Handbook of Econometrics Vols1-5 _ Chapter 7 pot

Developments in both identification and estimation of simultaneous equation models arise from the jointly endogenous feature of economic variables when they are treated from either a the

SPECIFICATION AND ESTIMATION

OF SIMULTANEOUS EQUATION MODELS

Handbook of Econometrics, Volume I, Edited by 2 Griliches and M.D Intriligator

0 North-Holland Publishing Company, 1983

1 Introduction

The simultaneous equation model is perhaps the most remarkable development in econometrics Many of the models used in the statistical analysis of economic data arose from previous work in statistics Econometric research has, of course, led to further developments and applications of these statistical models But in the case of the simultaneous equation problem, econometrics has provided unique insight And this insight arises from economic theory in terms of the operations of markets and the simultaneous determination of economic variables through an equilibrium model Consider a linear regression specification which relates the quantity purchased of a commodity to its price at time t:

where Z, is a k x 1 vector of other variables thought to affect the relationship What economic meaning can be given to the statistical specification of eq (l.l)? More explicitly, is eq (1.1) a demand curve or a supply curve or should we examine the least squares estimate a, to decide upon our answer?

The econometricians’ answer is that both quantity and price are simultaneously determined by the actions of the market so that to understand the quantity and price relationship we need to treat the two variables as jointly endogenous Thus,

eq (1.1) considered in isolation is not sufficient to determine the economic meaning of the statistical relationship Instead, we must consider a more complete model in which both quantity and price are determined simultaneously by the operation of economic markets With joint endogeneity, least squares becomes the wrong statistical technique to use in estimation of the unknown coefficients of eq (1.1) The implications of joint endogeneity of variables give the simultaneous equation model its unique characteristics which distinguishes it from most other regression type models in statistics Developments in both identification and estimation of simultaneous equation models arise from the jointly endogenous feature of economic variables when they are treated from either a theoretical or statistical viewpoint

The seminal papers from which the simultaneous equation model developed established the importance of joint endogeneity for statistical analysis of eco- nomic relationships Haavelmo (1943, 1944) realized that in the presence of jointly endogenous variables that a joint probability distribution was necessary to analyze the data He also distinguished the essence of the identification problem

We now return to our demand and supply example to see the importance of the concepts of identification and joint endogeneity for our statistical model Let us specify eq (1.1) to be the demand curve where we limit Z, to a single variable, consumer income, for simplicity We then specify the supply curve as

Again to simplify we let W, consist of a single variable, say the wage rate We

assume that both Z, and F are determined exogenously to our model in the sense that both are (asymptotically) uncorrelated with the stochastic residuals at and q,

But in general neither pt nor q, can be taken as exogenous even in the particular

equation in which they appear on the right-hand side because even if E( and vlt are

independent, pt is not uncorrelated with E, and likewise for qt and 9, The variables q, and p, are jointly endogeneous and require non-standard statistical

treatment

Before even considering questions of estimation, the problem of identification must be settled first We have agreed to call eq (1.1) the demand curve and eq (1.2) the supply curve, but can we actually distinguish them statistically? Without the presence of Z, and K our position is hopeless, because we would simply observe a scatter of points near the, intersection (equilibrium position) of the demand and supply curves The scatter would arise only because of the stochastic disturbance; and no statistical estimation procedure could establish the position

of the demand or supply curve in price-quantity space from the observation of a single point perturbed by stochastic disturbances But a change in Z, in eq (1 I), independent of E, and v~, causes the demand curve to shift and permits points on

the supply curve to be established Likewise, shifts in W,, again independent of E,

and qr, shift the supply curve so that points on the demand curve can be

established It is interesting to note that exogenous shifts in variables in the other

‘Of course, Haavelmo’s research had many antecedents Working (1927) gave an early account of the identification problem Joint endogeneity (although not caused by simultaneous determination of economic variables) might be said to have arisen first in the errors in variables problem in regression Adcock (1878) is the first reference that I know to the errors in variables problem

equation lead to identification of the equation in question This finding is the basis for the previous remark than an equation cannot be considered in isolation but that a more complete model is required Koopmans (1949) Koopmans and Reiersol(1950), and Koopmans, Rubin and Leipnik (1950) established conditions for identification in linear simultaneous models We will further consider the identification problem in Section 3

We now turn to statistical estimation What are the properties of our estimates

if we use least squares on eq (1 l)? Let us assume that we measure all variables in deviations from their means so that & and (~a are eliminated The least squares estimate of p, will be biased because of the correlation of pr with Ed and it has a probability limit

where Q, = Z - Z( Z’Z)- ‘Z’.2 The second term in eq (1.3) is not zero because solving eqs (1.1) and (1.2) in terms of the exogenous variables and the residuals yields

be consistently estimated We use W,, the identifying variable from the other equation in our model, as an instrumental variable to find

We now have a consistent estimator because

So long as the first term in brackets has a finite plim, consistent estimation occurs

‘Haavelmo (1944) was first to point out that least squares estimates of the coefficients of a structural equation are inconsistent For a simple example he derived the plim of the inconsistent estimation His argument against least squares is based on the correct point that the conditional expectation of the residuals given the right-hand-side variables is not zero, i.e some of the right-hand- side variables are jointly endogenous

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because the second term has a zero plim under the assumption that W and z are

exogenous Thus, just the variable that led to identification of the demand curve also provides us with the means to estimate consistently its parameters This point

is the basis for Haavelmo’s discussion of the simultaneous equation problem

I shall use the concept of instrumental variables to organize this survey The most important set of identification conditions, namely coefficient restrictions, involves determining whether a sufficient number of instruments are available [cf Fisher (1966)] Furthermore, it has recently been proven that the other type of identification restrictions used in linear simultaneous equation models, namely covariance restrictions, are also most easily understood in terms of instrumental variables [Hausman and Taylor (198Oa)l In terms of estimation almost all consistent estimators are either instrumental variables estimators or asymptotic approximations to them The original maximum likelihood estimator (FIML) proposed for the simultaneous equation model is an instrumental variable estima- tor [Hausman (1975)]; other estimators rely on asymptotic approximations to the basic likelihood equations [Hendry ( 1976)].3 Estimation is considered in Section 4 Two other interrelated notions that we consider are the endogeneity-exogeneity concept and tests of specification We have emphasized joint endogeneity as the principle behind simultaneous equations models Yet both identification and estimation rest on an exogeneity assumption, as our example indicates We attempt to explore this assumption from a statistical point of view The question naturally arises of whether the key exogeneity assumptions are testable, perhaps using recently developed techniques on causality from time-series analysis The answer is no If we have a surplus of exogenous variables, then a subset may be tested for endogeneity But these tests depend on the maintained assumption of exogeneity in other variables Specification tests look at questions of endogeneity and also at coefficient restrictions In empirical work they are not used as often as they should be In Section 5 we consider exogeneity tests and specification tests in reference to the simultaneous equation model

