A NOTE ON IDENTIFIABILITY IN THE LINEAR EXPENDITURE FAMILY*

G O R D O N FISHER, MICHAEL M c A L E E R and D I A N A W H I S T L E R

Queen’s University, Australian National University and London School of Economics and Political Science

I. INTRODUCTION

The theoretical simplicity and the ease with which the linear expenditure system (LES) of Stone [8] may be estimated have made it a popular vehicle of demand analysis for more than a quarter of a century. Yet it is not uncommon for significant residual serial correlation to be exhibited when the LES is estimated which, in view of the model’s simplicity, has gradually become recognised as evidence of specification error. For this and other reasons, the LES has been modified in various ways. Thus, Phlips [7] augmented the LES with a habit-formation hypothesis along the lines suggested by Houthakker and Taylor [3], who had previously introduced stock considerations into demand systems using a joint treatment of physical stocks of commodities and psychological stocks of habits. On the empirical side, Klevmarken [4] has produced evidence supporting the superiority of the LES with habit-formation over the other demand systems he investigated, such as the Rotterdam model and the Indirect Translog model. By introducing habit-formation, serial correlation in LES is typically rendered insignificant.

An alternative approach to modifying the LES is by way of the intertemporal consumption-savings choice originally examined by Tintner [9, 101. As applied to modifying the static utility maximising problem underlying the LES, the intertemporal problem incorporates two additional behavioural hypotheses. The first of these embodies the idea that consumption may be spread over time, thereby leading to the extended linear expenditure system (ELES) of Lluch [5 ] . The second hypothesis explicitly recognises the importance of physical stocks of goods which, together with the first hypothesis, results in the extended linear expenditure system with durables (DELES) of Dixon and Lluch [l]. DELES has the appeal of providing a more comprehensive model of consumer behaviour than either LES or ELES; it might therefore be expected to yield a better empirical explanation of consumer behaviour when applied to appropriate economic data. However, practical experience suggests that misspecification problems remain in the DELES model. For example, Fisher et al. [2] and McAleer et al. [6] have found that the serial correlation problem of ELES is not alleviated by the dynamic formulation of DELES. Moreover, estimation of DELES

*We are grateful to Denis Sargan for helpful discussion. The first author acknowledges the financial support of the Queen’s University Research Advisory Board.

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1982 IDENTIFIABILITY IN T H E LINEAR EXPENDITURE FAMILY 417

was achieved only by estimating some parameters extraneously. With this background, a more important question to consider is whether there are enough restrictions on the parameters of DELES to ensure its identifiability.

It is demonstrated in this note that the DELES model derived from economic theory is not identifiable. Alternative means that might be used to permit identifiability are discussed, as well as their implications.

11. THE LINEAR EXPENDITURE FAMILY

The notation to be used follows that of Fisher et a/. [2]. In deterministic form, the three models may be written as

= YIP(, + PPlZI" (3)

v , / = 8 / 4 / r , / + p q T * z : - / . I q ~ p n-. ;rl , (4) If

(DELES)

in which the subscripts i and j refer to the ith non-durable and the jth durable good, respectively. Per capita expenditures on commodities are represented by vi, and v//, prices by p i , and n-;,, and per capita total expenditure and permanent income by 171, = X\>,/ and z,, respectively. The utility function parameters for goods i and.jarep, and q,, with Xp, + Vq,= 1, while the corresponding subsistence parameters are y , and +r The decay rate for du rab le j is denoted by 8/. Supernumerary permanent income in DELES is given by

i

I /

z T = Z , - c Y ,Pi/ -c q4, T;, ( 5 )

and the marginal propensity to consume out of permanent income in both ELES and DELES is ~ = 6 / p , the ratio of the subjective time-preference discount rate (6) to the market interest rate (p). If a = Z )qj/ ( O j + p ) ( , then pr = p,/ ( 1 - 8a) is the marginal

budget share (MBS) of non-durable i; the MBS of du rab le j is q,?* = q; ( O , + p - 6 ) , where $=q,/](O,+p)( 1 - Sa)( and pq? is the shift-parameter for durable j . The subscript t denotes a sample realisation and a dot above a variable denotes its time derivative.

