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grounds, in particular by Blackorby, Primont and Russell [S] and by Simmons and. Weiserbs [9]. Some of these criticisms are reviewed in section IV. Below, it is ...
ESTIMATION OF TRANSLOG DEMAND SYSTEMS* KEITH R. M c L A R E N Monash University I.

INTRODUCTION

In a pathbreaking and highly influential paper, Christensen, Jorgenson and Lau [6] (henceforth CJL) introduced the transcendental logarithmic (translog) direct and indirect utility functions. These represent a considerable generalisation of the CobbDouglas, which they include as a special case, and hence allow the possibility of testing various restrictions such as additivity and homotheticity, which had previously been maintained hypotheses of applied demand analysis. Herein lies the importance of the CJL paper. However, the CJL methodology has since been criticised on a number of grounds, in particular by Blackorby, Primont and Russell [ S ] and by Simmons and Weiserbs [9]. Some of these criticisms are reviewed in section IV. Below, it is argued that the estimation and testing procedure of CJL should be further modified in two important respects, by relating approximations used to the sample data and by reexamining which variables are endogenous and which exogenous. After briefly examining duality theory in section I1 and the concept of approximation in section 111, the CJL procedure is reviewed and extended in section IV. The estimation procedure is described in section V, and estimates of the translog indirect utility function for Australian data are presented in section VI, along with the test results. Section VII describes a procedure which attempts to appropriately estimate the translog direct utility function. Our understanding of the CJL paper owes much to the recent book of Blackorby, Primont and Russell [4], (henceforth BPR). 11. DUALITY THEORY Represent the utility function by

u= U ( X , ,...,X,) . The basic maintained hypothesis of demand theory is that the consumer maximises U subject to the budget constraint CpiXi= M , with p i the price of the ith commodity and M total expenditure. Introducing normalised prices v i = p i / M , i = 1, ...,m, allows the budget constraint to be written more symmetrically as CviX,= 1. The indirect utility function is defined as

u(x,,... ,~ ~ 1 s C. t~. , X , , ... jf”,(x,,)> 3

which can be approximated by the first and second order terms of a Taylor series about an arbitrary point as $0 W(x>= (Yo+

C q ” f , ( x ,+> %

c c PI, I

PlJ=

f;(Xl,.qXJ,

(5)

J

PJJ

x=

(see BPR pp. 292-293 but note that 1 is redundant and partials on p. 292 should be with respect tof, rather than x I ) .Such a specification can represent any function W ( x ) “near” the point X,in the sense that the values of the functions and of the derivatives up to second order coincide at the point X,and in the sense that in any neighbourhood of X the error of approximation is bounded by the size of the higher order terms. BPR refer to this specification as a “generalised quadratic”. It is important t o distinguish between testing restrictions on the specification $0 W ( x )as such, and testing restrictions on $0 W ( x )a t 2. We refer to these as global and local restrictions, respectively.

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Letting JI =A=!.?n and W = V gives the transcendental logarithmic indirect utility function of CJL:

R n V = a. + CaiBnvi+‘/2 C

3 PiiBnviPnvj

I

J

and application of Roy’s Theorem gives a j + Z PiiRnvi ’ i ’ . . v.X.= 1, ...,m ) a,+? P,Rnvi

u=

(6)

(7)

J

I

where a,,,= C a ,p,=: a

(i= 1, ..., m ) and

pki

can be normalised to - 1. Similarly, W = Ugives the transcendental logarithmic direct utility function of CJL: -RnU=ao+ & t $ ! n X i ‘/2 +

3 1

p,!i!nXi!2n4

and application of Wold‘s Theorem gives: ai+ C piinnxi v.X.= ’ i ‘ 1, ..., m) a,+C PMiQnXi



(8)

J

u=

(9)

i

IV.

THECHRISTENSEN-JORGENSON-LAU TESTS

Given the theory presented in sections TI and Ill it is now possible to reconsider some of the tests considered by CJL. These will be grouped into three categories.

A.

