THE SIMULTANEOUS EQUATIONS MODEL

THE SIMULTANEOUS EQUATIONS MODEL

CHAPTER 25

The simultaneous equations model

25.1 Introduction

The simultaneous equations model was first proposed by Haavelmo (1943)

aS a way to provide a Statistical framework in the context of which a theoretical model which comprises a system of simultaneous interdependent equations can be analysed His suggestion was to provide the basis of a research programme undertaken by the Cowles Foundation during the late 1940s and early 50s Their results, published in two monographs (Koopmans (1950), Hood and Koopmans (1953)), dominated the econometric research agenda for the next three decades

In order to motivate the simultaneous equations formulation let us consider the following theoretical model:

Ip =H + 422M, + % 32), + Lary, (25.2) where ?m,, í, p, 1„ ø, refer to (the log„ of) the theoretical variables, money, interest rate, price level, income and government budget deficit, respectively For expositional purposes let us assume that there exist observed data series which correspond one-to-one to these theoretical variables That is, (1)-(2} is also an estimable model (see Chapter 1) The question which naturally arises is to what extent the estimable model (1)-(2) can be statistically analysed in the context of the multivariate linear regression model discussed in Chapter 24 A moment's reflection suggests that the presence of the so-called endogenous variables i, and m, on the RHS

of (1) and (2), respectively, raises new problems The alternative

608

25.1 Introduction 609 formulation:

i, = By 2+ BrP, + Bs2¥ t+ Bards (25.4) can be analysed in the context of the multivariate linear regression model because the f;s, i=1,2, i=1,2,3,4, j=1,2, are directly related to the statistical parameters B of the multivariate linear regression model (see Chapter 24)

This suggests that if we could find a way to relate the two parametrisations in (1)}-(2) and (3}{(4) we could interpret the theoretical parameters o;,S as a reparametrisation of the £;,s In what follows it is argued that this is possible as long as the a,;s can be uniquely defined in terms of the f;;s The way this is achieved is by ‘reparametrising’ (3)-{4) first into the formulation:

ip = Aya + Ug 9M, +39, + Ag2P, +4529) (25.6)

and then derive the a,;s by imposing restrictions on the a,,s such as as, =0, a3,=0 In view of this it is important to emphasise at the outset that the simultaneous equations formulation should be interpreted as a theoretical parametrisation of particular interest in econometric modelling because it

‘models’ the co-determination of behaviour, and not as a statistical model

In Section 25.2 the relationship between the multivariate linear regression and the simultaneous equation formulation is explicitly derived

in an attempt to introduce the problem of reparametrisation and overparametrisation The latter problem raises the issue of identification which is considered in Section 25.3 The specification of the simultaneous equation model as an extension of the multivariate linear regression model where the statistical parameters of interest do not coincide with the

theoretical (structural) parameters of interest is discussed in Section 25.4

The estimation of the theoretical parameters of interest by the method of maximum likelihood is considered in Section 25.5 Section 25.6 considers two least-squares estimators in an attempt to enhance our understanding of the problem of simultaneity and its implications These estimators are related to the instrumental variables method in Section 25.7 In Section 25.8

we consider misspecification testing at three different but interrelated levels

Section 25,9 discusses the issues of specification testing and model selection

In Section 25.10 the problem of prediction is briefly discussed

It is important to note at the outset that even though the dynamic

610 The simultaneous equations model

simultaneous equations model is not explicitly considered the results which follow can be extended to the more general case in the same way as in the context of the multivariate linear regression model (see Chapter 24) In particular, if we interpret X, as including all the predetermined variables,

25.2 The multivariate linear regression and simultaneous equations

models

‘The multivariate linear regression model discussed in Chapter 24 was based

‘on the statistical GM:

with 0=(B,Q), B=Xj/E,,, Q=2,,-X,,X7/L,,, the statistical parameters of interest As argued in Section 25.1, for certain estimable models in econometric modelling the theoretical parameters of interest do

not coincide with @ and we need to reparametrise (7) so as to accommodate such models

In order to motivate the reparametrisation needed to accommodate estimable models such as (1}{2) let us separate y,, from the other

endogenous variables yj") and decompose B and Q conformably in an obvious notation:

A natural way to get the endogenous variables yŸ into the systematic

component, purporting to explain y,,, is to condition on the o-field

generated by y<"), say a(y!), ie for

Hap EV: FO) STL YY + ALK, (25.10)

where P? = Q3,',, and A, = 8, —B,;,3'w,, The systematic component defined by (10) can be used to construct the statistical GM

25.2 The MLR and simultaneous equations models 61

where

f= Vir —E(y:,/#?)

