAR CONSTRAINTS ON B AND Γ ONLY

In this section, we shall assume that all prior information is in the form of linear restrictions on B and Γ, apart from the symmetry constraint on Σ.

5 IDENTIFICATION WITH LINEAR CONSTRAINTS ON B, Γ, AND Σ

In Theorem 16.2 we obtained a global result, but this is only possible if the constraint functions are linear in B and Γ and independent of Σ. The reason is that, even with linear constraints on B, Γ, and Σ, our problem becomes one of solving a system of nonlinear equations, for which in general only local results can be obtained.

Proof.

The identifiability of (B₀, Γ₀, Σ₀) depends on the uniqueness of solutions of

(21)

(22)

(23)

The symmetry of Σ follows again from the symmetry of Ω₀ and (22). Equations (21)–(23) form a system of nonlinear equations (because of ( 22 )) in B, Γ, and vech(Σ). Differentiating ( 21 )–( 23 ) gives

(24)

(25)

(26)

and hence, on taking vecs in (24) and (25),

(27)

(28)

(29)

Writing vec Σ = D_m vech(Σ), vec Γ′ = K_m vec Γ, and using Theorem 3.9(a), (28) becomes

(30)

From (27), (30), and (29), we obtain the Jacobian matrix

(31)

where we notice that J depends on Γ, but not on B and Σ. (This follows of course from the fact that the only nonlinearity in ( 21 )–( 23 ) is in Γ.) A sufficient condition for (B₀, Γ₀, Σ₀) to be locally identifiable is that J evaluated at Γ₀ has full column rank. (This follows essentially from the implicit function theorem.) For Γ = Γ₀, we can write

where

and

using the fact that , see Theorem 3.12(b). The matrix J₂(Γ₀) is nonsingular. Hence, J(Γ₀) has full column rank if and only if J₁ has full column rank. This, in turn, is the case if and only if W has full column rank.

6 NONLINEAR CONSTRAINTS

Exactly the same techniques as used in establishing Theorem 16.3 (linear constraints) enable us to establish Theorem 16.4 (nonlinear constraints).

7 FIML: THE INFORMATION MATRIX (GENERAL CASE)

We now turn to the problem of estimating simultaneous equations models, assuming that sufficient restrictions are present for identification. Maximum likelihood estimation of the structural parameters (B₀, Γ₀, Σ₀) calls for maximization of the loglikelihood function ( 8 ) subject to the a priori and identifying constraints. This method of estimation is known as full‐information maximum likelihood (FIML). Finding the FIML estimates involves nonlinear optimization and can be computationally burdensome. We shall first find the information matrix for the rather general case, where every element of B, Γ, and Σ can be expressed as a known function of some parameter vector θ.

8 FIML: ASYMPTOTIC VARIANCE MATRIX (SPECIAL CASE)

Theorem 16.5 provides us with the information matrix of the FIML estimator , assuming that B, Γ, and Σ can all be expressed as (nonlinear) functions of a parameter vector θ. Our real interest, however, lies not so much in the information matrix as in the inverse of its limit, known as the asymptotic variance matrix. To make further progress we need to assume more about the functions B, Γ, and Σ. Therefore, we shall assume that B and Γ depend on some parameter, say ζ, functionally independent of vech(Σ). If Σ is also constrained, say Σ = Σ(σ), where σ and ζ are independent, the results are less appealing (see Exercise 3).

Exercises

1. 1. In the special case of Theorem 16.6 where Γ₀ is a known matrix of constants and B = B(ζ), show that the asymptotic variance matrix of and vech is
2. 2. How does this result relate to the expression for ℱ⁻¹ in Theorem 15.4.
3. Assume, in addition to the setup of Theorem 16.6, that Σ is diagonal and let σ be the m × 1 vector of its diagonal elements. Obtain the asymptotic variance matrix of . In particular, show that , the asymptotic variance matrix of , equals

where E_ii denotes the m × m matrix with a one in the ith diagonal position and zeros elsewhere.

9 LIML: FIRST‐ORDER CONDITIONS

In contrast to the FIML method of estimation, the limited‐information maximum likelihood (LIML) method estimates the parameters of a single structural equation, say the first, subject only to those constraints that involve the coefficients of the equation being estimated. We shall only consider the standard case where all constraints are of the exclusion type. Then LIML can be represented as a special case of FIML where every equation (apart from the first) is just identified. Thus, we write

(42)

(43)

The matrices Π₀₁ and Π₀₂ are unrestricted. The LIML estimates of β₀ and γ₀ in Equation (42) are then defined as the ML estimates of β₀ and γ₀ in the system ( 42 )–(43).

We shall first obtain the first‐order conditions.

