Identification of Nonparametric Simultaneous Equations Models with a Residual Index Structure

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1 Identification of Nonparametric Simultaneous Equations Models with a Residual Index Structure Steven T. Berry Yale University Department of Economics Cowles Foundation and NBER Philip A. Haile Yale University Department of Economics Cowles Foundation and NBER December 12, 2013 Abstract We present new results on the identifiability of a class of nonseparable nonparametric simultaneous equations models introduced by Matzkin (2008, These models combine standard exclusion restrictions with a requirement that each structural error enter through a residual index. We provide three general results nesting a variety special cases that permit tradeoffs between the exogenous variation required of instruments and the structure placed on the joint density of the structural errors. These special cases include results with no density restriction and results allowing instruments with arbitrarily small support. The identification results for this class of models in Matzkin (2008, 2010 are also obtained as special cases. We show that our primary conditions for identification are verifiable, i.e., their satisfaction or failure is identified. The maintained assumptions of the model also imply testable restrictions. PRELIMINARY AND INCOMPLETE This paper supersedes our 2011 and 2013 working papers on this topic with the title Identification in a Class of Nonparametric Simultaneous Equations Models. The results given there are special cases of those shown here. Some of these results grew out of our related work on differentiated products markets and benefited from the comments of audiences at several university seminars, the 2008 World Congress of the Game Theory Society, 2008 LAMES, 2009 Econometrics of Demand Conference, 2009 FESAMES 2010 Guanghua-CEMMAP-Cowles Advancing Applied Microeconometrics Conference, 2010 French Econometrics Conference, 2011 LAMES/LACEA, and 2012 ESEM. We also received helpful comments from Elie Tamer, Alex Torgovitzky, and seminar participants at LSE. We thank the National Science Foundation for financial support.

2 1 Introduction Economic theory typically produces systems of equations characterizing the outcomes observable to empirical researchers. The classical supply and demand model is the most familiar example, but systems of simultaneous equations arise in a wide variety of contexts in which multiple agents interact or a single agent makes multiple interrelated choices. The identifiability of simultaneous equations models is therefore a fundamental question for a wide range of topics in empirical economics. Nonparametric identification is a necessary step toward development of consistent nonparametric estimation methods. Perhaps more important, even when sample sizes dictate use of semiparametric or parametric estimators, one would like to know whether parametric restrictions are merely finite sample approximations or maintained hypotheses essential to identification of the features of interest. Nonparametric identification results can also provide valuable guidance to empirical researchers by suggesting which features of the data or data generating process including the type and extent of exogenous variation available are essential. Early work on (parametric identification in econometrics treated systems of simultaneous equations as a primary focus. Many prominent examples can be found among the contributions to Cowles Monograph 10 (Koopmans (1950 and Cowles Monograph 14 (Hood and Koopmans (1953. Fisher s (1966 monograph, The Identification Problem in Econometrics, illustrates the extent of this focus: he considered only identification of simultaneous equations models, explaining (p. vii, Because the simultaneous equation context is by far the most important one in which the identification problem is encountered, the treatment is restricted to that context. 1 Although there has been substantial recent interest in the identification of nonparametric economic models that feature endogenous regressors and nonseparable errors, there remain remarkably few results for fully simultaneous systems. A fully general (and more general 1 See also the discussion in Manski (

3 than we will allow nonparametric simultaneous equations model can be written m j (Y, Z, U = 0 j = 1,..., J (1 where J 2, Y = (Y 1,..., Y J R J are the endogenous variables, U = (U 1,..., U J R J are the structural errors, and Z is a vector of exogenous variables. Assuming m is invertible in U, 2 this system of equations can be written in its residual form U j = ρ j (Y, Z j = 1,..., J. (2 Unfortunately, there are no known identification results for this fully general model, and most recent work has considered a triangular restriction of (1 that rules out many important economic applications. In this paper we consider identification in a class of fully simultaneous nonparametric models introduced by Matzkin (2008. These models take the form m j (Y, Z, δ = 0 j = 1,..., J. where δ = (δ 1 (Z, X 1, U 1,..., δ J (Z, X J, U J and δ j (Z, X j, U j = g j (Z, X j + U j. (3 Here X = (X 1,..., X J R J are observed exogenous variables (instruments specific to each equation, and each g j (Z, X j is assumed strictly increasing in X j. This formulation respects traditional exclusion restrictions in that X j is excluded from equations k j (e.g., a demand shifter enters only the demand equation. However, it restricts the fully general model (1 by requiring X j and U j to enter the nonseparable function m j through a residual index δ j (Z, X j, U j. If we again assume invertibility of m (now in δ see the examples 2 See, e.g., Palais (1959, Gale and Nikaido (1965, and Berry, Gandhi, and Haile (2013 for conditions that can be used to show invertibility in different contexts. 2

4 below, we obtain the analog of (2, δ j (Z, X j, U j = r j (Y, Z j = 1,..., J or, equivalently, r j (Y, Z = g j (Z, X j + U j j = 1,..., J. (4 Below we provide several examples of important economic applications in which this structure can arise. Matzkin (2008, section 4.2 considered a two-equation model of the form (4 and showed that it is identified when U is independent of X, (g 1 (X 1,..., g J (X J has large support, and the joint density of U satisfies certain shape restrictions. Matzkin (2010 develops an estimation approach for such models, focusing on the case in which each index function δ j is linear in X j (with coefficient normalized to 1, again assuming large support for X and restrictions on the joint density to ensure identification. 3 Matzkin (2010 also provided identification results for related models that could be extended to the model considered here. These results also require a combination of conditions on the support of instruments X and on the joint density of the structural errors. We provide new identification results for this class model models, generalizing those in Matzkin (2008 and Matzkin (2010. Our theorems nest a wide variety of special cases that permit tradeoffs between the exogenous variation in instruments required and the structure placed on the joint density. At one extreme, if one maintains the large support condition, identification of the model in Matzkin (2010 (where the residual index function is linear holds without any restriction on the joint density of structural errors U. Alternatively, one can relax the large support condition by placing restrictions on the joint density that hold for a wide range of known density functions. Among these results are cases in which the support of X can be arbitrarily small. The identification results for nonparametric fully 3 In Matzkin (2010 the index structure and restriction g j (X j = X j follow from Assumption 3.2 (see also equation T

