Nonparametric Demand Systems, Instrumental Variables and a Heterogeneous Population

Transcription

1 Nonparametric Demand Systems, Instrumental Variables and a Heterogeneous Population Stefan Hoderlein Manneim University September 2004 Abstract Tis paper is concerned wit empirically modelling te demand beavior of a population wit eterogeneous preferences under a weak conditional independence assumption. More specifically, we caracterize te testable implications of negative semidefiniteness and symmetry of te Slutsky matrix across a eterogeneous population witout assuming anyting on te functional form of individual preferences. In te same spirit, implications of a linear budget set are being considered. Since te conditional independence assumption is te only substantial restriction in tis model, we analyze possible alternatives and solutions if tis assumption is violated. In particular, we consider in detail te concept of instruments in tis framework. Besides being able to integrate econometric concepts, te same framework admits also economic extensions. As an example we consider welfare analysis. Finally, we provide asymptotic distribution teory for te new test statistics tat emerge out of tis framework, and apply tese to Canadian data. 1 Introduction Economic teory yields strong implications for te actual beavior of individuals. Tis is particularly true for demand teory, were a couple of wellknown restrictions arise. All restrictions imposed by rationality on demand 1

2 beavior are qualitative in nature, wic means tat tey do not predict a specific functional form for te demands of individuals. To test te implications of rational beavior, by and large two strands of literature ave emerged. Te first uses revealed preference teory, is nonparametric in nature and concentrates on violations of te Strong Axiom in observable data. Key contributions are Afriat (1967) and Varian (1982). More recently, a similar approac as been suggested by Blundell, Browning and Crawford (2002). Te second strand of literature tests a couple of restrictions on demand beavior, using fully specified parametric demand systems. Tis literature dates back to at least te fifties (Stone (1954)), but as really peaked wit te advent of fully flexible functional form demand systems. More recent examples are te Translog, Jorgenson et al. (1982), te AIDS, Deaton and Muellbauer (1980), Blundell et al (1993), or te exact QUAIDS, Banks et al.(1997), see also Lewbel (1999) for a compreensive survey. Obviously, bot approaces ave teir limitations: Te first usually leads to tests of low power, as price movements are dwarfed by movements in income, and concentrates on one specific property only. Te second suffers from te limitations tat demands take a certain functional form and tat te introduction of preference eterogeneity as not been solved convincingly (see, e.g. Brown and Walker (1989)). We aim in tis paper at laying te foundations for nonparametric demand systems, ideally combining te advantages of bot approaces: Being nonparametric in nature, i.e. not specifying any functional form, and still able to judge te restrictions imposed by rationality robustly as well as compreensively. Since we model individual eterogeneity by a nonseparable model, tere are some similarities to te work of Imbens and Newey (2003) and Matzkin (2003). However, teir general approac is different, as will become apparent in te following. Tis paper makes several contributions: First, we provide a framework for nonparametric demand analysis in a eterogeneous population. Second, witin tis framework we establis testable implications of te key elements of demand teory. In so far, tis approac as some parallels to tose of Lewbel (1990, but especially 2001, Teorem 1) and Brown and Walker (1989). However, tese approaces are significantly extended in tat new and muc more general testable consequences are derived. For instance, a very general test for Slutsky negative semidefiniteness is derived tat as no remote similarity to anyting in te literature. Tese two contributions are being 2

3 considered in te second section. Anoter important contribution is te generalization of te econometric concepts of endogeneity and instruments to tis framework, wic occupies te tird section. It is a major result of tis analysis tat all of te general results regarding te identifiability of structures ineriting properties from economic teory continue to old under tis weaker set of assumptions. As a fourt contribution we discuss welfare analysis, bot on te microand on te aggregate level, witin tis framework of a eterogeneous population. Tis will be our concern in te fourt section. All contributions made tus far ave been confined to te identification of economically relevant structures. As a second set of contributions we provide te econometric teory for te tests tat emerge out of te identification sections. Individual contributions include first te joint asymptotic distribution of estimators for levels and derivatives of te mean regression and te scedastic function in a systems locally linear model. One innovation ere is te inclusion of bot pre-estimated regressors and pre-estimated dependent variables. Building on tis result, we analyze te asymptotic beavior of nonparametric local tests for symmetry and omogeneity, bot in te baseline as well as in te endogeneity scenario. Moreover, we propose a test for endogeneity in tis nonparametric regression framework. Te fift section covers tis econometric part. Finally, a brief application and an outlook conclude tis paper. 2 Te Demand Beavior of a Heterogeneous Population 2.1 Structure of te Model Demand teory assumes tat te demand of individual i is te result of a well beaved utility maximization problem, yielding a demand function w i = φ(p, y i, u i ), (2.1) were w i, p and y i are budget sares, log prices and log total expenditure, vectors of lengt L, L and 1, respectively. Furtermore, u i = u i ( ) denotes te individual s utility function. Trougout, we restrict ourselves to continuously differentiable demand functions, wic restricts preferences to be itself 3

4 continuous, strictly convex and locally nonsatiated, wit utility function everywere twice differentiable. Moreover, we follow te demand literature in assuming tat preferences be additively separable over time, wic justifies te use of total expenditure instead of income. Te existence of te φ( ) functional (from now on called teoretical microrelation) can be derived from te argmax operator, i.e. a rule tat relates tese variables. Te teoretical properties of tis functional are as follows: For fixed u i, say u 0, φ(,, u 0 ) beaves like a standard rational demand function, wic obeys te usual conditions of rational beavior, e.g. te compensated price derivatives form te negative semidefinite and symmetric Slutsky matrix. In order to avoid tecnical difficulties arising wit te differentiation on function spaces, we sall assume encefort tat u i may be completely described by a finite vector v i = (v 1i,..., v Mi ) of parameters 1. Terefore we consider φ as a [0, 1] L valued function defined on R L R R M, continuously differentiable in p and y. Tis set of assumptions caracterizing te space of admissable preferences will be maintained trougout te paper, and denoted by (P ). We will also confine ourselves to observationally distinct preferences, i.e. if v j, v k R M + and v j v k, ten tere exist some p 0, y 0 R L R suc tat φ(p 0, y 0, v 1 ) φ(p 0, y 0, v 2 ). We strengten tis assumption by requiring tat te same inequality olds for te derivatives. If we interpret eac individual as a realization from an underlying population, we can give te equivalent formulation to (??) in terms of random variables. We assume tat (W i, Y i, V i ) and all oter random variables to appear below, denoted as random vector by G i, are iid wit (W i, Y i, V i, G i ) (W, Y, V, G), were te latter denote te population variables. Also, for simplicity of exposition, we consider p to be a positive nonrandom vector. Tis is immaterial for our argumentation as te same arguments go troug if prices depend on time series randomness alone, wile oter variables exibit cross-section variation. Also, te case of prices varying across te population can easily be accommodated witin tis framework, see Hoderlein (2002) on bot issues. Summarizing, we ave Assumption 2.1 Let all variables and functions be as defined above. Ten, 1 Tis does not mean tat te concepts can not be defined more generally, see Hoderlein (2002), wo uses Frecet-derivatives (see Luenberger (1997)). Little is, owever, gained in terms of Economic understanding. 4

