# Topics in Nonparametric Comparative Statics and Stability.

This paper presents the analysis of the Jacobian matrix of a small, macroeconomic model to determine the robustness of the model's comparative statics. The values of the model's coefficients are assessed with respect to seven alternative estimation strategies such as ordinary least squares or two-stage least squares. For each alternative version, the invertibility and stability of the Jacobian matrix is studied. The model is robust to the degree that these characteristics are shared by any matrix with the same sign pattern or other similar nonparametric conditions on its entries. One way to address the relative success of the different estimation strategies is through the robustness of the comparative statics of the resulting model.

(JEL D00, C10)

Background

In this paper, the term "nonparametric" refers to the degree to which a conceptual model expressed as a system of simultaneous equations can be used to reach conclusions when all of its components are not fully quantified. Technically, the issues at stake can often be reduced to the mathematical problem of determining when the signs of the entries of an inverse matrix can be determined from a less than full quantification of the entries of the matrix being inverted. There is a considerable range of issues to which this subject matter can be applied. Using economics as an example, the questions posed can relate to determining whether refutable hypotheses can be derived from the theory. Generally, economic theory specifies interrelationships and (at least sometimes) the direction of influence of one variable upon another, but it seldom specifies influences in terms of specific magnitudes.  Expressed this way, the theory is scientific, that is, falsifiable if and only if conclusions can be reached that could be rejected based on data [Popper, 1959]. Alternatively, for models with statistically estimated components, the accuracy of the values found are error-prone and the stability of the true values, or even the relationships themselves, can be transitory. Given this, conclusions derived from the model are robust to the degree that they can be based on something less than the full quantification of model components [Lady, 1995]. Finally, a model's nonparametric properties can be used to audit the performance of its computer implementation. For example, if the direction of changes in solution values corresponding to changes in assumption values can be specified in advance, then a new solution can be inspected with respect to these necessary characteristics [Lady, 1981; Hale and Lady, 1995]. This capability can be an important tool in managing computer-based systems with thousands, if not millions, of numerical elements in their solutions.

For economists, issues such as these were raised by Samuelson  for both comparative statics and stability. The specific problem addressed was a determination of the matrices' invertibility and stability based only upon information about the sign pattern of its entries. Samuelson felt that it would be extremely unlikely for such an analysis to be successful. He thought that other information would also be necessary to achieve results, for example, the curvature assumptions that satisfy the second-order conditions for a (perhaps constrained) maximum or minimum. Lancaster  proposed that systems submitting to a successful analysis based upon sign patterns alone could be transformed into a particular standard form. Gorman  showed that Lancaster's standard form was sufficient, but not necessary, by proposing an algorithm to construct a system that could be successfully analyzed based on sign patterns that did not satisfy the standard form. Lady  constructed a necessary and sufficient algori thm for constructing such a system (with a qualitatively invertible or QI matrix). Klee et al.  showed that these algorithms by Gorman and Lady were equivalent (Gorman had conjectured but did not prove that his algorithm was necessary).

Significantly, Lancaster  developed an algorithmic approach, the elimination method, for detecting QI matrices independent of the form in which they are presented. Ritschard  developed a more efficient version of Lancaster's algorithm and employed side conditions, ranking relationships among the matrices' entries, to resolve comparisons for which sign pattern information was not sufficient. Gillen and Guccione  provide the means of determining which entries in the matrix must be further assessed using ranking information. The elimination method is a computationally expensive and entirely practical method for detecting QI matrices. The algorithm does not inform about the characteristics of QI matrices. Algorithms implementing an algebraic approach are provided by Maybee  and Lady  which do detect necessary and sufficient characteristics for QI matrices.

The algebraic conditions for a QI matrix were first given in Bassett et al.  and are based upon an earlier analysis by Maybee . The basic methodology utilizes the technique of signed directed graphs (SDGs) and forms the basis for considerable literature in applied mathematics and, to a lesser extent, economics and other disciplines (for example, Quirk [1981, 1986, 1992, 1997], Lady , Lady and Maybee , and Hale et al. ). Whether or not the conditions for invertibiity and stability based upon sign pattern information are as unlikely as Samuelson  thought, they are nevertheless stringent. The modeling context of the system being studied has been known for some time to provide additional information about a model's comparative statics properties [Quirk, 1997]. Other approaches have been recently developed that relate to the nonparametric issue but also to existence and uniqueness (for example, Milgrom , Milgrom and Roberts [1990, 1994], and Milgrom and Shannon ). I nterest in the last decade has also turned toward the extension of results to include other types of information about Jacobian's entries, independent of the modeling context. The general approach has been to work through the various information categories provided by the theory of measurement scales, (for example, Stevens ). The general idea is to develop conditions for signing the entries of the matrices' inverse when, in addition to sign pattern information, a ranking of the absolute values of the matrices' entries is known or when it is known that the values of the matrices' entries lie within given intervals [Lady, 1996]. Applying such additional information can be considerably facilitated if certain structural characteristics of a matrix can be detected [Lady et al., 1995].

