Austral. J . Stat&, 15 (2) , 1973, 111-117

ESTIMATION OF LLUCH’S EXTENDED LINEAR

SECTIONAL DATA’ EXPENDITURE SYSTEM FROM CROSS-

ALAN POWELL* Momash University

Abstract Complete sets of demand relations may be fitted using varying

types of sample information and varying a priori specifications. In this paper the identification and estimation of Lluch’s extended linear expenditure system (ELES) from cross-sectional data alone is investi- gated. Under the most favourable conditions of data availability, all of the parameters of the ELES model are identified, and are estimable by the method of reduced form least squares. This is the case where observations on permanent income are available for the consuming units of the cross section and where, in addition, prices are recorded (even though they do not vary from one consuming unit t o the next). Under the least favourable conditions only the marginal budget shares are identified. This corresponds t o the case where no data on permanent income, or on savings, are available. The conventional ordinary least squares estimators of the marginal budget shares are, under these conditions, biased and inconsistent. Expressions are developed for the large-sample biases.

Acknowledgement Many of the ideas developed here are to be found in Belandria

([l], Oh. 4). The purpose of the present note is to put these results on a systematic baeis and to integrate them with some new results on the large sample biases inherent in common specifications (such as [S], 191, [11 I) of consumer expenditure equations.

The ELES Specification In references [7a, 7 b ] Lluch shows that the intertemporal

maximization of a Stone-Geary utility function [3] leads (under his behavioural specification) to the following set of commodity expenditure equations2 :

Manuscript received August 23, 1972 ; revised February 27, 1973. * This paper was written whilst the author, on leave from Monash University,

was an economist a t the Development Research Center, World Bank. He is very grateful to Constantino Lluch for patient conversations and stimulus, and to Irma Adelman, Haifa Al-Sherbati, Bela Balassa, Charles Blitzer, Steve Brown, John Chang, Ami Glasberg, Sandra Hadler, Nico Klijn, Roger Norton, Kerry Smith and Ross Williams for comments. The World Bank is not responsible for any views expressed in this paper, nor for any errors : the latter are the author’s alone.

‘In [S] Lluch and Morishima give an account of the more general control- theory formdation of ijhe consumer problem of which [7] is the special case corres- ponding to an instantaneous utility function of Stone-Geary form. The first general statement of the problem seems to have been made in 1938 by Tinter ([121, [131). Hie tmatment, unfortunately, did not lead to empirically fruitful specializations for more than 30 years.

112 ALAN POWELL

(1) ui:=pi,yi+yBi(z,-P;y) (i=1, . - 7 n), in which ett is the expenditure of a representative consumer at time t (or of the tth consumer at a given time) on the ith item of consumption expenditure (i=l, . . ., n ) ; pi, is the price of the ith item at time 1 (or confronting the tth consumer at a given point of time) ; zt is an appropriately defined " permanent income )' variable comprising the total income of the representative consumer at time t (or of the Ith consumer at a, given time) to which is added the present value of expected future changes in labour income as perceived by the repre- sentative consumer at time t (or by the tth consumer at a given point of time); that is

[Definition of '' Permanent Income "1 def

(2) 2, = Y,+PW,+L(Y,), where y t is labour income, wt is wealth (excluding human capital), p is the rate of return on wealth, y, is the sequence of expected changes in yt into the indefinite future as perceived from viewpoint t, and L is a present value operator which capitalizes the expectational series y, at the rate of discount p. The other symbols in (1) are interpreted as follows. The vector pt is an n-dimensional column showing the prices of all n consumption items relevant to t p r i m e denotes transposition. The parameters of (1) are the yi of the Stone-Geary utility function (sometimes interpreted as subsistence minima quantum indexes of the various consumption items-see, e.g., [ll]). Collected as a column vector, these rz coefficients are written y. The parameter p may be interpreted as the marginal propensity to consume out of permanent income 2,. The coefficients pi are the marginal budget shares of the n consumption items in total consumption expenditure.