Theorem 16.7

Consider a single equation from a simultaneous equations system,

(44)

completed by the reduced form of Y,

(45)

where y (n × 1) and Y (n × m) contain the observations on the endogenous variables, X₁ (n × k₁) and X₂ (n × k₂) are exogenous (nonstochastic), and u₀ (n × 1) and V₀ (n × m) are random disturbances. We assume that the n rows of (u₀ : V₀) are independent and identically distributed as N(0, Ψ₀), where Ψ₀ is a positive definite (m + 1) × (m + 1) matrix partitioned as

There are parameters to be estimated, namely γ₀ (m × 1), β₀ (k₁ × 1), Π₀₁ (k₁ × m), Π₀₂ (k₂ × m), , θ₀ (m × 1), and vech(Ω₀) . We define

(46)

(47)

If and are solutions of the equations

(48)

(49)

where (X : u) and (Z : V) are assumed to have full column rank, then the ML estimators of α₀, Π₀, and Ψ₀ are

Proof.

Given (46) and (47), we may rewrite (44) and (45) as

We define W = (u : V), where

Then we can write the loglikelihood function as

where π = vec Π′ and ψ = vech(Ψ). The first differential is

(50)

Since dW = −(Zdα, XdΠ) and

(51)

where η² = σ² − θ′Ω⁻¹θ, we obtain

and hence

The first‐order conditions are

(52)

(53)

(54)

Postmultiplying (54) by Ω⁻¹ − (1/σ²)θθ′ yields

(55)

Inserting σ² = u′u/n, Ω = V′V/n, and θ = V′u/n in (53) and (55) gives

and hence, since u = y − Zα and V = Y − XΠ,

Since (X : u) and (Z : V) have full column rank, the matrices Z′(I − VV⁺)Z and X′(I − uu⁺)X are nonsingular. This gives

(56)

(57)

Hence, we can express u in terms of V and V in terms of u as follows:

Given a solution of these equations, we obtain from (56), from (57), and from (52).

10 LIML: INFORMATION MATRIX

Having obtained the first‐order conditions for LIML estimation, we proceed to derive the information matrix.

11 LIML: ASYMPTOTIC VARIANCE MATRIX

Again, the derivation of the information matrix in Theorem 16.8 is only an intermediary result. Our real interest lies in the asymptotic variance matrix, which we shall now derive.

Proof.

Theorem 16.8 gives the information matrix. The asymptotic information matrix, denoted by ℱ, is obtained as the limit of (1/n)ℱ_n for n → ∞. We find

with

where A_zz and A_zx are now defined as the limits of (58):

and e, ,and S are defined in Theorem 16.8.

It follows from Theorem 1.3 that

with

To evaluate ℱ^αα, which is the asymptotic variance matrix of , we need some intermediary results:

and also, using Theorems 3.13(d), 3.13(b), 3.12(b), and 3.9(a),

since S′e = 0. Hence,

(63)

It is not difficult to partition the expression for ℱ^αα in (63). Since

we have

and

where P₁ and P₂ are defined in (61). Hence,

We now proceed to obtain expressions for the other blocks of ℱ⁻¹. We have

and

using Theorem 3.17. Further,

(64)

With Q and G as defined in (60) and (62), one easily verifies that

(65)

Also,

Hence,

(66)

where H is defined in ( 62 ). Inserting (65) and (66) in (64) gives (59).

Next,

and finally

Using Theorem 3.15, we find

and

because

This concludes the proof.

Exercises

1. 1.
1. Show that
2. 2.
1. Hence prove that
(see Holly and Magnus 1988).
3. 3. Let . What is the interpretation of the hypothesis θ₀₂ = 0?
4. Show that

where

is partitioned conformably to θ₀ (see Smith 1985). How would you test the hypothesis θ₀₂ = 0?
5. Show that H_* in Theorem 16.9 is positive semidefinite.
6. Hence show that

BIBLIOGRAPHICAL NOTES

3–6. The identification problem is thoroughly discussed by Fisher (1966). See also Koopmans, Rubin, and Leipnik (1950), Malinvaud (1966, Chapter 18), and Rothenberg (1971). The remark in Section 16.5 is based on Theorem 5. A.2 in Fisher (1966). See also Hsiao (1983).

7–8. See also Koopmans, Rubin, and Leipnik (1950) and Rothenberg and Leenders (1964).

9. The fact that LIML can be represented as a special case of FIML where every equation (apart from the first) is just identified is discussed by Godfrey and Wickens (1982). LIML is also close to instrumental variable estimation; see Kleibergen and Zivot (2003) for relationships between certain Bayesian and classical approaches to instrumental variable regression.

10–11. See Smith (1985) and Holly and Magnus (1988).

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Table of Contents for Chapter 16: Simultaneous equations

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1 INTRODUCTION

2 THE SIMULTANEOUS EQUATIONS MODEL

Exercise

3 THE IDENTIFICATION PROBLEM

4 IDENTIFICATION WITH LINEAR CONSTRAINTS ON B AND Γ ONLY

5 IDENTIFICATION WITH LINEAR CONSTRAINTS ON B, Γ, AND Σ

6 NONLINEAR CONSTRAINTS

7 FIML: THE INFORMATION MATRIX (GENERAL CASE)

8 FIML: ASYMPTOTIC VARIANCE MATRIX (SPECIAL CASE)

Exercises

9 LIML: FIRST‐ORDER CONDITIONS

10 LIML: INFORMATION MATRIX

11 LIML: ASYMPTOTIC VARIANCE MATRIX

Exercises

BIBLIOGRAPHICAL NOTES

Table of Contents for
Chapter 16: Simultaneous equations