5 demand is given by W = φ(p, Y, V ) (2.2) As our aim is to establis te link between te teoretical microrelation and empirically estimable quantities, we consider te conditional average 2. Te conditioning ere is on observables, were te set of observables obviously depends on te information at and. To relate preferences to observables, we assume tat every preference depends on te individuals current observable and unobservable attributes, denoted as random vectors by Z and A respectively. Here, Z denotes all observable ouseold attributes (like age, ouseold size, etc.). However, economic coice variables sould be excluded, because oterwise preferences would become endogenous, i.e. would depend on oter economic coices. Te variable A in turn is meant to capture individual specific unobservables. Tese could in principle be time-varying as well as infinite dimensional, owever, for simplicity of exposition we desist from tis greater generality and consider only te case of a finite (S-dimensional) and time invariant vector 3. Tis leads to te following Assumption 2.2 Let all variables be as defined above. Ten V = ϑ(z, A), (2.3) were ϑ is a fixed Borel-measurable R M -valued function defined on te set Z A of possible values of (Z, A). So far we ave defined all main components of our framework. To state te next assumption, wic ensures tat intercanging differentiation and integration is well defined, we need te following notation: Let F G be te cumulative distribution function of a random variable G, and denote by F G H te conditional cdf of G given H. Let m(p, y, z) = E[W Y = y, Z = z] = E[φ(p, Y, V ) Y = y, Z = z] denote te empirical regression function. Moreover, let x f denote te partial derivative of a vector valued function f wit respect to a scalar x, and D x f denote te 2 Te above mentioned observational distinctness is now formulated as follows: Tere exist two sets Ω 0, Ω 1 Ω wit P [Ω l ] > 0, l = 0, 1. For all ω j Ω 0 and ω k Ω 1, we ave Y (ω j ) = y 0 = Y (ω k ), V (ω j ) = v 0 v 1 = V (ω k ) and φ(p 0, Y (ω j ), V (ω j )) φ(p 0, Y (ω k ), V (ω k )) for some p 0 R L. 3 Bot complications can be andled by te metods below. 5

6 matrix of derivatives of a vector valued function f wit respect to a vector x. Finally, wenever convenient we suppress te arguments of te respective functions. Assumption 2.3: Tere exists a function g, s. t. ( y φ(p, y, ϑ(z, a)) vec [D p φ(p, y, ϑ(z, a))] ) g(a), wit g(a)f A (da) <, uniformly in (p, y, z). 4 Note tat tis assumption implies tat every element of te M M matrix y {φ(p, y, ϑ(z, a))φ(p, y, ϑ(z, a)) } is also uniformly bounded in absolute value 5. Finally, te last assumption is Assumption 2.4: F A Y,Z = F A Z. Basically, tis assumption states tat - conditional on Z - income and unobserved eterogeneity are distributed independently. To give an example, take a subgroup of te population, e.g. catolic female students living in small university towns. Suppose tere are only two income classes for tis group, ric and poor, and two types of preferences, type 1 and 2. Ten, for bot ric and poor individuals witin tis subgroup, te proportion of type 1 and 2 must be identical. Tis is obviously a substantial assumption, needed in tis strengt due to te generality of te oter assumptions. Neverteless, in every subgroup of te society tere is certainly a tendency towards social coesion, towards a relatively omogeneous preference structure. Hence, if te information set is very large and allows to identify tese more omogeneous subgroups, tis assumption may approximately old. Conversely, if our information set is small, we are likely to mix up subgroups wose distribution may well be correlated wit income. Tis would lead to a breakdown of A2.4. Finally, as we will demonstrate in section 3 below, assumption A2.4 may be relaxed along several lines. 4 Among te primitive economic conditions tat ensure tat tis assumption olds are: strict convexitiy, local nonsatiation and continuous differentiability of te preferences, a linear budget constraint and p >> 0. Here, vec denotes te operator tat stacks a matrix into a single vector columnwise. 5 To see tis, take j, k arbitrary and consider y φ j (p, y, ϑ(z, a))φ k (p, y, ϑ(z, a)). But tis is smaller tan y φ j (p, y, ϑ(z, a)) φk (p, y, ϑ(z, a)). Since te second term is smaller tan one for all p, y, z, a, and te first term is uniformly bounded by A2.3, te statement follows. 6

7 2.2 Implications for Conditional Moments Given tese assumptions and notations, we concentrate first on te relation of teoretical quantities and te conditional moments. Specifically, we focus on te following questions: 1. How are te empirically obtained derivatives ( y m, D p m) wit respect to prices and income related to te teoretical ones ( y φ, D p φ)? 2. How and under wat kind of assumptions do elements of observable beavior allow inference on key elements of economic teory. Especially, wat does observable beavior tell us about omogeneity, adding up as well as negative semidefiniteness and symmetry of te Slutsky-matrix S(p, y, v) = D p φ(p, y, v)+ y φ(p, y, v)φ(p, y, v) +φ(p, y, v)φ(p, y, v) diag {φ(p, y, v)}, were diag {m} denote te matrix aving te m j, j = 1,.., L on te diagonal and zero off te diagonal. Tese concepts are commonly known as rationality in tis scenario 6, and sall be subject of Proposition 2.2. Let us start wit te very trivial Proposition 2.1 wic establises te relationsip between te derivatives. In wat follows all equalities are meant to old almost surely. Proposition 2.1 Let all te variables and functions be as defined above. Assume tat (P ) and (A2.1) - (A2.3) old. Ten follows tat (i) D p m(p, Y, Z) = E[D p φ(p, Y, V ) Y, Z]. If in addition (A2.4) olds, we ave (ii) y m(p, Y, Z) = E[ y φ(p, Y, V ) Y, Z]. Moreover, iff V is Z-measurable, ten (iii) y m(p, Y, Z) = y φ(p, Y, V ) as well as D p m(p, Y, Z) = D p φ(p, Y, V ). Proof: Appendix. Parts (i) and (ii) of tis proposition state tat eac individual s empirically obtained marginal effect is te best approximation (in te sense of minimizing distance wit respect to L 2 -norm) to te individual s teoretical marginal effect. For price derivatives, tis olds under virtually no conditions at all, for income derivatives we ave to invoke te additional assumption A2.4, because te individually varying income effects are not to be confounded wit 6 We adopt tis language. For oter definitions of rationality, see Ciappori and Rocet (1987). 7

