A system comparison approach to distinguish two nonseparable and nonnested agricultural household models.

Next, the market price elasticities of the total labor supply are greater in absolute value than that of the farm labor supply when the internal wages are fixed, as shown in panel (i). These relations imply that a shift in the supply curve [L.sup.S.sub.t] of total labor is greater than the corresponding shift in the supply curve [L.sup.S.sub.f] of farm labor in figure 1. We note two points to interpret that difference. First, panel (i) of table 4 shows that the difference in shifts among the supply functions of farm, nonfarm and total labor clearly reflect the size of their respective income elasticities. (20) Second, the estimated income elasticity of the total labor supply -0.35 is inferred to be biased downward because it is greater in absolute value than both the income elasticities of farm and nonfarm labor supply -0.26 and -0.21. These points suggest that the greater shifts of the supply function of total labor result from the biased income elasticity of the total labor supply.

In view of (21) and (22) as well as figure 1, the steeper slope and the greater shifts of the supply function of total labor imply that the internal wage [w.sup.**] responds more sensitively to changes in market prices than the other internal wage [w.sup.*] does. Panels (ii) and (iii) of table 4 verify this relation.

In view of (23) and (24), such greater elasticities of internal wage [w.sup.**] cause greater internal wage effects on quantity variables. Actually, most elasticities of demand for farm labor and other variable inputs include much greater internal wage effects in panel (iii) than in panel (ii). The elasticity of output supply includes moderately greater internal wage effects in panel (iii) because internal wage effects on the demands for farm labor and other variable inputs are mutually offsetting to some degree.

Conclusion

This paper describes a system comparison approach to distinguish nonseparable AHM under the HET and RES hypotheses. Using a two-step estimation, we found their consumption side to be distinguished because they yield demand systems that not only have different dependent variables but also different numbers of equations. We proposed an empirical procedure to apply the Cox-type test of Smith (1992) to make an appropriate comparison of the nonnested systems.

Our specific comparison reveals the following three points. First, two nonseparable models are likely to be nonnested in general, and we can use a Cox-type test for their comparison. By choosing an appropriate matrix [A.sub.k,j] in the Appendix, this method is applicable to comparing two AHMs, which determine the same endogenous variables without specifying their types (separable or nonseparable), their expressions (reduced or structural form) or causes of their nonseparability.

Second, our Cox-type test can clearly show better applicability of the HET hypothesis over the RES hypothesis, although nonnested tests often encounter difficulties that either both hypotheses are rejected or do not reject either. Hence, our Cox-type test actually has sufficient validity in empirical analyses.

Finally, we find much smaller (but significant) elasticities of the internal wage and hence much smaller (but significant) internal wage effects on factor demand and output supply under the HET hypothesis. This finding shows that distinguishing two nonseparable models exhibits important implications in evaluating responses of farm households to changes in economic conditions or policy variables.

Appendix

This appendix explains the Cox-type statistic. Let E[[h.sub.k]([y.sub.k.i], [[theta].sub.k])] = 0 represent the orthogonality conditions under hypothesis k and let [[OMEGA].sub.k] represent the covariance matrix of [h.sub.k], where [[theta].sub.k] and [y.sub.k.i], respectively, denote the vectors of parameters and variables for observation i (= 1, ..., N). In our case, [h.sub.k]([y.sub.k,i], [[theta].sub.k]) = {[S.sub.k,i] -[S.sub.k]([y.sub.k,i], [[theta].sub.k])} [cross product] [Z.sub.k,i]; therein, [S.sub.k,i], [S.sub.k] () and [Z.sub.k.i], respectively, denote vectors of expenditure shares, expenditure share functions and instruments under hypothesis k. We also define [h.sub.kN] = [N.sup.-1] [[summation].sup.N.sub.i=1] [h.sub.k]([y.sub.k,i], [[theta].sub.k]) and [H.sub.kN] = [partial derivative][h.sub.kN]/[partial derivative][[theta].sup.T.sub.k], with [[upsilon].sup.T] denoting the transpose of [upsilon]. Then, the Cox-type test of a null hypothesis k against an alternative j uses the following statistic, which is asymptotically distributed as N(0, 1):

(A.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therein, I(m) represents an m-dimensional identity matrix; [??] denotes an estimator of [upsilon].

For practical use we must estimate the matrix [A.sub.k,j], noting that dim [h.sub.kN] [not equal to] dim [h.sub.jN] because we estimate two share equations under HET but only a single share equation under RES. If the vectors [h.sub.kN] and [h.sub.jN] have the same dimension, a locally optimal test is performed by choosing [[??].sub.k.j] = [[??].sub.j][[??].sup.-1.sub.k], as shown in Smith (1992). By referring to this choice and the lemma 2.2 of Smith, we can construct our matrix [A.sub.k,j].

We confine our analysis to cases in which we choose a common set z of instruments under null and alternative hypotheses because this choice seems to allow an appropriate comparison of the original systems. When we test the HET hypothesis against RES, we can define a p x 2p matrix J = [I (p): I (p)] (p = dimz) and devise a locally optimal test by choosing [[??].sub.k,j] [[??].sub.j] [[??].sup.-1.sub.k]. Conversely, when we test the RES hypothesis against HET, we choose [[??].sub.k,j] = [[??].sub.j][J.sup.T][[??].sup.-1.sub.k].

[Received September 22, 2006; accepted August 27, 2007.]

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