5 simultaneous models in Matzkin (2008 and Matzkin (2010 are also obtained as special cases of our results. For our results that do require a joint density restriction, our conditions hold for many standard joint densities. Perhaps more important, our primary sufficient conditions (including those on joint densities are verifiable, i.e., their satisfaction or failure is identified. Further, given these or other verifiable conditions, the maintained assumptions on the model imply testable (falsifiable restrictions. All our proofs are constructive; thus, in addition to providing a stronger foundation for the nonparametric estimation methods proposed by Matzkin (2010, our results may suggest new estimation strategies. Prior Work Brown (1983, Roehrig (1988, Brown and Matzkin (1998, and Brown and Wegkamp (2002 considered identification of simultaneous equations models, assuming one structural error per equation and focusing on cases where the structural model (1 can be inverted to solve for the residual equation (2. A claim made in Brown (1983 and relied upon by the others implied that traditional exclusion restrictions would identify the model when U is independent of Z. Benkard and Berry (2006 showed that this claim is incorrect, leaving uncertain the nonparametric identifiability of fully simultaneous models. A major breakthrough in this literature came in Matzkin ( For models of the form (2 with U independent of Z, Matzkin (2008 provided a new characterization of observational equivalence and showed how this could be used to prove identification in several special cases. These included a linear simultaneous equations model, a single equation model, a triangular (recursive model, and a fully simultaneous nonparametric model (her supply and demand example of the form (4 with J = 2. The last of these easy generalizes to J > 2, and, to our knowledge, this was the first result demonstrating identification in a fully simultaneous nonparametric model with nonseparable errors. As already noted, Matzkin s sufficient conditions combine a large support condition with a restriction on the joint density of the structural errors. More recently, Matzkin (2010, while focused on 4 See also Matzkin (

6 estimation, has included additional identification results for the case in which the residual index function is linear. Her main identification result follows Matzkin (2008, although the linearity restriction allows her to partially relax the density restriction used previously. Matzkin (2010 also offers constructive partial identification results for a two-equation model that could be extended to show identification of the full model we consider. 5 These results require joint restrictions on the support of the instruments and shape of the joint density, the latter ruling out some natural densities including the multivariate normal (see Berry and Haile (2013a. Relation to Transformation Models The model (4 considered here can be interpreted as a generalization of the transformation model to a system of simultaneous equations. The usual (single-equation semiparametric transformation model (e.g., Horowitz (1996 takes the form t (Y j = Z j β + U j (5 where Y i R, U i R, and the unknown transformation function t is strictly increasing. In addition to replacing Z j β with g j (Z, X j, 6 (4 generalizes (5 by dropping the requirement of a monotonic transformation function and, more fundamental, allowing a vector of outcomes Y to enter the unknown transformation function. Relation to Triangular Models Much recent work has focused on models with a triangular (recursive structure (see, e.g., Chesher (2003, Imbens and Newey (2009, and Torgovitsky (2010. A two-equation version of the triangular model is Y 1 = m 1 (Y 2, Z, X 1, U 1 Y 2 = m 2 (Z, X 1, X 2, U 2 5 We refer here to the results in her section 4.1. In an Appendix we show that such extensions, at least as we can anticipate them, would also be special cases of our theorems. 6 A recent paper by Chiappori and Komunjer (2009 considers a nonparametric version of the singleequation transformation model. See also the related paper by Berry and Haile (

7 with U 2 a scalar monotonic error and with X 2 excluded from the first equation. This structure often arises in a program evaluation setting, where Y 2 might denote a non-random treatment and Y 1 denotes an outcome of interest. To contrast this with a fully simultaneous system, consider the classical supply and demand system, where Y 1 might be the quantity of the good and Y 2 its price. Supposing that the first equation is the structural demand equation, the second equation would be the reduced form for price (quantity does not appear on the right, with X 2 as a supply shifter excluded from demand. However, in a supply and demand context as in many other traditional simultaneous equations settings the triangular structure is difficult to reconcile with economic theory. Typically both the demand error and the supply error will enter the reduced form for price. One obtains a triangular model only in the special case that the two structural errors monotonically enter the reduced form for price through a single index. This problem of having either multiple endogenous variables or multiple structural errors in each equation is the key challenge in a fully simultaneous system. The triangular framework requires that at least one of the reduced-form equations feature a monotone index of the all original structural errors. This is an index assumption, although one that is different from that of the model we consider. Our structure arises naturally from a fully simultaneous structural model with a nonseparable residual index; the triangular model will be generated by other kinds of restrictions on the functional form of simultaneous equations models. Examples of simultaneous models that do reduce to a triangular system can be found in Benkard and Berry (2006, Blundell and Matzkin (2010 and Torgovitsky (2010. Blundell and Matzkin (2010 have recently provided a necessary and sufficient condition for the simultaneous model to reduce to the triangular model, pointing out that this condition is quite restrictive. Completeness We consider a fully specified model of all endogenous variables and seek restrictions on primitives that allow us to demonstrate identification. An alternative approach is to combine conditional moment conditions with completeness assumptions (Lehmann and 6