Testing the Theory of Demand

The fundamental hypothesis of consumer demand theory is that the consumer’s preference ordering satisfies sufficient regularity conditions to allow its representation by a utility function U ( X ) which is a twice differentiable, non-decreasing and strictly quasi-concave function defined on the non-negative orthant. Given this utility function, and taking prices p and expenditure M a s exogenous, the maximisation of U ( X ) subject to the budget constraint CpiXi= M endogenises the quanitities X and leads to the set of demand functions X = +(v). The above optimisation process leads t o a set of restrictions on 4, and it is these restrictions that are usually referred to as the “theory of demand”. Using estimates of the parameters of equation sets (7) and (9), CJL are led to “reject” the theory of demand. At least two objections can be raised about this test procedure: (i) As Simmons and Weiserbs [9] point out, although (6) implies (7) with Pii=pji, the functional form of the demand system (7) could have been generated by an alternative utility function which did not imply symmetry of the particular parameters pii in the demand system. Thus it is improper to associate parameters estimated from a demand system with the parameters of a particular utility function and associate tests

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of the theory of demand with tests of the utility function. Rather, tests of demand theory should be based on the demand systems themselves (for example by directly testing the Slutsky symmetry conditions). (ii) In the theory of demand as outlined above, prices p and expenditure M are exogenous, and quantities demanded are endogenous. Thus if the indirect utility function is specified, application of Roy’s Theorem leads to the demand functions specifying quantity as a function of prices and expenditure. Thus a reduced form system results, and the interpretation is straightforward. However, if the direct utility function is specified, application of Wold’s Theorem gives the set of m inverse demand or shadow price functions, which describe for any X those values of v which would have led to such X being chosen by the utility maximisation. These inverse demand functions are merely implicit representations of the ordinary demand functions. While it is true that the implied shadow prices are the solution to optimisation problem (4), this is merely an aspect of duality theory. There is no suggestion that maximisation problem (4) is actually carried out by economic agents. The implication of this reasoning is that the usual outcome of duality theory, of simply swapping the roles of v and X , does not extend to the estimation procedure. If observed data is to be used to estimate the parameters of the system (9), thexi should be treated as endogenous and the vj exogenous. Since this is the exact opposite to the procedure followed by CJL, we suggest that half of their tests, those relating to the direct translog utility function, are meaningless given the behavioural assumptions of utility maximisation which apparently constitute their maintained hypothesis.] We return to the question of the estimation of system (9) in section VII. B.

Local and Global Tests

As noted in section 111, the true function $ o W ( x ) can be represented in the neighbourhood of an arbitrary point X- by the second order Taylor expansion about .F, $0 W ( x ) . With respect to the translog specification, three separate formulations of hypotheses might be considered:

B1. $ o W ( x ) is sub-globally translog. The adjective sub-globally is required as the translog specification cannot provide a global representation of either the direct or the indirect utility function, and refers to the sub-domain of W ( x ) on which the representation is valid in the sense of satisfying the usual postulates of utility theory, such as non-satiety, quasi-concavity. This sub-domain should encompass any data which is likely to arise in practice, including of course the sample.

m)

B2. provides a satisfactory approximation to $0 W ( x )at the point 2 in the sense that the use of statistical criteria leading to the acceptance/rejection of demandtheoretic restrictions on $0 W ( x ) at X provides “convincing evidence” that the true unknown @oW ( x ) possessesldoes not posses these properties at the point X. Since typical demand theoretic restrictions d o not involve derivatives of the utility function IIf, alternatively, the authors have in mind some alternative maintained hypothesis, such as the adjustment of prices to clear markets, then it is the results pertaining to the translog indirect utility function that are called into question.

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of higher than second order, knowledge of the derivatives of the utility function up to this order at some point X would guarantee that a second-order Taylor series approximation at this point would be indistinguishable according to demand theoretic criteria from the true utility function. The question of whether this statement has any operational significance is taken up below.