/

On the basis of the above, the specialisation of DELES to ELES occurs in three steps. First, replace T,, in equations (4) and (5) byp,,/ (0, + p). where p i , is the price of a unit of service from durable j . Second, let all durables become non-durables by the device of requiring 0,- 00 for all j . In these circumstances, given values of '7, ensure (Y - 0 and hence qJ - 0 and pq? - 0; similarly, fir --pi; theratio (Bi + p - S)/ (Oj+ p) ( - 1 and hence q,** - qf Finally, redefine viand +,as Piand y,, respectively. Durables in ELES are still included in the analysis, but they are treated as if they were non- durables.

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418 AUSTRALIAN ECONOMIC PAPERS DECEMBER

The restrictions on each of the Oi in DELES that lead to ELES cause 6 and p to appear only as the ratio p. Clearly, neither 6 nor p is identifiable in ELES. To analyse the identifiability of DELES, consider n non-durable goods ( i= I , 2 , ..., n ) and r durable goods (j= 1 , 2, ..., r ) , and rewrite equations (3) , (4) and ( 5 ) as

The parameter functions are

a l i = y i , i = 1 , 2 ,..., n, (6)

i = 2 , 3 , ..., n

and, f o r j = I , 2, ..., r ,

b,i = 6i4i , (9)

b,i = b , p i + P - 6) , (10)

b3i=6r l i : jP(Oi+P)(1 - 6 4 1 . (1 1)

There are (2n+ 3r + 1) parameters of interest: y i ( i = 1,2, ..., n) , pi(i= 2,3, ..., n); qtOi’4i(j=1,2,.. . ,r); p and 6. The fact that only ( n - 1) of the P i ( i # l ) need to be determined is a reflection of the adding-up restriction. The equations (6)-( 1 1) represent a system of ( 2 n + 3r) equations, and thus its solution can determine at most ( 2 n + 3r) of the parameters of interest. This suggests that a necessary condition for identifiability is for one parameter to be fixed uyriori . Given one parameter restriction, the solution of the system can be analysed to determine the existence of observationally equivalent models.

As an illustration, consider the solution of (6)-( 11) when one of the decay rates, say O , , is known.

Clearly, the y i are determined directly from equation (6). Then equations (9) and (10) may be solved for p , O j ( j = 2,3, ..., r ) and 4j(j= 1,2, ..., r ) as functions of 6. From equation (1 l), the qj may also be determined as functions of 6 by solving the linear system

to obtain

j = 1 , 2 ,..., r ,

1982 IDENTIFIABILITY I N THE LINEAR EXPENDITURE FAMILY 419

r

j= I where b3= C b3f

The next step is to use equation (8) to express the P i ( i = 2,3, ..., n ) as

Finally, the expressions for the q,(j= 1,2, ..., r ) and the P i ( i = 2 , 3, ..., n ) may be substituted into (7) and solved to yield

n r in which a = C a2, and b,= C b,.

2 j = l /= I

Thus, a restriction on one of the decay rates is sufficient for the identifiability of the remaining parameters. In a similar way, the solution of (6)-( 1 1) may be analysed when, say, p or 6 is fixed a priori.

111. DISCUSSION

In an attempt to improve the theoretical basis of the LES and ELES, Dixon and Lluch [l] have derived a dynamic system of expenditure equations (DELES) that explicitly incorporates durable goods. However, this note has demonstrated that DELES is not identifiable. Thus, more information is needed to evaluate the system empirically. There are at least three ways of providing such information.

