Handbook of Econometrics Vols1-5 _ Chapter 28 ppsx

DISEQUILIBRIUM, SELF-SELECTION, AND Estimation of the switching regression model: Sample separation known Estimation of the switching regression model: Sample separation unknown Estimati

DISEQUILIBRIUM, SELF-SELECTION, AND

Estimation of the switching regression model:

Sample separation known

Estimation of the switching regression model:

Sample separation unknown

Estimation of the switching regression model with imperfect

sample separation information

Switching simultaneous systems

Disequilibrium models: Different formulations of price adjustment

6.1 The meaning of the price adjustment equation

6.2 Modifications in the specification of the demand and supply functions

6.3 The validity of the “Mm” condition

Some other problems of specification in disequilibrium models

7.1 Problems of serial correlation

7.2 Tests for distributional assumptions

7.3 Tests for disequilibrium

7.4 Models with inventories

Multimarket disequilibrium models

Models with self-selection

Multiple criteria for selectivity

Handbook of Econometrics, Volume III, Edited by Z Griliches and M.D Intriligator

1634 G S Maddala

1 Introduction

The title of this chapter stems from the fact that there is an underlying similarity between econometric models involving disequilibrium and econometric models involving self-selection, the similarity being that both of them can be considered switching structural systems We will first consider the switching regression model and show how the simplest models involving disequilibrium and self-selection fit

in this framework We will then discuss switching simultaneous equation models, disequilibrium models and self-selection models

A few words on the history of these models might be in order at the outset Disequilibrium models have a long history In fact all the “partial adjustment” models are disequilibrium models.’ However, the disequilibrium models consid- ered here are different in the sense that they add the extra element of ‘quantity rationing’ The differences will be made clear later (in Section 6) As for self-selection models, one can quote an early study by Roy (1951) who considers

an example of two occupations: Hunting and fishing and individuals self-select based on their comparative advantage This example and models of self-selection are discussed later (in Section 9) Finally, as for switching models, almost all the models with discrete parameter changes fall in this category and thus they have a long history The models considered here are of course different in the sense that

we consider also “endogenous” switching We will first start with some examples

of switching regression models Switching simultaneous equations models are considered later (in Section 5)

Suppose the observations on a dependent variable Y can be classified into two regimes and are generated by different probability laws in the two regimes Define

and

X and Z are (possibly overlapping) sets of explanatory variables fil, p2 and (Y are sets of parameters to be estimated ul, u2 and u are residuals that are only contemporaneously correlated We will assume that (u,, u2, U) are jointly nor-

‘The disequilibrium model in continuous time analyzed by Bergstrom and Wymer (1976) is also a

mally distributed with mean vector 0, and covariance matrix

0: 012 (Jlu

We have set var(u) = 1 because, by the nature of the conditions (1.3) and (1.4) (Y

is estimable only up to a scale factor

The model given by eqs (1.1) to (1.4) is called a switching regression model If

% = (72” = 0 then we have a model with exogenous switching If uiU or u2U is non-zero, we have a model with endogenous switching This distinction between switching regression models with exogenous and endogenous switching has been discussed at length in Maddala and Nelson (1975)

We will also distinguish between two types of switching regression models Model A: Sample separation known

Model B: Sample separation unknown

In the former class we know whether each observed y is generated by (1.1) or (1.2) In the latter class we do not have this information Further, in the models with known sample separation we can consider two categories of models:

Model A-l: y observed in both regimes

Model A-2: y observed in only one of the two regimes

We will discuss the estimation of this type of models in the next section But first,

we will given some examples for the three different types of models

Fair and Jaffee (1972) consider a model of the housing market There is a demand function and a supply function but demand is not always equal to supply (As to why this happens is an important question which we will discuss in a later section.) The specification of the model is:

Demand function: D = XP, + u1

Supply function: S = X/3, + u2

The quantity transacted, Q, is given by

Q = Min( D, S) (the points on the thick lines in Figure 1)

Thus Q=X&+u, if D<S,

if D > S

1636 G S Maddala

P

Figure 1

The condition D < S can be written as:

where a2 = Var(u, - u2) = 0: + u; - 25 Thus the model is the same as the switching regression model in eqs (1.1) to (1.4) with 2 = X, (Y = ( p2 - Pr)/a and

u = (ur - u2)/u If sample separation is somehow known, i.e we know which observations correspond to excess demand and which correspond to excess supply, then we have Model A-l If sample separation is not known, we have Model B