B3. Same as B2, except that inference is made as to global properties of $0 W ( x )on the bases of tests made on $0 W ( x )at the point 2. Rejection of B2 implies rejection of B3, since if a property does not hold at a point, it cannot be satisfied globally. Now proposition B1 corresponds to the usual situation in demand analysis, where an explicit functional form is specified for either the direct or indirect utility function. If a certain test fails, blame can be attributed either to demand theory in general or to the specific functional form. However, it is difficult to see how either of the hypotheses B2 or B3 could be refuted since if the true W ( x )were known, Jlo W ( x )and $0 W ( x )would correspond exactly at 2. But W ( x )is unknown, and by estimating the parameters of Go W ( x )such that it is compatible with the data in some other space, Jlo W ( x )may or may not be made to coincide with Jlo W ( x )at 2. There seems to be no methodology for relating tests on JloW(x) to tests on JloW(x). Thus in the context of the question of symmetry, Simmons and Weiserbs conclude that, “... when p, = pii it is not possible to distinguish between the following two cases:

(1) the translog utility function is an approximation to an unknown true function that has a symmetric Hessian at a base point. (2) the translog utility function represents consumer preferences globally” [8, p. 8941.

This discussion raises two points. Firstly, there seems to have been little discussion of the relative merits of the following two approximation procedures: (a) approximating the utility function, and imposing the exact restrictions generated by the approximating function on the demand equations, as is done by CJL. (b) approximating the demand functions by some arbitrary but convenient form which is sufficiently flexible to allow the imposition of demand theoretic constraints at some chosen point, as in the Rotterdam demand model. The second of these approaches has the advantage of carrying out the approximation in the space of variables in which estimation is to take place, whereas it is hard to see the logic of attempting to approximate any function in an unobservable space. The second point, somewhat related to the first, is that there has been virtually no discussion of the choice of the point of approximation in the case of approximating either the direct or indirect utility functions. The only reference in the literature seems to be the following from BPR: “It is perhaps natural to interpret the “point of

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approximation” as the point at which aiand p, are estimates of the first and second derivatives of the true function” [4,p. 2991. This argument is hardly convincing, and in any event circular, since it was the equality of first and second derivatives between functions at the point of approximation that led to the use of a Taylor series approximation in the first place. While a rigorous approach to the choice of the approximation point X would require a general analysis involving the underlying loss function and the integration of the theoretical specification and the estimation method, it would seem that an arbitrary choice of X is unjustified. Since @ o W ( x ) approximates @ o W ( x )only in the neighbourhood of X,it would seem that a basic necessary condition would be that be central to the sample data, in some sense. Implicit in the translog literature is that X is chosen to be 1.0. But this means that the point of approximation is not even invariant to the units of measurement of the data which, as will be demonstrated below, can lead to tests of hypotheses in which the outcome depends on the units of measurement of the data. A preferred alternative would be to draw an analogy with the literature on approximating demand functions, in which the demand theoretic constraints are imposed at the sample means. Now since the translog functions are specified in terms of logarithms, an obvious choice for the point of approximation would seem to be the sample mean of the log ( X I ) for , the direct utility function, and the sample mean of log (v, ),for the indirect utility function. These points are used in the empirical work below. Of course, other “central” points could also be used, but the argument we are presenting is simply that the choice should be determined by the actual sample. This argument is to some extent supported by the work of Wales [lo], who in an experimental situation, found that the translog could not approximate the true function for data that ranged too far from the point of approximation.

x

C.

Tests of Structure

Given the distinction between local and global representation by a translog function, CJL then proceed to test for various combinations of local and global additivity and homotheticity. These tests have been criticised by Blackorby, Primont and Russell [S], who show that the translog is “separability inflexible”, i.e. once weak separability is imposed, the translog is .... “not capable of providing a second order approximation to an arbitrary weakly separable function in any neighbourhood of a given point” [5, p. 1961. This is a criticism of the sequence of tests on the global translog function. But the sequence of tests using the translog as an approximation to the true function can also be criticised. Consider, for example, the CJL derivation of their test for local additivity. If the indirect utility function V is additive, Q nV= F ( C Q nV ’ ( v, ) ) and the parameters of a translog approximation to an additive indirect utility function satisfy, at the point of approximation V where the first and second derivatives of the true function and the approximating function correspond,

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so that, by defining 8 = F'l(F')*,the restriction can be written as k PjkQni,) P rl. . = 8 ( a i +kC pikQnGk)(aj+C

.

