Individual Counterfactuals with Multidimensional Unobserved Heterogeneity

Transcription

1 Individual Counterfactuals with Multidimensional Unobserved Heterogeneity Richard Blundell Dennis Kristensen Rosa Matzkin Provisional Draft, February 015 Abstract New nonparametric methods for identifying and estimating counterfactuals on individuals, when each is characterized by a vector of unobserved characteristics, are developed and applied to estimate systems of consumer demand. The methods can be used in systems of equations with either continuous or discrete exogenous explanatory variables. Instead of requiring a particular structure in the model of interest, as in Matzkin (010, 015), the new method employs variables that are excluded from the model of interest and correlated with the vector of unobserved variables. The excluded variables can be either discrete or continuously distributed. Conditional on the excluded variables, the unobserved variables in the model are assumed to be independent of the other explanatory variables in the system. Identification of each individual s behavior is shown when one of two testable conditions is satisfied. Either a rank condition holds on the conditional density of the vector of unobserved heterogeneity given the excluded variable or a "best fit" condition between the unobserved heterogeneity and the excluded variables is satisfied. We analyze the asymptotic properties of these estimators. We illustrate the usefulness of the estimators through an empirical application using UK household demand data. JEL: C0, D1 Keywords: consumer behaviour, simultaneous equations, constructive identification, nonparametric methods, nonseparable models, revealed preference, bounds. Earlier drafts of this paper ciculated under the title "Consumer Demand with Unobserved Heterogeneity". We would like to thank participants at many seminars and conferences for helpful comments and suggestions. The research is part of the program of research of the ESRC Centre for the Microeconomic Analysis of Public Policy at IFS. Funding from the ESRC, grant number R is gratefully acknowleged. Kristensen acknowledges support from Kristensen gratefully acknowledges support from the Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation, and the European Research Council (grant no. ERC-01-StG 31474). Matzkin acknowledges support from NSF (grants BCS-08561, SES , and SES ). Material from the FES made available by the ONS through the ESRC Data Archive has been used by permission of the controller of HMSO. Neither the ONS nor the ESRC Data Archive bear responsibility for the analysis or the interpretation of the data reported here. The usual disclaimer applies. Department of Economics, UCL, and Institute for Fiscal Studies, r.blundell@ucl.ac.uk Department of Economics, UCL, Institute for Fiscal Studie, and CREATES. d.kristensen@ucl.ac.uk. Department of Economics, UCLA. matzkin@econ.ucla.edu.

2 1 Introduction We develop nonparametric methods for identifying and estimating counterfactuals on individuals, when each is characterized by a vector of unobserved characteristics. The counterfactuals may be generated either by discrete or by infinitesimal changes. The unobserved characteristics are allowed to enter in unrestricted ways. We apply the methods to consumer demand, where the system of demand functions of each individual is a nonparametric and nonadditive function of prices, income, and a vector of unobserved tastes. Given the choice that an individual makes on an observed budget set, our methods allow to point identify the choice that same individual would make on a different observed budget set. For budget sets that are not observed, our methods allow to obtain sharp bounds on the counterfactual demand of the individual. The model we consider corresponds to the reduced form of a nonparametric system of simultaneous equations, where each function depends on a vector of unobserved variables, characterizing the "multidimensional unobserved heterogeneity." Our identification strategy relies on three fundamental assumptions: First, the system is invertible in the vector of unobserved heterogeneity. Second, the unobserved heterogeneity variables are unknown functions of observed and unobserved covariates, not directly entering the system. We will refer to the observable covariates as "external covariates" since they do not appear inside the model of interest. These external covariates will allow us to trace the unobserved vector in the system if they are continuously distributed. Third, once one conditions on the external covariates, the vector of unobserved heterogeneity is independent of the observable covariates appearing in the system, the "internal covariates". Under these three assumptions, we show that each function in the system is nonparametrically identified. When the external covariates are discrete, we show that the derivatives of the functions in the system, with respect to the continuous arguments, are identified. Our identification results are constructive in the sense that they lead to natural estimators of the demand function. We analyze the properties of the estimators and show how the performance canbeimprovedinanumberofrelevant scenarios. These include: Availability of multiple sets of external covariates; shape constraints; semiparametric restrictions. The methods for simultaneous equations that we develop in this paper differ from those in Matzkin (010, 015) in several ways. First, instead of requiring a particular structure in the model of interest, the new methods introduce "external variables" that are correlated with the 1