8 Scheffe, 1950, 1955 sufficient for identification. This is a natural approach in many settings, and in Berry and Haile (2013b we have shown how the identification arguments of Newey and Powell (2003 or Chernozhukov and Hansen (2005 can be adapted to the class of models considered here. 7 However, independent of general concerns one might have with the interpretability of completeness conditions, such conditions may be particularly unsatisfactory in a simultaneous equations setting. A simultaneous equations model already specifies the structure that generates the joint distribution of the endogenous variables, exogenous variables, and structural errors. Imposing a high-level assumption like completeness implicitly places further restrictions on this structural model, although the nature of these restrictions is typically unclear. It may, therefore, by more helpful to take an alternative approach that makes explicit the types of restrictions on the system of equations that allow for identification. An additional motivation, expressed previously by Matzkin (2010, is that nonparametric estimation methods based on moment conditions and completeness assumptions can be difficult. An alternative identification strategy particularly one using constructive arguments may suggest new approaches to estimation. Outline We begin with some motivating examples in section 2. Section 3 then completes the setup of the model. In section 4 we consider identification of the model when the residual index is linear, as in Matzkin (2010. Section 5 provides extensions to the case of a nonlinear index. 2 Examples Example 1. Consider a nonparametric version of the classical simultaneous equations model, where the structural equations are given by Y j = Γ j (Y j, Z, X j, U j j = 1,..., J. 7 The identification results for nonparametric regression models are not directly applicable because the structural functions m j take multiple structural errors as arguments. Nonetheless, the extension is straightforward. 7

9 Examples include the classical supply and demand model or models of peer effects. The residual index structure is imposed by requiring Γ j (Y j, Z, X j, U j = γ j (Y j, Z, δ j (Z, X j, U j j where δ j (Z, X j, U j = g j (Z, X j + U j. This model features nonseparable structural errors but requires them to enter the nonseparable nonparametric function Γ j through the index δ j (Z, X j, U j. If each function γ j is invertible (e.g., strictly increasing in δ j (Z, X j, U j then one obtains (4 from the inverted structural equations by letting r j = γ 1 j. Identification of the functions r j and g j implies identification of Γ j. Example 2. Consider an imperfectly competitive market for differentiated products. Demand (expressed, e.g., in levels or shares for each product j = 1,..., J in market t is given by ( S jt = σ j Pt, δ d t (6 where P t R J is the price vector and δ d t is a vector of demand indexes δ d jt = g j (X jt + ξ jt with X jt R and ξ jt R reflecting, respectively observed and unobserved demand shifters (all other observed demand shifters have been conditioned out, treating them fully flexibly. Prices are determined through oligopoly competition, yielding a reduced form for equilibrium prices with the form ( P jt = π j δ d t, κ s t j = 1,..., J (7 where κ s t is a vector of cost indexes κ s jt = h j (W jt + ω jt with W jt R and ω jt R denoting, respectively, observed and unobserved cost shifters (all 8

10 other observed cost shifters have been conditioned out. The functions g j and h j are all assumed to be strictly increasing. Following Berry, Gandhi, and Haile (2013 and Berry and Haile (2010, one can show that this structure is obtained from many standard models of differentiated products demand and oligopoly supply under appropriate residual index restrictions on preferences and costs. Further, the system can be inverted, yielding a 2J 2J system of equations g j (X jt + ξ jt = σ 1 j (S t, P t h j (W jt + ω jt = π 1 j (S t, P t where S t = (S 1t,..., S Jt, P t = (P 1t,..., P Jt. This system takes the form of (4. Berry and Haile (2010 show that identification of the unknown functions in this system implies identification of demand, marginal costs, and firms marginal cost functions. Example 3. Consider identification of a production function in the presence of unobserved ( shocks to the marginal product of each input. Output is given by Q = F Y, Ũ, where Y R J + is a vector of input quantities and Ũ RJ is a vector of latent factor-specific productivity shocks. Let P and W denote the (exogenous prices of the output and inputs, respectively. The observables are (Q, P, W, Y. With this structure, cost-minimizing input demand is determined by a system of first-order conditions F (y, ũ p = w j j = 1,..., J (8 y j whose solution can be written y j = η j (p, w, ũ j = 1,..., J. Observe that the reduced form for each Y j depends on the entire vector of shocks U. The index structure can be imposed by assuming that each structural error U j enters as a multiplicative 9

11 shock to the marginal product of the associated input, i.e., F (y, ũ y j = f j (y ũ j for some function f j. The first-order conditions (8 then take the form (after taking logs ln (f j (y = ln ( wj ln (ũ j j = 1,..., J. p which have the form of our model (4 after defining X j = ln ( Wj, U P j = ln (Ũj. The results below will imply identification of the functions f j and, therefore, the realizations of each U j. Since Q is observed, this implies identification of the production function F. 3 The Model 3.1 Setup The observables are (Y, X, Z. The exogenous observables Z, while important in applications, add no complications to the analysis of identification. Thus, from now on we condition on an arbitrary value of Z and drop it from the notation. As usual, this treats Z in a fully flexible way. Stacking the equations in (4, we then consider the model r (Y = g (X + U (9 where r (Y = (r 1 (Y,..., r J (Y and g (X = (g 1 (X 1,..., g J (X J. We let X =int(supp(x and Y =int(supp (Y. We maintain the following assumptions on the model throughout. Assumption 1. (i X is nonempty; (ii g is differentiable, with g j (x j / x j > 0 for all 10

12 j, x j ; (iii r is one-to-one on Y, differentiable on Y, and has nonsingular Jacobian matrix J(y = r 1 (y r 1 (y y J y r J (y y 1... r J (y y J for all y Y; and (iv U is independent of X and has positive joint density f on R J. The following result documents two useful implications of Assumption 1. Lemma 1. Under Assumption 1, (i y Y, supp(x Y = y =supp(x; and (ii x X, supp(y X = x =supp(y. Proof. Both claims follow immediately from (9 and the assumption that U is independent of X with support R J. For some results we will strengthen the smoothness assumption on r, allowing us to exploit the following result. Lemma 2. Let Assumption 1 hold and suppose that r C 1. Then Y is path-connected. Proof. Because r is one-to-one, continuously differentiable, and has nonzero Jacobian determinant, it has a continuous inverse r 1 on Y such that Y = r 1 (g(x + U. Since supp(u X = R J, the result follows from the fact that the image of a path-connected set (here R J under a continuous mapping is path-connected. 3.2 Normalizations We impose three standard normalizations. 8 First, observe that all relationships between (Y, X, U would be unchanged if for some constant κ j, g j (X j were replaced by g j (X j + κ j 8 We follow Horowitz (1982, p , who makes equivalent normalizations in his semiparametric single-equation version of our model. Alternatively we could follow Matzkin (2008, who makes no normalizations in her supply and demand example and shows only that the derivatives of r and g are identified up to scale. 11

