- Email: [email protected]
The semiflexible almost ideal demand system
EUROPEAN K%%“‘c ELSEVIER
European Economic Review 42 (1998) 349-364
The semiflexible almost ideal demand system Giancarlo Moschini Department
of Economics, Iowa State University, Ames, IA 50011, USA
Received 1 January 1995; accepted 1 October 1996
Abstract The concept of a semiflexible functional form is applied to the Almost Ideal (AI) demand system. This yields a demand model that is more parsimonious than standard ones while preserving a degree of flexibility, that satisfies the curvature property of concavity of the underlying expenditure function (at least locally), and that preserves the desirable properties of the AI model (aggregation across consumers and nonlinearity of the Engel curves). The model is illustrated with an application to a relatively large demand system emphasizing food consumption. 0 1998 Elsevier Science B.V. All rights reserved Keywords:
Curvature conditions; Demand systems; Flexible functional forms
1. Introduction The Almost Ideal (AI) demand system of Deaton and Muellbauer (1980) provides one of the most widely used models in applied demand analysis. The popularity of this model is related to its properties. Like most demand systems, the integrability properties of symmetry and homogeneity can be handled by simple parametric restrictions. In addition, a distinguishing feature of this model is that the resulting demand equations possess nonlinear Engel curves while at the same time allowing for exact aggregation across consumers. This attribute is due to the fact that the preferences underlying the AI model are of the ‘Generalized Gorman Polar Form’ (Blackorby et al., 1978), a class that includes some other popular models (such as a version of the translog demand system; see Lewbel, 1987). Furthermore, the particular (nonlinear) Engel curves of the AI model imply that, if a good is income inelastic (a necessity), an increase in income not only decreases 0014-2921/98/$19.00
0 1998 Elsevier Science B.V. All rights reserved.
PII SOO14-2921(97)00060-3
350
G. Moschini/European
Economic Reciew 42 (1998) 349-364
the share of income allocated to that good, but also reduces the income elasticity of that good. This type of nonlinear Engel curve has worked well in empirical studies, particularly those modeling food demand. Empirical applications of the AI demand system often encounter two distinct problems. First, the curvature property (concavity of the expenditure function, which implies that the Slutsky matrix is negative semidefinite) is not explicitly built into the model and is often violated by estimation results. ’ Second, the model becomes prohibitively demanding in terms of data as the number of goods being modeled increases. Like other standard flexible demand systems, the AI model possesses $(~z - l)(n + 4) p arameters when the number of goods is II (maintaining homogeneity and symmetry). Hence, the number of parameters to be estimated increases quadratically as the number of goods increases, whereas the number of effective observations increases only linearly. For large demand systems, this leads to a degrees-of-freedom problem, which may severely affect the statistical properties of the estimated model. The purpose of this paper is to illustrate a procedure to solve these two seemingly unrelated problems by applying the concept of a semiflexible functional form (Diewert and Wales, 1988) to the AI demand system. This procedure allows saving degrees of freedom by restricting the substitution possibilities across goods. This is achieved by restricting the rank of the Slutsky substitution matrix, which in general is (n - 1) for the case of iz goods, to a number K < (n - 1). Furthermore, because this rank restriction is handled within a parametric specification of the Slutsky matrix that maintains negative semidefiniteness (at a point), the imposition of the curvature property of concavity is naturally handled in this context. The model obtained, termed the Semiflexible Almost Ideal (SAI) demand system, preserves the distinguishing features of the original AI system. Thus, the system maintains both its aggregation properties across individuals and the desirable nonlinear shape of Engel curves associated with the AI model. The SAI model presented here may be compared with the Normalized Quadratic (NQ) semiflexible model of Diewert and Wales (1988). Because the NQ is derived from a Gorman Polar Form specification of the expenditure function, it also permits exact aggregation across consumers. Its main advantage, relative to the SAI model, is that it allows one to maintain globally the required concavity of the expenditure function. Unfortunately, the quasi-homotheticity of the underlying preferences entails linear Engel curves. Although these Engel curves are not necessarily born in the origin, they do have restrictive implications. In particular, the income elasticity of goods with inelastic demand (necessities) is forced to rise
’ To this author’s knowledge, only one paper (Chalfant et al., 1991) has maintained the curvature property within the AI demand system. Their semi-Bayesian approach is quite different, and computationally more demanding, than the procedure followed here.
G. Moschini/ European Economic Review 42 (1998) 349-364
351
as the income level increases. This feature is undesirable when modeling food demand, especially if the estimated model is to be used to forecast consumption under scenarios of considerable income changes. By contrast, the apparent disadvantage of the SAI model, relative to NQ, is that it does not allow one to maintain concavity in prices globally (i.e., for any possible price level). It should be clear, however, that this is not a failure of the AI parameterization per se, but a consequence of the Generalized Gorman Polar Form assumption.
2. The AI demand system The AI demand share form as:
system of Deaton
and Muellbauer
(1980) can be written
in
n
wi = ai +
c yijlog
pj + pi
log ; (
j=l
1
(1)
where wi z piqi(p, x> denote budget shares, pi denote prices, qi( p, x) are Marshallian demand functions, x is total expenditure on the n goods (income for short), and P is a translog price aggregator satisfying: n
1””
l”g(P)=cu,+CailogPi+~,~,~YijlogPilogPj i= 1 r=l1=1
(2)
For this system, homogeneity, adding-up, and symmetry hold if Cj oi = 1, Cjyij = Cirij = Ci pi = 0, and yii = rji. Hence, the theoretical properties of homogeneity, adding-up, and symmetry can be imposed by parametric restrictions and will hold globally (that is, for any level of the exogenously given prices and income). The property of negativity (concavity in prices of the underlying expenditure function), on the other hand, requires that the matrix of Slutsky substitution terms Sij = L?hi(p, u)/dpj, where h,(.) denote Hicksian demands, be negative semidefinite. It is easily verified that the Slutsky substitution terms for the AI are not constant but depend on prices and income. Thus, as will be detailed below, for the AI model the inequality restrictions of negativity can only hold locally (for some values of the exogenously given prices and income). The system as it stands is nonlinear in the parameters, but Deaton and Muellbauer (1980) suggested an attractive linear version, which is obtained by replacing the price aggregator P in Eq. (1) with a price index P * constructed before estimation. The resulting linear AI model is straightforward to estimate, and because P * typically tracks P very well, the empirical results thus obtained are usually indistinguishable from those of the nonlinear AI model. However, because estimation of nonlinear demand systems does not present insurmountable computational problems, here I neglect this simplification and concentrate on the AI model
G. Moschini/
352
European Economic Review 42 (1998) 349-364
in its nonlinear form. To estimate this model, it is necessary to recognize that (~a cannot be estimated because it is unidentified. Given that (~a is not necessary for the local flexibility properties of the model, in what follows I will set cy,,= 0 throughout. 2 The Slutsky substitution terms for the AI model can be written as:
sij =
yij
+
wiwj -
sijwi+ pi pj log
; (
)I
where Sjj is the Kronecker delta (aij = 1 for i =j and ajj = 0 for i Zj). From Eq. (3) it is clear that the only set of parametric restrictions guaranteeing that the matrix of Slutsky terms is globally negative semidefinite is yij = 0 (Vi, j) and pi = 0 (Vi>. Unfortunately, these restrictions reduce the AI system to a constant (Cobb-Douglas) share model and thus are not appealing. Alternatively, because the substitution terms are approximately equal to [yiJ + wiwj - Sijwi] and observing that the matrix [wiwj - 6ijwil is negative semidefinite as long as shares are nonnegative, the desired curvature property will likely be guaranteed for all price and income levels that generate nonnegative shares if the matrix [rjj] is negative semidefinite. A similar approach was followed by Jorgenson and Fraumeni (1981) to impose curvature on a translog production model. As illustrated by Diewert and Wales (19871, however, this method ends up imposing ‘too much concavity’, thus destroying the flexibility property of the model. Thus, I do not pursue these avenues, and limit the scope to maintaining the curvature property locally. Without loss of generality, let the point at which concavity is maintained be pi = x = 1. Typically, one will choose the sample mean, that is the point with the highest sample ‘information’, as the point to maintain concavity, and units can be arbitrarily chosen (i.e., the data can be scaled) such that pi = x = 1 at the mean point. Given the assumption (~a = 0, then at this point wi = (Y;. Thus, the substitution term at the mean point, say Bij = Si, (p =x = 11, can be written as: 6ij = yij + cqa; - Sij’yi
Hence, for concavity semidefinite. 3
(4)
at the desired point to hold, the matrix [ oij] must be negative
a In other words, I require P = 1 when p, = 1 for all i. In this application it was found that the relevant estimation results are unaffected by the arbitrary choice of (Y,,. 3 Alternatively, as noted by a reviewer, one may start with the expenditure function underlying the AI model, which can be written as C(p, u) = P(p)eUB(J’), where u is the utility index, P(p) is the function defined by Eq. (21, and B(p) = n:= 1pai. With (~a = 0, C( p. u) will be concave in p at the chosen reference point if P(p) is concave at that point, which is guaranteed by the conditions just discussed.