Note:

(i) E(e,,)= ELE(e,,/F)?)) =9,

(ii) Eley€1.)= ELE £15/F¢)]

are simple functions of B and @ That ¡s, they constitute an alternative

parametrisation of 6, in D(y,/X,; 8), based on the decomposition

D(y,/X,5 0) = Dv y/yts Xs my) DỤC Xe May) (25.15)

Moreover, the normality assumption ensures that y!!) is indeed weakly exogenous with respect to the parameters (a,,v,,) The statistical GM (11) is

a hybrid of the linear and stochastic linear regression models These

comments suggest that the so-called simultaneity problem has nothing to

do with the presence of stochastic variables among the regressors The decomposition (15) holds for any one endogenous variable y,, Diy,/X,; Ø)= D(vu(vị), Xo tị): DU Xu đụ), (25.16) giving rise to a statistical GM:

612 The simultaneous equations model

The problem, however, is that no decomposition of D(y,/X,; 6) exists whict can sustain all m equations i=1,2, , m in (17) That is, the system

where

E=S(H;,I;, ,E„), A =(A/,A;, , A¿),

8, =S(Ete E2 Emy}

(I; is essentially F? with — 1 added as is ¡th element), is not a well-defined statistical GM because 1t consfitutes an over-reparametrisation of (7) For one equation, say the first, the reparametrisation

n, =(19,Ay,0;,) and 4a) = (Bi), Q22)

is well defined but

m

MXM XX ty = X 1

i=1

is not

A particular case where the cartesian product X?_, 4; is a proper

reparametrisation of 0 is when there exists a natural ordering of the y,,s such that },, depends only on the y,,s up to j—1, ie

E(¥_/ FO) = EV p/n f= 1,2, ,5- DX, =x), f= 1,2, ,m

(25.19)

In this case the distribution D(y,/X,; 8) can be decomposed in the form

D(y,/X5 = [] Dini Vie Var Vint Xe Hộ): (25.20)

i=1

This decomposition gives rise to a lower triangular matrix and the system

(18) is then a well-defined statistical GM known as a recursive system (see below)

In the non-recursive case the parametrisation given in (18) is defined in terms of n=m(m—1)+mk+4(m+ 1) unknown parameters 4=(T, A, V)

where V= E(e,¢,), and there are only mk +4m(m-+ 1) well-defined statistical

parameters in 6; a shortfall of m(m— 1) parameters Given that 4 constitutes

a reparametrisation of @ there can only be mk+4m(m+ 1) well-defined parameters in y In order to see the relationship between Ø and y let us premultiply (18) by (I”)~! (assumed to be non-singular):

If we compare (21) with (7) we deduce that @ and y are related via

25.2 The MLR and simultaneous equations models 613

The first thing to note about this system of equations is that they are not a priori restrictions of the form considered in Chapter 24 where F and A are assumed known The system of equations in (22) ‘defines’ the parameters y

in terms of @ As it stands, however, the system allows an infinity of

solutions for y given 6

The parameters 0 and y will be referred to as statistical and structural parameters, respectively The system of equations (22) enables us to determine only a subset 9, of (y= (4, :72)) for any given set of well-defined statistical parameters @ That is, (22) can be solved for mk+4m(m+ 1) structural parameters y, in the form

Hence, we need additional information to determine y, elsewhere

Note that 9, 1s a m(m— 1) x 1 vector of structural parameters Without any additional information the structural parameters 4 are said to be not identified The problem of identifying the structural parameters of interest using additional information will be considered formally in the next section

In the meantime 1t is sufficient to note that, if we supplement the system (22) with m(m— 1) additional independent restrictions, then a unique solution (implicit or explicit),

exists for the structural parameters of interest = L(y)