Finally, in Section 6 we briefly consider the non-linear simultaneous equation model In the general case it appears that identification ceases to be a problem Consistent estimation by instrumental variables or by minimum distance estima- tors is possible [Amemiya (1974b)] Yet at the present time problems which arise

3Another possible classification of estimators arises from a minimum distance (minimum chi square) interpretation Malinvaud (1970) and Rothenberg (1973) use this approach We consider this approach

in Section 4 The reason that I prefer the instrumental variable approach is because it carries over to the case of non-linear simultaneous equations As results in Section 6 demonstrate, the attraction of maximum likelihood estimation is reduced in the non-linear case because consistent estimation usually requires correct specification of the unknown stochastic distributions Instrumental variable estima- tion does not require knowledge of the distributions Furthermore, maximum likelihood is a particular application of instrumental variables, but not vice versa Minimum distance estimation is severely limited in the non-linear case by the non-existence of a convenient reduced-form expression

with estimation by maximum likelihood estimation have not been completely resolved

2 Model specification

Three basic specifications have been used in the interpretation of linear simulta- neous equation models: the structural form, the reduced form, and the recursive form The structural form has stochastic equations and sometimes accounting identities which correspond to the basic economic theory underlying the model It typically contains all the economic knowledge that we are able to include in the model The demand and supply example of the previous section is a simple example of a structural model The major difference between structural models and more traditional linear models in statistics is the presence of jointly endog- enous variables The reduced-form model can be obtained from the structural model by a non-singular linear transformation.4 The joint endogeneity is eliminated from the model by the reduced-form transformation as each endog- enous variable can be written as a linear function of only exogenous variables Thus, the reduced-form specification is similar to the well-known multivariate least squares regression specification (although non-linear parameter constraints are typically present) The question might well be asked: Why then do we need the structural form?5 Estimation and prediction might well proceed with the reduced form The traditional answer is that the change in one structural equation will change the entire (restricted) reduced form I do not find the answer particularly persuasive because we could re-estimate the reduced form after a structural change occurs

Can a case be made for structural estimation? First, structural models provide a crucial inductive method to increase our knowledge about economic relationships and to test hypotheses about economic behavior Almost all economic theory is concerned with structural models so that the unresolved questions of economics will usually be set within a structural framework Also, when one considers a reduced form it contains all the current and lagged exogenous and endogenous (or predetermined) variables in the model on the right-hand side We often will not have enough observations to estimate such a model in unrestricted form.6

41t is important to note that while the reduced form follows in a straightforward manner from the structural model in the linear case, usually no simple reduced-form specification exists in the non-linear case

5Transformation from a structural form to a reduced form with regression properties cannot in general be accomplished in the non-linear simultaneous equation model Important differences arise in identification and estimation from the presence of non-linearities

%ince all restrictions arise from the structural model, specification and estimation of reduced-form

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Arguments have been put forth [Liu (1960) and Sims (1980)] that in reality structural models would also contain all endogenous and exogenous variables without identification being possible Thus, they do not escape the problems inherent in reduced-form estimation Models are always at best approximations to reality It seems that whichever approach is taken, either structural specifications

or reduced-form specifications, restrictions on either the structural form or reduced form will be necessary for estimation to be possible.7 Economic theory provides some guidance about restrictions on the structural form which in part can be tested, while we have little theory to guide us about which variables to omit from a reduced-form specification Lastly, while we can always go from the structural form to the reduced form, the reverse transformation is impossible to perform when the unrestricted reduced form is used It is unclear how much we can learn about economic behavior by specification and estimation of a reduced form unless it is derived from a structural form.* This point may have important implications for the analysis of economic policy If a structural parameter were to change, perhaps because policymakers change their behavior, analysis via the structural form may be useful Possible analysis via the unrestricted reduced form seem quite limited here

The last specification, the recursive form, can also be derived from the structural form via a non-singular linear transformation In unrestricted form the recursive form can be estimated by least squares type techniques However, interpretation of the resulting parameter estimates is not straightforward More- over, severe restrictions have to be placed on the distribution of the stochastic disturbances for least squares to yield consistent estimates and for the parameters

to be readily interpretable Without the restrictions the recursive form has little to add to the analysis beyond the structural form and the reduced form Very little research is currently done on recursive-form specifications of simultaneous equa- tion models In previous work Wold (1964) and others have argued that the structural form is naturally recursive so that least square type techniques are appropriate because the structural specification takes the recursive form But the necessary assumptions for the recursive specification are usually beyond accep- tance However, the use of a block recursive specification is sometimes made so that analysis of a large econometric model may be simplified [Fisher (1966, ch 4) and Koopmans (1950)]

7Fisher (1961) considers the effects on identification and estimation when the restrictions are very close to true but not exactly met

*For forecasting purposes we have many unresolved questions about what type of model might be more aunronriate For further discussion, see Fair (Chanter 33 in this Handbook) To the extent the unres&ted*reduced form is used to test the structuri specification, it is an underutilized tool of econometric methodology We discuss such tests in Section 5 Use of the reduced form for tests of dynamic specification may also be quite important

The general specification of a linear structural model is

where Y is the T X A4 matrix of jointly endogenous variables, Z is the T x K

matrix of predetermined variables which contains both exogenous and lagged

endogenous variables, and U is a T X A4 matrix of the structural disturbances of

the system The matrices B and r consist of unknown parameters to be estimated

as well as known values (usually zeros) which arise from a priori economic knowledge We assume that all identities have been substituted out of the system which typically may lead to an arithmetic combination of some of the original economic variables We also assume that all variables are accurately measured because a consideration of errors in variables in simultaneous equation models would lead us too far afield.’

We now consider some assumptions that permit statistical analysis of the structural equation model of eq (2.1)

If contrary to assumption, B were singular, then the model does not provide a

complete theory of the determination-of the endogenous variables Also, eq (2.2) demonstrates that a small structural disturbance could lead to an infinite change

in some of the dependent variables Such an event is contrary to most economic theory

Assumption 2.2

Z has full column rank equal to k

We rule out linear dependence so that the reduced form has a unique interpreta- tion in terms of its unknown coefficients

Assumption 2.3

The rows of U are independent and identically distributed U has mean zero and non-singular covariance matrix 263 IT (Thus, the t th row of U, denoted U,, has mean zero and covariance matrix X.)