G. Moschini / European Economic Review 42 (1998) 349-364
353
3. The locally concave AI demand system To facilitate the imposition of concavity at the mean point, it is convenient to reparameterize the model by using Eq. (4) to rewrite the yij in terms of {e,,, ai}. This yields: n
wi = q
log $ (
+ cii
+ c 1
where P” is a price function,
eij log pj + pi log ; (
j=l
homogeneous
of degree plus one, defined by:
logP”=&_Yilogpi i=
16)
1
The homogeneity property of demand implies that cjyij maintained by writing the model as: wi = ai +
(Yi
1
log 5 (
+~$sijlog(~)
+BilOS($)
1
= cjBij = 0. This can be
(7)
where the translog price index can now be written as: log P=log
P” - k(log
P”)’
+ ; k cY;(log pJ2
(8) Concavity at a point is satisfied if the (n - 1) X (n - 1) matrix @ = [O,,] is negative semidefinite. This condition can be maintained by reparameterizing the matrix 0 with the Cholesky decomposition (Lau, 1978). According to the version of this decomposition adopted by Diewert and Wales (1987) a necessary and sufficient condition for the matrix 0 to be negative semidefinite is that it can be written as 0 = - T’T, where T = [7ij] is an (n - 1) X (n - 1) upper triangular matrix (that is, 7ij = 0 for i > j). Hence, the desired curvature property can be maintained at a point by rewriting the fIjj parameters in terms of the ri, parameters such that 0 = - T’T. For example, if n = 5, the 4 X 4 matrix T can be written as:
(9)
354
G. Moschini/European
such that the matrix
0 is:
2 711
TllQ-I2
e2 @=
Economic Review 42 (1998) 349-364
+
711713
62
‘12’13
+
711714
‘lZ714
‘22’23
+
r22r24
TF3
(
+
T2”3 +
T3’3
symmetric)
Tl3Tl4
+
‘23’24
+
T33T34
r:, + 72”,+ T3’4+ r4’,
(10) This reparameterized AI model is concave by construction at the point p = x = 1 (the mean point). The cost of this procedure is that the resulting system is highly nonlinear in the parameters. The implementation of the model is facilitated if it is explicitly rewritten in terms of the 7ij parameters. By utilizing the structure exemplified in Eq. (IO), and factoring out parameters, the locally concave AI system can be written as: t-vi =
(Yi +
a;
log (;~)-~rJOgP:+/3iIog(;), ; s=l
where the aggregator prices, satisfy:
functions
i=1,2,...,n-1
P,‘, which are homogeneous
of degree zero in
(12) Thus, the first share in Eq. (11) involves only P;, the second share uses P; and Pi, and so on. Of course, the parameterization of the yij first in terms of the olj and then in terms of the 7ij, entails a particular structure of the translog price index P that deflates income in the AI model. Specifically: log P = log Pa - $(log
Pa)’
+ ;,e q(log t=l
pJ2 - ;y(log
P;)2
s=l (13)
It should be observed, however, that the original suggestion of Deaton and Muellbauer (1980) still applies, in the sense that one can always approximate this index before estimation, say by the Stone price index, with some simplification in the estimation procedure.
G. Moschini/ European Economic Review 42 (1998) 349-364
4. The semiflexible
355
AI demand system
The version of the AI model derived above is concave by construction at the point p = x = 1 (the mean point). However, one may be interested in a restricted version of this model for at least two reasons. First, the model as it stands may still have too many parameters. As discussed earlier, an appealing procedure in such a case is to restrict the substitution possibilities across goods, which here entails restricting the Bij, and therefore the rij. A systematic way of doing so was suggested by Diewert and Wales (1988) which entails restricting the rank of T (and therefore the rank of 0 = - T’T). Note that it is not necessary to restrict the rank of 0 to restrict, even severely, the substitution across goods. For example, the substitution-independence parameterization of the AI model proposed by Keller (1984) does not restrict the rank of the substitution matrix although it reduces the number of parameters estimating Sij from in(n. - 1) to 12. The solution utilized here, though, represents a somewhat agnostic way of constraining the substitution possibilities and has the added advantage of handling curvature properties. Furthermore, by allowing one to choose any rank K where 1 I K I n - 1, the procedure also has the advantage of being adaptable to specific needs. A second reason to consider a restricted version of the locally concave model arises from estimation considerations. When estimation of the unrestricted AI model of Eq. (1) or Eq. (7) yields results that violate local concavity, then estimation of the full locally concave model of Eq. (11) may be difficult. Violation of concavity means that some of the eigenvalues are positive. In such a case, estimation of the locally concave model, say by least squares or maximum likelihood, will tend to drive the eigenvalues that are positive in the unrestricted model to a value close to zero. In other words, estimation under the restriction of concavity is likely to yield a substitution matrix of less than full rank. When this happens, estimation of the (otherwise unrestricted) locally concave model breaks down because the linearized version of the model entails a singular design matrix. Alternatively, as Eq. (12) illustrates, if rSj = 0 for s > K, then the corresponding regressors log P,’ vanish. Thus, when unrestricted parameter estimates violate concavity, considering a model with a substitution matrix of rank K < (n - 1) may be useful to achieve convergence of the parameters of the locally concave model. 4 Diewert and Wales (1988) have shown that a semidefinite matrix of rank K has a K-column triangular decomposition. Based on this result, the rank of the substitution matrix of the locally concave AI model is restricted to a number
4 As a rule of thumb, one does not encounter convergence problems if K does not exceed the number of negative eigenvalues of the unrestricted model. Another point worth noting is that the Cholesky matrix T is not unique, and this fact may also have bearings on convergence when some of the eigenvalues of the curvature-constrained model are close to zero.
G. Moschini/European
356
Economic Reuiew 42 (1998) 349-364
1) by setting rij = 0 for all i > K. For the earlier example of n = 5, choosing K = 2 entails setting to zero the last two rows of the T matrix in Eq. (9). This yields three parameter restrictions which simplify the @ matrix as follows:
K < (n -
2 Q-11
711712
62
+
711713
‘12’13
G2
711714
+
r22723
T12714
+
‘22’24
+
72’3
‘13’14
+
‘23’24
@= -
(14) 7-:3
symmetric
By recalling the particularly AI system, the Semiflexible as: y=~i+cXilog
useful quasi-recursive structure of the locally concave Almost Ideal (SAI) system of rank K can be written
fi (
)
~73ilogP:cpilog(;) s=l
where the indices P,’ are defined in Eq. (12). Note that the restrictions all s > K imply log P,’ ES0 for all s > K.
5. An application:
(15) rsi = 0 for
Food demand in Canada
To illustrate the SAI demand system just derived, we present the results of a lo-good complete demand system. The application emphasizes food consumption. As discussed earlier, the AI system is likely to be of particular interest when modeling food demand. Specifically, 8 of the 10 specified goods represent food and beverages consumed at home by Canadians, whereas the other 2 remaining goods represent meals consumed away from home and all other goods consumption. Specifically, the 10 consumption categories are: q, = beef and pork; q2 = poultry and fish; q3 = dairy products; q4 = fruits; qs = vegetables; q6 = bread and bakery; q, = other food; q8 = nonalcoholic beverages; q9 = food away from home; and, ql,, = nonfood. Although the extreme aggregation assumption about the nonfood sector may be viewed as too restrictive, when interest centers on food demand, a full system with such restrictive features is still preferable to a system that ignores the nonfood sector (such as a second-stage demand system conditional on food expenditures). The resulting complete demand system yields unconditional demand elasticities that, unlike those obtained from conditional demand systems, are typically more suitable for policy and welfare analysis. Data on consumption and prices are available for Canada in terms of annual food expenditures from the system of national accounts of Statistics Canada. This
G. Moschini/
European Economic Review 42 (1998) 349-364
3.51
data set provides both current and constant-price consumer expenditures for fairly disaggregated food categories consumed at home. These food expenditure data were supplemented with corresponding food away from home data, and nonfood expenditure data, from Agriculture Canada’s Handbook of Food Expenditures, Prices and Consumption. Implicit price indices for each category were constructed by dividing current expenditures by constant-price expenditures. The entire data set used for the period 1961- 1988, including both current and constant-price expenditures at 1986 prices, is reported in Moschini and Moro (1993), who also provide other details on data construction. In general, over the sample period, the ‘real income’ of Canadian consumers has increased continuously. This increased purchasing power has been devoted mostly to the consumption of goods other than food. Thus, as one would expect from the typical consumption patterns of developed countries, food consumption in Canada constitutes a relatively small and decreasing component of income allocation. Before estimation, all prices and per-capita income were normalized by dividing through by the respective (geometric) mean. To allow for some preference change over the estimation period, the shares were augmented by a term K~Q, where ~~ are preference change parameters satisfying cj K~ = 0, and Q is a dummy variable that equals -0.5 for the first half of the sample period and 0.5 for the second half of the sample period. 5 5.1. Estimation The stochastic system of demand equations, with or without the concavity conditions (possibly of less than full rank) imposed, can be written as a standard system of seemingly unrelated regressions: y,=f(
p, Zt) +e,,
t= 1, L...,
T
(16)
where yt is a vector of (n - 1) shares at time t, 6 p is the vector of all coefficients to be estimated, Z, is the vector of the corresponding exogenous variables at time t, and e, is a vector of error terms. The random terms are assumed to be contemporaneously correlated but serially uncorrelated, in other words E[ e,] = 0 and E[ e,e:] = 0 (Vt), and E[ e,e:] = 0 (V t # s). It is common at this stage to assume further that the e,‘s are multinotmally distributed, leading to maximum likelihood estimation, which is invariant with respect to which equation is omitted. The maximum likelihood estimator of p minimizes the log of the determinant of the variance-covariance matrix 0( p) (see, for example, Davidson and MacKin-
5 This guarantees that wi = (Y~at the mean of all explanatory variables. An alternative to using the dummy variable Q, adopted by a number of authors, is to include a time trend in the share equations or, as suggested by a reviewer, a logistic-type trend. 6 One of the shares is omitted because of the well-known singularity problem of share equation systems.