Consider the labor supply model considered by Gronau (1974) and Lewis (1974) The wages offered W, to an individual, and the reservation wages W, (the wages

at which the individual is willing to work) are given by the following equations:

Example 3: Demand for durable goods

This example is similar to the labor-force participation model in Example 2 Let

y, denote the expenditures the family can afford to make, and yz denote the value

of the minimum acceptable car to the family (the threshold value) The actual expenditures y will be defined as y = yi iff yi 2 y, and = 0 otherwise

Banks are reluctant to frequent the discount window too often for fear of adverse sanctions from the Federal Reserve One can define:

y, = Desired borrowings

y2 = Threshhold level below which banks will not use the discount window The structure of this model is somewhat different from that given in examples 2 and 3, because we observe yi all the time We do not observe y2 but we know for each observation whether y, I y, (the bank borrows in the Federal funds market)

or yi > y2 (the bank borrows from the discount window)

Some other examples of the type of switching regression model considered here are the unions and wages model by Lee (1978), the housing demand model by Lee and Trost (1978), and the education and self-selection model of Willis and Rosen (1979)

2 Estimation of the switching regression model: Sample separation known

Returning to the model given by eqs (1.1) to (1.4), we note that the likelihood function is given by (dropping the t subscripts on U, X, Z, y and I)

1638 G S Maddala

density gr( ur) and the conditional density fr( u[ui), with a similar factorization of the bivariate normal density of (u,, u) Note that ui2 does not occur at all in the likelihood function and thus is not estimable in this model Only urU and uzu are estimable In the special case u = (ur - ~*)/a where u2 = Var(u, - u2) as in the examples in the previous section, it can be easily verified that from the consistent estimates of a:, CT:, uiU and a,, we can get a consistent estimate of ur2

The maximum likelihood estimates can be obtained by an iterative solution of the likelihood equations using the Newton-Raphson method or the Berndt et al (1974) method The latter involves obtaining only the first derivatives of the likelilood function and has better convergence properties In Lee and Trost (1978) it is shown that the log-likelihood function for this model is uniformly bounded from above The maximum likelihood estimates of this model can be shown to be consistent and asymptotically efficient following the lines of proof that Amemiya (1973) gave for the Tobit model To start the iterative solution of the likelihood equations, one should use preliminary consistent estimates of the parameters which can be obtained by using a two-stage estimation method which

is described in Lee and Trost (197Q2 and will not be reproduced here

There are some variations of this switching regression model that are of considerable interest The first is the case of the labor supply model where y is observed in only one of the two regimes (Model A-2) The model is given by the following relationships:

J’ = Yl if Y, 2 y2

= 0 otherwise

For the group Z = 1, we know yi = y and y, 2 y

For the group Z = 0, all we know is yi < y,

Hence the likelihood function for this model can be written as:

where

u2 =Var(u, - u2) = 6~: + u: -2u,,,

‘This procedure first used by Heckman (1976) for the labor supply model was extended to a wide

@( ) is the distribution function of the standard normal and f is the joint density

of ( uit, uZt) Since y is observed only in one of the regimes, we need to impose some identifiability restrictions on the parameters of the model These restrictions are:

(a) There should be at least one explanatory variable in (1.1) not included in (1.2)

The estimation of this model proceeds as before We first write the criterion function in its reduced form and estimate the parameters by the probit method Note that, for normalization purposes, instead of imposing the condition Var( u)

=l, it is more convenient to impose the condition that the variance of the residual U* in the reduced form for (2.3) is unity

This means that Var( u) = u,’ is a parameter to be estimated But, in the switching regression model, the parameters that are estimable are: pi, &, u:, I$, ulU*, and (I~,,* where a& = Cov(u,, u*) and ulU * = Cov(u,, u*) The estimates of uiU* and u2U * together with the normalization eq (2.4) give us only 3 equations from which

we still have to estimate four parameters ui2, uiU, u2,, and u,‘ Thus, in this model

we have to impose the condition that one of the covariances Q, ulU, u2U is zero The most natural assumption is u12 = 0

1640

As for the estimation of the parameters in the choice function (2.3) again we have to impose some conditions on the explanatory variables in y, and y2 After obtaining estimates of the parameters 8% and &, we get the estimated values jjl and j$ or y, and y2 respectively and estimate the parameters in (2.3) by the probit method using these estimated values of y, and y2 The condition for the estimability of the parameters in (2.3) is clearly that there be no perfect multicol- linearity between j+, j$ and z