8 te individually varying preference eterogeneity tat is correlated wit income. In tis general scenario, tis is as close as current observables allow us to get to te true marginal effects. Usually, te empirical coefficients will still be an average across individuals wit te same realization of Z, and te preference-induced eterogeneity will still be bigger tan te observed eterogeneity. However, te tird part of te proposition gives a condition on te information needed for bot to coincide: all individual randomness tat affects demand must be fully captured by current observables. Note tat A2.4 could be relaxed to a local independence condition y F A Y,Z (a, y, z) = 0, y [y 0, y 1 ] for a certain fixed y 0 and y 1 if we were just interested in te marginal effects of a subgroup of te population defined by income. Regarding te average across a population or a subgroup, te following corollary olds: Corollary 2.2 Let all te variables and functions be as defined above. Suppose tat (P ) and (A2.1) - (A2.4) old. Ten follows E[ y m(p, Y, Z) F] = E[ y φ(p, Y, V ) F], for any F σ {Y, Z}. In particular E[ y m(p, Y, Z)] = E[ y φ(p, Y, V )]. A similar condition olds for D p under (P ), (A2.1) - (A2.3). Proof: Appendix. Tus, te average of te empirical marginal effects over te wole population or over a subgroup coincides almost surely wit te true average marginal effect across population or subpopulation. Hence, we could see te average marginal effect as a treatment effect under exogeneity. Anoter straigtforward corollary concerns te standard practice of inferring elasticities from te observed regression function. It is instructive to note tat te average elasticities ave to be calculated from te log budget sare regressions, and can in general not be obtained by a transformation of te coefficients of te budget sare regression (see Hoderlein (2004) for details). We turn now to te question wic economic properties in te eterogeneous population ave testable counterparts. Tis problem bears some similarities wit te literature on aggregation over agents in demand teory, because taking conditional expectations can be seen as an aggregation step, as long as te measurability condition of P 2.1 (iii) is not met. We introduce 8

9 te following notation: Let V [G, H F] denote te conditional covariance matrix between two random vectors G and H, conditional on some σ-algebra F, and V [H F] be te conditional covariance matrix of a random vector H. Moreover, let m 2 (p, y, z) = E[W W Y = y, Z = z]. Now we are in te position to state te following Proposition 2.3 Let all te variables and functions be as defined above, and suppose tat (P ), (A2.1) - (A2.3) old. (i) If φ fulfills ι φ = 1 (a.s.) ι m = 1 (a.s.). Let additionally (A2.4) old as well. Ten follows tat (ii) If φ fulfills φ(p + λ, Y + λ, V ) = φ(p, Y, V ) (a.s.) D p mι + y m = 0 and m(p + λ, Y + λ, Z) = m(p, Y, Z) (a.s.). (iii) If S is negative semidefinite (nsd) (a.s.) D p m + y m (m 2 diag {m}) is nsd (a.s.), were D p m = D p m + D p m. (iv) If S and V [ y φ, φ Y, Z] are symmetric (a.s.) D p m + y mm is symmetric (a.s.). (v) Let V be Z measurable {S is symmetric and nsd iff D p m + y mm + m 2 diag {m} is symmetric and nsd}. Moreover, if V is Z measurable, te converse olds in (i) and (ii). Proof: Appendix. Te importance of tis proposition lies in te fact tat it allows testing te key elements of rationality witout aving to specify te functional form of te individual demand function or teir distribution in a eterogeneous population. Suppose we see any of tese conditions rejected in te observable (generally nonparametric) regression at a position y, z, p. Recalling te interpretation of te conditional expectation as average (over a neigborood ) tis proposition tells us tat tere exists a set of positive measure of te population ( some individuals in tis neigborood ) wic does not 9

10 conform wit te postulates of rationality. Tis is te case regardless of ow ric our information about eterogeneity is: If our information set is poorly, and we are neverteless able to identify a local average for wic one of te conditions is violated, ten it must be a fortiori violated if our information set increases. If we believe te information to be complete - see case (v) - ten we may directly identify tese individuals, for ten tey are completely caracterized by teir observables. Moreover, te reverse implication is peraps even more significant. Statements linking te observed model D p m + y mm to individual beavior 7, namely te S, are only true if V is Z measurable, i.e. if all individual eterogeneity as been captured by observables. Tis is a fortiori true for te parametric literature: Appending an additive error capturing unobserved eterogeneity and proceeding as usual is not a solution to solve te problem of unobserved eterogeneity. Note tat we may always append an additive error, since m = φ + (m φ) = φ + ε. Te crux is now tat te error is generally a function of y and p, as was already noted by Brown and Walker (1989). For instance, te potentially nonsymmetric part of te Slutsky matrix becomes S = D p m + y mm + D p ε + ( y m) ε + ( y ε) m + ( y ε) ε, and te last four terms will not vanis in general. Returning to Proposition 2.2., one sould note a key difference between negative semidefiniteness and symmetry. For te former we may provide an if caracterization witout any assumptions oter tan te basic ones (see (iii)). To obtain a similar result for symmetry, we ave to invoke an additional assumption about te conditional covariance matrix. Tis matrix is unobservable - at least witout any furter identifying assumptions. Note tat tis assumption is (implicitly) implied in all of te demand literature, since symmetry is inerited by D p m + y mm only under tis assumption. Conversely, if tis additional assumptions does not old, we are at most able to test te first tree elements of rationality. It is well known tat omogeneity, adding up and Slutsky negative semidefiniteness alone amount to demand beavior generated by complete, but not necessarily transitive preferences. Details of tis demand teory of te weak axiom can be found in Kilstrom, Mas-Colell and Sonnenscein (1976), Kim and Ricter 7 For instance: All individuals display a negative semidefinite Slutsky matrix, as is evident from te empirical results. 10

