# Definitions and tests

Let X and Y be two income variables at either two different points in time, before and after taxes, or for different regions or countries. Let X1, X2,…, Xn be n not necessarily iid observations on X, and Y1, Y2,…, Ym be similar observations on Y. Let U1 denote the class of all utility functions u such that u’ > 0, (increasing). Also, let U2 denote the subset of all utility functions in U1 for which u" < 0 (strict concavity), and U3 denote a subset of U2 for which u”’ > 0. Let X0 and Y^ denote the ith order statistics, and assume F(x) and G(x) are continuous and monotonic cumulative distribution functions (cdfs) of X and Y, respectively. Let the quantile functions X(p) and Y(p) be defined by, for example, Y(p) = inf{y : F(y) > p}.

Proposition 1. X first-order stochastic dominates Y, denoted X FSD Y, if and only if any one of the following equivalent conditions holds:

1. E[u(X)] > E[u(Y)] for all u О Uv with strict inequality for some u. This is the classical definition.

2. F(x) < G(x) for all x in the support of X, with strict inequality for some x (e. g. see McFadden, 1989).

3. X(p) > Y(p) for all 0 < p < 1, with strict inequality for some p (e. g. see Xu et al.,

1995) .

Proposition 2. X second-order stochastic dominates Y, denoted X SSD Y, if and only if any of the following equivalent conditions holds:

1. E[u(X)] > E[u(Y)] for all u Є U2, with strict inequality for some u.

2. fx_^F(t)dt < fx_^G(t)dt for all x in the support of X and Y, with strict inequality for some x.

3. Фх(p) = /pX(t)dt > Фу(p) = fp0Y(t)dt, for all 0 < p < 1, with strict inequality for some value(s) p.

Weaker versions of these relations drop the requirement of strict inequality at some point. When either Lorenz or Generalized Lorenz Curves of two distributions

cross, unambiguous ranking by FSD and SSD may not be possible. Shorrocks and Foster (1987) show that the addition of a "transfer sensitivity" requirement leads to third-order stochastic dominance (TSD) ranking of income distributions. This requirement is stronger than the Pigou-Dalton principle of transfers since it makes regressive transfers less desirable at lower income levels. TSD is defined as follows:

Proposition 3. X third-order stochastic dominates Y, denoted X TSD Y, if any of the following equivalent conditions holds:

1. E[u(X)] > E[u(Y)] for all u Є U3, with strict inequality for some u.

2. /-„/L[F(t) – G(t)]dt dv < 0, for all x in the support, with strict inequality for some x, with the end-point condition:

[F(t) – G(t)]dt < 0.

3. When E[X] = E[Y], X TSD Y iff .X(q) < .2y(q), for all Lorenz curve crossing points q, i = 1, 2,…, (n + 1); where. X(q) denotes the "cumulative variance" for incomes upto the ith crossing point. See Davies and Hoy (1995).

When n = 1, Shorrocks and Foster (1987) show that X TSD Y if (i) the Lorenz curve of X cuts that of Y from above, and (ii) var(X) < var(Y). This situation seemingly revives the coefficient of variation as a useful statistical index for ranking distributions. But a distinction is needed between the well known (un­conditional) coefficient of variation for a distribution, on the one hand, and the sequence of several conditional coefficients of variation involved in the TSD.

The tests of FSD and SSD are based on empirical evaluations of conditions (2) or (3) in the above definitions. Mounting tests on conditions (3) typically relies on the fact that quantiles are consistently estimated by the corresponding order statistics at a finite number of sample points. Mounting tests on conditions (2) requires empirical cdfs and comparisons at a finite number of observed ordi­nates. Also, from Shorrocks (1983) it is clear that condition (3) of SSD is equivalent to the requirement of generalized Lorenz (GL) dominance. FSD implies SSD. The Lorenz and the generalized Lorenz curves are, respectively, defined by: L(p) = (1/p) /p0Y(u)du, and GL(p) = pL(p) = /Y(u)du, with GL(0) = 0, and GL(1) = p; see Shorrocks (1983).