This procedure, called the “two-stage probit method” gives consistent estimates

of the parameters of the choice function Note that since (yr - jr) and (y2 - j&) are heteroscedastic, the residuals in this two-stage probit method are hetero- scedastic, But this heteroscedasticity exists only in small samples and the residuals are homoscedastic asymptotically, thus preserving the consistency properties of the two-stage probit estimates For a proof of this proposition and the derivation

of the asymptotic covariance matrix of the two-stage probit estimates see Lee (1979)

3 Estimation of the switching regression model: Sample separation unknown

In this case we do not know whether each observation belongs to Regime 1 or Regime 2 The labor supply model clearly does not fall in this category because the sample separation is known automatically In the disequilibrium market model, where the assumption of unknown sample separation has been often made, what this implies is that given just the data on quantity transacted and the explanatory variables, we have to estimate the parameters of both the demand and supply functions Once we estimate these parameters, we can estimate the probability that each observation belongs to the demand and the supply function Consider the simplest disequilibrium model with sample separation unknown:

0, = X1,/?, + urr (Demand function),

S, = X& + uzl (Supply function),

Q, = Min(D,, S,)

The probability that observation I belongs to the demand function is:

X, = Prob( 0, < S,),

Let f(~r, u2) be the joint density of (ur, u2) and g(D, S) the joint density of D and S derived from it If observation t is on the demand function, we know that

on the supply function, we know that S, = Q, and 0, > Q, Hence,

1642 G S Maddala

discriminant analysis He shows that the classification rule suggested by Kiefer and Gersovitz is optimal in the sense that it minimizes the total probability of misclassification Even in a complicated model, these relationships hold good Note that in a more complicated model (say with stochastic price adjustment equations) to calculate h, as in (3.1) or to compute (3.7) we need to derive the marginal distribution of D, and S,

There are two major problems with the models with unknown sample sep- aration, one conceptual and the other statistical The conceptual problem is that

we are asking too much from the data when we do not know which observations are on the demand function and which are on the supply function The results cannot normally be expected to be very good though the frequency with which

‘good’ results are reported with this method are indeed surprising For instance,

in Sealey (1979) the standard errors for the disequilibrium model (with sample separation unknown) are in almost all cases lower than the corresponding standard errors for the equilibrium model! Goldfeld and Quandt (1975) analyze the value of sample separation information by Monte-Carlo methods and Kiefer (1979) analyzes analytically the value of such information by comparing the variances of the parameter estimates in a switching regression model from a joint density of ( y, D) and the marginal density of y (where y is a continuous variable and B is a discrete variable) These results show that there is considerable loss of information if sample separation is not known In view of this, some of the empirical results being reported from the estimation of disequilibrium models with unknown sample separation are surprisingly good Very often, if we look more closely into the reasons why disequilibrium exists, then we might be able to say something about the sample separation itself This point will be discussed later in our discussion of disequilibrium models

The statistical problem is that the likelihood functions for this class of models are usually unbounded unless some restrictions (usually unjustifiable) are imposed

on the error variances As an illustration, consider the model in eqs (1.1) to (1.4): Define

Prob( y = _~r) = r,

Prob(y=y,)=l-lr

The conditional density of y given y = yr is:

f(YlY = vr) =fr(_Y - J&)/r

Similarly,

f(rlv = Y2) =fz(Y - X&/<I - r>

Hence, the unconditional density of y is:

Ch 28: Disequilibrium, Self-selection, and Switching Models 1643

Where fi and f2 are the density functions of ui and a2 respectively Thus, the distribution of y is the mixture of two normal distributions Given n observations

yi, we can write the likelihood functions as:

where

and

I

Take u2 # 0 and consider the behaviour of L as u1 + 0 If Xi& = yi, then

A, -, 00 and A,, A,,-A, all + 0 But B,, B,,- B, are finite Hence L + co Thus,

as ui , 0 the likelihood function tends to infinity if Xi& = yi for any value of i

Similarly, if ui # 0, then as a2 -+ 0 the likelihood function tends to infinity if

Xiip2 = yi for any value of i

In more complicated models, this proof gets more complicated, but the struc- ture of the proof is the same as in the simple model above [See Goldfeld and Quandt (1975) and Quandt (1983, pp 13-16) for further discussion of the problem of unbounded likelihood functions in such models.]