There are several things worth summarising in the above discussion Firstly, given that the structural formulation (18) is a reparametrisation of the statistical GM for the multivariate linear regression model, the structural parameters of interest € (when identified) are no more well defined than the statistical parameters 9 This suggests that before any question about £ can be asked we need to ensure that @ is well defined (no misspecification test has shown any departures from the assumptions underlying the multivariate linear regression model) Secondly, (18) does

not constitute a well-defined statistical GM unless I is a triangular matrix;

the system of equations ts recursive This is because although each equation separately 1s well defined the system as a whole is not because of overparametrisation In the case ofa recursive system I is lower triangular

and V is diagonal with

det(Z,)

Ð, being the /th leading diagonal matrix This implies that X7_, 9;

614 The simultaneous equations model

constitutes a proper reparametrisation of @ with mk +4m(m-+ 1) structural

parameters in (I, A, V) Using the notation y?_.,=(Vjy Var «++ s Vous)’ We

can express the recursive system in the form

Vn = eye 4, +6:X,+6,, i=1,2, ,m, teT

We can estimate the structural parameters 4,=(y?, 6,, v,,) by

7? _ Yj-rVY ¡ VƑJX\” (VY? ay;

25.27

¡=> (y,—Y?_¡#?— Xô, (y,— Y?- ¡$? — Xô,) T i= 1,2, ,m (25.28)

It can be verified that these are indeed the MLE’s of tụ

25.3 Identification using linear homogeneous restrictions

As argued in the previous section, the reparametrisation of the statistical

No unique solution for q exists, corresponding to any set of well-defined

statistical parameters 0, unless the system of equations (31) is supplemented

with some additional a priori restrictions

In order to simplify the discussion of the identification problem let us

assume that the system V=I’QF determines V for given T and Q and

concentrate on the determination of T and A The system of equations

BI'+A=0 written in the form

Note that A* denotes the restricted structural coefficient parameters

Hence, the structural formulation (30) is said to be identified if and only if

we can supplement it with at least m(m— 1) additional restrictions More general restrictions as well as covariance restrictions are beyond the scope

of the present book (see Hsiao (1983) inter alia)

The identification problem in econometrics is usually tackled not in terms of the system (30) as a whole but equation by equation using a particular form of linear homogeneous restrictions, the so-called exclusion

(or zero-one) restrictions In order to motivate the problem let us consider the two equation estimable model introduced in Section 25.1 above The

616 The simultaneous equations model

unrestricted structural form (30) (compare with (5)-{6)) of this model is

M, = Yori, +511 +5219, +031P, +0419 + Eis (25.39) i= 712M, +012 +022), +0327, +0429, + € ny (25.40)

Ascan be seen, the two equations are indistinguishable given that they differ only by a normalisation condition The overparametrisation arises because the statistical GM underlying (39) and (40) is in effect

m= By, + Bary + B31 Pit Bar Git Uae (2541)

and the two parametrisations B and (I, A) are related via

Ba Baz Oat 042 ‘0 0:

which cannot be solved for Iand A uniquely, given that there are only eight

B,;8 and ten 7,;8 and 0,8

A natural way to ‘solve’ the identification problem is to impose exclusion restrictions on (39) and (40) such as 64, =0, d,,=0 These restrictions enable us to distinguish between the money and interest rate equations

Equivalently, the restrictions enable us to get a unique solution for (y, 2,721;

611,912, O13, Oya, O21, 031) given (B;;,i=1, ,4,7=1,2)

The exclusion restrictions on the ith structural equation can be expressed

in the form

where ®, is a ‘selector’ matrix of zeros and ones and ¢,; is the ith column of A

In the above example the selector matrices are

In the special case of exclusion restrictions the order condition can be

made even easier to apply Let us consider the first equation of the system

25.3 Identification 617 (31):

and impose (m—m,)+(k—k,) exclusion restrictions; omit (m—m,) endogenous and (k —k,) exogenous variables from the first equation In this case we do not need to define ®, explicitly and consider (46) because we can substitute the restrictions directly into (48) Re-arranging the variables so as

to have the excluded ones last, the restricted structural parameters † and A# are

ồ

I?=|?, Ì w-(§) where y,:(m,—1)x 1, 6y:k, x 1

(25.49) Partitioning B conformably the system (48) under the exclusion restrictions