G. Moschini/European
358
Economic Review 42 (1998) 349-364
non, 1993, p. 316). Thus, when any one equation contains more parameters than observations, the maximum likelihood estimator is not feasible because the log-likelihood function can be made arbitrarily large by making any one equation fit perfectly. ’ This problem typically occurs for our nonlinear AI model because all the substitution parameters enter all equations (through the translog price index that deflates income). A suitable procedure in this case is the minimum distance estimator originally analyzed by Malinvaud (1970). Given a sample of T observations, the method of minimum distance chooses p to minimize the least square deviations around a positive definite matrix SP’, that is:
The minimum distance estimator is consistent and asymptotically normal. In addition, if S is a consistent estimator of 0, the minimum distance estimator of p is asymptotically efficient as well, thus sharing the same properties as the maximum likelihood estimator (Chow, 1983). Minimizing the distance function of Eq. (17) boils down to minimizing the trace of a (transformed) variance-covariante matrix, rather than its determinant, and thus this estimator is suitable for large models with many parameters in all equations. Specifically, the following two-step procedure is used. First, choose a (n - 1) X (n - 1) matrix S by deleting one row and one column from the matrix S, defined as:
(18) where I,, is an n X n identity matrix, and I, is a n X 1 vector of ones. ’ Conditional on this matrix, the system of equations is estimated to find a vector of parameters 6 and of estimated residuals e”,= yr - f( ,LL,Z,). These residuals are used to construct an estimate of 0, say fi. The second step sets S = &! and minimizes the distance function conditional on this matrix. To economize on degrees of freedom, we adopt an estimator of the contemporaneous covariance matrix that depends only on n parameters (de Boer and Harkema, 1986). Specifically, the second step of the estimation procedure chooses S by deleting one row and one column from the following matrix, which is defined in terms of only IZ coefficients ai: S,=D,--
44 s
‘See the discussion in Deaton degrees-of-freedom problem because ’ Because the first step’s S matrix estimator is invariant with respect to
(19) (1986, pp. 1783-1786). Note that this is not necessarily a of the presence of cross-equations restrictions. treats all equations in a symmetric fashion, this minimum distance which share equation is omitted (Chavas and Segerson, 1987).
G. Moschini/ European Economic Review 42 (1998) 349-364 Table 1 Eigenvalues
359
of the Slutsky matrix at the mean point
SAI model of rank
Unrestricted AI model
1
2
3
4
5
6
7
- 0.075
- 0.080 -0.026 0 0 0 0 0 0 0
- 0.08 1 - 0.026 _ 0.014 0 0 0 0 0 0
- 0.082 - 0.026 -0.015 -0.012 0 0 0 0 0
- 0.083 - 0.027 -0.016 -0.013 -0.010 0 0 0 0
- 0.083 - 0.027 -0.017 -0.014 -0.010 - 0.003 0 0 0
- 0.084 - 0.027 -0.017 -0.015 - 0.010 - 0.006 - 0.003 0 0
-0.081 - 0.026 -0.015 -0.014 -0.010 - 0.006 - 0.002 0.003 0.008
where d,=[S,, S, ,..., S,y is an iz X 1 vector, 6 = c, 6, is a scalar, and D, is an n X n diagonal matrix with the coefficients ai on the diagonal. 9 To make the resulting matrix an appropriate covariance matrix, the Si parameters solve the nonlinear system of equations: +_
2
cy= ,si:
i= l,..., n (20) T ’ given the estimated Z residuals from the first step. de Boer and Harkema (1986) present a very efficient algorithm for the solution of Eq. (20) that relies on finding the unique real root of an equation in only one variable. Note that the diagonal elements of the matrix S, are simply the estimated variances from the first step least-squares residuals. 5.2. Results Estimation of the unrestricted AI model, carried out by using TSP 4.2, shows that the Slutsky matrix, at the mean point, does not satisfy the restrictions of concavity. Specifically, 7 eigenvalues are negative and 2 are positive. Table 1 reports the eigenvalues of the Slutsky matrix of all the SAI models up to rank 7, as well as those of the unrestricted model. Convergence could not be achieved for the rank 8 SAI model, nor for the full-rank locally concave AI model. From this table, it is clear that the SAI demand system of order K can approximate well the largest (in absolute value) negative eigenvalues of the Slutsky matrix of the unrestricted AI model. Table 2 reports the R* associated with each individual equation, for each model. Despite the fact that the model is expressed in share form, the fit is rather 9
The method used by Deaton and Muellbauer setting Si = CT’ for all i.
(1980) is a special case of this procedure,
obtained by
G. Moschini/ European Economic Review 42 (1998) 349-364
360
Table 2 R, for single eauations SAI models of rank
Unrestricted AI model
1
2
3
4
5
6
7
Beef and pork Poultry and fish Dairy Fruits Vegetables Bread and bakery Other food Beverages Food away from home Nonfood
0.96 0.71 0.94 0.97 0.71 0.99 0.99 0.46 0.86 0.98
0.98 0.54 0.94 0.97 0.89 0.99 0.99 0.55 0.87 0.98
0.98 0.58 0.97 0.97 0.90 0.99 0.99 0.59 0.89 0.98
0.98 0.61 0.97 0.97 0.93 0.99 0.99 0.70 0.89 0.98
0.98 0.58 0.97 0.97 0.93 0.99 0.99 0.72 0.89 0.98
0.98 0.59 0.97 0.97 0.93 0.99 0.99 0.76 0.89 0.98
0.98 0.59 0.97 0.97 0.93 0.99 0.99 0.76 0.89 0.98
0.98 0.60 0.98 0.97 0.94 0.99 0.99 0.80 0.89 0.98
Distance
330.7
245.6
210.0
187.8
180.8
178.5
178.3
169.8
36
44
51
57
62
66
69
72
No. of parameters
good for most of the reported models. In particular, the rank 4 SAI model seems to fit nearly as well as the rank 7 model. Table 2 also reports the minimized distance function and the number of parameters of each model. To make the distance function comparable, all models were estimated conditional on the same S matrix (specifically, the covariance matrix estimated from the first-step residuals of the unrestricted model). On the basis of the estimated minimized distance function, one can make some inference on the validity of the estimated models by using the quasi likelihood ratio (QLR) test. Consider the null hypothesis H,: ,g( p) = 0, where the function g( p) is (possibly) vector valued. Let fi denote the unrestricted minimum distance estimator and let p” denote the minimum distance estimator obtained under the restrictions g( p) = 0, where both estimators must be obtained conditional on the same matrix S (say, the estimated covariance matrix from the first-step of the unrestricted model). Denote the minimized distance for the two cases as L( @, S) and L( p*, S). Then the QLR, defined as:
QLR = L( PO,S) - L( ,G, s)
(21)
is asymptotically distributed as ,y* with r degrees of freedom under the null hypothesis, where Y is the number of restrictions imposed on p (Gallant and Jorgenson, 1979). The hypotheses of interest here are curvature and rank reduction. It is known that testing curvature per se is problematic because this condition does not shrink the parameter space (Lau, 1978). Hence, in this context it may be preferable to maintain the hypothesis of curvature, and test the hypothesis that the Slutsky substitution matrix is of less than full rank conditional on maintaining concavity of the expenditure function. Given the earlier discussion on the difficulty of estimat-
G. Moschini/ European Economic Review 42 (1998) 349-364 Table 3 Estimated
own-price
Hicksian
elasticities
at the mean point
SAI models of rank
Beef and pork Poultry and fish Dairy Fruits Vegetable Bread and bakery Other food Beverages Food away from home
361
Unrestricted AI model
1
2
3
4
5
6
7
- 0.06 - 0.01 -0.00 -0.01 -0.00 -0.01 - 0.03 -0.00 - 0.67 - 0.05
-0.21 - 1.01 - 0.02 -0.04 -0.51 - 0.03 - 0.05 -0.14 - 0.68 - 0.06
-0.19 -1.18 -0.27 - 0.07 - 0.60 - 0.23 -0.11 - 0.20 - 0.70 - 0.06
-0.21 -1.31 -0.28 - 0.08 - 0.69 - 0.29 -0.13 - 0.93 - 0.70 - 0.06
- 0.29 - 1.61 -0.27 -0.21 -0.71 - 0.30 - 0.28 - 0.99 -0.71 - 0.07
-0.32 - 1.64 -0.25 -0.22 - 0.73 -0.39 -0.31 - 1.19 -0.71 - 0.07
-0.33 - 1.64 - 0.47 - 0.23 - 0.73 - 0.43 -0.31 -1.19 - 0.72 - 0.07
-0.28 - 1.56 -0.39 - 0.01 - 0.64 -0.34 - 0.27 - 0.90 -0.70 - 0.06
ing a full rank locally concave model, in the present case one can take the rank 7 SAI model as the best representation of the locally concave AI model. Then, given the values of the minimized distance function reported in Table 2, the QLR criterion at the 5% significance level cannot reject the hypothesis of rank reduction for SAI models of rank 4, 5, and 6, whereas the hypothesis is rejected for the SAI models of rank less than 4. Of course, this testing strategy favors the null hypothesis of rank reduction, and other more neutral model selection criteria may be considered at this stage. However, it seems clear that in this application the SAI model provides an efficient and theoretically consistent way of reducing the parameter space of the AI demand system. Given that the semiflexible approach operates by reducing the rank of the
Table 4 Estimated
income elasticities