Although there are still many unresolved issues for the conduct of nonparametric analyses, a substantial body of results is now in hand. Generally, efforts to develop automated techniques for detecting conditions already known are as appropriate as extending the conditions that identify when a nonparametric analysis can be successful.  The point of this paper is to review many of the kinds of nonparametric techniques that are now possible through illustration. hi the second section, the framework for the analysis is presented and a number of standard forms are specified that facilitate the analysis. In the third through sixth sections, a roster of conditions that support the development of nonparametric conclusions is developed. In the seventh section, the conditions are applied to the nonparametric analysis of a small macroeconomic model sometimes called Klein's Model I. Seven versions of the same basic mathematical structure are assessed with different parameter values for the model's estimated componen ts based upon alternative estimation methodologies. As a result, the robustness of results can be studied with respect to each version of the model individually and across the different versions. The eighth section provides a brief summary.

Standard Forms for Analysis

It is assumed that the conceptual model at issue has the form:

[f.sup.i](x,y) = 0 , (1)

where x is an n-vector with entries (each corresponding to a distinct variable) to be evaluated by solving the model, y is an m-vector with entries (each corresponding to a distinct parameter) to be set prior to solving the model (as an expression of assumptions), and [f.sup.i](.) are functional relationships among the variables and parameters. A comparative statics analysis of the system concerns the influence of changes in parameter values on solution values. This is addressed through studying the linear system tangent to a referent solution:

[[[sigma].sup.n].sub.j=1] [delta][f.sup.i]/[delta][x.sub.j] d[x.sub.j] = - [[[sigma].sup.m].sub.k=1] [delta][f.sup.i]/[delta][y.sub.k] d[y.sub.k], i = 1,2,...,n (2)

For ease of expression and to focus attention on the mathematical issues at stake, the system in (2) can be rewritten as a standard linear system:

Aw = z , (2a)

where A is the Jacobian matrix of the system in (1), w is an n-vector with the j th entry corresponding to d[x.sub.j], and z is an n-vector with the i th entry corresponding to the i th right-hand side of (2). Now, for a given set of facts about the entries of A, what can be said about the invertibility and stability of A? Given this, what can be said about the signs of the entries of [A.sup.-1]?

Detecting a system's nonparametric properties (if any) is facilitated by first putting the system into one of several standard forms. The following chosen forms do not add to or detract from any characteristic of the system being studied.

Standard Form 1 (SF1)

Reindex the rows and columns of A such that [a.sub.ii] [not equal to] 0 for all i. For A to be possibly nonsingular, there must be at least one nonzero term in the expansion of the determinant. Putting A into SF1 brings one such term onto the main diagonal.

Standard Form 2 (SF2)

For A in SF1, multiply columns of A by -l as appropriate, such that [a.sub.ii] [less than] 0 for all i. Putting A into SF2 simply changes the sign convention for the variables corresponding to the columns multiplied by -1.

Standard Form 3 (SF3)

For A in SF2, divide the entries of the j th column, j = 1,2, ...,n, by abs([a.sub.jj]). Given this, [a.sub.jj] = -1 for all j. Putting A into SF3 changes units. If the entries of each column are ranked prior to putting A into SF3, then the prior ranking remains intact. Putting A into SF3 can disturb the ranking relationships of entries in different columns. Notice that the forms are hierarchical. If A is in SF3, then it is also in SF2 and SF1. Additionally, if A is in SF2, then it is also in SF1.

Once put into SF2 or SF3, the SDG corresponding to A, SDG(A), can be constructed. The algebraic methods to be applied work in terms of the cycles and paths present in SDG(A). Even for fairly large systems, SDG(A) could be constructed by hand if A has many zero entries.

SDG(A) can be constructed by hand as follows. First, write down the indices of the variables (1, 2, ..., n) on a piece of paper. The places on the paper occupied by the numerals are called vertices.

Second, if and only if variable j appears in equation i and i [not equal to] j, then draw an arrow on the page from vertex j to vertex i, that is, j - i (for example, if and only if the off-diagonal entry [a.sub.ij] [not equal to] 0). The arrow drawn is called a directed arc. From the standpoint of the inference structure of the model, the arc could be said to represent the relationship that variable i depends on variable j. For each arc, note the sign of the coefficient corresponding to the arc and subscript the arc with the sign:

j [[right arrow].sub.+] i if and only if [a.sub.ij] [greater than] 0 and j [[right arrow].sub.-] i if and only if [a.sub.ij] [less than] 0.

As constructed, the vertices and signed directed arcs are the SDG corresponding to A, SDG(A). Inspection of SDG(A) can reveal circumstances such that, starting at a given vertex, it is possible to traverse the graph, that is, move from vertex to vertex by following the arrows without passing through any other vertex more than once. This traversal between two distinct vertices is called a path, and from a given vertex back to itself, a cycle.  From the standpoint of the inference structure of the model, the significance of a cycle is that the variables corresponding to the vertices must be solved for simultaneously.