Bi: kị XI, Bị;: ki; xứữm —l), By3: ky x (m—my),

B;¡:(k—kj)xXI, B;;:(k—kj)x(m, — 1), Bạy: (k—ki) x (m—m)) Determining 6, from (51) presents no problems if y, is uniquely determined

in (52) For this to be the case we need the condition that

In view of the result that the rank(B,,)=min(k —k,,m, — 1) we can deduce that a necessury condition for identification under exclusion restrictions is that

618 The simultaneous equations model

This is known as the order condition for identification which is necessary but not sufficient This can be easily seen in the example (39), (40), above when the exclusion restrictions are 04, =0, 6,4=0 The selector matrices for the two equations are ®, =(0, 0, 0,0, 0, 1), B, = (0, 0, 0, 0, 0, 1) Clearly rank(®,) = rank(@®,)= 1 and thus the order condition 1s satisfied but the rank condition (47) fails because

This is because when g, is excluded from both equations the restriction is

‘phoney’ Such a situation arises when:

(1) all equations satisfy the same restriction; and

(1) some other equation satisfies all the restrictions of the ith equation

(see example below)

Let us introduce some nomenclature related to the identification of particular equations in the system (30)

Example

Consider the following structural form with the exclusion restrictions imposed:

25.4 Specification 619

0 1000 œ= " [0 010 0], (2° ` U \0 000 1 °° 17

0 0001 ,=(0, 1,0, 0, 0)

The first equation is overidentified since rank(@, A*)=2 but rank(®,)=3

The second equation is underidentified because rank(@,A*)=1 even

though rank(@®,)=2 (the order condition holds) This is because the first

equation satisfied all the restrictions of the second equation rendering these

conditions ‘phoney’ The third equation is underidentified as well, because

rank(®;A*)=1<2

It is important to note that when certain equations of the structural form (30) are overidentified then not only the structural parameters of interest are uniquely defined by (31) but the statistical parameters @ are themselves restricted That is, overidentifying restrictions imply that 6¢ @, where O, is

a subset of the parameter space © An important implication of this is that the identifying restrictions cannot be tested but the overidentifying ones

can (see Section 25.9)

The above discussion of the identification problem depends crucially on the assumption that the statistical parameters of interest @=(B, Q) are well defined; the assumptions [1]-[8] underlying the multivariate linear

regression model are valid However, in the case where some assumption is invalid and the parametrisation changes we need to reconsider the

identification of the system For example, in the case where the

independence assumption is invalid and the statistical GM takes the form

y¥,=Box,+ ¥ Aiy,-it ¥ Bix,-;+u,, “ r>i (25.63)

¡=1

t=

the identification problem has to be related to the statistical parameters

6* =(A,, , A), By B, , B,, 24), not 6 Hence, the identification of the system is not just a matter of a priori information only

25.4 Specification

The simultaneous equation formulation as a statistical model is viewed as a

620 The simultaneous equations model

reparametrisation of the multivariate linear regression model where the theoretical (structural) parameters of interests € do not coincide with the Statistical parameters of interest @ In particular the statistical model is specified as follows:

DD Statistical GM

[1] H, = È(y,/X,=x,) and u,=,— E(y,X,=x,) are the systematic and

non-systematic components, respectively

[2] () 0=(B, ©) are the statistical parameters of Interest where

B=27722,;, Q=L,,—L,2277E2,,06O=R™xC,,; (ii) €=L(T,A, V) are the theoretical parameters of interest

where [ =H,(0), A=H,(0) V =H, (6)

[3] X, 1s assumed to be weakly exogenous with respect to Ø (and &) [4] The theoretical parameters of interest €=H(0), €€E, are identified

[5] rank(X)=k where X =(x,, x,, , x7): T xk for T>k

(di) Probability model

@®= [Diu 0) “TS exp! —Hy, — B’x,)'Q” '(y, -—B’x,)},

6eR™x Cte if

(25.65)

[6] (1) D(y,/X,; 8) is normal;

(it) E(y,/X, = ,) = B’x, - linear in x,;

(til Cov(y,/X; = x,) = Q — homoskedastic (free of x,);