3 vector of unobservable variables in the system of equations. Second, we exploit rank conditions and averages over conditioning variables in a way different than in Matzkin (010, 015). Third, when the objects of interest are derivatives of the nonparametric functions in the system, the "external variables" may be discrete. The methods can be applied to a large set of scenarios, including equilibrium models of differentiated products, models of hedonic equilibrium, and models of multidimensional optimization. We focus on the case where the model under consideration is the demand for a vector of products of individual consumers, each characterized by a vector of unobserved tastes. For each of a finite number of observed vectors of prices, and for continuous income levels, we assume that a distribution of demand generated from a distribution of tastes is given. Our methods allow to trace out the demand corresponding to each vector of unobserved tastes. Our methods for consumer demand generalize Blundell, Browning, and Crawford (003, 008), who also considered the situation where for each of a finite number of observed vectors of prices, and for continuous income levels, a distribution of demand was given. Blundell, Browning and Crawford (003, 008) assumed that, on each budget set, the distribution of demand was generated from the demand of a single consumer plus measurement error. They imposed revealed preference restrictions on the demand of the single consumer. 1 In contrast, we assume that the distribution of demand is generated from a distribution of unobserved tastes. We impose revealed preference restrictions on the demand generated by each vector of unobserved tastes. Blundell, Kristensen, and Matzkin (014) studied a model where the distribution of demand is generated by a distribution of tastes, but their method was restricted to the case where demand and taste were scalars, and demand was increasing in taste. Estimation of multidimensional demands with multidimensional taste vectors requires methods appropriate for estimation of systems of simultaneous equations, which are much more complex than the conditional quantile based methods used in Blundell, Kristensen, and Matzkin (014). We use our new methods together with the inequalities generate from SARP to derive sharp bounds on predicted individual heterogeneous demands under the inequalities generated from SARP. To guarantee invertibility of demand, we impose conditions under which individual demand can be backed out from observation of repeated cross-sections. [See Matzkin (005, 007), Brown, Deb, and Wegkamp (006), Beckert and Blundell (007), Berry, Gandhi, and Haile (013), and references 1 The generalization of BBC to derive sharp bounds for SARP inequalities, rather than WARP, is given in Blundell, Browning, Cherchye, Crawford, de Rock, and Vermeulen (015).

4 therein.] Employing a stochastic revealed preference approach (McFadden (007),) Kitamura and Stoye (013) remove the use of invertibility but are only able to identify distributions of demands. For scalar unobserved heterogeneity, invertibility follows when the demand function is increasing in it. In the case of two goods, Stoye and Hoderlein (013) show that an invertibility assumption is without loss of generality given the data structure. Hoderlein and Stoye (013) use the same assumptions and show how to bound from above and below the fraction of the population who violate WARP. We present the framework in the next section. Identification is developed in Sections 3 and 4 and estimation in Sections 5-7. Extensions, applications, and simulations are developed in the later sections. Framework To focus on one model when developing the methodology, we consider a consumer characterized by income level R + and unobserved individual characteristics E. The consumer chooses quantities of +1 different goods. The ( +1) th good is chosen as numeraire and we set the price of this good to 1. Let =( 1 ) 0 R + denote the (relative) prices of the first goods. Given these prices, the consumer demands =( 1 ) 0 R of the first goods. The demand for the ( +1) th good is given as +1 = 0.Welet : R +1 + E 7 R + denote the demand function that maps prices and consumer characteristics into demands, = ( ) The aim is then to derive conditions under which the function ( ) is identified from individual consumer data. Although it is not necessary for all of our results, we will throughout assume that is continuously distributed. We wish to allow for both the case where (i) only discrete variation of prices is available in data and (ii) there is continuous variation in prices. To cover both cases in a simple fashion, we rewrite the model as = ( ) where W contains observables with continuous variation and the function absorbs the remaining variables that only exhibit discrete variation. For example, if we have both continuous variation in prices and income, we set = ( ) and ( ) = ( ). If, on the other 3