at the mean point
SAI models of rank
Beef and pork Poultry and fish Dairy Fruit Vegetables Bread and bakery Other food Beverages Food away from home Nonfood
Unrestricted AI model
1
2
3
4
5
6
I
0.19 0.39 0.01 0.16 0.43 _ -0.31 0.31 0.75 1.18 1.14
0.20 0.34 0.01 0.16 0.46 -0.30 0.30 0.77 1.18 1.14
0.19 0.25 0.13 0.11 0.36 -0.19 0.26 0.67 1.21 1.14
0.20 0.32 0.08 0.09 0.31 -0.16 0.26 0.88 1.20 1.14
0.18 0.39 0.07 0.13 0.33 -0.18 0.23 0.93 1.20 1.14
0.18 0.41 0.07 0.16 0.31 -0.14 0.22 0.84 1.20 1.14
0.19 0.41 0.10 0.15 0.32 -0.15 0.22 0.85 1.20 1.14
0.20 0.40 0.08 0.10 0.33 -0.18 0.21 0.94 1.20 1.14
G. Moschini/European
362
Table 5 Matrix of estimate Hicksian Elasticity of
41
q2 q3 q4
q5 q6 q7 q8 49 410 Mean share
elasticities
Economic Review 42 (1998) 349-364
at the mean point, rank 4 SAI model
With respect to the price
PI
P2
-0.21 0.59 0.18 - 0.05 -0.41 - 0.03 0.08 - 0.58 0.15 0.00
0.18 - 1.31 0.04 0.11 0.66 0.14 -0.11 0.06 0.04 0.00
0.03 1
0.010
0.12 0.08 -0.28 0.09 0.22 -0.11 0.03 0.60 -0.09 0.00 0.020
P4
P5
P6
P7
P8
P9
PI0
-0.02 0.13 0.05 -0.08 -0.19 0.06 -0.03 -0.01 -0.04 0.00
-0.15 0.82 0.13 -0.19 -0.69 0.10 0.00 - 0.09 0.00 0.00
-0.02 0.26 -0.10 0.09 0.15 -0.29 0.10 - 0.32 -0.10 0.01
0.10 - 0.45 0.06 -0.10 0.01 0.20 -0.13 0.23 - 0.08 0.00
-0.13 0.04 0.21 0.00 - 0.05 -0.12 0.04 -0.93 0.02 0.01
0.22 0.18 -0.20 -0.17 0.00 -0.24 -0.10 0.11 -0.70 0.05
-0.07 -0.36 -0.09 0.30 0.30 0.3 1 0.11 0.93 0.81 -0.06
0.012
0.012
0.018
0.037
0.007
0.045
0.808
Slutsky substitution matrix, it is of some interest to consider the effects of rank reduction on Hicksian elasticities. Table 3 reports all own-price Hicksian elasticities for all the estimated models. As in Diewert and Wales (1988), it seems that reducing the rank of the substitution matrix tends to reduce the (absolute) value of own-price elasticities. This is evident for the rank 1 model, which yields several own-price elasticities close to zero. However, the elasticities of the rank 5 SAI model are very close to those of the rank 7 model (the best representation of the locally concave AI model for this data set). It is also of some interest to see whether reducing the rank of the substitution matrix indirectly affects income elasticities. Table 4 reports income elasticities at the mean point associated with all the estimated models. Clearly, this indirect effect is minimal. This suggests that SAI models, possibly with small rank, may be desirable for applications in which the primary objective is estimating Engel curves (as is typically the case for studies of cross-section data, where price variation is limited). As explained earlier, because of its nonlinear Engel curves and aggregation properties, the AI model (and hence the SAI model) is particularly appealing for such applications. Finally, Table 5 reports the estimated Hicksian price elasticities, at the mean point, of the rank 4 SAI model (the most parsimonious semiflexible specification supported by the data). As expected, food demand is quite inelastic with respect to income. The average income elasticities of the first seven goods, which constitute all the food consumed at home, is 0.13. One of the commodities (bread and bakery) actually turns out to be an inferior good. Inspection of the price elasticities suggests that most goods are also inelastic with respect to own price. The only group to have elastic demand is that of poultry and fish. Compensated cross elasticities are small for most pairs of goods, indicating limited substitution possibilities in consumption. It is not clear how one classifies substitution possibil-
G. Moschini/
European Economic Review 42 (1998) 349-364
363
ities when many goods are involved. Blackorby and Russell (1989) suggest using the notion of Morishima elasticities of substitution. Calculation of such elasticities (not reported here, but easily computed from Table 5) indicates that most pairs of goods are substitutes.
6. Conclusion In this paper I have applied the notion of a ‘semiflexible’ functional form to the AI demand system of Deaton and Muellbauer (1980). In the process, I have illustrated a systematic way of maintaining concavity at a point for the AI demand system. Restricting the rank of the substitution matrix of such a locally concave AI demand model yields the SAI demand system. The gain in degrees of freedom permitted by this procedure inevitably comes at the cost of some restrictions on the substitution possibilities across goods. However, the resulting restricted substitution possibilities do not have to follow any obvious a priori pattern. Thus, the approach lets the data speak for themselves, subject to the constraint that the rank of the matrix be K < (n - 11. The SAI model was applied to a model of food demand in Canada, and the empirical results illustrate the usefulness of this demand system. Because of the appealing features and widespread popularity of the AI demand system, the procedure illustrated here is likely to be of interest in many empirical applications. In particular, the SAI demand system is recommended when one wants to maintain the theoretical property of concavity (albeit locally) in estimating a regular AI demand system. Such a problem is meaningful only if unrestricted estimation of the same violates concavity at the point of interest, in which case the necessity arises of constraining the ‘offending’ eigenvalues. Also, the SAI model may be useful when one wants to reduce the parametric space without abandoning completely the ‘flexibility’ properties of demand systems based on second order approximations, an objective that is likely to be desirable in relatively large demand systems. Finally, the SAI model may be of interest when one wants to account in a very parsimonious, but theoretically consistent, way for the effects of prices on consumption when the primary objective is to estimate Engel curves, as is often the case in demand analyses based on cross-section or panel data.
Acknowledgements This article was written while the author was visiting the University of Siena as part of the RAISA research project of Italy’s National Council of Research, the support of which is gratefully acknowledged. Thanks are due to Zuhair Hassan for his help with the data and to two anonymous referees for their constructive comments. This is Journal Paper No. J-16743 of the Iowa Agriculture and Home Economics Experiment Station, Project No. 3139.
364
G. Moschini/
European Economic Review 42 (1998) 349-364
References Blackorby, C., Russell, R.R., 1989. Will the real elasticity of substitution please stand up?. American Economic Review 79, 882-888. Blackorby, C., Primont, D., Russell, R.R., 1978. Duality, Separability, and Functional Structure: Theory and Economic Applications. North-Holland, New-York. Chalfant, J.A., Gray, R.S., White, K.J., 1991. Evaluating prior beliefs in a demand system: the case of meat demand in Canada. American Journal of Agricultural Economics 73, 4766490. Chavas, J.-P., Segerson, K., 1987. Stochastic specification and estimation of share equation systems. Journal of Econometrics 35, 337-358. Chow, G.C., 1983. Econometrics. McGraw-Hill, New-York. Davidson, R., MacKinnon, J.G., 1993. Estimation and Inference in Econometrics. Oxford University Press, Oxford. Deaton, A., 1986. Demand analysis. In: Griliches, Z., Intriligator, M.D. (Eds.), Handbook of Econometrics, Vol. III. Elsevier Science Publishers, New-York. Deaton, A., Muellbauer, J., 1980. An almost ideal demand system. American Economic Review 70, 312-326. de Boer, P.M.C., Harkema, R., 1986. Maximum likelihood estimation of sum-constrained linear models with insufficient observations. Economics Letters 20, 325-329. Diewert, W.E., Wales, T.J., 1987. Flexible functional forms and global curvature conditions. Econometrica 55, 43-68. Diewert, W.E., Wales, T.J., 1988. A normalized quadratic semiflexible functional form. Journal of Econometrics 37, 327-342. Gallant, A.R., Jorgenson, D.W., 1979. Statistical inference for a system of simultaneous, nonlinear, implicit equations in the context of instrumental variable estimation. Journal of Econometrics 11, 275-302. Jorgenson, D.W., Fraumeni, B.M., 1981. Relative prices and technical change. In: Bemdt, E., Field, B. (Eds.), Modeling and Measuring Natural Resource Substitution. MIT Press, Cambridge, MA. Keller, W.J., 1984. Some simple but flexible differential consumer demand systems, Economics Letters 16, 77-82. Lau, L.J., 1978. Testing and imposing monotonicity, convexity, and quasi-convexity constraints. In: Fuss, M., McFadden, D. (Eds.), Production Economics: A Dual Approach to Theory and Applications, Vol. 1. North-Holland, Amsterdam. Lewbel, A., 1987. AIDS, translog, and the Gorman polar form. Economics Letters 24, 161-163. Malinvaud, E., 1970. Statistical Methods of Econometrics, 2nd ed. North-Holland, Amsterdam. Moschini, G., Moro, D., 1993. A Food Demand System for Canada. Agriculture Canada Technical Report l/93, Ottawa, Canada.