[7] 0=(B,Q) is time invariant

(UI) Sampling model

[8] y=(¥,.¥2.- ¥,) 1s an independent sample sequentially drawn

from Dty,/X,; 0), f= 1, 2, T, respectively

The above specification suggests most clearly that before we can proceed to consider the theoretical parameters of interest € we need to ensure that the Statistical parameters of interest @ ure well defined That is, the misspecification testing discussed in Chapter 24 precedes any discussion of

either identification or statistical inference related to & Testing departures from multivariate normality, linearity, homoskedasticity, time invariance

25.55 Maximum likelihood estimation 621

and independence became of paramount importance in econometric modelling in the context of simultaneous equations

25.5 Maximum likelihood estimation

In view of the simultaneous equations model specification considered in the previous section and the discussion of the identification problem in Section 25.3 we can deduce that in the case where the theoretical (structural) parameters of interest & are just-identified the mapping H(-}: R™ x C, > =, where €=H(@), is one-to-one and onto This implies that the reparametrisation is invertible, @=H_~ '(€) and an obvious estimator of & is the indirect maximum likelihood estimator (MLE)

A more explicit formula for the IMLE 1s given in the case of one equation, say the first, when just-identification 1s achieved by linear homogeneous restrictions It is defined as the solution of

In the simpler case of exclusion restrictions the system (66) is

and the FMLE take the explicit form

given that B,, is a square (m, — 1)x (m, — 1) non-singular matrix The [MLE can be viewed as an unconstrained MLE of the theoretical parameters of interest which might provide the ‘benchmark’ for testing any

622 The simultaneous equations model

overidentifying restrictions As argued above, the identifying restriction are not testable because they provide the reparametrisation from @ to & bu: the overidentifying restrictions are testable because they imply restrictions for 6, the statistical parameters of interest

In order to simplify the derivation of general MLE’s of é let us utilise the

formulae derived in the context of constrained MLE’s of @ under a prior:

linear restrictions in Chapter 24

The constrained MLE’s of B and Q, in the context of the multivariate linear regression model, subject to the linear a priori restrictions

a=(4) and Z=(Y,X)

(see Hendry and Richard (1983)) Substituting (76) and (77) into the log likelihood function of the multivariate linear regression model log L(0; Y)

we get the ‘concentrated’ likelihood function log L(é; Y) defined by

log L(é; Y)=const 5 log(det Qa

~5log(det[(f'Y'M,YT) ~1(A'Z/ZA)]) (25.78)

This function can be viewed as the objective function for estimating & using

6 =(B, Q) as opposed to the direct estimation of € by solving (76) and (77) for

€ The log likelihood function (78) has the advantage that it provides us with

25.5 Maximum likelihood estimation 623

a natural objective function to ‘solve’ (76) and (77) In the case of general restrictions € = H(@) the ‘solution’ is rather prohibitive and thus we consider the case where the restrictions are exclusion restrictions

When the identification of € is achieved by exclusion restrictions the constrained structural parameter matrices I'(é) and A(@) are linear in This implies that the first-order conditions can be derived explicitly The conditions

The system of equations (82) could be used to derive the MLE’s of € and

hence f', A, B and Q An asymptotically equivalent form can be derived

using Q for Q in view of the fact that (Q—Q) 4 0 and

624 The simultaneous equations model

argued in Chapter 24, they are functions of the minimal sufficient statistics

Moreover, the orthogonality between the systematic and non-systematic components, preserved with their estimated counterparts by the MLE’s (discussed in Chapters 19 and 23) holds asymptotically for (84) In order to see this consider the systematic and non-systematic components related to the system of equations (73) defined by

B

k,

and ¢,, where ¢,= —A‘Z,, Z,=(y,, Xj)’ (25.88)

It can be shown (see Hendry (1976)) that

by Hendry the numerical optimisation rule for deriving the MLE’s é of € from (84) must be distinguished from the alternative approximate solutions

25.5 Maximum likelihood estimation 625

equation (EGE) The usefulness of the EGE is that it unifies and summarises

a huge literature in econometrics by showing that:

(a) Every solution is an approximation to the maximum likelihood estimator

obtained by variations in the ‘initial values’ selected for V and Mand in the number of steps taken through the cycle (90)}H1{92), including not iterating (b) All known econometric estimators for linear dynamic systems can be

obtained in this way, which provides a systematic approach to estimation

theory in an area where a very large number of methods have been

proposed

(c) Equation (90) classifies methods immediately into distinct groups of

varying asymptotic efficiency as follows:

(i) as efficient asymptotically as the maximum likelihood A if (V, I)

are estimated consistently;

(1) consistent for A if any convergent estimator is used for (V, IT); (ili) asymptotically as efficient as can be obtained for a single

equation out ofa system if V=I but IT is consistently estimated

(see Hendry and Richard (1983))

Definition 4

In the case where V is mx m and non-singular the MLE of 6, called the full information maximum likelihood (FI ML) estimator, is the solution of the system (84) for &

It is interesting to note that in this case the system of equations (76) determining & takes the simple form

when the theoretical parameters of interest & are just identified then the statistical parameters are not constrained, i.e B=B and Q=Q and thus (93) reduces to the indirect maximum likelihood estimator

(see Chapter 13), where E(-) is with respect to the underlying probability

model distribution Diy,’X,; 0) Using the asymptotic information matrix

626 The simultaneous equations model

defined by

1„ (ÿ)= lim (+ 149) (25.97)

Toys

we can deduce that

A more explicit form of I,,() will be given in the next section in the case

where only exclusion restrictions exist, for the 3SLS (three-stage least-

squares) estimator

One important feature of the above derivation of (84) and (90)-(92) is that

it does not depend on IT being m x m and non-singular These results hold true with P being m x g (g <m) In this case the structural formulation is said

to be incomplete (see Richard (1984)) For T mx m, (m, <m), ‘solving’ (84) gives rise to a direct sub-system generalisation of the limited information maximum likelihood (LIML) estimator (with m, = 1) (see Richard (1979))

25.6 Least-squares estimation

As argued in the previous section most of the conventional estimators of the structural parameters € can be profitably viewed as_ particular approximations of the FIML estimator Instead of deriving such estimators using the estimator generating equation (EGE) discussed above (see Hendry (1976) for a comprehensive discussion) we will consider the derivation of two of the most widely used (and discussed) least-squares estimators, the two-stage (2SLS) and three-stage least-squares (3SLS)

estimators, in order to illustrate some of the issues raised in the previous

sections In particular, the role of weak exogeneity and a priori restrictions

in the reparametrisation

(1) Two-stage least-squares (2SLS)

2SLS is by far the most widely used method of estimation for the structural parameters of interest in a particular equation For expositional purposes let us consider the first structural equation of the system

As argued in Section 25.2, equation (100) constitutes a proper statistical

25.6 Least-squares estimation 627

GM based on the following decomposition of the probability model:

Diy,/X,5 0) = DOvyi/¥e"! Xs my) Diy?/X,; m1) (25.101) with systematic component

where Z,,,=(Y'',X) This estimator has the same properties in the estimator of B* in the stochastic linear regression model given that (100) isa hybrid of the linear and stochastic linear regression models Moreover, the variance v,, can be estimated by

1

A AP A

The optimality of the estimators (104) and (105) arises because of the orthogonality between the systematic and non-systematic components

Eyer) = EVEL ere Oe} = EU Ele /olye 3 =O (25.106)

since E[e,,/a(y!’))] =0 Note that the expectation operator in E(3,£¡,) i5

defined relative to D(y,/X,; 6), the distribution underlying the probability model

The above scenario, however, changes drastically by the imposition of a priori restrictions on 4, In order to see this let us consider the simple case of exclusion restrictions Let us rearrange the vectors y, and x, so as to get the

m, and k, included endogenous (y,,) and exogenous (x,,) variables first, 1.e Y,=Ú Vier Yee) Xr = (Xie Xap)” In terms of the decomposition in (101)

this rearrangement suggests that

D(y,/X;; b= Dvir Yip Yoiye> X,; 1)

‘Dit Von Xe ni): DWuyXx Hã )- (25.107)

628 The simultaneous equations model

In terms of this decomposition the statistical GM (100) becomes

Vie =Vi¥ae FM Vege FOX $8) Xe +8 (25.108)

(108) takes the restricted structural form

Let us consider the implications of these restrictions for the rest of the system via