(10)

(iZj, i, j = 1, ..., m )

.

Thus the implied constraint depends on the chosen point of approximation, C, and different choices of V will lead to different restrictions, and hence different tests of additivity of the underlying true functions, i.e. tests of additivity at different points. CJL write the restriction (10) as Pij= 0 apj, and have thus implicitly chosen V = 1. As noted above, the outcome of this test may not be invariant to units of measurement. Similarly, the appropriate test for homotheticity at the point V is the set of restrictions

(i= 1, ..., m ) whereas CJL test the restrictions PMi=aai. V.

ESTIMATION PROCEDURE

In estimating equation system (7), vjXj= Sj was designated as endogenous, and v as exogenous. Since (7) is in share form and only exogenous variables appear on the r.h.s. it is reasonable to assume that an additive error structure would satisfy the usual assumptions for maximum likelihood estimation. The restriction CSj= 1allows the mth equation to be deleted, with maximum likelihood estimates invariant to the choice of equation deleted. In principle full information maximum likelihood estimates may be obtained, and tests of thevarious restrictions carried out, by means of likelihood ratio tests. If it were not for restrictions (10) and (11), Wymer's ASIMUL programme (Wymer [l 11) could be used. However, note that in practice vi will be price indices, for which the base is arbitrary. Thus one approach would be to scaleallvisothat the sample mean of In(vi) is approximately zero for all i by appropriately choosing the base of the price indices. In fact, this idea can be extended even further, since there is no reason why each price must have the same base, and thus it is possible to scale the data in such a way as to ensure that the sample mean of In vi=O, i= 1 ... m. We note that the variables Sj are unaffected by this transformation. In this way restrictions (10) and (1 1) can be imposed at the sample mean of the Qnvi in the forms

p,=

eff,aj

BMi= uai

(12)

(13)

which can easily be coded into ASIMUL. Although these restrictions look like those used by CJL, a careful reading of their paper suggests that the data was not SO transformed, and their point of approximation was not related to sample data.

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VI.

399

AN EMPIRICAL EXAMPLE

Our estimates are based on Australian per capita consumption for the period 1958 (3) to 1979 (1) for the categories food, clothing, and other. This decomposition was chosen instead of that used by CJL because of the availability of reliable price series in the Australian case. Although such a decomposition is far from satisfactory, three components were considered sufficient to illustrate the points of this paper. The data is defined as follows:

P

: population, millions; ABS [2]

CPTR : final consumption expenditure, private, total, real; ABS [I] CPT

: final consumption expenditure, private, total; ABS [ 13

CPF

: final consumption expenditure; food; ABS [I]

CPC

: final consumption expenditure; clothing, footwear, draperies; ABS [ I ]

CPO

: final consumption expenditure, other,

=CPT- CPF- CPC PCF

: consumer price index; food; 1966/67= 1; ABS [3]

PCC

: consumer price index; clothing; 1966/67= 1; ABS [3]

PCO

: implicit price index; other = CPO/ (CPTR - CPF/ PCF- CPC/ PCC)

x,

: CPF/(PCF.P)

x2

: CPC/(PCC.P)

: CPO/(PCO.P)

M

: CPT/(1000 P)

v '1

: PCF/M

v;

: PCC/M

v;

: PCO/M

i'

: =exp(Qnvi-Rn

s,

: CPF/CPT

s2

: CPCiCPT.

-

v3

i= 1,2,3.

Results are presented in Table I for the set of hypotheses tested by CJL. For comparison, Table I1 presents results based on virather than vi. In this case the point of approximation has no apriorijustification, and the data are simply as they came from the Monash database, except that M is scaled by .001 to bring v and M into comparable magnitudes in order to preserve accuracy. This is the type of adjustment

)

*

-

834.185 -0.2004 (268.73) -0.0951 (78.80) -0.0944 (8.44) 0.0105* (2.60) 0.0781* (2.58) -0.0349 (4.26) 0.0371' (2.58) -0.0057' (0.17) 0.0127' (0.78) 0.6443 (6.30) 0.5534 (2.57)

ADDITIVITY

-1.4063 (4.51)