This extension of Stone's linear expenditure system achieves the integration of the complete systems approach to consumer demand with a model (albeit a simple one) of accumulation. The permanent income stream is treated exogenously ; savings, however, are deter- mined endogenously from the consumption function. The latter is obtained by simply summing (1) over commodities:

~Consu~pt~on lhnction]

in which v , is total consumption expenditure, and where the identity forcing the marginal budget shares p i to sum identically to unity has been used.

The specification of an error structure in more conventional versions of linear expenditure systems has always been complicated by the operation of the budget identity,

[Budget Identity] (4) v,=i'v,, in which i is an n-component column vector of units (the " summation vector"), and v, is the column vector containing expenditures on the n items as appropriate to t . In the more conventional case (e.g. references [6], [9] and [ll]), the variable v , is predetermined and

(3 1 o,=(l -P)PiY +I%,

LLUCH’S EXTENDED LINEAR EXPENDITURE SYSTEM 113

appears on the right of the expenditure equations analogous to (1). The incorporation of a series of stochastic disturbances into these equations requires degeneracy in their joint distribution in order to preserve the budget identity.

I n the case of the present specification, it seems appropriate to introduce the disturbances into the systems equations (l), obtaining

[Reduced Porm of Chnmodity Egpenditure Xystem]

Since permanent rather than actual income is involved, there is no iron law of aggregation acting across these commodity expenditure equations. ’Further, the exogeneity to this model of the series {z t } guarantees the plausibility of the assumption that x , is independent of equation error disturbance. For future use we assume, therefore, that

(5 ) wit=pityi+~Bi(x,-P;y)+Eit (i=1, * * ‘ 9 .n).

Cov (zf, ~ ~ ~ ) = O = p l i m - C z f ~ i t (for all i ) . (z 1 Consider the case in which the sample is a time-series. If the cit are free from within.-equation and between-equations serial correlation, then the covariance structure is fully characterized by the contem- poraneous variance-covariance matrix of the E’S j that is, by

(7) def

EE,E; = Z (for all t ) ,

where E, is the n-component vector of stochastic disturbances in the commodity expenditure equations a t data point t . In the case of cross-sectional data, the matrix Z very likely gives an adequate representation of the error structure.l As indicated above, there is no reason to suppose that any restrictions operate across the rows or columns of Z j that is, we assume (8) rank (Z)=n.

n x n

Data on Savings and Permanent Income Available Assume for the moment that data on x f are available €or a cross-

sectional sample in which prices are unknown, but it is known that every consuming unit pays the same price. Then the it” commodity expenditure equation (5) may be rewritten (9.1) wit=ei-tclS,z,+Eil +I, . . ., n) in which (9.2) 8 , = p r y l - p p i p ’ y ( p i f = p , for all t , by assumption), and (9.3) @z=PBi. Equation (9.1) is t,he “ identical regressors ” problem in which every left-hand variable is regressed on the same set of exogenous variables [i, 21. To put it slightly differently, (9.1) is one of n reduced form equations from the same structural system. Under the assumption that the are joint normally distributed, the full information

1 If the data are listed according to some geographic or locational index, there may be instances in which the possibility of auto-correlation must be entertained.

D

114 ALAN POWELL

maximum likelihood estimates of (9.l), therefore, can be obtained by the use of ordinary least squares applied on a commodity-by- commodity basis ([4], pp. 207-212).