It is customary to consider K points on the L (or GL or the support) curves for empirical evaluation with 0 < p1 < p2 <… < pK = 1, and pi = i/K. Denote the corresponding quantiles by Y(p), and the conditional moments y; = E(Y | Y < Y(p,)), and.2 = E{(Y – у)21 Y < Y(pi)}. The vector of GL ordinates is given by n = (p1.1, p2 of,… pK. K)’. See Xu et al. (1995) who adopt the x2 approach described above to test quantile conditions (3) of FSD and SSD. A short description follows: Consider the random sequence {Zt} = {Xt, Yt}’, a stationary ф-mixing sequence of random vectors on a probability space (Q, ^, P). Similarly, denote the stacked vector of GL ordinates for the two variables as nZ = (nX, nY)’, and the stacked vector of quantiles of the two variables by qZ = (qX, qY)’, where qX = (X(p1), X(p2),… X(pK))’, and similarly for Y. In order to utilize the general theory given for the x 2-distribution, three ingredients are required. One is to show that the various hypotheses of interest in this context are representable as in (25.23) above. This is possible and simple. The second is to verify if and when the unrestricted estimators of the n and q functions satisfy the asymptotic representation given in

(25.22) . This is possible under conditions on the processes and their relationships, as we will summarize shortly. The third is to be able to empirically implement the / statistics that ensue. In this last step, resampling techniques are and will become even more prominent.

To see that hypotheses of interest are suitably representable, we note that for the case of conditions (3) of FSD and SSD, the testing problem is the following:

H0 : hK(qZ) > 0 against H1 : hK(qZ) / 0, where hK(qZ) = [IK : – IK]qZ = I*qZ, say, for FSD, and hK(qZ) = BI* x qZ, for the test of SSD, where, B = (Bij), Bij = 1, i > j, Bj = 0, otherwise, is the "summation" matrix which obtains the successive cumulated quantile (Ф) and other functions.

Tests for GL dominance (SSD) which are based on the ordinate vector nZ are also of the "linear inequality" form and require h(nZ) = I*nZ.

Sen (1972) gives a good account of the conditions under which sample quantiles are asymptotically normally distributed. Davidson and Duclos (1998) provide the most general treatment of the asymptotic normality of the nonparametric sample estimators of the ordinates in n. In both cases the asymptotic variance matrix, W, noted in the general setup (25.22) is derived. What is needed is to appropri­ately replace R in the formulations of Kodde and Palm (1986), or Gourieroux et al. (1982), and to implement the procedure with consistent estimates of Q in W = RQR’.

For sample order statistics, < f, it is well known that, if X and Y are independent,

(<Z – qZ) A N(0, Q)

Q = G-1VG’-1

G = diag[ fx(X(…; fy(Y(… L i = 1…, K

V = lim E( gg), g = П & gy),

%x = [{f (X(p0) – pa>,…, {F(X(pk)) – pk}], gy similarly defined.

As is generally appreciated, these density components are notoriously difficult to estimate. Kernel density methods can be used, as can Newey-West type robust estimators. But it is desirable to obtain bootstrap estimates based on block bootstrap and/or iterated bootstrap techniques. These are equally accessible computationally, but may perform much better in smaller samples and for larger numbers of ordinates K. Xu et al. (1995) demonstrate with an application to the hypothesis of term premia based on one – and two-month US Treasury bills. This application was based on the Kodde and Palm (1986) critical bounds and encountered some
realizations in the inconclusive region. Xu et al. (1995) employ Monte Carlo simulations to obtain the exact critical levels in those cases.