European Economic Review 42 (1998) 349-364
The semiflexible almost ideal demand system Giancarlo Moschini Department
of Economics, Iowa State University, Ames, IA 50011, USA
Received 1 January 1995; accepted 1 October 1996
Abstract The concept of a semiflexible functional form is applied to the Almost Ideal (AI) demand system. This yields a demand model that is more parsimonious than standard ones while preserving a degree of flexibility, that satisfies the curvature property of concavity of the underlying expenditure function (at least locally), and that preserves the desirable properties of the AI model (aggregation across consumers and nonlinearity of the Engel curves). The model is illustrated with an application to a relatively large demand system emphasizing food consumption. 0 1998 Elsevier Science B.V. All rights reserved Keywords:
Curvature conditions; Demand systems; Flexible functional forms
1. Introduction The Almost Ideal (AI) demand system of Deaton and Muellbauer (1980) provides one of the most widely used models in applied demand analysis. The popularity of this model is related to its properties. Like most demand systems, the integrability properties of symmetry and homogeneity can be handled by simple parametric restrictions. In addition, a distinguishing feature of this model is that the resulting demand equations possess nonlinear Engel curves while at the same time allowing for exact aggregation across consumers. This attribute is due to the fact that the preferences underlying the AI model are of the ‘Generalized Gorman Polar Form’ (Blackorby et al., 1978), a class that includes some other popular models (such as a version of the translog demand system; see Lewbel, 1987). Furthermore, the particular (nonlinear) Engel curves of the AI model imply that, if a good is income inelastic (a necessity), an increase in income not only decreases 0014-2921/98/$19.00
0 1998 Elsevier Science B.V. All rights reserved.
PII SOO14-2921(97)00060-3
350
G. Moschini/European
Economic Reciew 42 (1998) 349-364
the share of income allocated to that good, but also reduces the income elasticity of that good. This type of nonlinear Engel curve has worked well in empirical studies, particularly those modeling food demand. Empirical applications of the AI demand system often encounter two distinct problems. First, the curvature property (concavity of the expenditure function, which implies that the Slutsky matrix is negative semidefinite) is not explicitly built into the model and is often violated by estimation results. ’ Second, the model becomes prohibitively demanding in terms of data as the number of goods being modeled increases. Like other standard flexible demand systems, the AI model possesses $(~z - l)(n + 4) p arameters when the number of goods is II (maintaining homogeneity and symmetry). Hence, the number of parameters to be estimated increases quadratically as the number of goods increases, whereas the number of effective observations increases only linearly. For large demand systems, this leads to a degrees-of-freedom problem, which may severely affect the statistical properties of the estimated model. The purpose of this paper is to illustrate a procedure to solve these two seemingly unrelated problems by applying the concept of a semiflexible functional form (Diewert and Wales, 1988) to the AI demand system. This procedure allows saving degrees of freedom by restricting the substitution possibilities across goods. This is achieved by restricting the rank of the Slutsky substitution matrix, which in general is (n - 1) for the case of iz goods, to a number K < (n - 1). Furthermore, because this rank restriction is handled within a parametric specification of the Slutsky matrix that maintains negative semidefiniteness (at a point), the imposition of the curvature property of concavity is naturally handled in this context. The model obtained, termed the Semiflexible Almost Ideal (SAI) demand system, preserves the distinguishing features of the original AI system. Thus, the system maintains both its aggregation properties across individuals and the desirable nonlinear shape of Engel curves associated with the AI model. The SAI model presented here may be compared with the Normalized Quadratic (NQ) semiflexible model of Diewert and Wales (1988). Because the NQ is derived from a Gorman Polar Form specification of the expenditure function, it also permits exact aggregation across consumers. Its main advantage, relative to the SAI model, is that it allows one to maintain globally the required concavity of the expenditure function. Unfortunately, the quasi-homotheticity of the underlying preferences entails linear Engel curves. Although these Engel curves are not necessarily born in the origin, they do have restrictive implications. In particular, the income elasticity of goods with inelastic demand (necessities) is forced to rise
’ To this author’s knowledge, only one paper (Chalfant et al., 1991) has maintained the curvature property within the AI demand system. Their semi-Bayesian approach is quite different, and computationally more demanding, than the procedure followed here.
G. Moschini/ European Economic Review 42 (1998) 349-364
351
as the income level increases. This feature is undesirable when modeling food demand, especially if the estimated model is to be used to forecast consumption under scenarios of considerable income changes. By contrast, the apparent disadvantage of the SAI model, relative to NQ, is that it does not allow one to maintain concavity in prices globally (i.e., for any possible price level). It should be clear, however, that this is not a failure of the AI parameterization per se, but a consequence of the Generalized Gorman Polar Form assumption.
2. The AI demand system The AI demand share form as:
system of Deaton
and Muellbauer
(1980) can be written
in
n
wi = ai +
c yijlog
pj + pi
log ; (
j=l
1
(1)
where wi z piqi(p, x> denote budget shares, pi denote prices, qi( p, x) are Marshallian demand functions, x is total expenditure on the n goods (income for short), and P is a translog price aggregator satisfying: n
1””
l”g(P)=cu,+CailogPi+~,~,~YijlogPilogPj i= 1 r=l1=1
(2)
For this system, homogeneity, adding-up, and symmetry hold if Cj oi = 1, Cjyij = Cirij = Ci pi = 0, and yii = rji. Hence, the theoretical properties of homogeneity, adding-up, and symmetry can be imposed by parametric restrictions and will hold globally (that is, for any level of the exogenously given prices and income). The property of negativity (concavity in prices of the underlying expenditure function), on the other hand, requires that the matrix of Slutsky substitution terms Sij = L?hi(p, u)/dpj, where h,(.) denote Hicksian demands, be negative semidefinite. It is easily verified that the Slutsky substitution terms for the AI are not constant but depend on prices and income. Thus, as will be detailed below, for the AI model the inequality restrictions of negativity can only hold locally (for some values of the exogenously given prices and income). The system as it stands is nonlinear in the parameters, but Deaton and Muellbauer (1980) suggested an attractive linear version, which is obtained by replacing the price aggregator P in Eq. (1) with a price index P * constructed before estimation. The resulting linear AI model is straightforward to estimate, and because P * typically tracks P very well, the empirical results thus obtained are usually indistinguishable from those of the nonlinear AI model. However, because estimation of nonlinear demand systems does not present insurmountable computational problems, here I neglect this simplification and concentrate on the AI model
G. Moschini/
352
European Economic Review 42 (1998) 349-364
in its nonlinear form. To estimate this model, it is necessary to recognize that (~a cannot be estimated because it is unidentified. Given that (~a is not necessary for the local flexibility properties of the model, in what follows I will set cy,,= 0 throughout. 2 The Slutsky substitution terms for the AI model can be written as:
sij =
yij
+
wiwj -
sijwi+ pi pj log
; (
)I
where Sjj is the Kronecker delta (aij = 1 for i =j and ajj = 0 for i Zj). From Eq. (3) it is clear that the only set of parametric restrictions guaranteeing that the matrix of Slutsky terms is globally negative semidefinite is yij = 0 (Vi, j) and pi = 0 (Vi>. Unfortunately, these restrictions reduce the AI system to a constant (Cobb-Douglas) share model and thus are not appealing. Alternatively, because the substitution terms are approximately equal to [yiJ + wiwj - Sijwi] and observing that the matrix [wiwj - 6ijwil is negative semidefinite as long as shares are nonnegative, the desired curvature property will likely be guaranteed for all price and income levels that generate nonnegative shares if the matrix [rjj] is negative semidefinite. A similar approach was followed by Jorgenson and Fraumeni (1981) to impose curvature on a translog production model. As illustrated by Diewert and Wales (19871, however, this method ends up imposing ‘too much concavity’, thus destroying the flexibility property of the model. Thus, I do not pursue these avenues, and limit the scope to maintaining the curvature property locally. Without loss of generality, let the point at which concavity is maintained be pi = x = 1. Typically, one will choose the sample mean, that is the point with the highest sample ‘information’, as the point to maintain concavity, and units can be arbitrarily chosen (i.e., the data can be scaled) such that pi = x = 1 at the mean point. Given the assumption (~a = 0, then at this point wi = (Y;. Thus, the substitution term at the mean point, say Bij = Si, (p =x = 11, can be written as: 6ij = yij + cqa; - Sij’yi
Hence, for concavity semidefinite. 3
(4)
at the desired point to hold, the matrix [ oij] must be negative
a In other words, I require P = 1 when p, = 1 for all i. In this application it was found that the relevant estimation results are unaffected by the arbitrary choice of (Y,,. 3 Alternatively, as noted by a reviewer, one may start with the expenditure function underlying the AI model, which can be written as C(p, u) = P(p)eUB(J’), where u is the utility index, P(p) is the function defined by Eq. (21, and B(p) = n:= 1pai. With (~a = 0, C( p. u) will be concave in p at the chosen reference point if P(p) is concave at that point, which is guaranteed by the conditions just discussed.