13 while r j (Y is replaced by r j (Y + κ j. Thus, without loss, for an arbitrary point ẏ Y and an arbitrary vector τ = (τ 1,..., τ J we will set r j (ẏ = τ j j. (10 Given this restriction (and a choice of τ, we will still require normalizations on the location and scale of the unobservables U j, as usual. 9 Since (9 would continue to hold if both sides were multiplied by a nonzero constant, we normalize the scale of U j by taking an arbitrary ẋ X and setting g j (ẋ j x j = 1 j. (11 And since (9 would be unchanged if g j (X j were replaced by g j (X j + κ j for some constant κ j while U j were replaced by U j κ j, we fix the location of U j by setting g j (ẋ j = ẋ j j. (12 With (12, we will find a convenient choice of τ (recall (10 to be τ j = ẋ j j so that r j (ẏ g j (ẋ j = 0 j. ( Identifiability, Verifiability, and Falsifiability As usual, we will say that a structural feature is identified if it is uniquely determined by the joint distribution of the observables. The primary structural features of interest here are the functions r, f and, when we allow a nonlinear index function, g. However, we will also be interested in identification of our key conditions for identification. In particular, we will 9 Often these restrictions are without loss as well, although one can imagine applications in which the location and/or scale of U j has economic meaning. 12

14 say that a condition A is verifiable if the value of the indicator 1{A} is identified. When a condition is verifiable, its satisfaction or failure is an identified feature. An unusual property of our identification results is that our primary sufficient conditions (those beyond the model setup and maintained regularity conditions are verifiable. 10 Finally, we will say that a model is falsifiable it implies a verifiable condition B. Falsifiability is a standard notion of the testability of a model. Often falsifiability is shown by demonstrating existence of overidentifying restrictions on observables, and this will be the case with our results below. Such overidentifying restrictions are of particular interest when the maintained identifying assumptions are themselves verifiable, since this implies that (given verification of the identifying assumptions the hypothesis at risk under the falsifiable restriction is satisfaction of the basic modeling assumptions, not the joint hypothesis of the model and identifying conditions. 4 Identification with a Linear Index We begin with the case in which each g j is linear in X j. With the normalizations above, 11 this yields the model r j (Y = X j + U j j = 1,..., J. (14 By a standard change of variables, the conditional density of Y = y given X = x can then be written φ(y x = f(r(y x J (y. (15 We treat φ(y x as known for y Y, x X. 10 We are not aware of any prior use of the notion of verifiability in the econometrics literature although, as our definition makes clear, this is merely a special case of identifiability. Our use of the term borrows from philosophy (e.g., Ayer (1936, which makes similar distinctions but for different purposes. 11 Linearity of the index function does not alter the need for these normalizations or the extent to which they are without loss. 13

15 4.1 Identification without Density Restrictions Our first result shows that if one maintains Matzkin s (2008, 2010 large support assumption, there is no need for any restriction on f. 12 Theorem 1. Let Assumption 1 hold and suppose X = R J. Then r and f are identified. Proof. Since f (r (y x dx = 1, from (15 we obtain f (r (y x = φ (y x φ (y t dt. Thus the value of f (r (y x is uniquely determined by the observables for all x R J, y Y. Let F j denote the marginal CDF of U j. Since f (r (y ˆx dˆx = F j (r j (y x j (16 ˆx j x j,ˆx j the value of F j (r j (y x j is identified for x R J, y Y. By (12 and (13, F j (r j (ẏ ẋ j = F Uj (0. For any y Y we can then find the value x o (y such that F j (r j (y x o (y = F j (0, which reveals r j (y = o x (y. This identifies each function r j on the support of Y. Identification of f then follows from (14. Thus, given the maintained Assumption 1, large support for X is sufficient for identification of the model. Because this sufficient condition involves only the support of the observables X, its satisfaction (or failure is also identified, giving the following result. Remark 1. The condition X = R J is verifiable. 12 The argument used to show Theorem 1 was first used by Berry and Haile (2013b in combination with additional assumptions and arguments to demonstrate identification in models of differentiated products demand and supply. 14

16 4.2 Identification Combining Support and Density Conditions We obtained Theorem 1 without requiring differentiability of r or f, and without any restriction on the joint density f. However, this result required a large support condition. Adding a differentiability assumption and exploring restrictions on f will allow us to develop several alternative identification results, including some that permit more limited (even arbitrarily small support for X. We first develop a general result, then illustrate a variety of special cases. To do this we will add the following maintained hypotheses. Assumption 2. (i r is twice differentiable; (ii f is continuously differentiable.. With these additional regularity conditions, we can take logs of (14 and differentiate to obtain ln φ(y x ln f(r(y x = (17 x j u j and ln φ(y x y k = j ln f(r(y x r j (y ln J(y +. (18 u j y k y k Together (17 and (18 imply ln φ(y x y k = j ln φ(y x r j (y ln J(y +. (19 x j y k y k We rewrite (19 as a k (x, y = d (x, y b k (y (20 where a k (x, y = d (x, y = b k (x, y = ln φ(y x y ( k ln φ(y x ln φ(y x 1,,..., x 1 x J ( ln J(y, r 1(y,..., r J(y. y k y k y k 15