Another problem in this model, pointed out by Goldfeld and Quandt (1978) is the possibility of convergence to a point where the correlation between the residuals is either +l or -1 This problem, of course, does not arise if one assumes ui2 = 0 to start with

The disequilibrium model with unknown sample separation that we have been discussing is a switching regression model with endogenous switching The case of

a switching regression model with exogenous switching and unknown sample separation has been extensively discussed in Quandt and Ramsay (1978) and the discussion that followed their paper

The model in this case is:

Regime 1: yi = X;,& + &ri with probability X

Regime 2: yi = X2ifi2 + aZi with probability (1 - A)

&ii - IN(0, u,2) E2i - IN(0, u,2)

As noted earlier, the likelihood function for this model becomes unbounded for certain parameter values However, Kiefer (1978) has shown that a root of the

and G(y, xi, 9) is the value of the expression on the right hand side of (3.8) for

B = t$ and the ith observation

The normal equations obtained by minimizing (3.9) with respect to y are the same as those obtained by minimizing

(3.10)

;=I j=l

3Hartley and Mallela (1977) prove the strong consistency of the maximum likehood estimator but

on the assumption that q and 4 are bounded away from zero Amemiya and Sen (1977) show that even if the likelihood function is unbounded, a consistent estimator of the true parameter value in this

where

The normal equations in both cases are:

Schmidt (1982) shows that we get more efficient estimates if we minimize weighted sum of squares rather than the simple sum of squares (3.10), making use

of the covariance matrices 0, of (cil, si2, eik) for i = 1,2 n

Two major problems with the MGF estimator is the choice of the number of 8’s to be chosen (the choice of k) and the choice of the particular values of 6’, for

a given choice of k Schmidt (1982) shows that the asymptotic efficiency of the modified MGF estimator (the estimator corresponding to generalized least squares)

is a non-decreasing function of k and conjectures that the lower bound of the asymptotic variance is the asymptotic variance of the ML estimator Thus, the larger the k the better As for the choice of the particular values of ej for given k,

Kiefer, in his comment on Quandt and Ramsay’s paper notes that the 8’s determine the weights given to the moments of the raw data by the MGF estimator Small e’s imply heavy weight attached to low order moments He also suggests choosing B’S by minimizi ng some measure of the size of the asymptotic covariance matrix (say the generalized variance) But this depends on the values

of the unknown parameters, though some preliminary estimates can be sub- stituted Schmidt (1982) presents some Monte-Carlo evidence on this but it is inconclusive

The discussants of the Quandt and Ramsay paper pointed out that the authors had perhaps exaggerated the problems with the ML method, that they should compare their method with the ML method, and perhaps use the MGF estimates

as starting values for the iterative solution of likelihood equations

In summary, there are many problems with the estimation of switching models with unknown sample separation and much more work needs to be done before one can judge either the practical usefulness of the model or the empirical results already obtained in this area The literature on self-selection deals with switching models with known sample separation but the literature on disequilibrium models contains several examples of switching models with unknown sample separation [see Sealey (1979), Rosen and Quandt (1979) and Portes and Winter (1980)] Apart from the computational problems mentioned above, there is also the problem that these studies are all based on the hypothesis of the minimum condition holding on the aggregate so that the aggregate quantity transacted switches between being on the demand curve and the supply curve The validity

The problems of aggregation are as important as the problems of estimation with unknown sample separation discussed at length above The econometric problems posed by aggregation have also been discussed in Batchelor (1977), Kooiman and Kloek (1979), Malinvaud (1982) and Muellbauer and Winter (1980)

4 Estimation of the switching regression model with imperfect

sample separation information

The discussion in the previous two sections is based on two polar cases: sample separation completely known or unknown In actual practice there may be many cases where information about sample separation is imperfect rather than perfect

or completely unavailable Lee and Porter (1984) consider the case They consider the model:

for r =1,2, , T There is a dichotomous indicator IV; for each r which provides sample separation information for each t We define a latent dichotomous variable It where

1, = 1 if the sample observation Y, = Yr,

= 0 if the sample observation Y, = Y,,

The relation between 1, and W, can be best described by a transition probabil- ity matrix

and the marginal density of Y, is

If pl1 = pOl, then the joint density f(Y, W,) can be factored as:

(4.3)

(4.4)

and hence the indicators W, do not contain any information on the sample separation One can test the hypothesis p 11 = pOl in any actual empirical case, as shown by Lee and Porter Also, if 1 and pOl = 0, the indicator W, provides

1648 G S Maddala

perfect sample separation, and

Thus, both the cases considered earlier-sample separation known and sample separation unknown are particular cases of the model considered here

Lee and Porter also show that if pI1 # pal, then there is a gain in efficiency by using the indicator W, Lee and Porter show that the problem of unbounded likelihood functions encountered in switching regression models with unknown sample separation also exists in this case of imperfect sample separation As for