-

763.230 -0.1984 (131.53) -0.0940 (78.61) -0.2563 (8.59) -0.1118 (4.96) 0.6472* (19.25) -0.0497 (2.63) 0.2937* ( 13.16) 0.279 I * (4.61) 0.1322* (4.59) 0.995 1 * (4.47)

HOMOTHETICITY

714.705 -0.1987 (1 17.68) -0.0930 (63.16) -0.2693' (6.80) 0,0756' (19.12) 0.5759* (17.39) -0.1662' (8.96) 0.2694' (17.71) 0.3822' (6.52) 0.1788; (6.60) 1.3622* (6.34) 4.0914 (16.97) - I ,9232 (6.40)

ADDITIVITY AND HOMOTHETICITY

-

__

-0.0835' (7.61) -0.0245' (3.29) 0.4174 (8.23)

-

-0.0245 (3.29)

-

-

830.803 -0.2017 (35 1.22) -0.0955 (80.53) -0.0835 (7.61)

GLOBAL ADDITIVITY NOT HOMOTHETICITY

-

-

-

-

-

758.286 -0.2027 (1 66.96) -0.0959 (86.33) -0.3215 (14.25) -0.1584 (9.75) 0.4799* ( 18.15) -0.0707 (3.39) 0.229 I * (9.34)

HOMOGENEITY NOT ADDITIVITY

0.9232 (6.51)

-

-0.1878' (6.46) -0.0885' (6.47) -0.6468* (6.51)

-

-0.0885* (6.47)

-

-

701.244 -0.2034 (95.95) -0.0959 (66.88) -0.1878* (6.46)

GLOBAL ADDITIVITY A N D HOMOTHETICITY

indicates parameter value derived by restrictions. The restrictions referred to by the column headings are defined in the Appendix.

-

-

845.147 -0.2024 (243.14) -0.0958 (84.43) -0.1257 (10.56) -0.0634 (4.12) 0.0894 (3.02) -0.0704 (2.57) 0. I134 (2.95) -0.0997' (2.64) -0.0204' (1.11) 0.2250 (1.66)

EQUALITY AND SYMMETRY

-

1.8248 (9.95)

-

-

-

705.143 -0.2026 (1 13.52) -0.0955 (68.60) -0.2948. (9.96) 0.0353' (9.89) 0.2595* (9.95) -0.1576* (9.92) 0.1223' (9.91)

ADDITIVITY AND HOMOGENEITY

Estimates of the Parameters of the Indirect Translog Utility Function, Data Normalised to Give Mean (Log "'1 M) = 0

TABLE I(=)

LINE LOGARI

)

l

*

-

835.027 -0.1405 (32.24) -0.0745 (20.77) -0.0707 (7.42) 0.0059; (3.02) 0,625' (3.35) -0.0286 (5.15) 0.0331; (3.38) -0.0023' (0.10) 0.0104* (0.93) 0.4705 (11.07) 0.5663 (3.60)

ADDITIVITY

-0.8255 (15.14)

~

780.204 -0. I777 (120.82) -0.0860 (53.22) -0.1093 (6.03) -0.0354 (2.89) 0.2914' (16.57) -0.0207 (2.33) 0.1272' ( 1 I .70) 0.1467; (14.45) 0.0710' ( I 3.46) 0.6079' (15.44)

HOMOTHETICITY

747.897 -0.1805 ( I 12.69) -0.0886 (65.85) -0.0835* (6.09) 0.0 280 (30.14) 0.2313* (29.94) -0.0553' (7.61) 0.1135* (25.90) 0. I758* (31.38) 0.0863' (3 1.30) 0.7120 (36.65) 1.7533 (26.69) -0.9741 (35.91)

ADDITIVITY AND HOMOTHETICITY

-

-0,0714' (6.15) -0.02 10: (3.10) 0.3571 ( I 1.09)

-

-0.02 10 (3.10)

(6.15)

830.803 -0.1277 (59.20) -0.0691 (19.93) -0.0714

GLOBAL ADDITIVITY NOT HOMOTHETICITY

758.286 -0.1765 (9 I .99) -0.0833 (5 1.18) -0.3214 (14.25) -0.1584 (9.75) 0.4798* (18.15) -0.0707 (3.39) 0.2291' (9.34)