Third, enumerate the cycles of SDG(A). For big systems, this could be too hard to do by hand (although maybe not, if A has many zeros). For smaller systems, all of the cycles can be identified readily. Compute the value of each cycle by taking the product of the entries of A that correspond to the directed arcs which make up the cycle (the value of a path would be found in the same way if needed).

Fourth, partition the values of cycles into two groups, those with positive values and those with negative values. Total the values for each group separately.

A number of useful special cases can be revealed through inspecting the matrices' structures as expressed by SDG(A). Generally, many of the conditions for a successful nonparametric analysis require that the system in (2a) must be solved simultaneously for all of the variables. Such matrices are termed irreducible. For A in SF1, A is irreducible if and only if there is a path between every pair of vertices. Two cycles are called disjoint if they share no vertices in common. A cycle is a maximal cycle if it is not disjoint from any other cycle. Maybee's  early analysis revealed that for A in SF2, the nonzero terms in the expansion of det A were formed by the product of the matrices' entries corresponding to the arcs in some number of disjoint cycles with main diagonal terms included for vertices not present in the cycles, plus the product of the main diagonal entries. Further, any nonzero term in the expansion of det A has the sign [(-1).sup.n] if and only if it does not embody an odd number of positive cycles. Let [c.sub.q] denote the q th cycle, and v([c.sub.q]) is the value of the q th cycle. Let N be the index set for cycles with a negative value, P is the index set for cycles with a positive value, and Q is the index set of all cycles. The following are special cases that facilitate the analysis.

Intercyclic Matrix

For A in SF1, A is intercyclic [Metzlar, 1989] if no two cycles in SDG(A) are disjoint. For A in SF3, if A is intercyclic, then:

det A = [(-1).sup.n] + [(-1).sup.n-1] [[sigma].sub.q[epsilon]Q] v([c.sub.q]).

(3)

A convenient circumstance m assessing the invertibility of A (that is, establishing that det A [not equal to] 0) is finding that only one nonzero term in the expansion of det A has a sign different than the other nonzero terms. Such matrices are called nearly sign nonsingular [Lady et al., 1995] or, more simpiy, K matrices [Lady, 1995].

K Matrix

For A in SF2, A is a K matrix if and only if all positive cycles are maximal cycles and either Form A, where there is only one positive cycle (a KA matrix), or Form B, where all cycles are positive (a KB matrix).

Since the standard forms are arranged to enable a successful nonparametric analysis of invertibility to show that sgn (det A) = [(-1).sup.n], terms in the expansion of det A with the sign [(-1).sup.n] are sometimes called friendly [Maybee and Weiner, 1988].  Thus, a helpful special case would be a matrix structure with the unfriendly terms limited to those corresponding to each distinct positive cycle. Lady  called such matrices generalized K matrices or a GK matrix.

GK Matrix

For A in SF2, A is a GK matrix if and only if no positive and negative cycles are disjoint and no three positive cycles are disjoint.

For A, a GK matrix, in SF3, the formula for the expansion of its determinant is given by:

det A = [(-1).sup.n] + [(-1).sup.n-1] [[sigma].sub.q[epsilon]Q v([c.sub.q]) + (possibly other terms, all friendly)

For A in SF3, the following nonparametric equivalence classes will assist in characterizing the nonparametric properties of A:

1) For the qualitative image of A, let [Q.sub.A] = {[[b.sub.ij]]} such that sgn [b.sub.ij] = sgn [a.sub.ij].

2) For the normalized ranked image of A, let [NR.sub.A] = {[[b.sub.ij]]} such that [[b.sub.ij]][epsilon][Q.sub.A], [b.sub.ij] [greater than] [b.sub.gh] if and only if [a.sub.ij] [greater than] [a.sub.gh], and abs([b.sub.ij]) = 1 if and only if abs([a.sub.ij]) = 1.

3) For the normalized interval image of A, let U = [[u.sub.ij]] and V = [[v.sub.ij]] be bounds such that abs([u.sub.ij]) [greater than or equal to] abs([a.sub.ij]) [greater than or equal to] abs([v.sub.ij]). Then Nint[(U, V).sub.A] = {[[b.sub.ij]]} such that [[b.sub.ij]][epsilon][Q.sub.A], abs([b.sub.ij]) = 1 if and only if abs([a.sub.ij]) = 1. For abs([a.sub.ij]) [not equal to] 1, abs([u.sub.ij]) [greater than or equal to] abs([b.sub.ij]) [greater than or equal to] abs([v.sub.ij]) if and only if abs([u.sub.ij]) [greater than or equal to] abs([v.sub.ij]).