ML estimation, they suggest a suitable modification of the EM algorithm sug- gested by Hartley (1977, 1978) and Kiefer (1980b) for the switching regression model with unknown sample separation

The paper by Lee and Porter is concerned with a switching regression model with exogenous switching but it can be readily extended to a switching regression model with endogenous switching For instance, in the simple disequilibrium market model

Ch 28: Disequdibrium, Self -selecrion, and Switching Models 1649

the ‘directional method’ of Fair and Jaffee (1.972) in the sense that the sign of A P,

is taken to be a noisy indicator rather than a precise indicator as in Fair and Jaffee Further discussion of the estimation of disequilibrium models with noisy indicators can be found in Maddala (1984)

5 Switching simultaueous systems

We now consider generalizations of the model (1.1) to (1.4) to a simultaneous equation system Suppose the set of endogenous variables Y are generated by the following two probability laws:

and

If u is uncorrelated with Ui and U,, we have switching simultaneous systems with exogenous switching Goldfeld and Quandt (1976) consider models of this kind Davidson (1978) and Richard (1980) consider switching simultaneous systems where the number of endogenous variables could be different in the two regimes The switching is still exogenous An example of this type of model mentioned by Davidson is the estimation of a simultaneous equation model where exchange rates are fixed part of the time and floating the rest of the time Thus the exchange rate is endogenous in one regime and exogenous in the other regime

If the residual u is correlated with Vi and U, we have endogenous switching The analysis of such models proceeds the same way as Section 2 and the details, which merely involve algebra, will not be pursued here [See Lee (1979) for the details.] Problems arise, however, when the criterion function in (5.3) and (5.4) involves some of the endogenous variables in the structural system In this case we have to write the criterion function in its reduced form and make sure that the two reduced form expressions amount to the same condition As an illustration, consider the model

Y, = YiY, + &Xi + Uir

y2=Y2yl+&x2+u2 if Y,<c,

Unless (1 - y1y2) and (1 - yiy;) are of the same sign, there will be an inconsistency

in the conditions Yi < c and Y, > c from the two reduced forms Such conditions

1650 G S Maddala

for logical consistency have been pointed out by Amemiya (1974), Maddala and Lee (1976) and Heckman (1978) They need to be imposed in switching simulta- neous systems where the switch depends on some of the endogenous variables Gourieroux et al (1980b) have derived some general conditions which they call

“coherency conditions” and illustrate them with a number of examples These conditions are derived from a theorem by Samelson et al (1958) which gives a necessary and sufficient condition for a linear space to be partitioned in cones

We will not go into these conditions in detail here In the case of the switching simultaneous system considered here, the condition they derive is that the determinants of the matrices giving the mapping from the endogenous variables (Y,, Y,, , Y,) to the residuals (zdi, Us, , uk) are of the same sign, in the different regimes The two determinants under consideration are (1 - y1y2) and

(1 - yly$) The condition for logical consistency of the model is that they are of the same sign or (1 - y1y2)(1 - yly;) > 0 A question arises about what to do’with these conditions One can impose them and then estimate the model Alterna- tively, since the condition is algebraic, if it cannot be given an economic interpretation, it is important to check the basic structure of the model An illustration of this is the dummy endogenous variable model in Heckman (1976a) The model discusses the problem of estimation of the effect of fair employment laws on the wages of blacks relative to whites, when the passage of the law is endogenous The model as formulated by Heckman is a switching simultaneous equations model for XC& we have to impose a condition for “logical con- sistency” However, the condition does not have any meaningful economic interpretation and as pointed out in Maddala and Trost (1981) a careful examination of the arguments reveals that there are two sentiments, not one as assumed by Heckman, that lead to the passa,ge of the law, and when the model is reformulated, there is no condition for logical consistency that needs to be imposed

The simultaneous equations models with truncated dependent variables consid- ered by Amemiya (1974) are also switching simultaneous equations models which require conditions for logical consistency Again, one needs to examine whether these conditions need to be imposed exogenously or whether a more logical formulation of the problem leads to a model where these conditions are automati- cally satisfied For instance, Waldman (1981) gives an example of time allocation

of young men to school and work where the model is formulated in terms of underlying behavioural relations and the conditions derived by Amemiya follow naturally from economic theory On the other hand, these conditions have to be imposed exogenously (and are difficult to give an economic interpretation) if the model is formulated in a mechanical fashion where time allocated to’work was modelled as a linear function of school time and exogenous variables and time allocated to school was modelled as a linear function of work time and exogenous variables