HOMOGENEITY NOT ADDITIVITY

1.0116 (2.33)

-0.1994* (2.37) -0.0958' (2.35) -0.7164' (2.32)

-0.0958; (2.35)

-

697.993 -0.1971 (7 I .74) -0.0947 (63.87) -0.1994; (2.37)

GI.OBAL ADDITIVITY AND HOMOTHETICITY

indicates parameter value derived by restrictions. The restrictions referred to by the column headings are defined in the Appendix

-

845.147 -0.1289 (18.08) -0.0738 (14.49) -0. I195 (6.18) -0.0603 (3.03) 0.0850 (3.78) -0.0669 (2.38) 0. I078 (2.79) -0.0948; (2.08) -0.0194; (1.01) 0.2138 (2.00)

EQUALITY AND SYMMETRY

TABLEI I ( a ) Estimates of the Parameters of the Indirect Translog Utility Function

-

2.1553 (10.46)

-

705.849 -0.1836 (76.43) -0.091 I (63.89) -0.3231' (11.31) 0.0361' (12.10) 0.28 70 * ( 1 I .20) -0.1785; (10.88) 0.1425; (10.59)

ADDITIVITY AND HOMOGENEITY

LINE LOGARI

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that might be carried out regardless of approximation, and presumably may have been applied to the CJLdata. This scaling, ofcourse, implies that the means of the Pn v!are not too far from zero (in fact they are 0.62933, 0.60316 and 0.56628 respectively) so that the approximation points are not too far from those in Table I. However, the results do differ. An alternative scaling in which M was left in raw form, was also tried in order to accentuate the differences in points at which restrictions are imposed. This data usually gave poor results, and the algorithm failed to converge in those cases not related directly by translation to Table 11. Unrestricted parameter estimates for the two sets of data are presented separately in Table 111, to avoid repetition of constrained estimates. These results, like those of the linear logarithmic, homogeneity not additivity, and global additivity not homotheticity cases, do not depend on the point of approximation, and hence give identical likelihood values. TABLEI11 Unrestricted Estimates

Food

Clothing

t

Scaled

Unscaled

854.746 -0.2020 (233.78) -0.3531 (2.97) -0.1610 (1.20) 0.5315 (2.24) -1.2233 (1.99) -0.6344 (0.88) 2.5726 (2.03) -0.0956 (63.06) 0.0180 (0.07) 0.6604 (3.19) -0.9173 (2.36) 1.3704 (0.50) 8.1398 (3.60) -12.0376 (2.75)

854.746 -0.1410 (24.10) -0.2706 (3.09) -0.1234 (1.27) 0.4076 (2.69) -0.9374 (2.06) -0.4862 (0.92) 1.9721 (2.42) -0.3183

+ -0.4058 (0.09) -14.8052

+ 20.5728

+ -30.7461 (1.59) -132.4088

+

269.9006

+

t statistics not computed as corresponding estimated variances negative. Hessian matrix was not positive definite.

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In Tables I and I1 L represents the non-constant part of the maximised loglikelihood value. Parameter estimates designated by a * were derived by restrictions, the others were estimated directly. Recorded in parentheses under each parameter estimate are the so-called asymptotic t-values (the ratios of parameter estimates to the asymptotic standard errors). We should note that in this model t values are dependent on the normalisation chosen, so that for example in those cases where the results in Table I can be derived directly from Table I1 by translation, t-values d o not correspond. Tests are based on the test statistic -2 n h , where A is the ratio of the restricted over the unrestricted maximised likelihood. Under the assumptions of the model -2 Qnhis asymptotically distributed as x 2 with degrees of freedom equal to the number of restrictions to be tested. For comparability the same test procedure as in CJL is followed. Table IV reproduces the critical values of x2 for the relevant degrees of freedom. Results of tests are presented in Table V. It may be worth noting that in CJL Tables 1V and V, the test statistics have apparently been divided by their degrees of freedom. TABLE IV CriricaI Values of x 2