17 The following lemma provides the key result of this section. Lemma 3. Let Assumptions 1 and 2 hold. For a given y Y, suppose there exists no nonzero vector c = (c 0, c 1,..., c J such that d (x, y c = 0 x X. Then r(y/ y k is identified for all k. Proof. By the hypothesis, there must exist a set of points x = ( x (1,..., x (m with m J and each x (j X such that the m J matrix D ( x, y d ( x (1, y. d ( x (m, y has full (column rank. Take an arbitrary k {1,..., J} and let A k ( x, y = a k ( x (1, y. a k ( x (m, y. Stacking the equations obtained from (20 at x (1,..., x (m, we have A k ( x, y = D ( x, y b k (y. Because D ( x, y has full column rank, D ( x, y D ( x, y is invertible, so we obtain b k (y = ( D ( x, y D ( x, y 1 D ( x, y A k ( x,y. Since D ( x, y and A k ( x,y are observable, the result follows. Lemma 3 provides a sufficient condition for identification of the derivatives of r at a 16

18 point. This is easily extended to identification of the model using the following assumption. Assumption 3. For almost all y Y there is no c = (c 0, c 1,..., c J 0 such that ( ln f(r(y x 1,,..., x 1 ln f(r(y x c = 0 x X. x J Theorem 2. Let Assumptions 1, 2, and 3 hold. Then r and f are identified. Proof. By Lemma 3, r j (y/ y k is identified for all j, k, and y Y. With the boundary condition (10 and Lemma 2, we then obtain identification of r j (y for all y and j. Identification of f then follows from (14. The key requirement of Theorem 2 is the rank condition of Assumption 3. condition on the unknown joint density f. ln φ(y x x j However, because ln f(r(y x u j is observable, it is immediate that this condition is verifiable. Remark 2. Assumption 3 is verifiable. This is a ln φ(y x = x j and Under another verifiable condition, 13 the maintained assumptions of the model yield a set of testable restrictions. Remark 3. Let Assumptions 1 and 2 hold and suppose that, for some y Y, X contains two sets of points x = ( x 1,..., x m and x = ( x 1,..., x n such that x x and D ( x, y and D ( x, y have full column rank. Then the model (14 is falsifiable. Proof. By Lemma 3 r(y/ y k is identified for all k using just one of the sets of vectors x or x. Letting r(y/ y k [ x] and r(y/ y k [ x ] denote the implied values of r(y/ y k using each set, we obtain the verifiable restrictions r(y/ y k [ x] = r(y/ y k [ x ] for all k. 13 Compared to Assumption 3, the verifiable condition used in the hypothesis of Remark 3 involves only a single point y, but requires two distinct sets of points x and x such that D ( x,y and D ( x,y have full column rank (see the proof of Lemma 3. 17

19 4.3 Special Cases We now discuss several combinations of support and/or density restrictions that will ensure satisfaction of the rank condition Assumption 3, leading to several special cases of Theorem Critical Points and Tangencies We first consider conditions inspired by the proofs in Matzkin (2008, 2010, although we will be able to offer milder sufficient conditions for identification. Begin with the case J = 2. Suppose that for almost all y Y there exist points ( x (0 (y, x (1 (y, x (2 (y, each in X, with the following properties: ln f (r(y x 0 (y = 0 j u j ln f (r(y x 1 (y 0 = ln f (r(y x1 (y u 1 u 2 ln f (r(y x 2 (y 0. u 2 The point u 0 (y = r(y x 0 (y is a critical point of f. At u 1 (y = r(y x 1 (y, the derivative of f with respect to u 1 is nonzero, but that with respect to u 2 is zero. Thus, u 1 (y has a graphical interpretation as a vertical tangency with some (non-extremum level set of f. See Figure 1. Finally u 2 (y = f(r(y x 2 (y is any point such that the derivative of f(u 2 with respect to u 2 is nonzero (i.e., not also a point of vertical tangency. Letting x = ( x (0 (y, x (1 (y, x (2 (y, we then have D ( x, y = ln f(r(y x1 (y u ln f(r(y x2 (y u 1 ln(r(y x 2 (y u 2 Because this matrix is triangular with non-zero diagonal terms, it is full rank as required by Theorem 2. There is, of course, an analogous construction using a horizontal tangency to a 18

20 u 0 u 1 u2 u 2 u 1 Figure 1: Level sets of a log density with u 0 at maximum, u 1 at a vertical tangency and u 2 at any point where the density has non-zero second derivative. level set of f instead of a vertical tangency. Generalizing to J 2 is straightforward and yields the following result. Corollary 1. Let Assumptions 1 and 2 hold and suppose that for almost all y Y, there exist points ( x 0 (y, x 1 (y,..., x J (y in X such that (a ln f(r(y x 0 (y/ u j = 0 for all j, (b ln f(r(y x j (y/ u j 0 for all j, and (c the matrix ln f(r(y x 1 (y ln f(r(y x 1 (y u J u ln f(r(y x J (y u 1 ln f(r(y x J (y u J can be placed in triangular form through simultaneous permutation of rows and columns. Then r and f are identified. In general, existence of the points ( x 0 (y, x 1 (y,..., x J (y required by Corollary 1 for all y involves a joint requirement on the density f and the support of X. For example, if f has many critical values in different neighborhoods admitting tangencies to its level sets, 19