Sample analogs of n and similar functions for testing any stochastic order also have asymptotically normal distributions. Davidson and Duclos (1998) exploit the following interesting result which translates conditions (3) of the FSD and SSD into inequality restrictions among the members of the n functions defined above:

Let DX(x) = FX(x), and DY(y) = FY(y); then,   x

This last equality clearly shows that tests of any order stochastic dominance can be based on the conditional moments estimated at a suitable finite number of K ordinates as defined above. For instance, third order SD (s = 3) is seen to depend on the conditional/cumulative variance. Also, since poverty measures are often defined over lower subsets of the domain such that x < poverty line, dominance relations over poverty measures can also be tested in the same fashion. Using empirical distribution functions, Davidson and Duclos (1998) demonstrate with an example from the panels for six countries in the Luxembourg study. It should be appreciated, however, that these tests do not exploit the inequality nature of the alternative hypotheses. The union intersection method determines the critical level of the inference process here. The cases of unrankable distributions include both "equivalence" and crossing (non-dominant) distributions. A usual asymp­totic x2 test will have power in both directions. In order to improve upon this, therefore, one must employ the x 2-distribution technique.

Similarly, Kaur et al. (1994) propose a test for condition (2) of SSD when iid observations are assumed for independent prospects X and Y. Their null hypo­thesis is condition (2) of SSD for each x against the alternative of strict violation of the same condition for all x. The test of SSD then requires an appeal to a union intersection technique which results in a test procedure with maximum asymp­totic size of a if the test statistic at each x is compared with the critical value Za of the standard normal distribution. They showed their test is consistent. One rejects the null of dominance if any negative distances at the K ordinates is significant.

In contrast, McFadden (1989), and Klecan et al. (1991) test for dominance jointly for all x. McFadden’s analysis of the multivariate Kolmogorov-Smirnov type test is developed for a set of variables and requires a definition of "maximal" sets, as follows:

Definition 1. Let & = {X1, X2,…, XK} denote a set of K distinct random vari­ables. Let Fk denote the cdf of the kth variable. The set & is first- (second-)order maximal if no variable in & is first – (second-)order weakly dominated by another.

Let Xn = (x1n, x2n,…, xKn), n = 1, 2,…, N, be the observed data. We assume X. n is strictly stationary and а-mixing. As in Klecan et al., we also assume F;(X,), i = 1,

2,. .., K are exchangeable random variables, so that our resampling estimates of the test statistics converge appropriately. This is less demanding than the assumption of independence which is not realistic in many applications (as in before and after tax scenarios). We also assume Fk is unknown and estimated by the empirical distribution function FkN(Xk). Finally, we adopt Klecan et al.’s mathematical regularity conditions pertaining to von Neumann-Morgenstern (VNM) utility functions that generally underlie the expected utility maximization paradigm. The following theorem defines the tests and the hypotheses being tested:

Lemma Given the mathematical regularity conditions;

1. The variables in Ж are first-order stochastically maximal; i. e.

d = min max [F(x) – fj-(x)] > 0,

i* j x

if and only if for each i and j, there exists a continuous increasing function u such that Eu(X,) > Eu(Xj).

2. The variables in Ж are second-order stochastically maximal; i. e.

x

S = min max [F;(|j.) – F: (|л)]ф > 0,

i * j x

if and only if for each i and j, there exists a continuous increasing and strictly concave function u such that Eu(Xi) > Eu(Xj).

3. Assuming, (i) the stochastic process X. n, n = 1, 2,…, to be strictly stationary and а-mixing with a( j) = O( j~5), for some 5 > 1, and (ii) the variables in the set are exchangeable (relaxing independence in McFadden, 1989): d2N ^ d, and S2N ^ S, where d2N and S2N are the empirical test statistics defined as:

d2N = min max [Fn(x) – FNx)]

i * j x

and

x

S2N = min max [F;n(|J.) – F.-4ц)]ф

i * j x

0

Proof 1. See Theorems 1 and 5 of Klecan et al. (1991).

The null hypothesis tested by these two statistics is that, respectively, Ж is not first- (second-)order maximal – i. e. X, FSD(SSD) Xj for some i and j. We reject the null when the statistics are positive and large. Since the null hypothesis in each case is composite, power is conventionally determined in the least favorable case of identical marginals F; = Fj. As is shown in Kaur et al. (1994) and Klecan et al.