‘Some recent work in this area is found in Goldberger (1970), Geraci (1977, 1978), Hausman (1977), and Hsiao (1976) Also, see Aigner, Hsiao, Kapteyn and Wansbeek (Chapter 23 in this Handbook)

399

We assume independence of the structural disturbances across time and allow only for contemporaneous covariance in the case of time-series model specifica- tions Non-independence across time is briefly treated in subsequent sections, but

is left mainly to other chapters in this Handbook which deal with time-series problems For cross-section model specifications we are assuming independence

of the structural disturbances across individuals who would be indexed by t Note that with these three assumptions, all information contained in the structural model is also contained in the reduced-form model In particular, the reduced-form model determines the conditional distribution (on 2) of the endogenous variable since V has mean zero and non-singular covariance matrix 0@1,, where D =

B-“_ZB-‘ We now consider a more precise statistical definition of jointly endogenous and predetermined variables We separate the exogenous variables into two sets, truly exogenous variables R and lagged endogenous variables Ya which occur before the start of the sample period The variables Y, are treated as initial conditions for the analysis that follows which is done conditionally on Y,

We consider the joint probability distribution of U, R, and Y0 which by eq (2.2) determines the joint distribution of Y:

A decomposition of a joint distribution into conditional and marginal distribu- tions is always possible, but the importance of the exogenous variable assumption arises from

Assumption 2.4

Thus, the conditioning information adds no knowledge to the joint distribution of

U which is assumed independent of all fiast, current, and future realizations of the exogenous variables As Assumption 2.4 makes clear, the distribution of exoge- nous variables is independent of the structural population parameters This assumption corresponds to the Koopmans et al (1950, p 56) definition: “[exoge- nous variables] are defined as variables that influence the endogenous variables but are not themselves influenced by the endogenous variables.” In particular note that an implication of eq (2.4) is EU = E(UI R, Y,) = 0

We now turn to the other component of predetermined variables, namely lagged endogenous variables Clearly, the conditioning argument is nonsensical here because knowledge of a realization of a lagged endogenous variable together with the right-hand-side variables from the previous period certainly imparts information about the distribution of the stochastic disturbances in the previous period But an implication of our assumptions is that if we consider the marginal distribution of V,, which corresponds to a row of the U matrix, this marginal

distribution equals the conditional distribution given the knowledge of all past

realization of the endogenous variables

where Y( - ) denotes lagged endogenous variables Lagged endogenous variables can thus be treated along with R and Y, as predetermined at a particular point in time because they are not affected by the realization U, Thus, the assumption of temporal independence allows an important simplification since Y( -) can be solved for in terms of exogenous variables and lagged stochastic disturbances assumed independent of current stochastic disturbances

We have attempted a definition of variables which corresponds to the frame- work of Koopmans (1950) A time-series perspective on these issues is also present in Zellner and Pahn (1975) and Wallis (1977) Further understanding may

be gained from the following considerations Given our assumptions, suppose we want to estimate the model by maximum likelihood By Assumption 2.4 we need

to choose a parametric form for G Suppose for the moment that all prede- termined variables are exogenous with no lags Let t9 denote all unknown parameters We have the joint density:

fWW)=gW~z,~) 1 $ 1 =g,(u,e,lz)[~]g,(z,e,)

The Jacobian of the transformation for the linear case is lBlT which is non-zero by Assumption 2.1 So long as the unknown parameter vector 8 can be separated into two parts so that 8, and 0, are separate from an estimation point of view, then the exogenous variables can be taken as fixed numbers for purposes of estimation of 8, by ML.” Thus, f( Y, Z, 8) can be understood as two consecutive experiments (by nature) The first experiment chooses Z as a function of e, through g,(Z, t9,) This first experiment does not give any information regarding the parameters’ of interest, 8, Given the realized values of Z, the second experiment yields information on 8, only The additional knowledge of g,(Z, 0,)

is irrelevant since all information about 8, arises from the second experiment This setup corresponds to R A Fisher’s (1935, 1956) definition of ancillurity

Here, it is certainly the case that inference on 8, depends on the first experiment since inference is typically done conditional upon its outcome The first experi- ment affects the precision of our inference about 8,, but not the direct inference itself.” Furthermore, as I emphasized in Hausman (1975), it is the presence of the

“Note that inference regarding the parameters may well depend on the distribution g2 given the conditional form of equation (2.6)

“Of course this statement does not imply that the precision about 8, can be improved by further analysis of gz < 0,)

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Jacobian term that distinguishes the jointly endogenous variables in simultaneous equation estimation by ML and removes estimation from the least squares framework

When lagged endogenous variables are included, we shall assume that Y, is fixed in repeated samples The Jacobian of the transformation then becomes

Since _S is symmetric positive definite and B non-singular, at least one such matrix P exists.13 Postmultiplication yields

“Engle et al (1981) take a somewhat different approach to the definition of exogeneity It is important to note that a given variable may be jointly endogenous with respect to one equation in a structural system but predetermined with respect to another equation Examples of such “relative recursivity” are presented in the next section In these cases the factorization of (2.5) need not be for the entire structural system, but instead it is for a given equation being estimated which corresponds

to a subvector of B,

13P can be found by the following method Take the reduced-form system of eq (2.2), Y - ZII = V

Then take the Cholesky factorization 9-l = RR’, where R is a lower triangular matrix Thus,

YR - ZIlR = VR so that var(VR) = I, Then rescale so that 5) = 1 for the conventional normaliza-

C is now lower triangular and var( W) = A2, a diagonal matrix upon choice of renormalization It was once claimed that the recursive form has special ad- vantages in estimation However, in fact it has no particular advantage over either the structural form or reduced forms from which it can be derived Furthermore, the parameter matrices C and D are mixtures of structural slope coefficients and covariance coefficients as can be seen by the construction of P Given our usual almost complete lack of knowledge regarding 2, the recursive-form coefficients are even more difficult to interpret than are the reduced-form coefficients II Of course, if the structural specification took the special form of eq (2.8) without any needed transformation, i.e P is the identity matrix, then special characteristics do occur Basically, the simultaneous equation problem disappears because no spe- cial problems of identification or estimation beyond the usual least squares case occur, as we demonstrate in the next section However, the specification of B as triangular and Z as diagonal seems unacceptable in most model specifications While the recursive form offers few advantages and is now not often used, it does provide a useful reference point for special cases of the structural form We will see this distinction as we now turn to identification of simultaneous equation models