Degrees of Freedom

Level of Significance .I0

.05

,025

2.71 4.61 7.78 10.64

3.84 5.99 9.49 12.59

5.02 7.38 11.14 14.45

.01 6.63 9.21 13.28 16.81

Three points are worth making about the results presented in Table V. Firstly, all tests result in rejection at the 1 per cent level. Secondly, however, it is clear from the Table that all tests of structure (Le., other than equality and symmetry) depend on the point of approximation used. Thus although it is not the case with these results, it is clearly possible that a test could be rejected under one scaling of the data, yet accepted under another. (This was in fact the case with an earlier version of this paper, where the results were based on a smaller sample size). Thirdly, the rejection of hypotheses on the basis of an asymptotic test should perhaps be qualified in the light of recent work on the bias of such tests by Laitinen [7] and Meisner [8]. VII.

USEOF

THE

TRANSLOG DIRECT UTILITYFUNCTION

Specification of the translog direct utility function gives (8), and application of Wold’s theorem gives the set of m equations (9). However, it has already been noted that Wold’s theorem generates shadow price functions, which describe for any Xthose v that would have led such X to be chosen by the optimisation procedure (1) i.e. the

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TABLE V(a) Test Statistics

Hypothesis Equality and Symmetry Given the Theory of Demand Additivity Homotheticity Additivity and Homotheticity Given Additivity Homotheticity Global Additivity Given Homotheticity Additivity Homogeneity Given Additivity and Homotheticity Global Additivity Homogeneity Linear Logarithmic Given Global Additivity and Homotheticity Linear Logarithmic Given Additivity and Homogeneity Linear Logarithmic (a)

*

Degrees of Freedom

Scaled

Unscaled

6

19.20*

19.20*

2 2 4

21.92" 163.83* 260.89*

20.24* 129.89* 194.50*

2 1

238.96* 6.76*

174.26" 8.45*

2 1

97.05* 9.89*

43.84*

1 1 2

26.92* 19.12; 47.57*

99.81* 84.10* 113.96*

1

20.65*

14.148*

1

28.45*

29.86*

64.61*

indicates significant at the 1 per cent level.

demand functions are defined implicitly by (9), with the quantities endogenous and the prices and expenditure exogenous. Turning now t o the appropriate estimation of (9), again note that, because CSj= 1, if an additive error term is included on the r.h.s. of (9) only m- 1 error terms can be independent. However, one cannot simply delete an equation since m Xjappear in each equation of the system. The appropriate procedure is to specify m- 1 stochastic equations plus the identity CvjXj= 1 . The various restrictions applied to the translog indirect utility function can also be applied to the translog direct utility function, and the question of the appropriate point of approximation again arises. Since the translog direct utility function is specified in terms of QnX,, the appropriate point would seem to be that f i such that sample mean (QnXj)=Q n X jAgain, restrictions of the form (lo) and ( 1 1) can be reduced to the form (12) and (13) by normalising the data on X j to give sample mean (QnXj)=0, and modifying the vj consistently. In this form system (9) should be able to be estimated by ASIMUL. However, the programme failed to converge in the computation of the implicit reduced form, and SO parameter estimates could not be obtained. In an attempt to overcome this problem, the r.h.s. of (9) was linearised. To maintain consistency between the approximation of the translog, and the approximation in estimation, the linearisation was carried out in terms of the QnXi,and the identity was forced to hold in terms of !2nXiby the inclusion of an appropriate remainder as an exogenous variable. In this case ASIMUL ran, but failed to converge t o sensible parameter estimates. Alternative linearisations also failed.

I982

ESTIMATION OF TRANSLOC DEMAND SYSTEMS

405

In one sense, failure to obtain estimates of the translog direct utility function is disappointing, since it does not allow comparison with the CJL results. On the other hand, one may question whether such an attempt is worthwhile in the first place. Examining system (9), the appropriateness of an additive error term can certainly be questioned. A more desirable procedure would be to solve (9) for the implied demand functions, expressing each X, in terms of v only, and add an error term to these. But of course this is impossible, and avoiding the need to solve such problems is precisely the reason for using an indirect utility function in the first place. Since Uand Vare dual to each other, if one wishes to test restrictions on the true Ufunction a more appropriate procedure is to translate these restrictions into restrictions on V, and hence onto the demand functions rather than the shadow price functions. Many of the equivalences between restrictions on U and restrictions on V are explored in BPR. VIII.