The point of this lengthy discussion is that in switching simultaneous equation models, we often have to impose some conditions for the logical consistency of the model If these conditions cannot be given a meaningful economic interpreta- tion, it is worthwhile checking the original formulation of the model rather than imposing these conditions exogenously and estimating the parameters in the model subject to these conditions

An interesting feature of the switching simultaneous systems is that it is possible to have underidentified systems in one of the regimes As an illustration, consider the following model estimated by Avery (1982):

D = &Xi + a,Y + ui Demand for Durables

Y, =&Xi + a2D + u2 Demand for Debt

Y, =&X3 + a,D + uj Supply of Debt

Y = min( Yr, Y,) Actual quantity of Debt

D, Y,, Y, are the endogenous variables and X, and X3 are sets of exogenous variables Note that the exogenous variables in the demand for durables equation and the demand for debt equation are the same

The model is a switching simultaneous equations model with endogenous switching We can write the model as follows:

Thus, the condition for the logical consistency of this model is that (1 - (Y~oL~) and

(1 - (Y& are of the same sign - a condition that can also be derived by using the theorems in Gourieroux et al (1980b)

The interesting thing to note is that the simultaneous equation system in Regime 1 is under-identified However, if the system of equations in Regime 2 is identified, the fact that we can get consistent estimates of the parameters in the demand equation for durables from Regime 2, enables us to get consistent estimates of the parameters in the Y, equation Thus the parameters in the simultaneous equations system in Regime 1 are identified One can construct a formal and rigorous proof but this will not be attempted here Avery (1982) found

that he could not estimate the parameters of the structural equation for Y, but this is possibly due to the estimation methods used

In summary, switching simultaneous equations models often involve the im- position of constraints on parameters so as to avoid some internal inconsistencies

in the model But it is also very often the case that such logical inconsistencies arise when the formulation of the model is mechanical In many cases, it has been found that a re-examination and a more careful formulation leads to an alterna- tive model where such constraints need not be imposed

There are also some switching simultaneous equations models where a variable

is endogenous in one regime and exogenous in another and, unlike the cases considered by Richard (1980) and Davidson (1978), the switching is endogenous

An example is the disequilibrium model in Maddala (1983b)

6 Disequilibrium models: Different formulations of price adjustment

Econometric estimation of disequilibrium models has a long history The partial adjustment models are all disequilibrium models and in fact this is the type of model that the authors had in mind when they talked of ‘“disequilibrium model.” Some illustrative examples of this are Rosen and Nadiri (1974) arid Jonson and Taylor (1977)

The recent literature on disequilibrium econometrics considers a different class

of models and has a different structure These models are more properly called

“rationing models.” This literature started with the paper by Fair and Jaffee (1972) The basic equation in their models is

Fair and Jaffee considered two classes of models

(i) Directional models: In these we mfer whether Q, is equal to D, or S, based

on the direction of price movement, i.e

D, > St and hence Q, = S, if AP, > 0,

D, < St and hence Q, = D, if AP, c 0,

Ch 28: Disequd~bnum, Seij-sekrtton and Switchrnp Mod& 1653

The maximum likelihood estimation of the quantitative model is discussed in Amemiya (1974a) The maximum likelihood estimation of the directional model, and models with stochastic sample separation (i.e where only (6.1) is used or (6.2)

is stochastic) is discussed in Maddala and Nelson (1974)

The directional method is logically inconsistent since the condition that AP,

case there are not enough equations to determine the endogenous variables Q,

(4.2) included

There are three important problems with the specification of this model that need some discussion These are:

(i) The meaning of the price adjustment eq (6.2)

(ii) The modification in the specification of the demand and supply functions that need to be made because of the existence of the disequilibrium, and (iii) The vahdity of the min condition (6.1)

We will discuss these problems in turn

6.1 The meaning of the price adjustment equation

The disequilibrium market model usually considered consists of the following demand and supply functions:

and the eqs (6.1) and (6.2) To interpret the “price adjustment” eq (6.2) we have

to ask the basic question of why disequilibrium exists One interpretation is that prices are fixed by someone The model is thus a j&price model The disequi-

librium exists because price is fixed at a level different from the market equilibrat- ing level (as is often the case in centrally planned economies) In this case the

41~e directional method makes sense only for the estimation of the reduced form equations for 0, and S, in a model with a price adjustment equation There are cases where this is needed The likelihood function for the estimation of the parameters in this model is derived in Maddala and Nelson (1974) It is:

where g( D, S) is the joint density of D and S (from the reduced form equations) When A P < 0 we

have D = Q and S > Q and when AP > 0 we have S = Q and D > Q Note that the expression given

in Fair and Kelejian (1974) as the likelihood function for this model is not correct though it gives