11 (1986) or Qua (2000). Note furter some parallels wit te aggregation literature in economic teory: Only adding up and omogeneity carry immediately troug to te mean regression. Tis result is similar in spirit to te Mantel-Sonnenscein teorem, were only tese two properties are inerited by aggregate demand. Furtermore, it is also well known in tis literature tat te aggregation of negative semidefiniteness (usually sown for te Weak Axiom) is more straigtforward tan tat of symmetry. Also, a matrix similar to V [ y φ, φ] as been used in tis literature (as increasing dispersion, see Jerison (1984)). As a final note, te assumption tat V [ y φ, φ Y, Z] be symmetric may be relaxed in te control function spirit, see again Hoderlein (2004) for details. 3 Endogeneity and Exclusion Restrictions At te center of our argumentation in te second section stood te conditional independence assumption F A Y,Z = F A Z. Tis assumption may well be violated, in particular if our information set is small, as argued above. In traditional econometrics, te breakdown of A2.4 is called an endogeneity. Possible solutions to tis problem arise if we ave additional variables, wic were excluded from te original model. Trougout tis section, we assume to possess only an additional random scalar, denoted by X. Recall in tis respect tat economic coice variables ad to be excluded from Z, and remain terefore natural candidates. In particular, like in te traditional demand system literature we may use current labor income as an instrument for total expenditure. In a world of rational, but potentially eterogeneous agents, labor income is te result of maximizing beavior by individual agents (consumers and firms). Muc as in te second section, we can model X = υ(s, Z, A 2 ),were υ is a fixed Borel-measurable scalar valued function defined on te set S Z A 2. Here, s denotes macroeconomic variables like wage rates and interest rates, wic are again for simplicity assumed to be nonrandom. A 2 is a set of unobservables, wic may contain unobserved caracteristics of te production sector, but more importantly contain unobservables tat govern te decision of te individual s intertemporal optimal labor supply problem. Examples include te attitude towards risk, but also elements in te possibly idiosyncratic information sets. Wit tis additional variable, two possible solutions may be devised in tis framework. Tey differ in te respective independence assumption, and 11

12 eac of tem will occupy a subsection. Te two solutions can be seen as generalizations of te concepts of instruments and proxies. Eac of tese two solutions will admit a number of subcases, depending on wic kind of variables are employed for conditioning. 3.1 Conditioning on Instruments instead of Endogeneous Regressors As in section 2, te structural model is given by assumptions A2.1 and A2.2. However, additionally we make te following Assumption 3.1: Let tere exist a random scalar X, suc tat Y = µ(x, Z) + σ(x, Z)U, were µ and σ are fixed Borel-measurable scalar valued functions defined on te set X Z of possible values of (X, Z) wit E [U X, Z] = 0 and V [U X, Z] = 1. Hence, µ and σ are te nonparametric mean regression and scedastic function respectively. Neglecting marginal modifications of te boundedness condition A2.3, wic will be denoted as A2.3, te second material modification concerns te dependence structure. Instead of A2.4 we assume Assumption 3.2 F A X,U,Z = F A U,Z. Note tat tis implies F A Y,U,Z = F A U,Z. Te question we ave to answer now is weter A3.2 plausible for te variables as defined above. In particular, is labor income defined as X = υ(s, Z, A 2 ) likely to fulfill A3.2? Take first te extreme case tat A and A 2 are independent conditional on Z, and we require not only tat E [U X, Z] = 0 and V [U X, Z] = 1, but in addition also tat U is not a function of X, i.e. U = ϕ(a, Z). Tis property olds, if U is conditionally normal. Ten, assumption 3.2 is quite plausible as in tis case te condition F A X,U,Z = F A U,Z is equivalent to 12

13 F A υ(a2,z),ϕ(a,z),z = F A ϕ(a,z),z, and can be derived as follows: F A υ(a2,z),ϕ(a,z),z = F A,υ(A 2,Z),ϕ(A,Z) Z F υ(a2,z),ϕ(a,z) Z = F A,ϕ(A,Z) υ(a 2,Z),ZF υ(a2,z) Z F ϕ(a,z) Z F υ(a2,z) Z = F A,ϕ(A,Z) Z F ϕ(a,z) Z = F A ϕ(a,z),z were only te conditional independence between A and A 2 as been used 8. At te oter extreme, assume tat A = A 2, i.e. te unobservable caracteristics of te ouseold tat govern te two decisions are exactly te same. Ten we ave tat bot X and U are functions of A, altoug almost independent ones. Neverteless, F A X,U,Z = F A U,Z is still not completely impossible as U already reflects some of te influence of A. Hence, even in tis case A3.2 is still more likely to be fulfilled tan A2.4. Te reality is of course a mixture of bot: For instance, labor income may depend on te attitude towards risk tat does not affect te preferences for apples vs. bananas. Conversely, conservatism may be reflected in bot work coice and goods cosen. As argued, te case for A3.2 gets stronger, te more te first scenario dominates. Along similar lines as in te second section, we explore now te implications of tis new concept for te derivatives before turning to economic structures. As mentioned above, tere are tree possibilities of conditioning, namely σ {µ(x), Z}, σ {X, Z} and σ {Y, Z, U}. Note tat if µ( ) is bijective ten σ {µ(x), Z} σ {Y, Z, U} and σ {X, Z} σ {Y, Z, U} olds, but not in general (in particular not in te multivariate X case). In wat follows, we concentrate on σ {X, Z} and σ {Y, Z, U}. Te latter case will turn out to be te most convenient. Since it is a obvious generalization, we will call it te 8 To see te tird equality, consider P [A I 1, ϕ(a, Z) I 2 υ(a 2, Z) I 3, Z I 4 ] = P [ A { I 1 ϕ 1 (Z, I 2 ) } υ(a 2, Z) I 3, Z I 4 ] = P [ A { I 1 ϕ 1 (Z, I 2 ) } Z I 4 ], were ϕ 1 is te partial inverse of ϕ wit respect to A. Tis is true for any Borel sets I 1, I 2, I 3 and I 4. 13