CONCLUSION

Two recent developments in the systems of demand equations literature are the theory of duality, and the introduction of flexible functional forms. These ideas were briefly reviewed in sections I1 and 111. A particularly influential application of these methods was the introduction of the translog direct and indirect utility functions by Christensen, Jorgenson and Lau, and the use of these functions to test various restrictions on the underlying preference ordering. The CJL methodology was briefly reviewed in section IV, where it was suggested that there are at least two major shortcomings of their approach to the use of duality and flexible functional forms in the context of estimation and testing. The first point relates to the use of the translog as a n approximation to some arbitrary underlying (unknown) utility function. It was suggested that there are serious methodological questions about the appropriateness of attempting to approximate any unobservable function, but that if such an approximation is attempted, the point of approximation should be related to the observable sample data. When applied to Australian data, it was found that many test results were sensitive to the point of approximation. The second point relates to the appropriate estimation procedure for the demand equations generated by the translog direct utility function, regardless of whether this function is viewed as the true utility function or as an approximation. An alternative estimation procedure was presented, but parameter estimates could not be obtained. However, it was argued that it is the indirect utility function which is more useful as a functional form for the representation of preferences if the aim of the specification is the derivation of systems of demand equations. First version received December 1980 Final version accepted April 1981 (Editors)

406

AUSTRALIAN ECONOMIC PAPERS

DECEMBER

REFERENCES 1.

2. 3. 4.

5. 6. 7. 8. 9. 10.

1 1.

Australian Statistician, Quarterly Estimates of National Income and Expenditure, No. 5206.0, Australian Bureau of Statistics, Canberra. Australian Statistician, Monthly Review of Business Statistics, No. 13040, Australian Bureau of Statistics, Canberra. Australian Statistician, Consumer Price Index, No. 6402, Australian Bureau of Statistics, Canberra. C. Blackorby, D. Primont and R.R. Russell, Duality, Separability and Functional Structure: Theory and Economic Applications (New York: Elsevier North-Holland, 1978). C. Blackorby, D. Primont and R.R. Russell, “On Testing Separability Restrictions with Flexible Functional Forms”, Journal of Econometrics, vol. 5 , 1977. L.R. Christensen, D.W. Jorgenson and L.J. Lau, “Transcendental Logarithmic Utility Functions”, American Economic Review, vol. 65, 1975. K. Laitinen, “Why is Demand Homogeneity So Often Rejected’’, Economic Letters, vol. 1, 1978. J. Meisner, “The Sad Fate of the Asymptotic Slutsky Symmetry Test for Large Samples”, Economic Letters, vol. 2, 1979. P. Simmons and D. Weiserbs, “Translog Flexible Functional Forms and Associated Demand Systems”, American Economic Review, vol. 69, 1979. T.J. Wales, “On the Flexibility of Flexible Functional Forms: An Empirical Approach”, Journal of Econometrics, vol. 5 , 1977. C.R. Wymer, “Computer Programs: ASIMUL manual”, International Monetary Fund, Washington.

APPENDIX Specifiraiion of the Models Estimated

The translog indirect utility function generates the set of demand equations

1

Since 2 Si= I , the third equation is discarded and the constraints used to recover a3and I=

I

PJj,i = 1,2,3, from

the estimated parameters. The models estimated are then defined by the following constraints: (a)

Equality:

p,t,i

are constrained to equality across equations,

(b)

Symmetry: Po= P,,,

(c)

Additivity: p0=0(a,+

1

h=I

3

PiA Q n G l ) ( a , + h s I P , r Q n c A )

where G, is the point of approximation,

1 pik Q n G , ) ,

(d)

Homotheticity: p,,=u(a,+

(e)

Global Additivity: 0 =0,

(f)

Homogeneity: u = 0,

(g)

Linear Logarithmic: 0 = 0 and o = O

P=l