1654 G S Maddula

price adjustment eq (6.2) can be interpreted as the rule by which the price-fixing authority is changing the price However, there is the problem that the price-fixing

authority does not know D, and S, since they are determined only after P, is

fixed Thus, the eq (6.2) cannot make any sense in the fix-price model Laffont and Garcia (1977) suggested a modification of the price adjustment equation which is:

P t+l- P, = Y(D, - s,)

In this case the price fixing

(6.2’) authority uses information on the past period’s demand and supply to adjust prices upwards or downwards In this case the price-fixing rule is an operational one but one is still left wondering why the price-fixing authority follows such a dumb rule as (6.2’) A more reasonable thing

to do is to fix the price at a level that equates expected demand and supply One such rule is to determine price by equating the components of (6.3) and (6.4) after ignoring the stochastic disturbance terms This gives

source of disequilibrium in this formulation is stickiness of prices (due to some institutional constraints or other factors) Let P,* be the market equilibrating

price However, prices do not adjust fully to the market equilibrating level and we specify the “partial adjustment” model:

Note that in this case it is AP, (not AP,+l as in the Laffont-Garcia case) that

gives the sample separation But the interpretation is not that prices rise in response

to excess demand (as implicitly argued by Fair and Jaffee) but that there is excess

demand (or excess supply) because prices do not fully adjust to the equilibrating values.5

Equation (6.7) can also be written as

if we assume that the excess demand (0, - S,) is proportional to the difference

(P,* - P,), i.e the difference between the equilibrating price and the actual price The interpretation of the coefficient y in (6.8) is of course different from what Fair and Jaffee gave to the same equation

One can also allow for different speeds of upward and downward partial adjustment Consider the following formulation:

Again note that we get A P, and not A P,+ 1 in these equations

Ito and Ueda (1979) use Bowden’s formulation with different speeds of adjustment as given by (6.9) to estimate the rates of adjustment in interest rates for business loans in the U.S and Japan They prefer this formulation to that of Fair and Jaffee or Laffont and Garcia because in eq (6.9), A, and X2 are pure numbers which can be compared across countries The same cannot be said about the parameters yi and y2 in eq (6.11)

‘The formulation in terms of partial adjustment towards P* was suggested by Bowden (1978a) though he does not use the interpretation of the Fair-Jafiee equation given here Bowden (1978b)

There is still one disturbing feature about the partial adjustment eq (6.6) that Bowden adopts and under which we have given a justification for the Fair and Jaffee directional and quantitative methods This is that AP, unambiguously gives

us an idea about whether there is excess demand or excess supply As mentioned earlier this does not make intuitive sense On closer examination one sees that the problem is with eq (6.6), in particular the assumption that X lies between 0 and

1 This is indeed a very strong assumption and implies that prices are sluggish but never change to overshoot P,* the equilibrium prices There is? however, no

a priori reason why this should happen 6 Once we drop the assumption that h should lie between 0 and 1, it is no longer true that we can use AP, to classify

observations as belonging to excess demand or excess supply As noted earlier the assumption 0 < h < 1 implies that the conditions P,* > Pt ,, P, > PC_,, P,* > P,

and D, > S, are all equivalent With A > 1, this no longer holds good

In summary, we considered two models of disequilibrium the fix-price model and the partial adjustment model In the f&price model,the price adjustment eq (6.2) is non-operational The modification (6.2’) suggested by Laffont and Garcia

is an operational rule but really does not make much sense A more reasonable formula for a price-setting rule is the anticipatory pricing rule (6.5) But this implies that a price-adjustment equation like (6.2) or (62’j is not valid

In the case of the partial adjustment model one ca.n derive an equation of the form (6.2) though its meaning is different from the one given by Fair and Jaffee and many others using this price adjustment equation The meaning is not that prices adjust in response to excess demand or supply but that excess demand and supply exist because prices do not adjust to the market equilibrating level However, as discussed earlier, eq (6.2) can be derived from the partial adjustment model (6.6) only under a restrictive set of assumptions

The preceding arguments hold good when eq (6.2) is made stochastic with the addition of a disturbance term In this case there is not much use for the price-adjustment equation The main use of eq (6.2) is that it gives a sample separation, and estimation with sample separation known is much simpler than estimation with sample separation unknown If one is anyhow going to estimate a model with sample separation unknown, then one can as well eliminate eq (6.2) For fix-price models, one substitutes the anticipatory price eq (6.5) and for partial adjustment models one uses eq (6.6) directly