G. Moschini / European Economic Review 42 (1998) 349-364
353
3. The locally concave AI demand system To facilitate the imposition of concavity at the mean point, it is convenient to reparameterize the model by using Eq. (4) to rewrite the yij in terms of {e,,, ai}. This yields: n
wi = q
log $ (
+ cii
+ c 1
where P” is a price function,
eij log pj + pi log ; (
j=l
homogeneous
of degree plus one, defined by:
logP”=&_Yilogpi i=
16)
1
The homogeneity property of demand implies that cjyij maintained by writing the model as: wi = ai +
(Yi
1
log 5 (
+~$sijlog(~)
+BilOS($)
1
= cjBij = 0. This can be
(7)
where the translog price index can now be written as: log P=log
P” - k(log
P”)’
+ ; k cY;(log pJ2
(8) Concavity at a point is satisfied if the (n - 1) X (n - 1) matrix @ = [O,,] is negative semidefinite. This condition can be maintained by reparameterizing the matrix 0 with the Cholesky decomposition (Lau, 1978). According to the version of this decomposition adopted by Diewert and Wales (1987) a necessary and sufficient condition for the matrix 0 to be negative semidefinite is that it can be written as 0 = - T’T, where T = [7ij] is an (n - 1) X (n - 1) upper triangular matrix (that is, 7ij = 0 for i > j). Hence, the desired curvature property can be maintained at a point by rewriting the fIjj parameters in terms of the ri, parameters such that 0 = - T’T. For example, if n = 5, the 4 X 4 matrix T can be written as:
(9)
354
G. Moschini/European
such that the matrix
0 is:
2 711
TllQ-I2
e2 @=
Economic Review 42 (1998) 349-364
+
711713
62
‘12’13
+
711714
‘lZ714
‘22’23
+
r22r24
TF3
(
+
T2”3 +
T3’3
symmetric)
Tl3Tl4
+
‘23’24
+
T33T34
r:, + 72”,+ T3’4+ r4’,
(10) This reparameterized AI model is concave by construction at the point p = x = 1 (the mean point). The cost of this procedure is that the resulting system is highly nonlinear in the parameters. The implementation of the model is facilitated if it is explicitly rewritten in terms of the 7ij parameters. By utilizing the structure exemplified in Eq. (IO), and factoring out parameters, the locally concave AI system can be written as: t-vi =
(Yi +
a;
log (;~)-~rJOgP:+/3iIog(;), ; s=l
where the aggregator prices, satisfy:
functions
i=1,2,...,n-1
P,‘, which are homogeneous
of degree zero in
(12) Thus, the first share in Eq. (11) involves only P;, the second share uses P; and Pi, and so on. Of course, the parameterization of the yij first in terms of the olj and then in terms of the 7ij, entails a particular structure of the translog price index P that deflates income in the AI model. Specifically: log P = log Pa - $(log
Pa)’
+ ;,e q(log t=l
pJ2 - ;y(log
P;)2
s=l (13)
It should be observed, however, that the original suggestion of Deaton and Muellbauer (1980) still applies, in the sense that one can always approximate this index before estimation, say by the Stone price index, with some simplification in the estimation procedure.
G. Moschini/ European Economic Review 42 (1998) 349-364
4. The semiflexible
355
AI demand system
The version of the AI model derived above is concave by construction at the point p = x = 1 (the mean point). However, one may be interested in a restricted version of this model for at least two reasons. First, the model as it stands may still have too many parameters. As discussed earlier, an appealing procedure in such a case is to restrict the substitution possibilities across goods, which here entails restricting the Bij, and therefore the rij. A systematic way of doing so was suggested by Diewert and Wales (1988) which entails restricting the rank of T (and therefore the rank of 0 = - T’T). Note that it is not necessary to restrict the rank of 0 to restrict, even severely, the substitution across goods. For example, the substitution-independence parameterization of the AI model proposed by Keller (1984) does not restrict the rank of the substitution matrix although it reduces the number of parameters estimating Sij from in(n. - 1) to 12. The solution utilized here, though, represents a somewhat agnostic way of constraining the substitution possibilities and has the added advantage of handling curvature properties. Furthermore, by allowing one to choose any rank K where 1 I K I n - 1, the procedure also has the advantage of being adaptable to specific needs. A second reason to consider a restricted version of the locally concave model arises from estimation considerations. When estimation of the unrestricted AI model of Eq. (1) or Eq. (7) yields results that violate local concavity, then estimation of the full locally concave model of Eq. (11) may be difficult. Violation of concavity means that some of the eigenvalues are positive. In such a case, estimation of the locally concave model, say by least squares or maximum likelihood, will tend to drive the eigenvalues that are positive in the unrestricted model to a value close to zero. In other words, estimation under the restriction of concavity is likely to yield a substitution matrix of less than full rank. When this happens, estimation of the (otherwise unrestricted) locally concave model breaks down because the linearized version of the model entails a singular design matrix. Alternatively, as Eq. (12) illustrates, if rSj = 0 for s > K, then the corresponding regressors log P,’ vanish. Thus, when unrestricted parameter estimates violate concavity, considering a model with a substitution matrix of rank K < (n - 1) may be useful to achieve convergence of the parameters of the locally concave model. 4 Diewert and Wales (1988) have shown that a semidefinite matrix of rank K has a K-column triangular decomposition. Based on this result, the rank of the substitution matrix of the locally concave AI model is restricted to a number
4 As a rule of thumb, one does not encounter convergence problems if K does not exceed the number of negative eigenvalues of the unrestricted model. Another point worth noting is that the Cholesky matrix T is not unique, and this fact may also have bearings on convergence when some of the eigenvalues of the curvature-constrained model are close to zero.
G. Moschini/European
356
Economic Reuiew 42 (1998) 349-364
1) by setting rij = 0 for all i > K. For the earlier example of n = 5, choosing K = 2 entails setting to zero the last two rows of the T matrix in Eq. (9). This yields three parameter restrictions which simplify the @ matrix as follows:
K < (n -
2 Q-11
711712
62
+
711713
‘12’13
G2
711714
+
r22723
T12714
+
‘22’24
+
72’3
‘13’14
+
‘23’24
@= -
(14) 7-:3
symmetric
By recalling the particularly AI system, the Semiflexible as: y=~i+cXilog
useful quasi-recursive structure of the locally concave Almost Ideal (SAI) system of rank K can be written
fi (
)
~73ilogP:cpilog(;) s=l
where the indices P,’ are defined in Eq. (12). Note that the restrictions all s > K imply log P,’ ES0 for all s > K.
5. An application:
(15) rsi = 0 for
Food demand in Canada
To illustrate the SAI demand system just derived, we present the results of a lo-good complete demand system. The application emphasizes food consumption. As discussed earlier, the AI system is likely to be of particular interest when modeling food demand. Specifically, 8 of the 10 specified goods represent food and beverages consumed at home by Canadians, whereas the other 2 remaining goods represent meals consumed away from home and all other goods consumption. Specifically, the 10 consumption categories are: q, = beef and pork; q2 = poultry and fish; q3 = dairy products; q4 = fruits; qs = vegetables; q6 = bread and bakery; q, = other food; q8 = nonalcoholic beverages; q9 = food away from home; and, ql,, = nonfood. Although the extreme aggregation assumption about the nonfood sector may be viewed as too restrictive, when interest centers on food demand, a full system with such restrictive features is still preferable to a system that ignores the nonfood sector (such as a second-stage demand system conditional on food expenditures). The resulting complete demand system yields unconditional demand elasticities that, unlike those obtained from conditional demand systems, are typically more suitable for policy and welfare analysis. Data on consumption and prices are available for Canada in terms of annual food expenditures from the system of national accounts of Statistics Canada. This
G. Moschini/
European Economic Review 42 (1998) 349-364
3.51
data set provides both current and constant-price consumer expenditures for fairly disaggregated food categories consumed at home. These food expenditure data were supplemented with corresponding food away from home data, and nonfood expenditure data, from Agriculture Canada’s Handbook of Food Expenditures, Prices and Consumption. Implicit price indices for each category were constructed by dividing current expenditures by constant-price expenditures. The entire data set used for the period 1961- 1988, including both current and constant-price expenditures at 1986 prices, is reported in Moschini and Moro (1993), who also provide other details on data construction. In general, over the sample period, the ‘real income’ of Canadian consumers has increased continuously. This increased purchasing power has been devoted mostly to the consumption of goods other than food. Thus, as one would expect from the typical consumption patterns of developed countries, food consumption in Canada constitutes a relatively small and decreasing component of income allocation. Before estimation, all prices and per-capita income were normalized by dividing through by the respective (geometric) mean. To allow for some preference change over the estimation period, the shares were augmented by a term K~Q, where ~~ are preference change parameters satisfying cj K~ = 0, and Q is a dummy variable that equals -0.5 for the first half of the sample period and 0.5 for the second half of the sample period. 5 5.1. Estimation The stochastic system of demand equations, with or without the concavity conditions (possibly of less than full rank) imposed, can be written as a standard system of seemingly unrelated regressions: y,=f(
p, Zt) +e,,
t= 1, L...,
T
(16)
where yt is a vector of (n - 1) shares at time t, 6 p is the vector of all coefficients to be estimated, Z, is the vector of the corresponding exogenous variables at time t, and e, is a vector of error terms. The random terms are assumed to be contemporaneously correlated but serially uncorrelated, in other words E[ e,] = 0 and E[ e,e:] = 0 (Vt), and E[ e,e:] = 0 (V t # s). It is common at this stage to assume further that the e,‘s are multinotmally distributed, leading to maximum likelihood estimation, which is invariant with respect to which equation is omitted. The maximum likelihood estimator of p minimizes the log of the determinant of the variance-covariance matrix 0( p) (see, for example, Davidson and MacKin-
5 This guarantees that wi = (Y~at the mean of all explanatory variables. An alternative to using the dummy variable Q, adopted by a number of authors, is to include a time trend in the share equations or, as suggested by a reviewer, a logistic-type trend. 6 One of the shares is omitted because of the well-known singularity problem of share equation systems.