21 identification can be obtained from Corollary 1 with limited variation in X enough for the support of r (y X to cover one such neighborhood for almost all y Y. Alternatively, with large support for X, Corollary 1 would require existence of only a single set of points ( u 0, u 1,..., u J such that ln f(u0 u j = 0 for all j while ln f(u 1 ln f(u 1 u J u ln f(u J u 1 ln f(u J u J (21 is triangular (or, more generally, invertible. We state this result in the following corollary to Corollary 1. Corollary 2. Let Assumptions 1 and 2 hold. Suppose that X = R J and that there exist points ( u 0, u 1,..., u J, each in R J, such that (a ln f(u 0 / u j = 0 for all j, (b ln f(u j / u j 0 for all j, and (c the matrix ln f(u 1 ln f(u 1 u J u ln f(u J u 1 ln f(u J u J can be placed in triangular form through simultaneous permutation of rows and columns. Then r and f are identified. An even stronger restriction on f, but one satisfied by many densities on R J, is that f have at least one critical point and at least one point of tangency in each dimension (e.g., taking u 1 = û (u 1 and u 2 = û (u 2 in Figure 2. In that case the matrix in (21 will be diagonal. Combining this property with a large support condition yields the following special case of Corollary 2, first given by Matzkin (2010, Theorem 3.1. Corollary 3. Let Assumptions 1 and 2 hold. Suppose that X = R J and that there exist points ( u 0, u 1,..., u J, each in R J, such that (a ln f(u 0 / u j = 0 for all j, (b ln f(u j / u j 0 20

22 for all j, and (c the matrix ln f(u 1 ln f(u 1 u J u ln f(u J u 1 ln f(u J u J is diagonal. Then r and f are identified. Of course, large support is not essential for the special case of a diagonal first-derivative matrix. What is essential is that for each value of y, X contain J + 1 points ( x 0,..., x J such that the points u 0 = r (y x 0 u 1 = r (y x 1. u J = r (y x J correspond, respectively, to a critical value and tangencies in each dimension 1,..., J. Depending on the shape of f, this can hold even when X has limited support Second Derivatives The results in the previous section ensured that the matrix ln f(r(y x 1 u. ln f(r(y x J u is invertible for some ( x 1,..., x J X J only for certain j. by requiring points u at which ln f(u u j is nonzero Obviously this is not necessary. In fact, given additional smoothness on f, we can provide a necessary and sufficient condition for the rank condition of 3 in terms 21

23 of the second-derivative matrix H φ (x, y = 2 ln φ(y x x ln φ(y x x 1 x J 2 ln φ(y x x J x ln φ(y x x 2 J. (22 Lemma 4. Let f be twice differentiable. For a given y Y, there is a nonzero vector c = (c 0, c 1,..., c J such that d (x, y c = 0 x X (23 if and only if for c = (c 1,..., c J H φ (x, y c = 0 x X. (24 Proof. ((23 = (24 Suppose (23 holds. Differentiating (23 with respect to x yields (24, with c = (c 1,..., c J. So we need only show that c is nonzero. If c 0 = 0 then the fact that c 0 immediately implies c j 0 for some j > 0. If c 0 0, then since the first component of d (x, y is nonzero and d (x, y c = 0, we must have c j 0 for some j > 0. ((24 = (23 Now suppose (24 holds for c = (c 1,..., c J. Take an arbitrary point x 0 and let c 0 = J ln φ(y x 0 j=1 x j c j so that d (x 0, y c = 0 by construction. Recalling that the first component of d (x, y equals 1 for all (x, y, observe that (24 implies that x j [ d (x, y c ] = 0 for all j and every x X. Thus (23 holds. This equivalence allows us to provide a sufficient condition for identification in terms of the Hessian matrix 2 ln f(r(y x u u. Corollary 4. Let Assumptions 1 and 2 hold and assume that f is twice differentiable. Suppose that, for almost all y Y, there is no nonzero J-vector c such that 2 ln f (r (y x u u c = 0 x X. Then r and f are identified. 22

24 Proof. This follows immediately from (17, (22, Lemma 4, and Theorem 2. By this result, a sufficient condition for identification is that lnf (u have nonsingular Hessian matrix at points u = r(y x reachable through the support of X. A strong sufficient condition is that 2 ln f(u u u be nonsingular almost everywhere; in that case, the support of X can be arbitrarily small. This strong sufficient condition is satisfied by many standard joint probability distributions. For example, it holds under when 2 ln f(u u u is negative definite almost everywhere a property of the multivariate normal and many other log-concave densities (see, e.g., Bagnoli and Bergstrom (2005 and Cule, Samworth, and Stewart (2010. Examples of densities that violate the sufficient condition for the requirements of Corollary 4 are those with flat (uniform or log-linear (exponential regions. At the opposite extreme, if X has large support, it is sufficient that 2 ln f(u u u be invertible at a single point. This condition fails for uniform or exponential densities, although neither of these offers a proper density on R J. Corollary 4 can be used to provide conditions on the second derivative matrix of the density f, rather than the log-density, that are sufficient for identification: Corollary 5. Let Assumptions 1 and 2 hold and assume that f is twice differentiable. Suppose that for almost all y Y there exists x (y X such that for u (y = r(y x (y (a f(u(y u j = 0 for all j, and (b 2 f(u(y u u is nonsingular. Then r and f are identified. Proof. By (a, 2 ln f (u (y u u = 1 f (u (y 2 f(u(y u f(u(y u J u 1 2 f(u(y u 1 u J f(u(y u 2 J which is nonsingular by (b. The result then follows from Corollary 4. In some applications, it may be reasonable to assume that U j and U k are independent for all k j. In that case 2 ln f(u u u is diagonal. The following result, whose proof is immediate from Corollary 4, shows that this can allow identification with arbitrarily small X under only 23