3 Identification

Identification in parametric statistical models has an extremely simple form Suppose we intend to estimate the unknown parameter 0 from eq (2.6) The identification assumption, which is a regularity condition for the consistency of maximum likelihood, states there cannot exist 8 * 8 such that f(Y, 2, 0’) = f(Y, 2, 0) for all Y and Z In the linear simultaneous equation model, where we estimate the conditional form of eq (2.6), the parameter vector of interest

8, = (B, r, A’) The identification assumption then determines whether the a priori

structural assumptions are sufficient to guarantee the uniqueness of (B, r, 2)

Often we may similarly be interested in a subset of the parameters, say B, and r,, the first columns of B and r, which correspond to the parameters of the first equation We would then partition 8, and discuss identification of the subset of parameters in terms of the non-existence of alternative subvectors which lead to the same density function A considerable body of literature has arisen on this topic Koopmans (1949) and Koopmans, Rubin and Leipnik (1950) solved the identification problem for the case of linear restrictions on B and r Fisher (1966) reinterpreted the Koopmans’ conditions and also considered linear restrictions on the elements of E Wegge (1965) also considered covariance restrictions Recently, Hausman and Taylor (1980) have provided convenient sufficient conditions for the case of restrictions on Xl4

14Hsiao (Chapter 4 in this Handbook) also discusses identification of simultaneous equation

We briefly repeat our assumptions of Section 2: B is non-singular, 2 has full

column rank, and the rows of U are ii-d As we mentioned above, since the reduced-form model determines the conditional distribution of the endogenous variables, all the parameters (II, Q) are identified Identification of the structural parameters thus is equivalent to the question of whether any other structural model can have the same reduced form model Define A = [B IJ Then a

structural model is not identified if there exists a non-singular linear transforma-

tion H such that A’ = AH and U’ = UH is i.i.d with mean zero and covariance matrix 2 The possible existence of such a matrix H is closely tied with Fisher’s

(1966) approach to identification via admissible transformations

We first consider the case of a priori restrictions on B and r while 2 is left

unrestricted It is important to note that we use no information which might arise from possible knowledge of the form of the probability distribution G(U) If we

required the matrix H to have the property that G(U) = G(UH), then in many

cases the necessary and sufficient conditions for identification would be very much weaker, except in the case when G( ) is M-variate normal.‘5 We want to

interpret the identification conditions on B and r in instrumental variable form since we emphasize this approach to estimation For the case of restrictions on B

and r we limit attention to the case of linear restrictions These restrictions arise

from a priori economic theory and usually correspond to omission of elements Y

and Z from a given equation We impose the exclusion restrictions and a normalization Bji=l (i=l, , M) and rewrite the equations in regression form

as

yi = Eli + Ziyi + Ui = XiSj + ui (i=l , ,M), (3.1) where X, = [Y Z,], S! = [& $1 and eq (3.1) contains ri jointly endogenous

variables on its right-hand side and si predetermined variables for a total of

ki = ri + si right-hand-side variables In the current case, without loss of general-

ity, we are concerned with the identification of the unknown parameter vector (/?, , y,, u, , ) in the first structural equation Thus, the identification problem is to derive necessary and sufficient conditions so that the equations II = - TB- ’ and

D = (B’)-‘ZB-’ yield a unique solution for [B,, r,, a,,] given (II, 52) and prior

information on [B, r, 21, where B, is the first column of B and r, is the first

when @ is a g X (M + K) matrix of known constants and $I is a known g vector Since at present we take 2 as entirely unknown, the only restrictions which arise from the reduced-form equations on (B,, r,) take the form IIIB, = - r, together with eq (3.2) Therefore (B,, F,) is identified if and only if

has a unique solution for (B,, T,) A necessary and sufficient condition is that

rank II ’

[ @ 1 =hI+Ic (3.4)

Equation (3.4) is the rank condition for identification and is proven in Hood and Koopmans (1953), Fisher (1966), and most econometrics textbooks The necessary order condition is that g z M so that eq (3.3) has at least M + K rows Then for our normalized equation (3.1), we see that (p,, y,) are identified if and only if

has a unique solution for fi, = (- 1, /3,), where II,, is the submatrix of II which relates the endogenous variables included in the first equation ( y,, Y,) with the excluded predetermined variables The order condition is then k, = r, + s, < K or

r, d K - s, That is, the number of included jointly endogenous right-hand-side variables must be no greater in number than the excluded predetermined varia- bles These excluded predetermined variables are used to form the instruments for consistent estimation, as our example in Section 1 demonstrated We discuss instruments further in the next section on estimation Letting W, be the matrix of instruments, the rank condition takes the form that identification is present if and only if (W;X,) is non-singular

How do conditions change if Z, is known? We then have the additional equation to LlB, = -F, that QB, = (B’)-‘1, For identification of the first equation the only useful restriction is u,, = 0, so that the first structural dis- turbance is identically zero We then have the result that (B,, T,) are identified, using only prior restrictions on (B,, r,) and [I,, = 0 if and only if

405

Partition 52 = [a, : f2,] as we did II and we find the necessary condition that rank(S2,)>,(r,-l)‘(K-s,) If the rank of 2 and thus D is M-l, the order condition is fulfilled even without coefficient restrictions This rather peculiar result for the non-stochastic situation arises because if u,, = 0, plim(l/T)Y,‘U, = 0

for i = 1 , , M, so that every jointly endogenous variable is predetermined in the

first equation and can be used as an instrument so long as the rank condition is satisfied The case of variance restrictions, uji = 0, is not of very much importance

in econometric applications

Lastly, we consider covariance restrictions It turns out that covariance restric- tions can yield identification in one of two ways: an otherwise jointly endogenous variable can be made predetermined, somewhat like the last example, or an estimated residual from an otherwise identified equation can serve as an instru- ment We report results from Hausman and Taylor (1980a) where proofs are

given Besides linear restrictions on (B,, r,) and zero restrictions on 2, we also use exclusion (zero) restrictions on certain other elements of B We begin with two

definitions:

Definition

For a G x G matrix B, a chain product corresponding to the ith row and the jth column is a product of no more than G - 1 elements of B of the form &PabflbC p,,, where all indices are distinct The set of all such chain products is denoted B[i, jl’