Given these equivalence classes, a successful nonparametric analysis amounts to finding that A is such that its invertibility, stability, or characteristics of its inverse hold for all members of one (or all) of the classes, [Q.sub.A], [NR.sub.A], and Nint[(U,V).sub.A].

Finally, comparing terms in the expansion of det A based upon ranking relationship among the entries for A requires that the individual entries in each term can be arranged to conform to the same ranking outcome as proposed for the term itself. Specifically, for t1 and t2, two terms in the expansion of det A, if the comparative outcome abs (t1) [greater than] abs(t2) is invariant for all members of [NR.sub.A], then it must be possible to arrange the entries of A that comprise each term in the order of their absolute size, the largest to the smallest. For a1(r) and a2(r), the rth entries in each term are so arranged that abs(a1(r)) [greater than or equal to] abs (a2 (r)) for all r, and abs(a 1(r))[greater than] abs (a2(r)) for some r. Keeping track of such relationships generally is a substantial burden. Here, such comparisons will be attempted only for the terms in the expansion of det A that correspond to each single cycle. For A put into SF3, the relation BT (for bigger than) between the values of two cycl es, v([c.sub.1]) BT v([c.sub.2]) will represent the finding that for all members of [NR.sub.A], the two terms in the expansion of det A, t1 and t2, that correspond to these single cycles are such that abs (t1) [greater than] abs (t2) as given above. By the same reasoning, let BTE (for bigger than or equal to) be applied between two cycles, corresponding to the case abs (tl) [greater than or equal to] abs (t2).

Conditions for Nonparametric Properties of A

The Jacobian matrix will be assessed for three characteristics, based upon nonparametric assumptions about the state of knowledge about its entries. Nonparametric analysis is successful to the degree that the characteristics can be shown to hold for any or all of the three equivalence classes. The characteristics are:

1) invertibility, where A is invertible if and only if det A [not equal to] 0 ; 

2) Hicksian stability, where A is a Hicksian stable matrix if odd-order principal minors of A are negative and if even-order principal minors are positive [Hicks, 1946]; and

3) true stability, where A is a stable matrix if the real parts of the characteristic roots of A are negative [Samuelson, 1941]. 

Case A: Conditions for Characteristics of A Common to the Class [Q.sub.A]

A.1 : Invertibility

Let A be an irreducible matrix put into SF2. All B[epsilon][Q.sub.A] are such that sgn(det B) = [(-1.).sup.n] if and only if all cycles in SDG(A) have negative values. An irreducible matrix in SF2 with all negative cycles is a qualitatively invertible matrix, or QI matrix [Bassett et al., 1968].

A.1.1: Necessarily Determinable Entries in sgn [A.sup.-1]

A is an irreducible matrix in SF1. If A is a QI matrix when put into SF2 and if sgn ([a.sup.ij]) [not equal to] 0, then sgn [[b.sup.-1].sub.ji] = sgn [a.sub.ij] for all B[epsilon][Q.sub.A] [Lady, 1983].

A.1.2: Possibly Determinable Entries in sgn [A.sup.-1]

A is an irreducible QI matrix in SF2. If [a.sub.ij] = 0, then sgn [[b.sup.-1].sub.ji] = -sgn (path(i-j)) for all B[epsilon][Q.sub.A] if and only if all of the sgn (path (i-j)) in SDG(A) are the same. If the sgn(path(i-j)) in SDG(A) are not all the same, then the sgn [[b.sup.-1].sub.ji] are not the same for all B[epsilon][Q.sub.A] [Maybee and Quirk, 1969].

A.2: Hicksian Stability (Corollary of A.1.1)

If A is an irreducible QI matrix in SF2, then all B[epsilon][Q.sub.A] are Hicksian stable.

A.3: True Stability

If A is an irreducible QI matrix in SF2, then all B[epsilon][Q.sub.A] are stable matrices if and only if all cycles in SDG(A) involve (no more than) two vertices [Quirk and Ruppert, l965]. 

Case B: Conditions for Characteristics of A Common to the Class [NR.sub.A]

Let A be an irreducible matrix put into SF3. Let N be the index set for terms in the expansion of det A that embody one cycle with a negative value, N+ = N plus the main diagonal term, and P is the index set of the terms that embody one positive cycle. Let [vc.sub.q] be the index set of the vertices embodied in the cycle [c.sub.q].

B.1.1: Invertibility of GK Matrices

Let A be an irreducible GK matrix in SF3. If for every g[epsilon]P there is a distinct h[epsilon]N+ such that v([c.sub.h])BTEv([c.sub.g]) and at least one (g, h) pair such that BT holds, then sgn (det B) = [(-1).sup.n] for all B[epsilon][NR.sub.A] [Lady, 1995].

B.1.2: Invertibilily

Let A be a matrix in SF3. If for every g[epsilon]P there is a distinct h[epsilon]N, such that v([c.sub.h])BTEv([c.sub.g]) and [vc.sub.h] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [vc.sub.g], then sgn(det B) = [(-l).s [epsilon] [NR.sub.A]. [Hale et al., 1999]. 