Given that the restricted coefficient vectors take the form

I†ƒ=(—l.y/.0 AƑ=lð,.UV, (25.112) (111) is

In the case where the equation (110) is just identified (k —k, =m, — 1) B,, 1s (m, — 1) x (m, — 1) square matrix with rank m, — 1 Hence, (114) and (115) can be solved uniquely for y, and 6,:

Looking at (116) we can see that in this case 4, and q,,) are still variation free Moreover, no structural parameter estimator is needed because these

can be estimated via the indirect MLE’s:

25.6 Least-squares estimation 629

However, when (k—k,)>m,—1, the first equation is overidentified, the equations in (114), (115), impose restrictions on B,,, and thus the variation free condition between 4, and f0; 1s invalidated In an attempt to enhance our understanding of this condition let us return to the decomposition in (107) and define the structural parameters involved

844) = B21 — Bay, —Boatay Ur S11 —@1271 — Mra

An alternative way to view the above argument is in terms of the

systematic and non-systematic components If we define yz}, in (110) by

ut, = Ely, OV 1) X= X44) (25.124) then

where E(-) is defined in terms of D(y,/X,; 0) However, it can be verified

directly that under the restrictions (114) and (115)

630 The simultaneous equations model

where

en=Vit — E(yy,/o¥ 44), X1,=X1,)

(see Spanos (1985d)) This suggests that the natural way to proceed in order

to estimate (y,,6,,,,) is to find a way to impose the restrictions (114)-(115) and then construct an estimator which preserves the orthogonality between

ut, and ef The LIML estimator briefly considered in the previous section is indeed such an estimator (see Theil (197 1)) Another estimator based on the same argument is the so-called two-stage least-squares (2SLS) estimator Let us consider its derivation in some detail

The orthogonality in (126) suggests that

is equivalent to

subject to (114) and (115) In order to see this let us substitute

Vir = (By 2X1, + Bo oxy + Uy.) +0, x1, + ef, (25.130)

= (7, By +61)X 1, +6) By 2X)1), + 7 Uy, + Fy (25.131)

Given ef, = —T#'u,=u,,—yu,,, (130) becomes

Yi = (71 By +8)X 1 +9 Boo X iy, Ha (25.132) The LIML estimator can be viewed as a constrained MLE of B,, and B,, in (128) subject to the restrictions (114)-(115) as shown in (132) (see Theil (1971)) Similarly, the 2SLS estimator of a* =(y’, 5’) can be interpreted as

achieving the same effect by a two-stage procedure The method is based on

a re-arranged formulation of (130) for t=1,2, , T:

in an attempt to impose (114) and (115) in stage one by estimating (X,B,.+ X,,)B,,) separately using (129) Once the restrictions are imposed the next step is to construct an estimator of a* which preserves the orthogonality between the systematic and non-systematic component of (133) More explicitly:

Stage one: using the formulation provided by (133) estimate

25.6 Least-squares estimation 631

in the context of (129) or

Stage two: using Y, =XB., and the equality Y,;=Y, +U,, where

(2) -Ívy VI pm (25 140)

OrJosts AXIY, XÍX:? UX,

In the context of the model (1}12) estimating the parameters of (1) by 2SLS amounts to estimating the statistical form in equation (4):

i,= Boy + Boop, + Bas¥i+ BaaGi + Yar

by OLS and substitute the fitted values i, =, + Bp, + Br3V,+ B29, into the structural form of (1) to get:

The 2SLS estimator can be compared directly with the LIML estimator

(3) “(yy ree ee) (ey) 05.143)

632 The simultaneous equations model

which, in view of the fact that

it differs from the 2SLS estimator in so far as /* is not given the value one but

ism, +k, —1 Given that rank(X{X,)=k, by assumption [5], Section 25.3,

this rank condition 1s satisfied if rank(Y,Y,)=m, — 1 The latter condition holds when rank(B,,)>m, — 1,1 the order condition for the first equation

is satisfied It must be noted that in the case where this equation is exactly

The solution of (147) for a# is the indirect MLE

In order to discuss the properties of the 2SLS estimator we need to derive its distribution From (140) we can deduce that

Dosis =(¥(P, — PY.) 'Y,(P, — Py ly, (25.149)

where P,=X(XX) 'X, P, =X,(XX,)~'X} These results show that the distribution of 6,5,, can be derived directly from that of 7,5 Concentrating