The use of conditional cost functions to Generate estimable mixed demand systems.

and the term [partial derivative] [U.sup.m] /[partial derivative] [p'.sub.Ai] can be rewritten as -([partial derivative] [C.sup.h] / [partial derivative] [p'.sub.Ai]) / [partial derivative] [C.sup.h] / [partial derivative] u), which is obtained by differentiating the identity c = [C.sup.h] [[p.sub.A], [x.sub.B], [U.sup.m] ([p.sub.A], [x.sub.B], c)] with respect to [p'.sub.Ai]. Multiplying through (iii) by [p'.sub.Ai] / [x.sub.Ai] and rearranging gives:

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is the Marshallian cross price elasticity equation.

(5) In Moschini and Rizzi (2006), the second terms of F([p.sub.A], [x.sub.B]) and G([p.sub.A], [x.sub.B]) are written as ([[summation]'.sub.i], [[alpha]'.sub.i] [p'.sub.Ai]). [[summation]'.sub.j], [[mu]'.sub.j] [x'.sub.Bj]) and [[summation]'.sub.i], [[alpha]'.sub.i] [p'.sub.Ai]). [[summation]'.sub.j], [[phi]'.sub.j] [x'.sub.Bj]), respectively, where [[phi].sub.j] ([not equal to][[mu].sub.j]) are parameters.

(6) According to Blackorby, Davidson, and Schworm (1991), the direct utility function U([x.sub.A], [x.sub.B]) is implicitly separable with respect to Partition [??] if and only if there exists an implicit representation of U as [T.sub.A]([x.sub.A], u) = [T.sub.B]([x.sub.B], u), which implies that the corresponding conditional cost function can be written in the form

(v) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Clearly an implicitly separable mixed demand functions depend directly on u through their utility argument, but they also depend indirectly on u through the [T.sub.B] functions in their quantity index argument. Furthermore, this structure is identical to Deaton and Muellbauer's (1980b) implicitly separable preference structure when goods are patitioned into S groups with price subvectors ([p.sup.1.sub.7], ..., [p.sup.S]).

(7) These empirical mixed demand systems were constructed under the condition that the fish and meat product groups are directly and weakly separable from other commodities. This separability assumption needs to be held with an aim of keeping the estimation process manageable by merely dealing with certain aspects of the static demand model.

(8) See Holt (1998) for alternative autocorrelation parameterizations.

(9) For reasons of brevity, the detailed parameter estimates of the NQM, QAIMDS, and ISNCES are not reported below but are available on line as Readers' Appendix A at http://au.geocities. com/garywong21/parameter.pdf.

(10) This article adopts Fisher, Fleissig, and Serletis' (2001) approach to calculate the approximate LM test statistics (equivalent to score test statistics in the case of linear models) for autocorrelation. As pointed out by a reviewer, the models being examined here are non-linear in the variables, and thus the adopted approach used to construct the LM test statistics may not be fully appropriate (see Eitrheim and Terasvirta 1996). Future research on deriving and estimating mixed demand systems may focus on this issue, but such an investigation goes well beyond the scope of this study.

(11) The curvature conditions of the mixed demand models were checked by calculating the eigenvalues of the matrices of compensated price and quantity effects. Results indicate that the NQM and QAIMDS fail to satisfy the curvature conditions for some observations in the sample period, while the ISNCES satisfies the curvature properties over the whole sample period. This leads to the conclusion that the ISNCES is regular and therefore may be used to compute the exact welfare measures. Though constrained estimation (as has been done by Moschini [1998]) could be a simple option for NQM and QAIMDS, it is of course possible that the regularity problem may be due to the level of aggregation of the data, and more investigation along these lines would be justified in searching for a regular yet flexible representation of the data.

(12) Similar interpretations apply for fresh fish and shellfish.

K. K. Gary Wong is assistant professor, Department of Economics, The University of Macau, Macau SAR, China. Hoanjae Park is associate professor, Department of Economics, Catholic University of Daegu, Daegu, South Korea. The authors thank two anonymous reviewers for helpful comments. Any remaining errors and omissions are the sole responsibility of the authors. Table 1. Single Equation and System Measures of Fit

NQM QAIMDS ISNCES No. of free parameters 24 17 12 [R.sup.2] Salted fish 0.949 0.954 0.960 Processed meat 0.946 0.947 0.943 Fillet 0.444 0.385 0.367 Fresh meat 0.707 0.741 0.700 Fresh fish 0.858 0.896 0.864 Shellfish 0.982 0.978 0.980 L 1,631.320 1,641.731 1,631.303 SC -40.445 -40.795 -40.874 AIC -41.204 -41.333 -41.349 HQC -41.386 -41.462 -41.462

Residual diagnostics Durbin-Watson statistics Salted fish 2.528 2.359 2.692 Processed meat 2.849 2.865 2.735 Fillet 1.910 2.567 2.287 Fresh meat 2.619 2.417 2.639 Fresh fish 2.633 2.391 2.664 Shellfish 2.191 2.253 2.328 ALM test statistics for autocorrelation ([[chi square].sub.4,0.05] = 9.488) Salted fish 2.368 2.399 1.686 Processed meat 2.954 3.282 2.540 Fillet 3.025 1.999 4.047 Fresh meat 4.682 4.100 4.958 Fresh fish 6.058 3.457 6.470 Shellfish 2.438 1.450 3.157 Table 2. Summary Statistics for Nonnested Comparisons Comparison Test LDC Critical Values

Statistic (5% Significance Level) ISNCES--null model (rejected) V.S. 10.428 (3.254, 4.375) QAIMDS--alternative model QAIMDS--null model V.S. -10.411 (4.392, 5.833) NQM--alternative model (rejected) ISNCES--null model V.S. 0.017 (7.645, 9.260) NQM--alternative model (rejected) Table 3. Compensating and Equivalent Variations for a 10% Reduction in Supply of Fresh Meat, Fresh Fish, and Shellfish (Yens for Annual) Fish Category CV (Yens) %CV EV (Yens) %EV 1985 Fresh meat 4,591.148 6.7% 4,387.749 6.4% Fresh fish 7,918.718 10.1% 7,350.310 9.4% Shellfish 750.633 0.96% 745.162 0.95% 1994 Fresh meat 4,212.659 6.1% 4,029.895 5.8% Fresh fish 7,598.123 10.7% 7,052.650 10.0% Shellfish 678.717 0.96% 673.917 0.96% 2003 Fresh meat 3,330.433 6.1% 3,187.869 5.9% Fresh fish 5,952.340 10.2% 5,532.382 9.5% Shellfish 544.480 0.93% 540.671 0.91% Average Fresh meat 4,147.784 6.2% 3,967.787 5.9% Fresh fish 7,403.172 10.5% 6,874.234 9.7% Shellfish 690.548 1.1% 685.581 1% Note: The column titled % CV denotes compensating variation as a percent of total expenditure on meat and fish. while the column headed %EV is similarly defined for equivalent variation.