B.2: Hicksian Stability (Corollary of B.1.2)

Let A be an irreducible matrix in SF3 that satisfies the conditions of invertibility. Then B is Hicksian stable for all B [epsilon] [NR.sub.A].

B.3: True Stability

Let A be an irreducible matrix in SF3 that satisfies the conditions for both a KA and a KB matrix. If 1 BT v(c), then A is a stable matrix [Lady, 1996]. 

Case C: Conditions for Characteristics of A Common to the Class Nint[(U,V).sub.A]

Let 0 [less than] [alpha] [less than] 1 and [delta]1, [delta]2 [greater than] 1 with the specific values to be determined. For abs([a.sub.ij]) [not equal to] 1, let the lower bounds [v.sub.ij] = [alpha][a.sub.ij], and the upper bounds [u.sub.ij] = [delta]1[a.sub.ij] or [delta]2[a.sub.ij] as appropriate (with upper and lower referring to the absolute values of the entries of A). Let PCS be the sum of the values of the positive cycles in SDG(A), NCS is the sum of the values of the negative cycles in SDG(A), z1 is the largest number of entries in a positive cycle with an absolute value other than 1, and z2 is the largest number of entries in any cycle with an absolute value other than 1.

C.1. Invertibility

Let A be in SF3. If PCS [less than] 1, then for 1 [less than] [delta]1 [less than] [(1/PCS).sup.1/zl] , sgn(det B) = [(-1).sup.n] for all B [epsilon] Nint[(U, V).sub.A] [Hale et al., 1999; Lady, 1995, 1996].

C.2. Hicksian Stability

Let A satisfy the conditions of C.1. Then A is Hicksian stable for all B [epsilon] Nint[(U, V).sub.A] [Hale et al., 1999; Lady, 1996].

C.3: True Stability

Let A be in SF3. If PCS - NCS [less than] 1, then for 1 [less than] [delta]2 [less than] [(1/(PCS - NCS)).sup.1/z2], B is a stable matrix for all B [epsilon] Nint[(U, V).sub.A] [Hale, et al., 1999; Lady, 1996]. 

It can be noted that the issue of the signs of entries in the inverse Jacobian is not automatically resolved (except sometimes for the main diagonal entries) for invertible matrices less nonparametrically robust than QI matrices. To assess these entries, analysis must proceed on a cofactor-by-cofactor basis. Even so, although a matrix may not be a QI matrix, many of the arrays corresponding to its cofactors can be. For example, none of the versions of Klein's Model I discussed in the next section were QI matrices. However, one of them had 31 of its 49 cofactors corresponding to QI matrices. As will be seen, a given system can have individual components having any of the nonparametric characteristics enumerated above. Robustness at different levels of generality can apply across the same system.

An Example Using Klein's Model I 

Klein's  aggregative macroeconomic model is often utilized pedagogically to illustrate alternative methodologies for parameter estimation (shown in Table 1). As specified (for the current period), the model involves eight endogenous variables of which seven constitute an irreducible component (that is, they must be solved for simultaneously).  The Jacobian matrix, A, and associated vertices of the corresponding SDG(A) are given for the seven endogenous variables in the following order (for the current period):

1) C is private consumption;

2) I is investment;

3) [W.sub.1] is private wave bill;

4) Y is income;

5) P is profits (nonwage income);

6) W is total (private plus government) wage bill; and

7) E is private product.

In Berndt , the model's parameters are estimated via seven alternative methodologies. The Jacobian matrix, A, for the system being estimated has the following form: 

[MATHEMATICAL EXPERSSION NOT REPRODUCIBLE IN ASCII]

The directed graph corresponding to this matrix has five cycles, enumerated below. The arcs corresponding to coefficients in definitional equations are subscripted with the signs of the corresponding coefficients. Arcs corresponding to estimated coefficients are subscripted with a question mark since the sign of the coefficient can differ depending upon the strategy used to estimate the coefficient's value:

Cycle 1: 1 [right arrow]+ 4 [right arrow]+ 5 [right arrow]? 1 with value = (1)(1)([[alpha].sub.2]) = v(c(1))

Cycle 2: 1 [right arrow]+ 4 [right arrow]+ 7 [right arrow]? 3 [right arrow]? 1 with value = (1)(1)([[gamma].sub.1])([[alpha].sub.2]) = v(c(2))

Cycle 3: 1 [right arrow]+ 4 [right arrow]+ 7 [right arrow]? 3 [right arrow]+ 6 [right arrow]- 5 [right arrow]? 1 with value = (1)(1)([[gamma].sub.1])(1)(-1)([[alpha].sub.2]) = v(c(3))

Cycle 4: 2 [right arrow]+ 4 [right arrow]+ 5 [right arrow]? 2 with value = (1)(1)([[beta].sub.1]) = v(c(4))