(1991), when X and Y are independent, tests based on d^ and Sm are consistent. Furthermore, the asymptotic distribution of these statistics are non-degenerate in the least favorable case, being Gaussian (see Klecan et al., 1991, Theorems 6-7).

As is pointed out by Klecan et al. (1991), for non-independent variables, the statistic S2N has, in general, neither a tractable distribution, nor an asymptotic distribution for which there are convenient computational approximations. The situation for d2N is similar except for some special cases – see Durbin (1973, 1985), and McFadden (1989) who assume iid observations (not crucial), and indepen­dent variables in Ж (consequential). Unequal sample sizes may be handled as in Kaur et al. (1994).

Klecan et al. (1991) suggest Monte Carlo procedures for computing the signifi­cance levels of these tests. This forces a dependence on an assumed parametric distribution for generating MC iterations, but is otherwise quite appealing for very large iterations. Maasoumi et al. (1997) employ the bootstrap method to obtain the empirical distributions of the test statistics and of p-values. Pilot stud­ies show that their computations obtain similar results to the algorithm proposed in Klecan et al. (1991).

In the bootstrap procedure we compute d2N and S2N for a finite number K of the income ordinates. This requires a computation of sample frequencies, cdfs and sums of cdfs, as well as the differences of the last two quantities at all the K points. Bootstrap samples are generated from which empirical distributions of the differences, of the d2N and S2N statistics, and their bootstrap confidence intervals are determined. The bootstrap probability of these statistics being positive and/ or falling outside these intervals leads to rejection of the hypotheses. Maasoumi et al. (1997) demonstrate by several applications to the US income distributions based on the Current Population Survey (CPS) and the panel data from the Michigan study. In contrast to the sometimes confusing picture drawn by com­parisons based on inequality indices, they find frequent SSD relations, including between population subgroups, that suggest a "welfare" deterioration in the 1980s compared to the previous two decades.

2 Conclusion

Taking the one-sided nature of some linear and nonlinear hypotheses is both desirable and practical. It can improve power and lead to the improved computa­tion of the critical levels. A X2 and a multivariate KS testing strategy were described and contrasted with some alternatives, either the less powerful two-sided methods, or the union intersection procedures. The latter deserves to be studied further in comparison to the methods that are expected to have better power. Computa­tional issues involve having to solve QP problems to obtain inequality restricted estimators, and numerical techniques for computation of the weights in the X statistic. Bounds tests for the latter are available and may be sufficient in many cases.

Applications in the parametric/semiparametric, and the nonparametric testing area have been cited. They tend to occur in substantive attempts at empirical evaluation and incorporation of economic theories.

Note

* Comments from the editor and an anonymous referee helped improve this chapter. A more extensive bibliography and discussion is contained in the preliminary version of this chapter which is available upon request.

References

Anderson, G. J. (1996). Nonparametric tests of stochastic dominance in income distribu­tions. Econometrica 64, 1183-93.

Barlow, R. E., D. J. Bartholomew, J. N. Bremner, and H. D. Brunk (1972). Statistical Inference under Order Restrictions: The Theory and Applications of Isotonic Regression. New York: John Wiley and Sons.

Bartholomew, D. J. (1959a). A test of homogeneity for ordered alternatives. Biometrica 46, 36-48.

Bartholomew, D. J. (1959b). A test of homogeneity for ordered alternatives. Biometrica 46, 328-35.

Bohrer, R., and W. Chow (1978). Weights of one-sided multivariate inference. Applied Statistics 27, 100-4.

Davidson, R., and J.-Y. Duclos (1998). Statistical inference for stochastic dominance and for the measurement of poverty and inequality. GREQAM, Doc. de Travail, no. 98A14.

Davies, J., and M. Hoy (1995). Making inequality comparisons when Lorenz curves inter­sect. American Economic Review 85, 980-6.