Equations (i, j) are relatively triangular if and only if Bfi, jl = {O} Equations (i, j) relatively triangular does not imply that equations (j, i) are relatively triangular

It turns out to be the case that the relative triangularity of equations (i, j) is equivalent to a zero in the (i, j)th position of (B’)- ‘ The relative triangularity of equations (i, j) is a necessary condition for yj to be uncorrelated with ui and thus

to be predetermined in the ith equation We now need to consider zero restric-

tions on (B, Z,) which are useful for identification

We have the result that ((B’)-‘2,)j = 0 if and only if equations (j, 1) are relatively triangular and U, is uncorrelated with uk for equations (k, 1) which are not relatively triangular This condition is less restrictive than (I,~ = 0 for i = 2, ,

M We now give our final definition:

Definition

Equations (1, j) are relatively recursive if and only if ((B’)-‘Z,)j = 0

Then yj is uncorrelated with u, (because vj is uncorrelated with ui) and can be considered predetermined in the first equation along the lines of eq (2.4)

Relative recursion occurs if and only if yj is predetermined in the first equation Thus, we have demonstrated our first method by which zero restrictions on

(B,, 2,) are useful for identification They cause the jointly endogenous variable

3 to be predetermined in the first equation so that it can be used to form the mstrument matrix IV, Writing the zero restrictions as !P( B’)- ‘2, = 0 we note that this equation yields WIB, = 0 Then Hausman and Taylor (1980, p 27) prove that a necessary and sufficient condition for the identification of (B,, T,)

using linear restrictions on (B,, r,) and zero restrictions on (B, 2,) is

The necessary order condition takes the form that (B,, T,, El) is identified given

@ and !P so long as the number of unconstrained coefficients k, does not exceed the number of instrumental variables which can include all predetermined vari- ables for the first equation The necessary condition for the instruments is

is uncorrelated.with ui because ((B’)-‘Xi)j = 0 and vj is uncorrelated with ui Therefore 5 is predetermined in the ith equation Two implications arise from this result The rank condition of eq (3.7) is satisfied Furthermore, since all right-hand-side variables are predetermined, least squares on each equation is the appropriate estimator But, as we discussed above, the assumptions necessary for

a recursive system are unacceptable, especially that Z is diagonal If B is specified

to be lower triangular, but no assumptions on _Z are made beyond the usual conditions, the system is called triangular Interesting differences in estimation still arise from the non-triangular specification, but no special issues in identifica- tion arise We consider estimation of triangular systems in the next section [eq (4.20)]

The previous discussion of (relative) recursiveness raises the important question

of when can a variable, say rj, be treated as predetermined in a particular

407

equation? An analogous concern is whether yi can be used as an instrumental variable for estimation As the discussion demonstrated, in the presence of covariance restrictions yi will not necessarily be predetermined with respect to all equations as we assumed the z’s are For r, to be predetermined with respect to

equation i, it is necessary that plim(l/T)$ui = 0 Two important consequences

of this condition are as follow (1) We cannot generally say that yr is pre- determined (exogenous) with respect to another variable y, apart from the specification of the complete system of equations Bivariate relations which attempt to test for “causality” cannot be used to decide whether yi can be used as

an instrumental variable in equation i We discuss this point further in Section 5

(2) For any economic variable to be predetermined its reduced-form residual must

be uncorrelated with ui This event may often seem unlikely in many models But

as a model approximation, it is sometimes assumed that a variable is relatively recursive with respect to an equation because there is no structural feedback from variable yi to the predetermined variable yj But note that covariance restrictions are also necessary, as our definition of relative recursivity demonstrates.16 There- fore it is the relationship of uj and uj which is critical; conditions on both B and Z are required for a variable to be predetermined with respect to another equation The second form of covariance identification occurs when u,~ = 0 but equation

j is otherwise identified Consistent estimates of the residuals iij can then be used for identification We thus need to consider the more general case of linear restrictions on all of (B, r, 2) The proposition is that if u, j = 0 and Bj is either known or estimable, then Yij is predetermined in the first structural equation where ij = (1 - pj) Then either YBj or the residual lij can be used interchange- ably as an instrumental variable in the first equation The rank condition arises from the proposition that if y, is predetermined in the jth equation, then iii cannot be used as an instrument for the first equation Likewise, if y, is predetermined in thejth equation, then tij cannot be used as an instrument for yk

in the first equation Otherwise, estimated residuals can be used to form instru- ments We give an example in the next section

To summarize: four forms of linear restrictions can lead to identification of

(B,, r,, Z,), all of which have straightforward interpretations via instrumental variable interpretation (1) Coefficient restrictions of the Koopmans type which state that the number of included right-hand-side jointly endogenous variables must be no greater in number than the excluded predetermined variables which can serve as instruments (2) The Fisher condition u, , = 0 so that no simultaneous equation problem occurs (3) The Hausman-Taylor condition so that jointly

16Fisher (1966, ch 4) discusses “block recursive” systems where it is assumed that B is block triangular and B is block diagonal Then endogenous variables from a block of B are predetermined with respect to the equations of a higher numbered block because the condition of relative recursive-

endogenous variables become predetermined in the first equation (4) Estimated residuals from another equation can serve as instruments in the first equation.17 This list exhausts all linear identification restrictions for the linear simultaneous model which have been considered in the literature

4 Estimation

Estimation of simultaneous equation models is the central focus of this chapter

We have considered questions of specification and identification We now assume that sufficient restrictions are present for identification and that all restrictions are linear restrictions on the parameters of a given equation (Bi, c) Below, we will consider covariance restrictions on the elements of 2 The early work by Haavelmo (1944) and the Cowles Commission, i.e Koopmans (1950) and Hood and Koopmans (1953), emphasized maximum likelihood estimation of simulta- neous equation models The probability distribution of Assumption 2.4 was taken

to be multivariate normal, G(U) = N( 0, Z@Ir) At that time it was realized [Koopmans et al (1950, section 3)] that consistency of maximum likelihood was maintained even if G(U) was not normal so long as Assumption 2.3 concerning the moments of U was satisfied Still, in the pre-computer age of estimation, maximum likelihood estimation represented a laborious task Not until the two-stage least squares (2SLS) procedure of Basmann (1957) and Theil(1958) was invented did consistent estimation of simultaneous equation models require only the same order of magnitude of computation as least squares for regression specifications 2SLS is an example of instrumental variable (IV) estimation Sargan (1958) introduced instrumental variable estimation for simultaneous equa- tion models It is interesting to note that IV estimation for errors in variables models was known at the time of the Cowles Commission studies, i.e Gear-y (1949) and Reiersol (1945) But application of IV estimation to simultaneous equation models did not occur until after a further decade had passed