14 control function approac, and treat it first. Proposition 3.1 Let all te variables and functions be as defined above. Suppose (P ), (A2.2), (A2.3), (A3.1) and (A3.2) old. Ten, E[ y φ Y, Z, U] = y E[φ Y, Z, U]. Proof: By similar arguments as P 2.1. In addition, all of te results of te second section continue to old, simply conditioning on te augmented set of random variables. Moreover, in tis scenario we may come up wit a test for exogeneity. Te following lemma is central to tis: Lemma 3.2 Let all te variables and functions be as defined above. Suppose (P ), (A2.2), (A2.3), (A3.1) and (A3.2) old. Ten follows tat y E[φ Y, Z] = E[ y φ Y, Z] + E[E[φ Y, Z, U] y log f U Y,Z Y, Z]. Proof: Appendix. Recall from P 2.1 tat under exogeneity y E[φ Y, Z] = E[ y φ Y, Z]. Hence, if we assume tat F A Y,U,Z = F A U,Z olds in any case, we may base a test for endogeneity on, for instance, weter E [ E[E[φ Y, Z, U] y log f U Y,Z Y, Z] 2] or sup x,z E[E[φ Y, Z, U] y log f U Y,Z Y = y, Z = z] is bigger tan zero, or, in te local spirit of tis paper, weter E[E[φ Y, Z, U] y log f U Y,Z Y = y, Z = z] is bigger tan zero for any y, z. As mentioned, if σ {X, Z} σ {Y, Z, U} te conditional independence assumption generates an additional set of implications 9. Again we start wit te relationsip among te derivatives. Let m(p, x, z) and m 2 (p, x, z) denote E[W X = x, Z = z] and E[W W X = x, Z = z]. Ten, Proposition 3.3 Let all te variables and functions be as defined above. 9 If σ {X, Z} σ {Y, Z, U} olds, ten E[ y φ X, Z] = E [ y E[φ Y, Z, U] X, Z], by iterated expectations. Hence, no really new implications arise, as tese are only nonparametric regressions on σ {X, Z}. 14

15 Suppose (P ), (A2.2), (A2.3), (A3.1) and (A3.2) old Ten follows tat E[ y φ X, Z] = x m(x, Z) x µ(x, Z) 1 E {ξ(x, Z, U) X, Z} x σ(x, Z) x µ(x, Z) 1 E { W x log f U X,Z (U, X, Z) X, Z } x µ(x, Z) 1 (a.s)., were ξ(x, Z, U) x E[W X, Z, U] [ x µ(x, Z) + x σ(x, Z)U] 1 U. Proof: Appendix. Tis proposition provides again a relationsip between te average teoretical derivative and observable quantities, albeit a more complicated one. Te leading term can be seen as te most important element in tis formula, since it contains te empirical derivative wit respect to te instrument, x m. Note ere in particular te weigting by x µ(x, z) = x E[Y X = x, Z = z]. If te instrument is weak, ten x µ(x, z) will be very small. Hence, in any application, were we replace population quantities by sample analogs, any of te terms on te rs will be very imprecise. Te second and tird term present correction expressions: Te second is a correction for eteroscedasticity in te regression of Y on X and Z. Te last captures iger order effects and vanises if U and X are conditionally independent. Let us now focus on te economic implications, in particular on te Slutsky properties. Te following proposition summarizes te results: Proposition 3.4 Let all te variables and functions be as defined above, and let (P ), (A2.2), (A2.3), (A3.1) and (A3.2) old. (i) If φ fulfills φ(p + λ, Y + λ, V ) = φ(p, Y, V ) (a.s.) D p mι+ [ x m V {ξ, U X, Z} x σ E { W x log f U X,Z X, Z }] x µ 1 = 0 (a.s.). (ii) If S is negative semidefinite (nsd) (a.s.) D p m + x m 2 x µ 1 E {π(x, Z, U) X, Z} x σ x µ 1 E[W W x log f U X X, Z] x µ (m 2 diag {m}) is nsd (a.s.), were D p m = D p m + D p m and π(x, Z, U) = x E[W W X, Z, U] { x µ(x, Z) + x σ(x, Z)U} 1 U 15

16 (iii) If S and V [ y φ, φ X, Z] are symmetric (a.s.) D p m + [ x m V {ξ, U X, Z} x σ E { W x log f U X,Z X, Z }] x µ 1 m, is symmetric (a.s.). Here, all variables are as defined in Proposition 3.3. Proof: Appendix. Te essence of tis proposition is tat te results of te second section continue to old, wit derivatives wit respect to labor income replacing derivatives wit respect to (endogenous) total expenditure. All remarks after P 2.3 remain in place, in particular it is still possible to test for negative semidefiniteness witout furter assumptions. Are tere reasons to prefer one projection over te oter? As mentioned, if µ is bijective, σ {X, Z} σ {Y, Z, U}. Hence, we sould use te larger sigma algebra, as furter averaging only deprives us of possibilities to test. If σ {X, Z} σ {Y, Z, U}, te problem of weak instruments may be crucial for te first, as x µ may be close to zero. However, it is also problematic for te second, as ten - in terms of te influence of A - U and Y may be quite similar, meaning tat in any application some type of comovement ( collinearity ) may make estimation problematic 10. Te upsot is tat in te σ {X, Z} σ {Y, Z, U} scenario we sould use projections in all spaces, not just in one. 3.2 Conditioning on Endogenous Regressors and Excluded Variables - a Proxy Solution Tus far exploited te exclusion restriction wit a metod tat worked particularly well if, loosely speaking, te endogenous variable and te excluded variable ave little in common (at least conditionally). In tis section we will consider a metod tat works better for te reverse case. Tis second scenario employs a different independence assumption, namely Assumption 3.3 F A Y,X,Z = F A X,Z. If Y and X are generated by te same preference parameters, ten tis assumption may likely old. Tis is te case, because ten, loosely speaking, 10 Peraps te strongest argument in favor of σ {X, Z} is tradition. 16

17 te additional variable already contains a lot of te influence of A. However, te similarity in tis respect may ave a negative effects on te estimation, muc as in te above control function IV approac. It is almost needless to mention tat tis assumption is likely to break down if A and A 2 are independent or nearly so. Te consequences for testing economic teory are straigtforward: Most of te statements of section 2 continue to old, wit te extended σ-algebra σ {Y, X, Z} in place of σ {Y, Z}. A projection tat as not been used in te literature would involve σ {U, Z}. However, tis is under our assumption always a subalgebra of σ {Y, X, Z}. Hence, no new testable implications may be derived. 4 Implications for Welfare Analysis to be completed. 5 Econometric Specification In tis section we examine te asymptotic properties of te empirical nonparametric demand systems, wic is going to involve a system of local polynomial regression, and te associated test statistics. More importantly, we propose local test statistics for te symmetry and te omogeneity ypotesis. Recall tat - as a consequence of our model - it are te local properties of te empirical demand function tat ad an interpretation as conditional averages. Neverteless, global statistics may be of interest as well since one aim of te analysis migt be to determine weter omogeneity is rejected for te population as wole. Global tests are also going to be implemented in te empirical part. However, teir econometric teory involves functionals and is very different. It is being derived in a companion paper (Hoderlein, Haag and Pendakur (2004)). Finally, for testing negative semidefiniteness a bootstrap test will be applied, to wic no teory will be presented in tis paper. In te following subsection, we will analyze te exogenous, as well as te endogenous scenario wit IV. To treat tese scenarios under te same format, we consider te regression on W on P, Y and Z, were, Y may now denote eiter total expenditure (under exogeneity), wic was already denoted as Y, or labor income, wic was previously denoted as X under exogeneity. Z in 17