6.2 Modifications in the speci$cution of the demand and supp(v functions

The preceding discussion refers to alternative formulations of the price adjust- ment equation One can also question the specification of the other equations as

6Since no economic model has been specified, there is no reaon to make any aItematr\e assumption

well We will now discuss alternative specifications of the demand and supply functions

The probability that there would be rationing should affect the demand and supply functions There are two ways of taking account of this One procedure suggested by Eaton and Quandt (1983) is to introduce the probability of rationing

as an explanatory variable in the demand and/or supply functions (6.3) and (6.4)

A re-specification of eq (6.3), they consider is

As an illustration of this approach we will re-formulate the supply function by introducing expected prices We leave eqs (6.1) (6.2) and (6.3) as they are and re-define (6.4) as

where Pte is the expected price, i.e the price the suppliers expect to prevail in period t, the expectation being formed at time t - 1 (we will assume a one period lag between production decisions and supply) Regarding the expected price P,', if

we use some naive extrapolative or the adaptive expectations formulae, then the estimation proceeds as in earlier models with no price expectations, with minor modifications For instance, with the adaptive expectations formula, one would

‘Though the analysis is similar, the computations are more complex because of the presence of q in

Equation (6.12) implies that we can write

where u, is uncorrelated with all the variables in the information set It-l If the information set Z,_, includes the exogenous variables X,, and Xzt, i.e if these exogenous variables are known at time I - 1, then we can substitute P,’ = P, - u,

in eq (6.12) We can re-define a residual U$ = uzt - (Y*u, and u;, has the same properties as Us, Thus the estimation of the model simplifies to the case considered by Fair and Jaffee

If, on the other hand, X,, and X,, are not known at time (t - 1) we cannot treat u, the same way as we treat uzl since u, can be correlated with Xi, and X,,

In this case we proceed as follows

From eqs (6.2) (6.3), and (6.4’) we have

and XI7 and Xi, are the expected values of X,, and X,, (Note that this equation

is valid even if the price adjustment eq (6.2) is made stochastic.)

To obtain Xz and Xz we have to make some assumptions about how these exogenous variables are generated A common assumption is that they follow vector autoregressive processes Let us for the sake of simplicity of notation assume a first order autoregressive process

Yet another modification in the specification of the demand and supply function that one needs to make is that of ‘spillovers’ The unsatisfied demand and excess supply from the previous period will spill over to current demand and supply The demand and supply functions (6.3) and (6.4) are now reformulated respectively as:

0, = X,,& + alp, + &@,-, - Q,-,>+G

with 6, > 0, S, > 0, and S,6, ~1 [See Orsi (1982) for this last condition.]

At time (t - l), Q,_, is equal to D,_r or S,_, Thus, one of these is not observed However, if the price adjustment eq (6.2) is not stochastic, one has a four-way regime classification depending on excess demand or excess supply at time periods (t - 1) and t Thus, the method of estimation suggested by Amemiya (1974a) for the Fair and Jaffee model can be extended to this case Such extension

1660 G S Maddala

is done in Laffont and Monfort (1979) Orsi (1982) applied this model to the Italian labor market but the estimates of the spill-over coefficients were not significantly different from zero This method is further extended by Chanda (1984) to the case where the supply function depends on expected prices and expectations are formed rationally

6.3 The validity of the ‘Min’ condition

As mentioned in the introduction, the main element that distinguishes the recent econometric literature on disequilibrium models from the earlier literature is the

‘Mm’ condition’ (6.1) This condition has been criticized on the grounds that: (a) Though it can be justified at the micro-levei, it cannot be valid at the aggregate level where it has been very often used

(b) It introduces unnecessary computational problems which can be avoided by replacing it with

at the aggregate level

Regarding criticism (b), Richard (1980b) and Hendry and Spanos (1980) argue against the use of the ‘Min’ condition as formulated in (6.1) Sneessens (1981, 1983) adopts the condition (6.1’) However, eq (6.1’) is hard to justify as a behavioural equation Even the computational advantages are questionable [see Quandt (1983) pp 25-261 The criticism of Hendry and Spanos is also not valid

on closer scrutiny [see Maddala (1983a), pp 34-35 for details]

Criticism (c) is elaborated in Maddala (1983a,b), where a distinction is made between “Rationing models” and “Trading Models”, the former term applying to models for which the quantity transacted is determined by the condition (6.1), and the latter term applying to models where no transaction takes place if 0, # S, Condition (6.1) is thus replaced by