25 a mild additional restriction on f. Corollary 6. Let Assumptions 1 and 2 hold. Suppose that f (u = j f j (u j for all u and that 2 ln f j (u j u 2 j exists and is nonzero almost surely for all j. Then r and f are identified. We pause to emphasize that although Corollaries 5 and 6 have provided sufficient conditions involving nonsingularity of 2 ln f(u u u obtain identification through Corollary 4. at some points u, this is not necessary for one to For a given pair (x, y, 2 ln f(r(y x u u is singular if and only if there exists a nonzero vector c such that 2 ln f(r(y x u u c = 0. However, Corollary 4 shows that only when (for values of y with positive measure the same vector c solves this equation for every x X does the sufficient condition for identification in Theorem 2 fail. When this happens, the columns of 2 ln f(r(y x u u each x: they exhibit the same linear dependence for all x. do not merely exhibit linear dependence at Differenced Derivatives Although we exploited the assumed twice-differentiability of f in the previous section, it is straightforward to extend the arguments there to cases without such differentiability by replacing the matrix of second derivatives with differences in the first derivatives. Suppose that (23 holds. This implies that d (y, x c is constant across all x X; i.e., for any x and x in X, [d (y, x d (y, x ] c = 0. Since the first component of d (y, x d (y, x is zero, this is equivalent to the condition ln φ(y x x 1 ln φ(y x x 1. ln φ(y x x J ln φ(y x x J c = 0 x X, x X. (25 Thus, only when there exists a nonzero vector c satisfying (25 does the rank condition of Theorem 2 fail. 24

26 5 Identification with a Nonlinear Index We now move to the more general model (9, where the change of variables (cf. equation (15 now yields φ(y x = f(r(y g(x J (y for all y Y, x X. Thus ln φ(y x x j ln f(r(y g(x g j (x j = (26 u j x j and ln φ(y x y k = j ln f(r(y g(x r j (y ln J(y + (27 u j y k y k which together imply ln φ(y x y k = j ln φ(y x x j r j (y/ y k + g j (x j / x j ln J(y y k. ( Rectangle Regularity Relative to the linear index model considered in section 4, the complication introduced by the more general model (9 is the need to identify the index functions g j. Our strategy for doing so will build on an important insight in Matzkin (2008 that again relies on critical points of f and tangencies to its level sets. As we did in section 4.2, we will provide a general sufficient condition that involves joint restrictions on the density f and support of X. We will then show how this allows identification under several specific combinations of restrictions on f and X. We begin with some definitions. Definition 1. A J-dimensional rectangle is a Cartesian product of J nonempty open intervals. Definition 2. Let M J j=1 ( mj, m j and N J j=1 ( nj, n j denote two J-dimensional 25

27 rectangles. M is smaller than N if m j m j n j n j for all j. Definition 3. Given a J-dimensional rectangle U J j=1 ( uj, u j, the joint density f is regular on U if (i u j U such that f(u u j u j ( u j, u j, there exists û ( u j U satisfying = 0 for all j; and (ii for all j and almost all û j ( u j = u j f(û(u j u j 0 and f(û(u j u k = 0 k j. Definitions 1 and 2 are provided only to avoid ambiguity. Definition 3 introduces a notion regularity for the density f. It requires that f have a critical point in a rectangular neighborhood U in which the level sets of f are nice in a sense defined by part (ii. There, û(u j has a geometric interpretation as a point of tangency between a level set of f and the (J 1-dimensional plane defined by fixing U j = u j. Figure 2 provides an example in which J = 2 and u is a local extremum. Here, within a neighborhood of u, the level sets of f (or ln f are connected, smooth, and strictly increasing toward u. Therefore, each level set is horizontal at (at least one point above u and one point below u. Similarly, each level set is vertical at least once each to the right and to the left of u. There are many J-dimensional rectangles on which the illustrated density is regular. One such rectangle is defined in the figure using a single level set. The upper limit u 2 is defined by the largest horizontal point on this level set, while u 1 is defined by the rightmost vertical point and so forth. For each u 1 (u 1, u 1, the point û 2 (u 1 is the value of U 2 at a tangency between the vertical line U 1 = u 1 and a level set of f closer to u than that defining U. Since level sets within U are smooth, the tangency cannot be at a corner of the rectangle U; therefore, u 2 < û 2 (u 1 < u 2, implying (u 1, û 2 (u 1 U. Regularity on U requires that there exist a vertical tangency within U for all u 1 (u 1, u 1, as well as a horizontal tangency within U for all u 2 (u 2, u 2. The generalization to J > 2 is straightforward. 26

28 u 2 u2 û 2 (u 1 u u 2 u 1 u 1 u 1 u 1 Figure 2: The solid curves are the level sets of a log-density with a regular hill leading up to a local maximum at u, but with a less useful shape in other areas. The rectangle U (u 1, u 1 (u 2, u 2. For each u 1 (u 1, u 1 the point û 2 (u 1 is the value of U 2 at a tangency between the vertical line U 1 = u 1 and a level set. x 1 x 1 x 1 x 2 x2 x ˆx 2 (x 1 x 2 x 1 Figure 3: For arbitrary x X, the rectangle U (see Figure 2 is mapped to a rectangle X (x by first defining y to satisfy r j (y = g j (x + u j for all j, then defining x and x by r j (y = g j ( xj + uj = g j (x j + u j. 27

29 Assumption 4. For all x X there exists a point u R J and a J-dimensional rectangle ( X (x = j xj (x, x j (x X containing x such that (a f (u / u j = 0 for all j and (b f is regular on ( U (x = j uj (x, u j (x ( ( ( = j u j + g j x j gj (x (x, u j + g j x j gj (x (x. (29 Assumption 4 requires, for any given x, that f be regular on a rectangular neighborhood around a critical point u that maps through (9 to a rectangular neighborhood in X around x. Of course, if a set X is a rectangle, so is g (X. Further, if X includes any rectangle M around the point x, X also includes all smaller rectangles X M. In this case, as long as f is regular on some rectangle that is not too big relative to the support of X around x, the set X (x required by Assumption 4 must exist. Figure 3 illustrates, taking the arbitrary point x and the rectangle U in Figure 2 and mapping them to the rectangle X (x. The following results provide two alternative sufficient conditions for Assumption 4. The first combines a large support condition with regularity of f on R J. This was the combination of conditions proposed by Matzkin ( The second (which also clarifies the construction of the set U (x in (29 allows limited support and requires regularity only in sufficiently small rectangular neighborhoods around some critical point(s u. Outside such neighborhoods, f is unrestricted. Lemma 5. Suppose that g (X = R J and that f is regular on R J. Then Assumption 4 holds. ( Proof. Let X (x = j g 1 j (, g 1 j ( for all x. By (29, U (x = R J, yielding the result. 14 The regularity assumption stated in Matzkin (2008 is actually stronger, equivalent to assuming regularity of f on R J but replacing almost all u j ( u j, u j in the definition of regularity with all u j ( u j, u j. The latter is unnecessarily strong and rules out many standard densities, including the multivariate normal. Throughout we interpret the weaker condition as that intended by Matzkin (