18 turn remains uncanged. However, recall tat te control function approac involved te additional regressor U. In principle, tis means tat our set Z of nuisance regressors increases by one. But tis extension is far from trivial, as tis regressor as to be replaced by a pre-estimated version, wic involves, for instance, a Nadaraya Watson pre-estimator ˆµ for µ. We will treat te issue of a pre-estimated regressor in general fasion in te appendix. 5.1 Local Linear Systems of Equations Again we model te dependence of te sares of total expenditure commanded by eac of L goods on te log-prices of all goods and on te log of total ouseold expenditure. Denote by P i = [Pi 1,..., Pi L ] te log-price vector, by Y i log-expenditure, by Z i = [Zi 1,..., Zi K ] te vector of individual caracteristics and W i = [Wi 1,..., Wi L ] te expenditure sare vector of te i t expenditure unit - individuals in tis application - were i = 1,..., N. Let X i = [P i, Y i, Z i] and d = K + L + 1. Tis notation is not be confounded wit te notation X to denote instruments in te tird section. It is introduced to andle te exogenous as well as te endogenous case under te same format: In te exogenous case X i is simply te vector of all regressors, in te endogenous case it is te vector of exogenous regressors and instruments. In wat follows, i indexes individuals and j indexes goods (equations). For most parts of tis paper we sall assume to ave an iid sample {(W i, X i )} n, suc tat (W i, X i ) (W, X), but see te note in te appendix for extensions to models tat fulfill a mixing condition. Te task of estimating an empirical object, wic as some economically interpretable structure, involves now in bot scenarios te model W i = m(x i ) + Σ 1 (X i )η i, were m( ) and Σ 1 ( ) are now assumed to be Borel-measurable, smoot, L 1 vector valued and L 1 L 1 matrix valued functions, respectively. Moreover, tey admit a second order Taylor expansion 11. We assume tat m(x 0 ) = E(W i X i = x 0 ) at any fixed vector x 0 = [p 1 0,..., p L 0, y 0, z 1 0,..., z K 0 ], Σ 1 (x 0 )Σ 1 (x 0 ) = V ar(w i X i = x 0 ) wit finite elements for any x 0, and tat te η i are mutually independent and identically distributed zero mean random vectors wit an identity covariance matrix, independent of te X i s. A 11 We impose te adding up constraint tat expenditure sares add up to 1. 18

19 complete list of all assumptions involved, including regularity conditions, can be found in te appendix. Since W is assumed to be bounded, all iger moments - and tus also conditional moments - exist, making tis assumptions unrestrictive. Now, in any of te two scenarios, we consider additional dependent variables, denoted as R ni. Obviously, we sould not accept te exogeneity assumption witout aving a tool to test for tis property witin an encompassing set of assumptions. Tis tool was provided by Lemma 3.2, were te regression E [ ] E [W i Y i, Z i, U i ] y log f U Y,Z (U i ; Y i, Z i ) Y i, Z i played a central role. Terefore we ave R ni = Ê [W i Y i, Z i, U i ] y log ˆf U Y,Z (U i ; Y i, Z i ), wic is again a L 1 vector, and involves pre-estimated quantities, Ê [W i Y i, Z i, U i ] and ˆf U Y,Z (ence te subscript n). However, in te remainder of tis paper we focus on te endogenous scenario wit instruments. Here R ni consists of two components: First we ave te endogenous regressor (in demand: total expenditure denoted by D i ).on instruments regression, i.e. D i = µ(x i, Z i )+σ 2 (X i, Z i )ε i. Here, ε i is a mean zero, unit variance random scalar, jointly (wit η i ) independent of X i, Z i. For compactness of notation, we denote µ as m L, i.e. te L-t element of m. Second, recall from Lemma 3.1 tat we need to estimate bot te levels and derivatives of te functions µ and σ 2, and to determine te joint distribution of estimators for m, µ and σ 2 and derivatives tereof. Hence, R ni contains total expenditure as well as te estimated squared residuals from te regression of total expenditure on instruments, formally U ni = D i ˆm L (X i ), were ˆm L denotes a Nadaraya Watson pre-estimator for te conditional expectation. Let S i = [W i, D i, Uni] 2 = [W i, R ni], denote te vector of dependent variables, wic is, as a consequence, of lengt L = L + 1. Te estimation of te function m teir derivatives at a particular point x 0 as well as te conditional scedastic function σ 2 is based on local polynomial modelling. Altoug te asymptotics of local polynomial estimators (LPEs, for sort) for single equation models ave been derived, among oters, by Härdle and Tsybakov (1997), we give a full proof of asymptotic normality for te estimators in our model. Tis is done for te following reasons: First, we extend te result to systems of equations, analogous to SURE in te linear case. Second and more importantly, we derive te joint distribution of estimators of te mean regression m and te scedastic function σ 2, as well as teir derivatives. In particular, on our way we ave to tackle problems involving te uniform convergence of te Nadaraya Watson pre-estimator. Tird, we will treat te 19

20 problem of pre-estimation of some of te regressors, as is for instance needed wen te residuals of te regression of endogenous regressors on instruments are employed. And finally, since te focus in tis paper is on pointwise testing, we need tese results in a sligtly different formulation tan usually considered, in order to empasize te differences in speed of convergence. To tis end, consider te multivariate local linear (LL) model were we solve te following weigted least squares minimization problem were for all j = 1,.., L, ξ j i = Sj i αj (x 0 ) min n 1 α j (x 0 ),β j l (x 0),β j y(x 0 ),β j l (x 0) L l=1 n K Hn (X i x 0 ) ξ iξ i β j l (x 0) ( P l i p l) β j y(x 0 ) (Y i y) K β j k (x 0) ( Zi k z k), ξ i = [ξ 1 i,..., ξ L i ], K is an L-variate kernel suc tat K(ψ)dψ = 1, K Hn (ψ) = H n 1/2 K(Hn 1/2 ψ) and H n is an L L symmetric positive definite bandwidt matrix depending on n. Here, K Hn (X i z) is a weigt wic penalizes te distance of te observation X i from x 0 so tat observations near x 0 get more weigt tan tose distant from x 0. Te kernel function depends on te bandwidt matrix H n wic puts scale on te distances of te various components te independent variable vector X i. For simplicity of exposition, we sall use a product Kernel and a diagonal bandwidt matrix, wit H n = 2 ni L. Moreover, to keep track of te difference in speed of convergence, we consider n min n 1 K Hn (X i x 0 ) ξ iξ i, (5.1) θ(z) were, for all j = 1,.., L, l=1 L ξ j i = Sj i αj (x 0 ) β j l (x 0) P i l p l β j y(x 0 ) Y K i y ( ) Z β j k k (x i z k 0), and we denote te list of all parameters as θ j (x 0 ) = {α j (x 0 ), β j l (x 0), β j y(x 0 ), β j k (x 0), }, j = 1,.., L, l = 1,.., L, k = 1,.., K, and denote te parameters wic minimize (5.1) as ˆθ j (x 0 ) = { α j (x 0 ), ĥβj l (x 0 ), ĥβj y(x 0 ), ĥβj k(x 0 ), }, j = 1,.., L, 20 k=1 k=1