Cycle 5: 2 [right arrow]+ 4 [right arrow]+ 7 [right arrow]? 3 [right arrow]+ 6 [right arrow]- 5 [right arrow]? 2 with value = (1)(1)([[gamma].sub.1])(1)(-1)([[beta].sub.1]) = v(c(5))

The coefficient values given in Berndt [1991, p. 553] were estimated using each of seven different methodologies: equation by equation ordinary least squares (OLS), equation by equation two-stage least squares (2SLS), three-stage least squares system method (3SLS), full information maximum likelihood (FIML), iterative three-stage least squares system method (I3SLS), equation by equation generalized least squares assuming first-order autocorrelation (OLS-AR1), and equation by equation 2SLS assuming first-order autocorrelation (2SLS-AR1).

In Table 2, the coefficient estimates found using each method are provided. In Table 3, the values of the determinant, positive, and negative cycle sums are given for the Jacobian matrix corresponding to each method.

Inspection of the enumerated cycles in Table 3 reveals that all cycles involve vertex 4. As a result, no two cycles are disjoint and the Jacobian matrix is intercyclic. Given this, the value of the determinant can be easily calculated using (3):

Det A = (-1) + NCS + PCS

None of the Jacobian matrices estimated were QI matrices. However, many of their cofactors were based upon QI matrices. For example, the outcome of the estimation for OLS, 2SLS, OLS-AR1, and 2SLS-AR1 all showed the same sign pattern (positive) for the estimated coefficients. The 49 cofactors were found to correspond to 21 QI matrices, 26 K matrices, and 2 GK matrices. The distribution of these arrays corresponding to elements in the inverse Jacobian matrix is shown in Table 4. The sign (1 for + and -1 for (-)) of the corresponding element in the inverse is given for the entries corresponding to QI matrices, assuming that det A [less than] 0 (as was the case for all of the systems estimated).

As discussed, K matrices have only one term in the expansion of the determinant different than the rest. This circumstance portends a determination of robustness. Particularly for the OLS estimation, the cofactors corresponding to the main diagonal entries of the inverse were inspected for nonparametric properties. All but one were signable with respect to one of the categories given in the third through sixth sections. The results are given in Table 5.

The point of the analysis is that the cofactors of the Jacobian matrix can have their own, sometimes decisive, nonparametric properties. In fact, it is tempting for this model to accept the fact that the (4,4) cofactor is positive and the corresponding array is a QI matrix (in fact, a triangular matrix), assuring that det A [less than] 0. This follows from the irresistible, received macroeconomic wisdom that comparative-statically (and ceteris pan bus) dY/dTX [less than] 0 and dY/dG [greater than] 0. More could be made possibly of this kind of finding, but it is not pursued here. A summary of the nonparametric characteristics of the Jacobian matrix corresponding to each estimation methodology is given in Table 6.

It is interesting that seven alternative versions of the same model are available for analysis. Studying the arrays corresponding to the cofactors across all of the systems provides an additional dimension to the concept of robustness. In some cases, all of the arrays across all of the methodologies are QI matrices. In other cases, alternative methods lead to QI matrices but with different signs. In Table 7, entries are shown in the inverse, showing this degree of agreement or disagreement across methods.

Summary

The scope of this paper is to identify and illustrate principles of nonparametric analysis. As such, it is not a study of Klein's Model I. Still, insight about a model is part of the point of the analysis. Without more experience with results such as these, it is difficult to interpret the results across different estimation methodologies. For comparative statics, the system based upon the FIML methodology is the most robust (both in terms of invertibility and the count of QI arrays corresponding to cofactors). Alternatively, the FIML-based system does not satisfy the (sufficient only) conditions for true stability. It is an interesting trade-off to ponder when comparing the robustness of a system's invertibility and stability to that of the individual entries of the adjoint. In any case, 2SLS seems to provide more robust results than OLS, and it is this particular comparison that can often prompt the pedagogical use of this model (for example, as in Pindyck and Rubinfeld ).

Table 7 provides a summary of results for the systems taken together. The right-hand side of (Ml), (M2), and (M4) provide the most robust drivers of the model's solution. The relationship between the robustness of the drivers for (M3) and (M7) and the FIML estimation methodology is an interesting issue though not pursued further here. In Table 4, the disarray of results for Columns 5 and 6 of the inverse may be linked to the knife-edge properties of the balance conditions called for. The lack of any QI arrays corresponding to entries in Rows 2 and 5 of the adjoint suggest that predictions of investment and profit are the least robust outputs of the model (not entirely a surprise).

The success of some of the nonparametric analysis here is no doubt due to the fact that the model approached is small. Still, the Jacobian matrices corresponding to actual systems are often sparse.  Opportunities for successful analyses and the accumulation of experience and insight, therefore, await applications with existing software and further development of computer-based algorithms to support the analysis.

(*.) Temple University--U.S.A.