(i) One method is to fix a priori a reasonable value for one parameter. It was demonstrated in the previous section, for example, that prior information on one decay rate is sufficient to render the system identifiable. Alternatively, p or 6 may be fixed. For example, p might be set at the mean of interest rates observed over the sample period. An obvious drawback of this approach is that a factor which in reality is variable over the sample period is forced to be a constant parameter. However, this is a drawback of the model itself and is not inherently a criticism of the method of identification.

(ii) A variant of the first method is to consider several points within a range of possible values for one of the parameters, and to select that set of estimates which is judged ‘most reasonable’. See McAleer et al. [6] for an application of this kind, using a range of extraneously determined values of Of This method has the drawback of subjectivity in selecting the extraneous values, but is actually no more subjective than method (i).

(iii) A more ambitious approach is to seek to reformulate the model by allowing some parameters to become variables. This would decrease the number of parameters to be estimated while requiring an enlarged data set. The interest rate p, for example, which is really a supply-side variable, could be incorporated as an additional exogenous variable.

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In respect of methods (i) and (ii) above, it is worth emphasising that in seeking one representative value for what is inherently an exogenous variable may render the whole model impracticable. This could arise when the one representative value is not a good approximation to economic reality, whence the model would be identifiable but may not ‘fit’ the data very well.

Going further (and particularly in respect of method (ii) above) it will be found, in practice, that as (say) p is varied, the value of the likelihood will not change but the other estimates will vary according to the value assigned to p. This reflects the fact that the model is made identifiable by assigning a value to one parameter which, in this case, happens to be p. To put the matter another way, when all the parameters are made identifiable in terms of p, p itself is not identifiable. These different estimates of the model may all ‘fit’ the data rather well, in which case we would expect interest rates to have remained fairly constant throughout the sample period. On the contrary, if the ‘fit’ is poor, we would expect interest rates to have varied a good deal and the model to be in need of reformulation. If interest rates have not varied much, there may be some other reason for the poor performance of the model; in particular, it is possible that another parameter such as 6 should not be treated as constant.

In this way we see that the problem of identifiability may be exploitedjudiciously to the advantage of empirical analysis. Normally, we should expect market interest rates to play an important role in determining variations in consumer expenditures, especially those on durable goods. In this light, the procedure of using assigned values of p to identify the model may be seen as focusing attention on the role of p in the model, and on whether or not its treatment as a constant parameter is practically viable. But the same may also be said of 6.

First version received April 1981 Final version accepted February 1982

(Editors)

REFERENCES 1.

2.

3.

4.

5. 6.

P.B. Dixon and C. Lluch, “Durable Goods in the Extended Linear Expenditure System”, Review of Economic Studies, vol. 44, 1977. G.R. Fisher, M. McAleer and D. Whistler, “Interest Rates and Durability in the Linear Expenditure Family”, Canadian Journal of Economics, vol. 14, 1981. H. Houthakker and L.D. Taylor, Consumer Demand in the United States: Analyses and Projections, 2nd edn (Cambridge, Mass.: Harvard University Press, 1970). N.A. Klevmarken, “A Comparative Study of Complete Systems of Demand Functions”, Journal of Econometrics, vol. 10, 1979. C. Lluch, “The Extended Linear Expenditure System”, European Economic Review, vol. 4, 1973. M. McAleer, A.A. Powell, P.B. a x o n and A. Lawson, “Estimation of the Consumption Function: A Systems Approach to Employment Effects on the Purchase of Durables”, Chapter 7 in E.G. Charatsis (ed.), Selected Econometric Papers in Memory of Stefan Valavanis (Amsterdam: North Holland, 1981). L. Phlips, “A Dynamic Version of the Linear Expenditure Model”, Review of Economics and Statistics, vol. 64, 1972. R. Stone, “Linear Expenditure Systems and Demand Analysis: An Application to the Pattern of British Demand”, Economic Journal, vol. 64, 1954. G. Tintner, “The Maximization of Utility Over Time”, Economerrica, vol. 6, 1938.

7.

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9. 10. G. Tintner, “The Theoretical Derivation of Dynamic Demand Curves”, Econometrico, vol. 6, 1938.