21 l = 1,.., L, k = 1,.., K. Finally, let θ(x 0 ) = {θ 1 (x 0 ),..., θ L (x 0 )) and ˆθ(x 0 ) = (ˆθ 1 (x 0 ),..., ˆθ L (x 0 )). Tese are our parameters of interest. As sall become clear from te proof in te appendix, tey ave te following nice properties: Estimators for te levels as well as derivatives of te budget sare regression as well as te instrument regression are, for j = 1,.., L : m j (x 0 ) = α j (x 0 ), m j (x 0 )/ p l = β j l (x 0 ), l = 1,.., L, m j (x 0 )/ y = β j y(x 0 ), m j (x 0 )/ z k = β j k(x 0 ), k = 1,.., K. In turn, estimators for te covariance are given by σ 2 2(x 0 ) = α L (x 0 ), and for teir derivatives by σ 2 2(x 0 )/ p l = β L l (x 0 ), l = 1,.., L, σ 2 2(x 0 )/ y = β L y (x 0 ), and σ 2 2(x 0 )/ z k = β L k (x 0 ), k = 1,.., K. It is a comparably trivial exercise to te sow tat - analogously to te SUR literature - te parameters θ(x 0 ) can be estimated by locally weigted equation-by-equation OLS of expenditure sares, te endogenous regressors, and te squared empirical residuals on a constant, log-prices divided by, log-expenditure divided by and individual specific caracteristics divided by, if te regressors are te same across equations. Te formal result, wic is establised in te appendix, is as follows: Propostion 5.1: Let te model be as defined above, and let A1-A8 given in te Appendix old. Ten follows tat ) n d (ˆθ(x0 ) θ(x 0 ) 2 d bias(x 0 ) N (0, Ξ(x 0 ) A), were d denotes te number of regressors excluding te constant, A is a fixed (d + 1) (d + 1) matrix given by A = f X (x 0 ) 1 B 1 CB 1, f X (x 0 ) > 0 denotes te joint distribution of all regressors, and te fixed matrices B and C are defined as B = (d+1) (d+1) µ and C 0 0 µ 2 (d+1) (d+1) = κ d+1 0 κ d 0κ 1 κ d 0κ 1 κ d 0κ 1 κ d 0κ 2 κ d 0κ κ d 0κ 1 κ d 0κ 2 κ d 0κ 2 were µ 2 = ψ 2 K(ψ)dψ, and κ l = ψ l K 2 (ψ)dψ, l = 0, 1, 2. Finally Σ 1 (x 0 )Σ 1 (x 0 ) σ 2 (x 0 )Σ 1 (x 0 )µ ηε σ 2 2(x 0 )Σ 1 (x 0 )µ ηε 2 Ξ(x 0 ) = σ 2 (x 0 )µ ηεσ 1 (x 0 ) σ 2 2(x 0 ) µ ε 3σ 3 2(x 0 ), σ 2 2(x 0 )µ ηε Σ 2 1 (x 0 ) µ ε 3σ 3 2(x 0 ) µ ε 4σ 4 2(x 0 ) 21,

22 wit Σ(x 0 ) and σ 2 (x 0 ) as defined above, and µ η k ε l = E [ η k i ε l i], k = 0, 1, l = 1, 2, 3. Proof: Appendix. Remarks: 1. As is well known, but crucial for our analysis, te speed of convergence is not te same for te parameters of interest. Only te premultiplied derivative coefficients converge at te same speed as te ˆα j (z) j = 1,.., L. As a consequence, some test statistic may ave peraps not immediately expected properties. As an example consider te nonlinear test for symmetry under exogeneity. It will turn out tat tis test simplifies dramatically in tis scenario. 2. Te proof given in appendix A1 extends immediately to te case of all random variables being α-mixing stocastic processes. Tis class covers most commonly used stationary stocastic processes, e.g. of te AR(p)- type. Similarly, but not formally sown, a deterministic time trend may be included. However, like bot te parametric demand system literature (exception: Lewbel and Ng (2003)) and te nonparametric literature, wit nonparametric metods we can, as of yet, not andle nonstationary regressors. 3. If te pre-estimated residuals are included as additional regressors, we need for tis result to old in addition tat µ be four times differentiable and tat we employ a fourt order Kernel. Details are available from te autor upon request. 5.2 Local Homogeneity Recall te testable implications of te assumptions tat omogeneity olds in a eterogeneous population, as given in P 2.3 and P 3.3. Under exogeneity, as stated in P 2.3 (ii), D p mι + y m = 0 (a.s.), wile under endogeneity, P 3.3 (i) stipulates tat D p mι + [ y m E {ξ Y, Z} y σ E { W y log f U Y,Z Y, Z }] y µ 1 = 0 (a.s.), using te cange notation from x to y compared to section 3. Te first ypotesis is straigtforwardly testable given P 5.1. However, we will treat te 22