The consumption function obtained by summing the commodity expenditure equations (5) is

The variance of the disturbance in this equation is, from (7), (11) Var (i‘q) =i’Zi. The maximum likelihood estimate of u. is obtainable by adding the

(10) ~v,=(l--)P;y+~xt+i’E*.

estimated Qi’s : [where the

The maximum likelihood estimates of the are then obtainable as (13 1 pi =(Di/(i’*). I n the cross-sectional context, adding equations gives

A h A

restriction

has been used].’

marginal budget shares pi

the estimated 8’s across

(14) MLE of ((I--p)p’y}=i‘8. The maximum likelihood estimate of (p’y) is obtained as (15) MLE of (p’y) =i’Q/(1 -irk). Finally, estimates of ( p i y i ) are obtained from expression (9.2) as (16) MLE of ( p i y i ) =Oi +(Dii’8/(l -i’a). Notice that it is not necessary to have price variation over the subscript t in order to identify the yi’s j it is necessary to have measurements on the p i (that is, of the prices of the n different consumption items).

For the above method to be operational, cross-sectional data on permanent income are needed. One approach is that of Belandria [l], who uses socio-economic variables available in his cross-sectional sample data in order to develop proxy data for x t .

A A A A

Data Available on Savings, but Not on Permanent Income Define actual income as

def (17) $ 1 = Yt+P’wt,

the sum of labour and non-labour income. I n some cross-sectional surveys, data on mt are collected. I n some situations it may be reasonable to suppose that the present value of expected future gains 5n non-labour income bears a constant ratio to actual income for all consumption units surveyed ; that is, it may be reasonable to assume

where a is not a function of t . (18) L(Yt) =axt,

Then, from (2) and (17), 1 This $ is identically the estimate of p obtained by the single equation regression

(10) of wt on zt.(which justifies the ML interpretation of (12)). Similarly, the expression given in (14) is identically the estimate of (if)’) which would be obtained from the single equation regression (10). In equations (13), (15) and (16), on the other hand, 1 : 1 transformations from one parameter set into another are involved and hence the invariance theorem on ML estimators may be used.

LLUCH’S EXTENDED LINEAR EXPENDITURE SYSTEM 115

(19) x , =(1 +a)%:,.

(20) V , = U -~)P;y+~(l+a)a,+i‘Et ,

(21)

The consumption function, (lo), becomes

whilst the commodity expenditure equations (9.1) become

v,, =e, +(I +a)@.,$, + E , t

i f =$c/(i’$), [q = (%, - - * 9 $,,I,

=Oa+qlxt+E,t (i=1, . . ’, 1))

(say).

none of u , p, or the &et {yz} are identifiable without further information.

Only Expenditure Data Available1

Then whilst, the (3, are identified, and may be estimated as h def A

(22)

For many cross-sectional sets of data, only information on expenditures is collected. I n this case one can no longer work with reduced form equations such as (9.1) and ( lo) , but must work instead in equations which replace x , on the RHS of (9.1) by total expenditure v, (for which data are available).

(23 ) x ,=(v , -(1 -p)p;y-i‘e,)/p. Substituting from (23) into the RHS of ( 5 ) ,

From (10) we see that

(24) ~ z t =(Ptya - P ~ P ; Y ) +Plo t + ( ~ z t --Pai’Et).

Keeping in mind Lhat pt does not vary across t for cross-sectional data, (24) may be written

This is the form in which linear expenditure functions are commonly fitted to data lacking price variation. The budget identity (4) is sufficient to ensure that the ordinary least squares estimators of the ct’s and 13’s satisfy2

(25) ~ , t = ~ i + P : ~ t + e t , .

n~ n A c R i E O ; 2: pj - - l . i = l i = l

This i s fortunate since the theory requires both of these results ; the error structure of the system (25) is far from classical, however, and the regressor vt is endogenous. Summation of e a t over commodity equations i s instructive :

Thus it is see11 that the development of the linear expenditure equations (25) from (5) provides an economic and a statistical rationale for the degeneracy in the joint distribution of e,, (whereas in the standard treatment of (5) this degeneracy comes about from accounting necessity only).

Because (26) is guaranteed by ordinary least squares, the sums across equations of biases in the estimates of the tc’s and the P’s vanish identically. This is of little comfort, however, since its operational implication is that a large positive bias in the estimate

1 In the development below, whilst prices are treated as fixed in repeated samples, the remaining exogenous variable, permanent income, is treated as stochastic.