4.1 Single equation estimation

First we consider estimation of a single equation, say the demand equation from our initial example, eq (1.1) We denote it as the first equation and rewrite eq (3.1) as

Ch 7: Simultaneow Equation Models

Least squares is inconsistent because

If lagged endogenous variables are present as predetermined variables, all roots

of the characteristic equation l&B + a”- ‘H, + - - + aH,_ , + H, I= 0 lie within the unit circle where Z, are exogenous variables and yt B + gtG + cf=, y,_, H, = ut,

i.e the system is stable

We can now evaluate eq (4.2): ‘*

Instrumental variable estimation provides a consistent estimator for 6, For consistent IV estimation we require a T X K, matrix W, of instruments to estimate

So long as plim T( W{X,)- ’ exists and is finite, we require that plim(l/T)W;u, =

0 That is, the instruments must not be contemporaneously correlated with the stochastic disturbances In general, finite sample bias will not be zero due to the stochastic nature of Y,

Where do instruments arise in the linear simultaneous equation problem? The reduced form of eq (2.2) determines the conditional distribution of Y, Therefore, the predetermined variables Z provide a source of instrumental variables since Y,

is a stochastic linear function of Z,, while the predetermined variables are uncorrelated with U, by assumption Therefore, we consider linear combinations

of the predetermined variables IV, = ZA,, where A, is a K X K, matrix of rank K,

to form the matrix of instruments (Note the distinction between the instruments IV,, which have column rank K,, and the instrumental variables Z, which have

column rank > K, and are used to form IV,.) A, can either be known or estimated

as a, To determine the first-order asymptotic approximation to the distribution

of 8, IV, we consider only the case of Z to be exogenous For lagged endogenous variables, the results are identical but the issue of appropriate central limit theorems in the time-series case arises [see, for example, Mann and Wald (1943) and Fuller (1976)] Then,

The first matrix on the right-hand side of eq (4.6) has plim equal to A;MD,,

which is non-singular where D, = [II, : I,] with I, a selection matrix which chooses Z, The vector (l/@)A;Z’u, ‘forms a sequence of independent and non-identically distributed random variables We can apply either the Liapounoff version of the central limit theorem or the slightly weaker Lindberg-Feller version to claim that the vector converges in distribution to a normal random vector with distribution N(0, a,,A;MA,) [Rao (1973, p 128)].19 Then using the rules on products of random variables where we have a finite plim and the other converges in distribution [Rao (1973, p 122)], we find the asymptotic distribution

JT(4,Iv - 6,) fi N(O,o,,[(A;MD,)-‘A;MA,(D;MA,)-‘I) (4.7)

If A, is replaced by an estimate a, which has plim A,, we obtain identical asymptotic results again because of the product formulae for random variables with limiting distributions

Given the formula for the asymptotic covariance matrix for 8,,Iv, we would like

to find the best choice of A, to form the matrix of instruments W, That is, we

“Assumptions need to be made either about third moments of the random variable or about

want to choose A, to minimize, in a matrix sense, the asymptotic covariance.*’ It turns out that the best 4, is not unique but any optimum choice must satisfy the condition that plim A, = D, We can show an optimum choice is a, = (2’2)) ‘Z’X, For this choice of a, we calculate the asymptotic covariance matrix from eq (4.7):

V(i,,,v) = o,,plim[( X;Z(ZZ-’ MD,)_‘( x;z(z~z)-‘A4(z~z)-‘z~x,)

Factor M = NN’ by the symmetric factorization and define h = N- ‘Z’X,g

Thus, the plim h’h = g’[D;MD,]g, the inner term of which comes from eq (4.8) For the comparison estimator g’[(E;MD,)( E;ME,)- ‘( D;ME,)]g =

plim h’G(G’G)-‘G’h, where G = NE, Therefore the difference of the inverse covariance matrices is

since by the generalized Pythagorean theorem, h’h a h’P,h, where PC = G(G’G)-‘G’, the orthogonal projection That is, PC is an orthogonal projection so all its characteristic roots are either one or zero [Rao (1973, p 72)] Somewhat analogously to the Gauss-Markov theorem, we have shown that among all instrument matrices IV, formed by linear combinations of the predetermined variables, that the best choice is a matrix which has probability limit equal to

D, =[Il, I,] For a, then, IV, = Za, = Z[fi, I,]

*OWhen the terminology “asymptotic covariance matnx ” is used, we mean, more precisely, the covariance matrix of the asymptotic distribution Minimization of the asymptotic covariance matrix means that the matrix difference between it and the asymptotic covariance matrix of comparison estimator is negative semi-definite Equivalently, for any vector P, P’8,,tv has minimum variance in

It is now straightforward to demonstrate that the two-stage least squares estimator (2SLS) is numerically identical to the optimal IV estimator with

2, = (Z’Z))‘Z’X, 2SLS “purges” all jointly endogenous variables in eq (4.1) and replaces them with their conditional expectations estimated from the unre- stricted reduced form: Y, = ZI?, = Z( Z’Z)) ‘Z’Y, = P,Y, The second stage con- sists of replacing the jointly endogenous variables in eq (4.1):

1

where v, = Y, - Y, and is orthogonal to Y, and Z, Then least squares is done:

(4.12) 2SLS is thus identical to the IV estimator using A^, because (&Xl) = (X;P,P,X,)

= (RX,) = (a;Z’Xl).21 The two estimators remain identical only so long as all

predetermined variables are used to form 2, In fact, in purging Y, to form Y,, Z, must be included among the regressors If not, v, is not necessarily orthogonal to

Z, and inconsistency may result This mistake has been made by numerous researchers However, if the IV estimator of eq (4.4) is used, this problem does not occur since W, is a linear combination of predetermined variables Thus, the

IV estimator will continue to be consistent for any matrix A,, but will have a larger asymptotic covariance than the IV estimator with W, = ZD, Also, con- sistent IV estimation is computable for the case K > T > K, As we previously discussed in Section 2, in many cases the number of predetermined variables K will exceed T, the number of observations 2SLS is no longer computable because the unrestricted reduced-form estimates are not computable But IV estimation can still be used in this case.22