Footnotes

(1.) Actually, the situation is not always so austere in terms of given quantities. Klein's Model I used as the illustrative example later in the paper is expressed as a 7-equation system. The Jacobian matrix corresponding to the model has 17 nonzeros. Of these, only 4 are estimated from the data. The remaining 13 equal 1 in absolute value and correspond to the entries in accounting definitions. Four of the 7 equations are accounting relationships among the system's endogenous and exogenous variables.

(2.) PC-based software (for DOS) that enables much of the analysis conducted here can be downloaded from http://nimbus.ocis.temple.edu/[tilde]glady/helpweb/signsoft.htm.

(3.) There can be many distinct paths between vertices or cycles from a given vertex back to itself so long as for each one, no vertex traversed in the path or cycle appears more than once (except for the initial vertex in the case of a cycle).

(4.) sgn a = 1, 0, -1 as a [greater than] 0, a = 0, a [less than] 0, respectively.

(5.) Generally, if A is invertible, it will be shown that sgn(det A) = [(-1).sup.n]. In the case of QI matrices, invertibility leads to immediate conclusions about certain entries of [A.sup.-1]. Otherwise, further conclusions about entries of the inverse Jacobian matrix could require analysis of each cofactor, that is, entry of the adjoint individually.

(6.) Stability refers to movements of adjustment of a disturbed solution due to a specified dynamical process. The particulars of such will not be pursued here. In general, Hicksian stability insures that disturbed solutions adjust toward the reference solution but not necessarily along a convergent path. True stability implies adjustment along a convergent path. The two concepts are sometimes, but not always, equivalent. For the nonparametric cases, invertibility and Hicksian stability are often equivalent.

(7.) The main diagonal entries can be said to identify an arc from a vertex back to itself--a cycle with only one vertex.

(8.) This is a true theorem, but an algorithm needs to be written to make it practical for large systems.

(9.) The severe structural conditions make this a simple example of the principle of a dominant diagonal [McKenzie, 1960].

(10.) Lady  called a matrix satisfying this condition an SLO matrix (sum less than 1). The construction of ranges using [delta]1 and [delta]2 as defined here can clearly be strengthed. For example, write an algorithm that finds the maximum values to the nearest preset tolerance through inspection.

(11.) The model specification and parameter values used in this section are from Berndt [1991, pp. 550-3]. A six-equation version of this model was analyzed in Maybee and Weiner  although an error in transcribing SDG(A) was made. That same system was reconsidered in Hale et al. . In both of these works, only a model specification for one estimation methodology was studied.

(12.) The eighth endogenous variable, the current period capital stock, is the sum of current period net investment (one of the other seven endogenous variables) and the capital stock surviving from the immediately prior period.

(13.) The coefficients [[alpha].sub.3], [[beta].sub.2], [[beta].sub.3], [[gamma].sub.2], and [[gamma].sub.3] associated with the right-hand-side parameters are also estimated. The outcome for all but one had the same sign, positive, for each of the seven methodologies. The single exception was [[beta].sub.3]. For this coefficient, all of the estimated values were negative. Hence, the multipliers in the second column of the inverted system would have the opposite sign for K(-) than that perceived from inspection of the system above. Otherwise, the outcomes of these estimates do not influence the signs of the multipliers.

(14.) Hale and Lady  found that all of the entries of a 19 x 19 irreducible Jacobian matrix, corresponding to a model of international oil supply and demand, could be signed based upon sign pattern information alone. The model had a small number of cyclic patterns that involved all of the variables, but the Jacobian matrix, otherwise, had many zero entries.