5 hand, we only have continuous variation in income while prices are discrete, we set = and ( ) = ( ) where is some particular set of prices that we have observed in the data. In the latter case, we will only be able to identify demand as a function of income at this particular set of prices, and all conditions and results will be made conditionally on =. Theauxiliarymodel also allows for additional consumer specific observed co-variates to enter the demand function if desired; if they are continuous, they are simply added to ; if they are discrete they are absorbed in. We will be interested in identifying and estimating finite differences of the form ( 0 ) ( ), and for any continuous coordinates of, e.g., we will be interested in infinitesimal differences, oftheform ( ) Our method will be based on external variables, Z R, which do not enter directly into. For identification of finite differences, will be assumed to be continuously distributed. For identification of infinitesimal differences, may be discrete. The next two sections develop identification results for finite differences and infinitesimal differences, respectively. 3 Identification of finite differences The following identification argument will rely on two main requirements: First, the demand function ( ) has to be invertible in. Second, we need access to additional consumer specific information that will be assumed to be informative about in a suitable manner as explained in more detail below. The assumption that the auxiliary model is invertible and smooth in the unobserved component is formally stated as: A.1 For all W R, the function 7 ( ) is invertible w.r.t. E R with inverse ( ), = ( ) The function is twice continuously differentiable in all its arguments. Note that this assumption restricts the unobserved consumer characteristics to be of the same dimension as the number of "effective" goods,. Fortheidentification results, we only need 4

6 to be continuously differentiable w.r.t., but for the nonparametric estimation we will later require existence of second derivatives. For simplicity, we therefore throughout assume the demand function to be twice continuously differentiable. Next, we assume that, in addition to and, we have observed additional consumer specific information which we collect in the vector Z R. The s are external in the sense that they do not appear in the demand system of interest. The role of is two-fold: First, conditional on the additional covariates, and are assumed to be independent. Second, we assume that the conditional distribution of satisfies a certain restriction which will allow us to trace out the unobserved variation in. Formally, we assume that: A. is distributed independently of conditional on and ( ) = has a continuous distribution characterized by a density ( ) which is twice continuously differentiable. A.3 The conditional density ( ) satisfies the following "best fit" restriction: ( ) =0 = Note that Assumption A.3 implicitly requires to be of the same dimension as, =dim() = dim () =, and that they have common support, E = Z. Moreover, it also implicitly assumes that the s are continuously distributed. The best-fit restriction in A.3 can be seen as a requirement that the s allow us to trace and learn about the unobserved heterogeneity. This also implies that they should be chosen according to the type of unobserved heterogeneity that enters the demand model. One could think of each of the components of as capturing a particular type of tastes/preferences of the consumer. We then need to identify corresponding socio-economic characteristics in data that we expect are capturing variation in these unobserved tastes. One can weaken Assumption A.3 to: A.3 For all, 7 ( ) has a unique maximum. That is there exists some invertible function Λ () such that: ( ) =0 = Λ () (1) We see that Assumption A.3 implies A.3 with Λ () =. The above assumption, however, does not require knowledge of the precise form of the mapping Λ, only that such exists. This is We can weaken the last requirement to Z E, but in this case we can only identify ( ) for E. 5

7 an attractive feature compared to A.3 since the latter requires us, amongst other things, to have chosen such that it is on the same scale as the unobserved component. On the other hand, we are now only able to identify the demand functions up to the unknown transformation Λ: The restrictions in A.3 and A.3 may appear somewhat abstract and one may ask what type of data-generating mechanisms lead to this type of restriction. A set of sufficient conditions for Assumption A.3 to hold are stated in the following theorem: Theorem 1 A set of sufficient conditions for Assumption A.3 to hold are: = ( ()+) where : R 7 R and : R 7 R are invertible; R and R are mutually independent; and has a continuous distribution whose density has a unique mode at =0. Proof. Under the stated conditions, ( ) = ( () ()) () () where is the inverse of Then, with Λ () := ( ()), log ( ) = () log ( () ()) =0 = Λ () An important corollary to the above result is that Assumption A.3 does not require the existence of a deterministic relationship between and. We are now ready to establish the identification result. We first show the result under Assumptions A.1-A.3 and then extend the result to the case where A.3 is replaced by A.3. In both cases, observe that Assumptions A.1-A. allow us to express the conditional density of demand as a functional of and, ( ) = ( ( ) ) ( ) () where ( ) = ( ) R is the matrix of partial derivatives of the inverse demand function. This follows by standard results for densities of invertible transformations of random variables. 6