Dufour, J.-M. (1989). Nonlinear hypotheses, inequality restrictions and nonnested hypo­theses: Exact simultaneous tests in linear regression. Econometrica 57, 335-55.

Dufour, J.-M., and L. Khalaf (1995). Finite sample inference methods in seemingly unre­lated regressions and simultaneous equations. Technical report, University of Montreal.

Durbin, J. (1973). Distribution theory for tests based on the sample distribution function. Philadelphia, SIAM.

Durbin, J. (1985). The first passage density of a continuous Gaussian process to a general boundary. Journal of Applied Probability 22, 99-122.

Gourieroux, C., A. Holly, and A. Monfort (1980). Kuhn-Tucker, likelihood ratio and Wald tests for nonlinear models with inequality constraints on the parameters. Harvard Insti­tute of Economic Research, Mimeographed paper no. 770.

Gourieroux, C., A. Holly, and A. Monfort (1982). Likelihood ratio test, Wald test and Kuhn-Tucker test in linear models with inequality constraints in the regression para­meters. Econometrica 50, 63-80.

King, M. L., and Ping X. Wu (1997). Locally optimal one-sided tests for multiparameter hypotheses. Econometric Reviews 16, 131-56.

Kaur, A., B. L.S. Prakasa Rao, and H. Singh (1994). Testing for second-order stochastic dominance of two distributions. Econometric Theory 10, 849-66.

Klecan, L., R. McFadden, and D. McFadden (1991). A robust test for stochastic dominance. Working paper, Economics Dept., MIT.

Kodde, D. A., and F. C. Palm (1986). Wald criteria for jointly testing equality and inequality restrictions. Econometrica 50, 1243-8.

Kodde, D. A., and F. Palm (1987). A parametric test of the negativity of the substitution matrix. Journal of Applied Econometrics 2, 227-35.

Kudo, A. (1963). A multivariate analogue of the one-sided test. Biometrica 50, 403-18.

Maasoumi, E., J. Mills, and S. Zandvakili (1997). Consensus ranking of US income dis­tributions: A bootstrap application of stochastic dominance tests. SMU, Economics.

McFadden, D. (1989). Testing for stochastic dominance. In Part II of T. Fomby and T. K. Seo (eds.) Studies in the Economics of Uncertainty (in honor of J. Hadar). Springer-Verlag.

Perlman, M. D. (1969). One-sided testing problems in multivariate analysis. Annals of Mathe­matics and Statistics 40, 549-62.

Potscher, B. M., and I. R. Prucha (1991a). Basic structure of the asymptotic theory in dynamic nonlinear econometric models: I. Consistency and approximation concepts. Econometric Reviews 10, 125-216.

Potscher, B. M., and I. R. Prucha (1991b). Basic structure of the asymptotic theory in dynamic nonlinear econometric models: II. Asymptotic Normality. Econometric Reviews 10, 253-326 (with comments).

Sen, P. K. (1972). On the Bahadur representation of sample quantiles for sequences of ф-mixing random variables. Journal of Multivariate Analysis 2, 77-95.

Shorrocks, A. F. (1983). Ranking income distributions. Economica 50, 3-17.

Shorrocks, A., and J. Foster (1987). Transfer sensitive inequality measures. Review of Eco­nomic Studies 54, 485-97.

Stewart, K. G. (1997). Exact testing in multivariate regression. Econometric Reviews 16, 321-52.

Wolak, F. (1989). Local and global testing of linear and nonlinear inequality constraints in nonlinear econometric models. Econometric Theory 5, 1-35.

Wolak, F. (1991). The local nature of hypothesis tests involving inequality constraints in nonlinear models. Econometrica 981-95.

Xu, K., G. Fisher, and D. Wilson (1995). New distribution-free tests for stochastic domi­nance. Working paper No. 95-02, February, Dept. of Economics, Dalhousie University, Halifax, Nova Scotia. 