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```                              Klein's Model I
1   2  3                           4  5                   6
- C     + [[alpha].sub.1] [W.sub.1]    + [[alpha].sub.2] P
- I                                + [[beta].sub.1] P
- [W.sub.1]
C + I                             -Y
Y - P                 - W
[W.sub.1]                                          - W
Y
1  7
- C                  = -[[alpha].sub.1] [W.sub.2] - [[alpha].sub.3] P (-)
= -[[beta].sub.2]P(-) - [[beta].sub.3]K(-)
[[gamma].sub.1]E = -[[gamma].sub.2]E(-) - [[gamma].sub.3] YEAR
C                  = TX - G
= 0
= - [W.sub.2]
- E              = - TX + [W.sub.2]
1
- C (M1)
(M2)
(M3)
C (M4)
(M5)
(M6)
(M7)
Notes: [W.sub.2] denotes government wage bill, (-) denotes prior period
value, K denotes end of period capital stock, YEAR denotes years since
1931, G denotes government consumption, and TX denotes taxes.
Parameter Estimates
Method          OLS  2SLS  3SLS  FIML I3SLS OLS-ARI 2SLS-AR1
[[alpha].sub.1] .796 .810  .790  .791  .766  .478     .644
[[alpha].sub.2] .193 .017  .125 -.062  .164  .419     .301
[[beta].sub.1]  .480 .150 -.013 -.625 -.373  .490     .425
[[gamma].sub.1] .439 .439  .400  .278  .375  .430     .433
Cycle Sums and Determinant Value
Sums     OLS    2SLS   3SLS   FIML  I3SLS  OLS-AR1 2SLS-AR1
PCS-NCS  1.318  0.596  0.509  1.098  0.998  1.505    1.319
PCS      1.022  0.523  0.446  0.411  0.584  1.115    1.005
NCS     -0.295 -0.073 -0.063 -0.687 -0.414 -0.391   -0.314
det A   -0.273 -0.551 -0.617 -1.276 -0.831 -0.276   -0.310
Distribution of Cofactor Arrays for [A.sup.-1]
OLS, 2SLS, OLS-AR1, and 2SLS-AR1
Columns:    1       2    3     4       5      6    7
Row 1      KA      KA    GK   KA    QI (-1) QI (1) KB
Row 2      KA      GK    KA   KA      KA      KA   KA
Row 3    QI (-1) QI (-1) KB QI (-1) QI (-1) QI (1) KB
Row 4    QI (-1) QI (-1) KA QI (-1) QI (-1) QI (1) KA
Row 5      KA      KA    KA   KA      KA      KA   KA
Row 6    QI (-1) QI (-1) KB QI (-1) QI (-1)   KB   KB
Row 7    QI (-1) QI (-1) KA QI (-1) QI (-1) QI (1) KB
Notes: QI denotes QI matrix, KA denotes KA matrix, KB denotes
KB matrix, and GK denotes GK matrix. The cofactors at issue
are from the adjoint A and are thus transposed with respect
to the elements of A to which they correspond.
Main Diagonal Cofactors for OLS
i for Cofactor (i, i) Type of Array
1             KA matrix
2             GK matrix
3             KB matrix
4             QI matrix
5             KA matrix
6             KB matrix
7             KB matrix
i for Cofactor (i, i)            Nonparametric Category
1                            [NR.sub.A]
2           [Nint(U, V).sub.A] for 1 [less than] [delta]
1 [less than] 1.004
3           [Nint(U, V).sub.A] for 1 [less than] [delta]
1 [less than] 1.22
4                            [Q.sub.A]
5                            [NR.sub.A]
6                               none
7           [Nint(U, V).sub.A] for 1 [less than] [delta]
1 [less than] 1.22
Nonparametric Properties of the Jacobian Matrices
Method                Invertibility
OLS                        none
2SLS                Nint[(U, V).sub.A]
1 [less than] [delta] 1 [less than] 1.38
3SLS                Nint[(U, V).sub.A]
1 [less than] [delta] 1 [less than] 1.50
FIML                    [NR.sub.A]
I3SLS               Nint[(U, V).sub.A]
1 [less than] [delta] 1 [less than] 1.31
OLS-AR1                    none
2SLS-AR1                   none
Hicksian
Method                  Stability
OLS                        none
2SLS                Nint[(U, V).sub.A]
1 [less than] [delta] 1 [less than] 1.38
3SLS                Nint[(U, V).sub.A]
1 [less than] [delta] 1 [less than] 1.50
FIML                Nint[(U, V).sub.4]
1 [less than] [delta] 1 [less than] 1.56
I3SLS               Nint[(U, V).sub.A]
1 [less than] [delta] 1 [less than] 1.31
OLS-AR1                    none
2SLS-AR1                   none
Number of
True                      Cofactors
Method                   Stability                 from QI Arrays
OLS                        none                          21
2SLS                Nint[(U, V).sub.A]                   21
1 [less than] [delta] 2 [less than] 1.30
3SLS                Nint[(U, V).sub.A]                   12
1 [less than] [delta] 2 [less than] 1.40
FIML                       none                          31
I3SLS               Nint[(U, V).sub.A]                   12
1 [less than] [delta] 2 [less than] 1.001
OLS-AR1                    none                          21
2SLS-AR1                   none                          21
Columns:    1       2       3        4        5           6         7
Row 1       ?       ?    -1: FIML    ?     -1: OLS,   -1: OLS,   -1: FIML
1: FIML, ! 1: FIML, !
Row 2       ?       ?       ?        ?        ?           ?         ?
Row 3    -1, all -1, all -1: FIML -1, all  -1: OLS,    1: OLS,   -1: FIML
1: FIML, ! -1: FIML, !
Row 4    -1, all -1, all -1: FIML -1, all  -1: OLS,    1: OLS,   -1: FIML
1: FIML, ! -1: FIML, !
Row 5       ?       ?       ?        ?        ?           ?         ?
Row 6    -1, all -1, all -1: FIML -1, all  -1: OLS,       ?      -1: FIML
1: FIML, !
Row 7    -1, all -1, all -1: FIML -1, all  -1: OLS,    1: OLS,   -1: FIML
1: FIML, ! -1: FIML, !
Notes: "All" denotes sign of entry common to all systems, and ! denotes