G. Moschini/European
358
Economic Review 42 (1998) 349-364
non, 1993, p. 316). Thus, when any one equation contains more parameters than observations, the maximum likelihood estimator is not feasible because the log-likelihood function can be made arbitrarily large by making any one equation fit perfectly. ’ This problem typically occurs for our nonlinear AI model because all the substitution parameters enter all equations (through the translog price index that deflates income). A suitable procedure in this case is the minimum distance estimator originally analyzed by Malinvaud (1970). Given a sample of T observations, the method of minimum distance chooses p to minimize the least square deviations around a positive definite matrix SP’, that is:
The minimum distance estimator is consistent and asymptotically normal. In addition, if S is a consistent estimator of 0, the minimum distance estimator of p is asymptotically efficient as well, thus sharing the same properties as the maximum likelihood estimator (Chow, 1983). Minimizing the distance function of Eq. (17) boils down to minimizing the trace of a (transformed) variance-covariante matrix, rather than its determinant, and thus this estimator is suitable for large models with many parameters in all equations. Specifically, the following two-step procedure is used. First, choose a (n - 1) X (n - 1) matrix S by deleting one row and one column from the matrix S, defined as:
(18) where I,, is an n X n identity matrix, and I, is a n X 1 vector of ones. ’ Conditional on this matrix, the system of equations is estimated to find a vector of parameters 6 and of estimated residuals e”,= yr - f( ,LL,Z,). These residuals are used to construct an estimate of 0, say fi. The second step sets S = &! and minimizes the distance function conditional on this matrix. To economize on degrees of freedom, we adopt an estimator of the contemporaneous covariance matrix that depends only on n parameters (de Boer and Harkema, 1986). Specifically, the second step of the estimation procedure chooses S by deleting one row and one column from the following matrix, which is defined in terms of only IZ coefficients ai: S,=D,--
44 s
‘See the discussion in Deaton degrees-of-freedom problem because ’ Because the first step’s S matrix estimator is invariant with respect to
(19) (1986, pp. 1783-1786). Note that this is not necessarily a of the presence of cross-equations restrictions. treats all equations in a symmetric fashion, this minimum distance which share equation is omitted (Chavas and Segerson, 1987).
G. Moschini/ European Economic Review 42 (1998) 349-364 Table 1 Eigenvalues
359
of the Slutsky matrix at the mean point
SAI model of rank
Unrestricted AI model
1
2
3
4
5
6
7
- 0.075
- 0.080 -0.026 0 0 0 0 0 0 0
- 0.08 1 - 0.026 _ 0.014 0 0 0 0 0 0
- 0.082 - 0.026 -0.015 -0.012 0 0 0 0 0
- 0.083 - 0.027 -0.016 -0.013 -0.010 0 0 0 0
- 0.083 - 0.027 -0.017 -0.014 -0.010 - 0.003 0 0 0
- 0.084 - 0.027 -0.017 -0.015 - 0.010 - 0.006 - 0.003 0 0
-0.081 - 0.026 -0.015 -0.014 -0.010 - 0.006 - 0.002 0.003 0.008
where d,=[S,, S, ,..., S,y is an iz X 1 vector, 6 = c, 6, is a scalar, and D, is an n X n diagonal matrix with the coefficients ai on the diagonal. 9 To make the resulting matrix an appropriate covariance matrix, the Si parameters solve the nonlinear system of equations: +_
2
cy= ,si:
i= l,..., n (20) T ’ given the estimated Z residuals from the first step. de Boer and Harkema (1986) present a very efficient algorithm for the solution of Eq. (20) that relies on finding the unique real root of an equation in only one variable. Note that the diagonal elements of the matrix S, are simply the estimated variances from the first step least-squares residuals. 5.2. Results Estimation of the unrestricted AI model, carried out by using TSP 4.2, shows that the Slutsky matrix, at the mean point, does not satisfy the restrictions of concavity. Specifically, 7 eigenvalues are negative and 2 are positive. Table 1 reports the eigenvalues of the Slutsky matrix of all the SAI models up to rank 7, as well as those of the unrestricted model. Convergence could not be achieved for the rank 8 SAI model, nor for the full-rank locally concave AI model. From this table, it is clear that the SAI demand system of order K can approximate well the largest (in absolute value) negative eigenvalues of the Slutsky matrix of the unrestricted AI model. Table 2 reports the R* associated with each individual equation, for each model. Despite the fact that the model is expressed in share form, the fit is rather 9
The method used by Deaton and Muellbauer setting Si = CT’ for all i.
(1980) is a special case of this procedure,
obtained by
G. Moschini/ European Economic Review 42 (1998) 349-364
360
Table 2 R, for single eauations SAI models of rank
Unrestricted AI model
1
2
3
4
5
6
7
Beef and pork Poultry and fish Dairy Fruits Vegetables Bread and bakery Other food Beverages Food away from home Nonfood
0.96 0.71 0.94 0.97 0.71 0.99 0.99 0.46 0.86 0.98
0.98 0.54 0.94 0.97 0.89 0.99 0.99 0.55 0.87 0.98
0.98 0.58 0.97 0.97 0.90 0.99 0.99 0.59 0.89 0.98
0.98 0.61 0.97 0.97 0.93 0.99 0.99 0.70 0.89 0.98
0.98 0.58 0.97 0.97 0.93 0.99 0.99 0.72 0.89 0.98
0.98 0.59 0.97 0.97 0.93 0.99 0.99 0.76 0.89 0.98
0.98 0.59 0.97 0.97 0.93 0.99 0.99 0.76 0.89 0.98
0.98 0.60 0.98 0.97 0.94 0.99 0.99 0.80 0.89 0.98
Distance
330.7
245.6
210.0
187.8
180.8
178.5
178.3
169.8
36
44
51
57
62
66
69
72
No. of parameters
good for most of the reported models. In particular, the rank 4 SAI model seems to fit nearly as well as the rank 7 model. Table 2 also reports the minimized distance function and the number of parameters of each model. To make the distance function comparable, all models were estimated conditional on the same S matrix (specifically, the covariance matrix estimated from the first-step residuals of the unrestricted model). On the basis of the estimated minimized distance function, one can make some inference on the validity of the estimated models by using the quasi likelihood ratio (QLR) test. Consider the null hypothesis H,: ,g( p) = 0, where the function g( p) is (possibly) vector valued. Let fi denote the unrestricted minimum distance estimator and let p” denote the minimum distance estimator obtained under the restrictions g( p) = 0, where both estimators must be obtained conditional on the same matrix S (say, the estimated covariance matrix from the first-step of the unrestricted model). Denote the minimized distance for the two cases as L( @, S) and L( p*, S). Then the QLR, defined as:
QLR = L( PO,S) - L( ,G, s)
(21)
is asymptotically distributed as ,y* with r degrees of freedom under the null hypothesis, where Y is the number of restrictions imposed on p (Gallant and Jorgenson, 1979). The hypotheses of interest here are curvature and rank reduction. It is known that testing curvature per se is problematic because this condition does not shrink the parameter space (Lau, 1978). Hence, in this context it may be preferable to maintain the hypothesis of curvature, and test the hypothesis that the Slutsky substitution matrix is of less than full rank conditional on maintaining concavity of the expenditure function. Given the earlier discussion on the difficulty of estimat-
G. Moschini/ European Economic Review 42 (1998) 349-364 Table 3 Estimated
own-price
Hicksian
elasticities
at the mean point
SAI models of rank
Beef and pork Poultry and fish Dairy Fruits Vegetable Bread and bakery Other food Beverages Food away from home
361
Unrestricted AI model
1
2
3
4
5
6
7
- 0.06 - 0.01 -0.00 -0.01 -0.00 -0.01 - 0.03 -0.00 - 0.67 - 0.05
-0.21 - 1.01 - 0.02 -0.04 -0.51 - 0.03 - 0.05 -0.14 - 0.68 - 0.06
-0.19 -1.18 -0.27 - 0.07 - 0.60 - 0.23 -0.11 - 0.20 - 0.70 - 0.06
-0.21 -1.31 -0.28 - 0.08 - 0.69 - 0.29 -0.13 - 0.93 - 0.70 - 0.06
- 0.29 - 1.61 -0.27 -0.21 -0.71 - 0.30 - 0.28 - 0.99 -0.71 - 0.07
-0.32 - 1.64 -0.25 -0.22 - 0.73 -0.39 -0.31 - 1.19 -0.71 - 0.07
-0.33 - 1.64 - 0.47 - 0.23 - 0.73 - 0.43 -0.31 -1.19 - 0.72 - 0.07
-0.28 - 1.56 -0.39 - 0.01 - 0.64 -0.34 - 0.27 - 0.90 -0.70 - 0.06
ing a full rank locally concave model, in the present case one can take the rank 7 SAI model as the best representation of the locally concave AI model. Then, given the values of the minimized distance function reported in Table 2, the QLR criterion at the 5% significance level cannot reject the hypothesis of rank reduction for SAI models of rank 4, 5, and 6, whereas the hypothesis is rejected for the SAI models of rank less than 4. Of course, this testing strategy favors the null hypothesis of rank reduction, and other more neutral model selection criteria may be considered at this stage. However, it seems clear that in this application the SAI model provides an efficient and theoretically consistent way of reducing the parameter space of the AI demand system. Given that the semiflexible approach operates by reducing the rank of the
Table 4 Estimated
income elasticities