See, e.g., [lo].

116 ALAN POWELL

of the marginal budget share of one item must be off set by collectively large negative biases for other items. The large sample biases of the ordinary least squares estimates of the marginal budget shares can be analysed along conventional lines (see, e.g., [5], p. 150) as follows

in which !l’ is the sample size and 6 is the sample mean of total expenditure. Substituting from (25) into (28) we see that

1 T 1’

I - 1 t = l t = 1 2 Zlil(Of --B)/T = M i c (bt - 6 ) p +pi c vt(wt - 5 ) p (29)

IP + X ei,(v,-B)/T. I = 1

The first term on the RHS of (29) vanishes identically, whilst

1 T (30.1) wt(vt--6)/!l’ =Var ( v f ) ,

and

(30.2)

Consequently

(31 1

=Covar (vt, eif) .

Now, from (10) and (24),

since the E~~ and zt are, by assumption, independent. variance-covariance matrix of the E’S as

Rewrite the

(33)

I_ .C _I Then (32) reduces to (34) Covar (wt, ei,) =i’cri --pii’Zi. From (lo), the variance of w, is (35) Var (v,) =p2 Var ( X i ) +i’Zi. For the purpose of this discussion, define the large sample bias (LSB) of bi by

From (31), (34) and (35) , the large sample bias:isl

A A

(36) LSB (pi)=plim ( p i ) - P i .

1 If prices are regarded as stochastic, the denominator of (37) needs an additional term ; namely,

where V is the variance-covariance matrix of p l . + ( 1 - pL)2y’Vy,

LLUCH’S E~XTENDED LINEAR EXPENDITURE SYSTEM 11 7 A

(37)

Because the p i and ^pi both add across i to unity, the sum of these biases should be zero.

LSB (pi;) =(i‘ci --pii‘Zi)/(p2 Var (2,) +i’Zi).

Checking, we see that since

C i‘cj=i’Ci, i = l

(38)

the requirement n A

C IJSB ( p i ) z o i = l

(39 )

does, in fact, hold. If values can be found for these biases, then the ordinary least

squares estimators can be corrected to yield estimators which are consistent. As the marginal propensity to consume, p, is not identi- fiable from the type of data we are discussing, an extraneous estimate would be needed, as would also be the case with the variance of permanent income Var (x , ) . From the nature of the problem, only inconsistent estimakes of the p i and C would be available initially. Hence only rather rough (and inconsistent) estimates of the LSB’s would be available-the correction, however, may still be worth making.

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Unpublished Ph.D. dissertation, North- (Mimeo.)

[S] Dhrymes, Phoebus (1970) : Econometrics-Statistical Foundnfions and Applzca-

[3] Geary, R. 0. (1949-50) : “ A note on ‘ a constant-utility index of the cost of

[4] Goldberger, A. S. (1964) : Econometric Theory. New York : John Wiley & Sons. [5] Johnston, J. (1963) : Econometric Methods. New York : McGraw-Hill. [6] Leser, C. E. V. (1960) : “Demand functions for nine commodity groups in

Australia.” [7a] Lluch, Constantino (1970) : ‘‘ The extended linear expenditnre system.”

TJniversity of Esuex, Department of Economics Discussion Paper No. 6 (February), pp. 8.

European

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[8] Lluch, Constantino, and Morishima, Michio (1972) : ‘‘ Demand for com- modities under uncertain expectations.” Ch. 5 in M. Morishima (Ed.), Theory of Demand, Real and Monetary.

[9] Powell, Alan (1966) : “ A complete system of demand equations for the Australian economv fitted bv a model of additive Dreferences.” Econo -

Oxford University Press.

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[12] Tintner, Gerhard (1938) : “ The maximization of utility over time.”

[13] Tintner, Gtfhard (1938) :