Another member of the IV estimator class for single equation models is the limited information maximum likelihood (LIML) estimator which arises when a normality assumption is made for the distribution G(U) We will defer considera- tion of LIML until we discuss maximum likelihood estimation However, a related class of estimators to LIML, referred to as k-class estimators, deserve a brief mention [Theil(1961)] They can also be derived as IV estimators, and can

be considered analogous to the 2SLS estimators Define the instrument W, = [Y, Z,], where we now use Y, = Y, - ICY, for K, a scalar Clearly 2SLS has K = 1

*‘Another possible interpretation of 2SLS is as a minimum distance estimator where min,,(y, - X,S,)‘Z(Z’Z)-‘Z’(y, - X,6,) We thus project the model into the subspace spanned by the columns of the predetermined variables to obtain orthogonality

**An extensive literature exists on which IV estimator should be used when T > K The problem is difficult to resolve because it is small sample in nature, while our current optimality conditions depend

on asymptotic approximations as 7’ grows large Swamy (1980) reviews the ‘I undersized sample”

pbmfi( h., - &SLS> = Qi'ph ‘m(-+[Y;(I-~z)(l-K) O]ui), (4.13) where Q,, = plim( l/T) $2, The term containing K can be written as

plim(D(1 - K))( + f;Ut) = plim(\/T(l- K))((B’)-‘Zt),,

which yields the requirement plim J~;(K - 1) = 0 for asymptotic equivalence Thus, to be an IV estimator we need plim(rc - 1) = 0; to be an optimal IV estimator, we need plim \/?;(K - 1) = 0 Nagar (1959), Sawa (1973) and Fuller (1977) have considered the choice of K with the Nagar K = 1 + (K - k,)/T sometimes used because it eliminates bias in the first term of the asymptotic expansion

4.2 System estimation

We now consider estimation of the entire system of equations rather than only a single equation Under correct specification of the other equations, estimates of the coefficients of the first equation will then have a smaller asymptotic covari- ante matrix so long as Z, has u,~ f 0 for some j * 1 and the jth equation is overidentified The term “overidentification” refers to the case where there are more than enough restrictions so that the rank conditions of Section 2 are satisfied even if one or more prior restrictions are disregarded.23 Again we will only consider linear restrictions for a single equation so that a necessary condition for equationj to be overidentified is that the number of right-hand-side variables included (after the normalization) is strictly less than the number of prede- termined variables, 5 + sj = kj < K However, the gain of the reduction in the

23Altemative notations of overidentification exist When only exclusion restrictions are present, in terms of the order condition overidentification can be defined when K > k, = r, + s, [Hood and Koopmans ( 1953, p 139) and Theil(197 1, p 449)] The precise definition of over-identification is that

at least two sets of exact identifying restrictions exist which are not identical and the deletion of any restriction loses identification if they are the only restrictions Overidentification can also be defined

in terms of restrictions on the reduced form [Malinvaud (1970, p 663)] However, Mahnvaud’s definition can lead to problems in that an equation defined to be overidentified may not, in fact, be

asymptotic covariance matrix brings with it an increased risk of inconsistent estimation We emphasized in single equation estimation that IV estimators required only that the two properties of instruments W, be satisfied for consistent estimation With system estimation misspecification of any equation in the system will generally lead to inconsistent estimation of all equations in the system.24

In discussing system estimation, it is often easier to stack the system of - _ equations Each equation has the form

The general system IV estimator may be defined as

The particular form that W takes for system estimation is II” = -%‘(~‘e1,)-‘, where 8= diag(X,, X2 , , X,,,) and L? is a consistent estimate of 1 Note that each Xi = ZA, ‘must satisfy the two properties for instruments as well as the additional property that plim(l/T)X~uj = 0 for all i and j, i.e Xi must be asymptotically uncorrelated with all the disturbance vectors in the system instead

of only ui as in single equation estimation If 2 is replaced by another matrix with plim not equal to Z, consistent estimation still follows, but a larger asymptotic covariance matrix for the estimator results Derivation of the asymptotic distribu- tion is very similar to the single equation case:

415

The first matrix on the right-hand side of eq (4.17) has plim equal to A’Nfi, where a = diag( A ,, , A,), N=,Y’@M, and fi=diag(D ,, , D,) The second term has an asymptotic normal distribution with mean zero and covariance A’NA Therefore, the asymptotic distribution for the IV estimator is

JT(&v-6) “N(O,[( ~~~)_‘(A’Nk)(B’~~)_‘]) (4.18)

Again, Ai can be replaced by ai which has plim Ai with no change in the limiting distribution

Choice of the optimal matrix 2 follows exactly as in the single equation case and is not repeated here The best choice of A has each Ai satisfy the condition that plim Ai = Di = [Iii Ii] The asymptotic covariance matrix of the optimal system IV estimator is then

(4.19)

We now consider IV estimators which have an optimal Ai:

(1) Three-Stage Least Squares (3SLS) [Zellner and Theil, (1962)] The 3SLS estimator takes ai = (Z’Z)-‘Z’Xi Its estimate of 2 is 3 derived from the residuals of the structural equations estimated by 2SLS For 3SLS W takes the particularly simple form W’ = X’( s- ’ 8 Z( Z’Z)- ‘Z’) Note that if 9 is replaced

by the identity matrix IM we have 2SLS done on each equation The term 3SLS again arises because in the Zellner-Theil formulation yI was replaced by R in each equation and “seemingly unrelated regression” was done on the system.25 Iterated 3SLS has been considered [Dhrymes (1973)] where 3 is updated at each iteration The asymptotic distribution is not changed by the iterative procedure (2) Iterated Instrumental Variables The 3SLS estimator requires T > K just like the 2SLS estimator Brundy and Jorgenson (1971) and Dhrymes (1971) propose

an estimator which only requires that T > ri + si for all i = 1, , M The procedure first estimates bi for each equation by an IV estimator These consistent, but inefficient, estimates are used to form 3, a consistent estimate of Z System instruments are formed with p = _?‘($@Ir)-‘, where Xi = [ - Z(f’&); ’ Z,],

2SAgain differences can arise between the “repeated least squares” form and the IV form The optimal IV estimator requires only a consistent estimator of the 0, However, the repeated least squares form requires an estimate of II, at least as efficient as Iii = P,y In particular, if 2SLS estimates are used to form an estimate of Iii for the “purged” variable y, then a 3SLS-like estimator

no longer has an asymptotic covariance matrix as small as that of the optimal IV estimator Also, if 3SLS is done as a seemingly unrelated regression, the terms which arise from the first stage residuals

where (f%l)i is formed from the consistent estimates bi = [&fi]’ and the prior (zero) restrictions Then drv = (W’X)) ‘w’y This estimator has the identical asymptotic distribution to the optimal IV estimator since plin-(f&l) = II (so long as B is non-singular) However, since T < K is a “small sample” problem, it

is unclear how much the asymptotic argument can be relied on Small sample approximations will be required to evaluate the IV estimators better Also, knowledge about the effect of the initial consistent estimator on the small sample properties of b,, remains to be established