8 Given the expression of in eq. () and the fact that ( ) 6= 0for all ( ) due to Assumption A.1, we see that Assumption A.3 implies ( ) =0 ( ) =0 = (3) Given that is observable, Assumption A.3 therefore allows us to trace out the unobserved variation in through the observables. This in turn allows us to identify the demand function and its inverse: Theorem Under Assumptions A.1-A.3, ( ) is identified as ( ) =argmin ( ) Z (4) or, equivalently, ( ) = arg max log ( ) (5) Z Correspondingly, ( ) is identified as ( ) =argmin ( ) (6) = R + Proof. Take derivatives w.r.t. on both sides of eq. (), ( ) = ( ( ) ) ( ) (7) Now, consider a fixed value of ( ): Suppose that = ( ) satisfies ( ) =0 (8) Since ( ) has full rank, then ( ) 6= 0and so eqs. (7)-(8) imply that ( ( ) ) () = 0 ( ) =, where the last equivalence follows from A.3. In particular, eq. (4) holds. The second expression (5) follows from the fact that if satisfies (8) then =argmax Z log ( ) Finally, to show eq. (6), consider a fixed value of ( ) and suppose that = ( ) satisfies ( ) () =0. As before, this is equivalent to ( ( ) ) =0 ( )= = ( ) Next, we show identification of the normalized function under Assumption A.3 : 7

9 Theorem 3 Under Assumptions A.1-A. and A.3, and are identified up to the (unknown) transformation Λ. That is, ( ) := ( Λ ()) and ( ) is the inverse of are identified as and ( ) =argmin ( ) Z (9) ( ) = arg max ( ) (10) = R + Proof. Define := Λ 1 (), and observe that ( ) = (Λ ( ) ) Λ ( ) We can then express the original model, = ( ), in terms of the "rescaled" errors, = ( ), where is defined in the theorem. Given A.3, we have that ( ) =0 (Λ ( ) ) =0 Λ ( ) =Λ () = Thus, ( ) and ( ) are identified by Theorem. The above result is similar to the one found in the univariate case, where, under A.1-A., ( ) is identified up to an unknown transformation of ; see, e.g., Matzkin (007). Note however that, in our more general, multivariate setting, we require in addition that Assumption A.3 holds in order to obtain identification. This is not needed in the univariate case. We end this section by noting that eq. (3) also shows that A.3 (and A.3*) is testable from data: We can examine whether the conditional density indeed satisfies the restriction which is equivalent to A.3 being satisfied (under the maintained assumptions of A.1-A.). 4 Identification of infinitesimal changes The identification of infinitesimal differences will be based on the existence of +1values of the covariates at which a rank condition can be established. To state the condition, let (1) (+1) denote +1such different values of the vector and define () () = log () R () ( ) = log () R 8

10 and () ( ) = log () R The following result establishes that the derivative of with respect to is identified as long as a testable rank condition is satisfied. Theorem 4 Fix ( ) and let denote the (unique) value of the vector such that = ( ) Suppose that there exist +1 values, (1) (+1) of such that the matrix (1) () R(+1) (+1) (11) (+1) () 0 1 has rank +1 Then, ( ) =( ) () is identified as 1 ( ) (1) ( ) ( ) =.. (+1) ( ) 0 1 ( ) where is some function of derivatives of. (1) (). (+1) ( ) Proof. Consider the transformation of variables equation ( ) = ( ( ) ) ( ) Taking derivatives with respect to give µ log ( ) 0 µ µ ( ) log ( ( ) ) 0= + Note that ( ) () and log ( ( ) ) () do not depend on. Hence, under rank conditions on the matrix with rows, each corresponding to a different value of log ( ) log ( ) each corresponding to a different value of the constants can be found by solving the system of linear equalities. Since ( ) = ( ( )) ( ) a necessary and sufficient condition for the matrix to be full rank for +1values, (1) (+1) of is that for those values the rank of the matrix given in eq. (11) is +1. 9