30 Lemma 6. Suppose that (i for all x X, X contains a J-dimensional rectangle X (x x, and (ii for any J-dimensional rectangle M, there exists a J-dimensional rectangle N smaller than M such that f is regular on N. Then Assumption 4 holds. ( Proof. Take an arbitrary x X. Let M =g X (x and let N be a smaller rectangle j ( uj, u j on which f is regular (guaranteed to exist by part (ii of the hypothesis. Because f is regular on N, it has a critical point u N. Taking this u, define y by the equation r (y = g (x + u. Now define x j and x j for all j by r j (y = g j ( xj + uj r j (y = g j (x j + u j. Let X (x = j ( xj, x j. Then by (29, U (x = N, so f is regular on U (x. 5.2 Identification of the Index Function The following lemma is our key result regarding identification of g. Lemma 7. Let Assumptions 1-4 hold. Then for every x X there exists a rectangle X (x x such that for all x 0 X (x \x and x X (x \x, the ratio g j (x j/ x j g j (x 0 j / x j is identified for all j = 1,..., J. Proof. Take an arbitrary x X and let u and U (x = j ( uj, u j be as defined in 29

31 Assumption 4. Define y R J by 15 r(y = g(x + u. (30 By Assumption 4 and equation (26 ln φ(y x x j = 0 j. (31 Further, there exists X = i (x i, x i X such that r(y = g j (x j + u j, j = 1,..., J (32 r(y = g j (x j + u j, j = 1,..., J. (33 and such that for any j and almost every x j ( x j, x j there is a J-vector ˆx ( x j X satisfying ( ˆx j x j = x j ln φ ( y ˆx(x j 0 x j and (34 ln φ ( y ˆx(x j x k = 0 k j. (35 By construction, y is in the interior of the support of Y conditional on any x X (x. So φ (y x and its derivatives are observed for all x X (x. Thus the point y is identified, as are the pairs ( x j, x j and the point ˆx ( x j for any j and x j ( x j, x j. 16 Further, by (31, 15 To simplify notation, we suppress dependence of y on the arbitrary point x. Below we do the same for the set X, and the points x j, x j, u j, u j. 16 We do not require uniqueness of u or the set U. Rather, we use only the fact that for a given x there exist both a value y mapping through (14 to any critical point u and a rectangle around x mapping through (32 and (33 to a rectangle around u on which f is regular. Through (26, such a y and a rectangle around x are both identified. 30

32 at (y, x equation (28 yields ln J(y y k = ln φ(y x y k k. (36 This allows us to rewrite (28 at y and arbitrary x as j ln φ(y x x j r j (y / y k = ln φ(y x g j (x j / x j y k ln φ(y x y k (37 with both terms on the right-hand side known. Taking arbitrary j, arbitrary x j ( x j, x j and the known vector ˆx ( x j defined above, (37 becomes ln φ(y ˆx(x j x j r j (y / y k g j (x j / x j = ln φ(y x y k ln φ(y ˆx(x j y k. By (34, we may rewrite this as r j (y / y k g j (x j / x j = ln φ(y x y k ln φ(y ˆx(x j y k ln φ(y ˆx(x j. (38 x j Since the right-hand side is known, r j(y / y k g j (x j / x j is identified for almost all x j [ x j, x j ]. By the same arguments leading up to (38, but with x 0 j taking the role of x j, we obtain r j (y / y k g j (x 0 j / x j = ln φ(y x y k ln φ(y ˆx(x 0 j y k ln φ(y ˆx(x 0 j. (39 x j Because the Jacobian determinant J (y is nonzero, r j (y / y k cannot be zero for all k. Thus for some k the ratio of (39 to (38 must be well defined for each j = 1,..., J. This establishes the result. 17 This leads to the following result. 17 Since the argument can be repeated for any k such that r j (y / y k 0, the ratios in the lemma may typically be overidentified. 31

33 Theorem 3. Let Assumptions 1, 2, and 4 hold and suppose that X is path-connected. Then g is identified on X. Proof. We first claim that Lemma 7 implies identification of g j(x j / x j g j (x 0 j / x j for all x j X and a single x 0 j. If X (x = X in Lemma 7, this is immediate. Otherwise, consider any two points x and x in X. Because X is path-connected, there exists a continuous function ρ : [0, 1] X such that ρ (0 = x and ρ (1 = x. Because each rectangle X (ρ (λ guaranteed to exist by Lemma 7 is open, we know that for sufficiently small ɛ X (x ρ (ɛ X (ρ (λ ρ (λ ɛ λ (0, 1 X (ρ (λ ρ (λ + ɛ λ (0, 1 X (x ρ (1 ɛ. Thus, there exists an ordered set of sequentially overlapping rectangles {X (ρ (λ, λ [0, 1]} linking x to x. Since the ratios g j(x 1 j / x j g j (x 2 j / x j rectangles, it follows that the ratios g j(x j / x j g j (x j / x j then follows. known for all points ( x 1 j, x 2 j in each of these are known for all x and x in X. The claim Finally, taking x 0 j = ẋ j for all j, the conclusion of the Theorem follows from the normalization (11 and boundary condition ( Identification of the Model Theorem 3 is important because once g is known, identification results for the linear index model extend immediately to the setting with a nonlinear index. This follows by letting X j = g j (X j 32