Dhrymes (1971) proposed to iterate the process by replacing (pi-‘) at each iteration and 3 with the new estimates Hausman (1975) demonstrated that if the iterative process converged, then it would yield the maximum likelihood (FIML) estimates as is demonstrated when FIML estimation is considered, so long as

T > K + M Lyttkens (1970) considered iteration with S replaced by I,+, as did Brundy and Jorgenson While this estimator is properly regarded as a full information system estimator, since all equations must be specified in structural form, the asymptotic distribution is the same as the (system) 2SLS estimator (3) System k-class [Srivastava (1971) and Savin (1973)] This estimatpr is a straightforward generalization of the single equation case Replace Y by Y’ = [(I

- IZ)Y’+ &I’Z’] for a matrix R Then a system IV type estimator is used Again consistency requires plim R = I, while asymptotic efficiency requires plim @(I? - I) = 0 The proof of these requirements is the same as in the single equation case

An interesting special case of system estimation arises when the system is triangular [Lahiri and Schmidt (1978)] This specification occurs when B is lower triangular after the a priori restrictions have been applied.26 If 2 is also specified

to be diagonal, we then have the recursive specification [Wold (1964)] All right-hand-side variables are predetermined so least squares on each equation is the optimal estimator But if 2 is not diagonal, least squares is inconsistent If 2 were known, the system could be transformed and generalized least squares (GLS) used With E unknown, it can be estimated and 3SLS provides an optimal estimator in the sense of having an asymptotic distribution identical to the optimal IV estimator so long as the system is identified (apart from restrictions on 2) But a relevant question is whether GLS with a consistently estimated covariance matrix also has an identical asymptotic distribution The answer is no, although the estimator is consistent, because

plimJT( &v - 43~s)

(4.20) 26This specification arises in the path analysis model often used in other social sciences [see

417

While plim(l/T)Z’u = 0, plim(l/T)X’u 1 f 0 so that eq (4.20) would equal zero only if plim@(T - S) = 0 Thus, to apply GLS to a triangular system an efficient estimate of 2 is required or the estimator will have a larger asymptotic covariance matrix than the optimal IV estimator

4.3 Reduced-form estimation

So far we have directed our attention to estimation of the structural parameters (B, r, 2) Estimation of the reduced-form parameters (II, a) can be important to derive forecasts or for optimal control procedures If the entire system is just identified so that the appropriate rank conditions of eqs (3.4) and (3.6) are equalities, then II is unrestricted This result follows from the first block of eq (3.3), IIB, + rl = 0, which can be uniquely solved for (B,, r,) for any Il.27 On the other hand, when any structural equation is overidentified, then 27 is subject

to restrictions In the case of linear equation restrictions, the total number of restrictions on II, called the degree of system overidentification, is determined by the expression &( K - /c~).~*

Again let us stack the reduced-form equation (2.2) into a MT X 1 vector system:

where r’ = (r, ,, r2,, ,rKM) and v’ = (v, ,, v2,, , vTM) Note that V(v) = OS I,

so that eq (4.21) represents a multivariate least squares problem If rr is unre- stricted, then the GLS estimator and the OLS estimator are identical

In the case of over-identification with restrictions on r, then non-linear GLS is used with an estimated b [Zellner (1962)] This approach is very close to the minimum distance methods of Malinvaud (1970) and Rothenberg (1973, ch 4) Let +ij = - (I’B-‘)ii be subject to the a priori restrictions, while iiij represents the unrestricted OLS estimates Then put these elements into vectors and the

27This result also leads to the structural estimation model of indirect least squares (ILS) for just identified systems Estimates of the reduced-form parameters, II, are used to solve for (B,, I’,) ILS has been superseded by the IV estimators The shortcoming of ILS is its inapplicability to overidenti- fied models A generalization of ILS to this case is proposed by Khazzom (1976)

281f any equation is underidentified so the rank is less than M + K, then this equation adds no

minimum distance estimation is

n

where b = (l/T)y’(l- P,-))y for z= (1@Z) Solution of eq (4.23) represents a non-linear problem because of the restrictions on ii.29

Other estimators of II are possible An obvious suggestion would be to estimate

II from the structural parameter estimates, I?,, = - fr;,s, = h(arv) Since h is differentiable (totally) if pn, and Brv are optimal system IV estimators, e.g 3SLS,

we might expect fir, to have good asymptotic properties [Rao (1973, ch 6)] The asymptotic covariance matrix for an efficient system IV estimator, e.g 3SLS, is30

In fact, the & from eq (4.23) and fir, are asymptotically equivalent, plim fi( & - fitv) = 0 [Rothenberg (1973, ch 4)] This result follows because both estimators are asymptotically equivalent to FIML which we will examine momentarily However, if non-optimal system IV estimators are used to form I?, then no optimal asymptotic results hold Dhrymes (1973b) demonstrated that fi2sLs is not necessarily better than unrestricted least squares estimation of II, even though the overidentifying restrictions have been imposed

4.4 Maximum likelihood estimation

ML estimation formed the original approach to estimation of linear simultaneous equation models by Haavelmo (1944) and Koopmans et al (1950) The likelihood function follows from an additional assumption:

Assumption 4.3

The structural disturbances follow a non-singular normal distribution U - N(0, Z

@‘IT)

29Brown (1960) considered setting D = I which leads to a loss of asymptotic efficiency

301f an efficient single equation estimator is used, e.g 2SLS, the asymptotic covariance matrix is

Note this covariance matrix is larger than that of eq (4.23a) unless all equations are just identified A

det( B) = 1, and we have already seen how this case represents a special situation

[(B')-'2 -(l/T)Y’U)]” = 0 We know that the plim of this equation must be zero for ML to be consistent In fact, the plim is zero by our earlier calculation that plim(l/T)Y’U= (B')-'3 Therefore, it is the presence of the Jacobian term det(B) in the likelihood function which “corrects” for the correlation of the jointly endogenous variables and the structural disturbances which is the essential feature of the simultaneous equation specification Hausman (1975) combines eqs

Zr)X']" = 0 to derive the first-order conditions with respect to the unknown

elements of B and r:

(4.26)