at the mean point
SAI models of rank
Beef and pork Poultry and fish Dairy Fruit Vegetables Bread and bakery Other food Beverages Food away from home Nonfood
Unrestricted AI model
1
2
3
4
5
6
I
0.19 0.39 0.01 0.16 0.43 _ -0.31 0.31 0.75 1.18 1.14
0.20 0.34 0.01 0.16 0.46 -0.30 0.30 0.77 1.18 1.14
0.19 0.25 0.13 0.11 0.36 -0.19 0.26 0.67 1.21 1.14
0.20 0.32 0.08 0.09 0.31 -0.16 0.26 0.88 1.20 1.14
0.18 0.39 0.07 0.13 0.33 -0.18 0.23 0.93 1.20 1.14
0.18 0.41 0.07 0.16 0.31 -0.14 0.22 0.84 1.20 1.14
0.19 0.41 0.10 0.15 0.32 -0.15 0.22 0.85 1.20 1.14
0.20 0.40 0.08 0.10 0.33 -0.18 0.21 0.94 1.20 1.14
G. Moschini/European
362
Table 5 Matrix of estimate Hicksian Elasticity of
41
q2 q3 q4
q5 q6 q7 q8 49 410 Mean share
elasticities
Economic Review 42 (1998) 349-364
at the mean point, rank 4 SAI model
With respect to the price
PI
P2
-0.21 0.59 0.18 - 0.05 -0.41 - 0.03 0.08 - 0.58 0.15 0.00
0.18 - 1.31 0.04 0.11 0.66 0.14 -0.11 0.06 0.04 0.00
0.03 1
0.010
0.12 0.08 -0.28 0.09 0.22 -0.11 0.03 0.60 -0.09 0.00 0.020
P4
P5
P6
P7
P8
P9
PI0
-0.02 0.13 0.05 -0.08 -0.19 0.06 -0.03 -0.01 -0.04 0.00
-0.15 0.82 0.13 -0.19 -0.69 0.10 0.00 - 0.09 0.00 0.00
-0.02 0.26 -0.10 0.09 0.15 -0.29 0.10 - 0.32 -0.10 0.01
0.10 - 0.45 0.06 -0.10 0.01 0.20 -0.13 0.23 - 0.08 0.00
-0.13 0.04 0.21 0.00 - 0.05 -0.12 0.04 -0.93 0.02 0.01
0.22 0.18 -0.20 -0.17 0.00 -0.24 -0.10 0.11 -0.70 0.05
-0.07 -0.36 -0.09 0.30 0.30 0.3 1 0.11 0.93 0.81 -0.06
0.012
0.012
0.018
0.037
0.007
0.045
0.808
Slutsky substitution matrix, it is of some interest to consider the effects of rank reduction on Hicksian elasticities. Table 3 reports all own-price Hicksian elasticities for all the estimated models. As in Diewert and Wales (1988), it seems that reducing the rank of the substitution matrix tends to reduce the (absolute) value of own-price elasticities. This is evident for the rank 1 model, which yields several own-price elasticities close to zero. However, the elasticities of the rank 5 SAI model are very close to those of the rank 7 model (the best representation of the locally concave AI model for this data set). It is also of some interest to see whether reducing the rank of the substitution matrix indirectly affects income elasticities. Table 4 reports income elasticities at the mean point associated with all the estimated models. Clearly, this indirect effect is minimal. This suggests that SAI models, possibly with small rank, may be desirable for applications in which the primary objective is estimating Engel curves (as is typically the case for studies of cross-section data, where price variation is limited). As explained earlier, because of its nonlinear Engel curves and aggregation properties, the AI model (and hence the SAI model) is particularly appealing for such applications. Finally, Table 5 reports the estimated Hicksian price elasticities, at the mean point, of the rank 4 SAI model (the most parsimonious semiflexible specification supported by the data). As expected, food demand is quite inelastic with respect to income. The average income elasticities of the first seven goods, which constitute all the food consumed at home, is 0.13. One of the commodities (bread and bakery) actually turns out to be an inferior good. Inspection of the price elasticities suggests that most goods are also inelastic with respect to own price. The only group to have elastic demand is that of poultry and fish. Compensated cross elasticities are small for most pairs of goods, indicating limited substitution possibilities in consumption. It is not clear how one classifies substitution possibil-
G. Moschini/
European Economic Review 42 (1998) 349-364
363
ities when many goods are involved. Blackorby and Russell (1989) suggest using the notion of Morishima elasticities of substitution. Calculation of such elasticities (not reported here, but easily computed from Table 5) indicates that most pairs of goods are substitutes.
6. Conclusion In this paper I have applied the notion of a ‘semiflexible’ functional form to the AI demand system of Deaton and Muellbauer (1980). In the process, I have illustrated a systematic way of maintaining concavity at a point for the AI demand system. Restricting the rank of the substitution matrix of such a locally concave AI demand model yields the SAI demand system. The gain in degrees of freedom permitted by this procedure inevitably comes at the cost of some restrictions on the substitution possibilities across goods. However, the resulting restricted substitution possibilities do not have to follow any obvious a priori pattern. Thus, the approach lets the data speak for themselves, subject to the constraint that the rank of the matrix be K < (n - 11. The SAI model was applied to a model of food demand in Canada, and the empirical results illustrate the usefulness of this demand system. Because of the appealing features and widespread popularity of the AI demand system, the procedure illustrated here is likely to be of interest in many empirical applications. In particular, the SAI demand system is recommended when one wants to maintain the theoretical property of concavity (albeit locally) in estimating a regular AI demand system. Such a problem is meaningful only if unrestricted estimation of the same violates concavity at the point of interest, in which case the necessity arises of constraining the ‘offending’ eigenvalues. Also, the SAI model may be useful when one wants to reduce the parametric space without abandoning completely the ‘flexibility’ properties of demand systems based on second order approximations, an objective that is likely to be desirable in relatively large demand systems. Finally, the SAI model may be of interest when one wants to account in a very parsimonious, but theoretically consistent, way for the effects of prices on consumption when the primary objective is to estimate Engel curves, as is often the case in demand analyses based on cross-section or panel data.
Acknowledgements This article was written while the author was visiting the University of Siena as part of the RAISA research project of Italy’s National Council of Research, the support of which is gratefully acknowledged. Thanks are due to Zuhair Hassan for his help with the data and to two anonymous referees for their constructive comments. This is Journal Paper No. J-16743 of the Iowa Agriculture and Home Economics Experiment Station, Project No. 3139.
364
G. Moschini/
European Economic Review 42 (1998) 349-364
References Blackorby, C., Russell, R.R., 1989. Will the real elasticity of substitution please stand up?. American Economic Review 79, 882-888. Blackorby, C., Primont, D., Russell, R.R., 1978. Duality, Separability, and Functional Structure: Theory and Economic Applications. North-Holland, New-York. Chalfant, J.A., Gray, R.S., White, K.J., 1991. Evaluating prior beliefs in a demand system: the case of meat demand in Canada. American Journal of Agricultural Economics 73, 4766490. Chavas, J.-P., Segerson, K., 1987. Stochastic specification and estimation of share equation systems. Journal of Econometrics 35, 337-358. Chow, G.C., 1983. Econometrics. McGraw-Hill, New-York. Davidson, R., MacKinnon, J.G., 1993. Estimation and Inference in Econometrics. Oxford University Press, Oxford. Deaton, A., 1986. Demand analysis. In: Griliches, Z., Intriligator, M.D. (Eds.), Handbook of Econometrics, Vol. III. Elsevier Science Publishers, New-York. Deaton, A., Muellbauer, J., 1980. An almost ideal demand system. American Economic Review 70, 312-326. de Boer, P.M.C., Harkema, R., 1986. Maximum likelihood estimation of sum-constrained linear models with insufficient observations. Economics Letters 20, 325-329. Diewert, W.E., Wales, T.J., 1987. Flexible functional forms and global curvature conditions. Econometrica 55, 43-68. Diewert, W.E., Wales, T.J., 1988. A normalized quadratic semiflexible functional form. Journal of Econometrics 37, 327-342. Gallant, A.R., Jorgenson, D.W., 1979. Statistical inference for a system of simultaneous, nonlinear, implicit equations in the context of instrumental variable estimation. Journal of Econometrics 11, 275-302. Jorgenson, D.W., Fraumeni, B.M., 1981. Relative prices and technical change. In: Bemdt, E., Field, B. (Eds.), Modeling and Measuring Natural Resource Substitution. MIT Press, Cambridge, MA. Keller, W.J., 1984. Some simple but flexible differential consumer demand systems, Economics Letters 16, 77-82. Lau, L.J., 1978. Testing and imposing monotonicity, convexity, and quasi-convexity constraints. In: Fuss, M., McFadden, D. (Eds.), Production Economics: A Dual Approach to Theory and Applications, Vol. 1. North-Holland, Amsterdam. Lewbel, A., 1987. AIDS, translog, and the Gorman polar form. Economics Letters 24, 161-163. Malinvaud, E., 1970. Statistical Methods of Econometrics, 2nd ed. North-Holland, Amsterdam. Moschini, G., Moro, D., 1993. A Food Demand System for Canada. Agriculture Canada Technical Report l/93, Ottawa, Canada.