23 second ypotesis only in a simplified version, in give some indications below ow te test of te not simplified ypotesis may beave asymptotically. First, in te estimation part we already assumed tat U is conditionally independent of X i. Hence, E { W x log f U X,Z X, Z } = 0, by assumption. Hence D p mι + x m x µ 1 E {ξ X, Z} x σ x µ 1 = 0. As we sall se below, te asymptotic beavior of multiplicative statistics is determined by te speed of convergence of te slowest part. Hence, te asymptotic distribution of an estimator for E {ξ X, Z} does not matter. As a consequence, te fact tat ξ as to be pre-estimated is also irrelevant. Tis is because even if it were to impact te asymptotic distribution of an estimator of E {ξ X, Z} tis would not matter for te beavior of te test statistic (unless it does not cause divergence). From te asymptotic teory of te preceding subsection, te exogenous case is easily covered. Let R denote te L 1 d(l 1) matrix, R = I L 1 [ ] 0 ι L+1 0 K, and consider te test statistic ˆτ = [ ]] [ [ ] ] 1 ] R [ˆθ 2 bias R Σ1 Σ 1(x)  R R [ˆθ 2 bias, were  = ˆf X (x) 1 B 1 CB 1 element of Σ1 Σ 1(x), given by ˆσ 2 kl(x) = i K ( X i x and ˆσ 2 kl(x), k, l = 1,.., L 1 is te k, l-t ) Ûij Û il / i K ( ) X i x, were Ûij are te fitted residuals in te j-t regression. Moreover, bias is a pre-estimator for te bias. Since te bias contains largely second derivatives, we may use a local quadratic or cubic estimator for te second derivative, wit a substantial amount of undersmooting. Ten, by a trivial corollary to proposition 5.1, ˆτ d χ 2 L 1. Te issue becomes more involved, if we want to test for omogeneity in te exogenous case. First, note tat y σ 2 = y σ 2 2 [ 2 (σ 2 2) 1/2] 1. Ten, G (ψ) = (G 1 (ψ),.., G L 1 (ψ)) = 0, were for all j = 1,.., L 1, G j (ψ) = l βj l (x 0)+β j y(x 0 )β L y (x 0 ) α ξ,j (x 0 ) β L y (x 0 )β L y (x 0 ) 1 α L (x 0 ) 1/2, were α ξ,j (x 0 ) is te j-t element of α ξ (x 0 ) = E {ξ X = x 0 }. Tis leads to te test statistic, τ = [ )] [ (ˆψ) ) G (ˆψ D ψ G Σ(ˆψ)Dψ G (ˆψ ] 1 ) G (ˆψ, were ˆψ = (ˆθ, ˆα ξ ), ˆα ξ is any n d consistent estimator of α ξ (x 0 ), and 23

24 Σ(ˆψ) is te asymptotic covariance matrix. Ten due to ) G j (ˆψ = l ˆβ j l (x 0 ) + ˆβ j y(x 0 )ˆβ L y (x 0 ) 1 + ˆα ξ ˆβL y (x 0 )ˆβ L y (x 0 ) 1ˆα L (x 0 ) 1/2 = l ˆβ j l (x 0 ) + ˆβ j y(x 0 )ˆβ L y (x 0 ) 1 + α ξ ˆβL y (x 0 )ˆβ L y (x 0 ) 1ˆα L (x 0 ) 1/2 = l + (ˆα ξ α ξ) ˆβL y (x 0 )ˆβ L y (x 0 ) 1ˆα L (x 0 ) 1/2 ˆθj l (x 0 ) + ˆθ j y(x 0 )ˆθ L y (x 0 ) 1 + α ξˆθl y (x 0 )ˆθ L y (x 0 ) 1ˆθL (x0 ) 1/2 + (ˆα ξ α ξ) ˆθL y (x 0 )ˆθ L y (x 0 ) 1ˆθL (x0 ) 1/2, we know tat ) n d G (ˆψ = ) ( ) n d G (ˆθ, α ξ +o p n d+4 = ) n d G (ˆθ, α ξ + o p (1). Moreover, 1 n d ˆθ j ) l l (x 0 ) diverges, and ence we ave to pre-multiply G (ˆψ by. Tis implies tat tis first linear combination asymptotically dominates te test statistic, as it s variance is O( 2 ) bigger tan te variance of te second and tird expression. Neverteless, we sould keep te variance of te last two terms, as in any finite sample tey can be expected to be of some importance. Employing te same bias correction as above, ) ( d nd G ( ψ N 0, D θ G ( θ, α ξ) [Ξ(x0 ) A] D θ G ( θ, α ξ)), were ψ = ˆψ 2 bias, and τ = [ )] [ ( ψ) ) G ( ψ D θ G [Ξ(x0 ) A] D θ G ( ψ ] 1 ) d G ( ψ χ 2 L Local Symmetry Now te testable implications of symmetry in a eterogeneous population, as given in P 2.3 and P 3.3, are being scrutinized. Under exogeneity, we know tat te matrix D p m + y mm is almost surely symmetric. wile under endogeneity, P 3.3 (iii) establises tat te matrix D p m + [ y m E {ξ Y, Z} y σ] y µ 1 m, 24

25 is symmetric under te assumptions of section 5. Using again te cange of notation from x to y compared to section 3, we obtain tat under exogeneity we ave te 1/2L(L 1) restrictions β l k + β l yα k β k l β k yα l = 0, k, l = 1,.., L 1, k > l. Altoug tis is again a nonlinear test statistic, te fact te we multiply estimators wit different speeds of convergence makes life easier. Indeed, G s,x (ˆθ) given by tis system of restrictions, beaves asymptotically as if we ad only a linear restrictions among te derivatives, i.e. te variance of te estimators for α does not enter te test statistic. Here, te subscript s, x stands for symmetry under exogeneity. Hence, [ ] [ ] 1 ˆτ s,x = G s,x ( θ) D β G s,x ( θ) [Ξ(x 0 ) A] D β G s,x ( θ) Gs,x ( θ) d χ 2 1/2L(L 1), were D β denotes only te vector aving derivatives only wit respect to some β s and zeros in place of te derivatives wit respect to α, and θ = ˆθ 2 bias. Finally, under endogeneity life becomes even simpler asymptotically. By similar remarks as above, it is really only te variance of estimators for te price derivatives β l k and β k l tat matter in eac restriction asymptotically. Neverteless, as in te omogeneity test, we keep te multiplied ratio of income derivatives. Hence, wit te subsript s, d denoting for symmetry under endogeneity [ ] [ ] 1 ˆτ s,d = G s,d ( θ) D β G s,d ( θ) [Ξ(x 0 ) A] D β G s,d ( θ) Gs,d ( θ) d χ 2 1/2L(L 1). Tis almost summarizes te beavior of te test statistics, wose asymptotic distribution may straigtforwardly be derived from P 5.1. It s implementation, as well as te above mentioned bootstrap based test for Slutsky negative semidefiniteness, is discussed in te following section. 6 Empirical Implementation to be completed 7 Summary Unifying te treatment of preference eterogeneity in applied and teoretical work as been a long unresolved issue. Tis paper tries to fill tis gap by 25