11 5 Nonparametric Estimation Using Best-Fit Condition Eq. (4) suggests the following two estimators of ( ) for any given values of ( ) under Assumption A.3: or ˆ ( ) =arg ( ) = arg min Z 0 () max Z 0 () ˆ ( ) (1) ˆ ( ) (13) where ˆ is a nonparametric estimator of and Z 0 ( ) Z is some compact subset that the true function value ( ) lies in. Ideally we would like to set Z 0 ( ) =Z, but for proof technical reasons we have to restrict the set of candidate values to be compact. If the support of is in fact compact, we can set Z 0 ( ) =Z. If the interest lies in estimating ( ), this can be done in two different ways: A direct estimator is obtained by ˆ ( ) =arg min Q 0 () ˆ ( ) = (14) where Q 0 ( ) Q is some compact subset of the support of, Q, which contains ( ). Alternatively, one can first compute ˆ ( ) (or ( )) asdefined above and then solve ( ) = arg max kˆ ( ) k (15) R + If we replace Assumption A.3 by A.3, the proposed estimators will estimate the normalized demand function, ( ) and its inverse ( ) as defined in Theorem 3. We here only state results for ( ) (or, equivalently, ( )) under A.3. All subsequent arguments and results still hold with A.3 being replaced by A.3 and ( ) ( ( )) replacedby ( ) ( ( )). In principle, any nonparametric conditional density estimator could be employed in the computation of the above estimator. For the theoretical results, we here focus on the case where ˆ is chosen as a kernel density estimator, P =1 ˆ ( ) = K ( ) ( ) K ( ) P =1 (16) ( ) K ( ) where 0, 0 and 0 are the bandwidths for the three variables and : R 7 R and K : R 7 R are two kernel functions. We make the following assumptions to establish first-order asymptotic properties of the estimators: 10

12 A.4 The functions ( ) and ( ) are twice continuously differentiable with bounded derivatives. The density of satisfies inf Z0 () 0. A.5 The kernel functions are twice continuously differentiable. Furthermore, R R () =1, R R () =0, R R kk (), R R K () =1, R R K () =0,and R R kk K (). The analysis of the estimators follow along the same lines as the one for standard GMMestimators with the one exception that the sample "moments" here take the form of first-order derivatives of kernel density estimators. In particular, the estimators of and will converge with rates determined by these density derivative estimators: Theorem 5 Assume that A.1-A.5 hold together with the following rank condition, ( ) := ( ) 0 R has full rank. Suppose furthermore that + =() 4 0 for =,. Then,ˆ ( ), asdefined by eq. (1), is consistent and satisfies where q + ( ) = ( ) ( ) {ˆ ( ) ( )} 0 1 =() Z Z Z K () R () R The estimator ( ) is first-order equivalent to ˆ ( ). +4 and + ( ) ( ) 1 ( ) K () R + K () 0 R Proof. For any given W, we have (under standard regularity conditions) ˆ ( ) sup sup Ã! ( ) Z 0 = ; R + see, for example, Hansen (008). sup R + Following the arguments of Newey and McFadden (1994) for extremum estimators, where identification follows from Theorem, it then follows that Ã! kˆ ( ) ( )k =

13 The asymptotic distribution can be derived using standard arguments for GMM estimators based on a Taylor expansion: With ˆ := ˆ ( ) and and := ( ) denoting estimator and population value, respectively, 0= ˆ ( ˆ ) = ˆ ( ) + ˆ ( ) 0 (ˆ ) where is situated on the line segment connecting ˆ and. Under the bandwidth conditions, Lemma 11 implies that q + ( ˆ ( ) ) ( ) (0 ( )) ˆ ( ) 0 ( ) where ( ) R and ( ) R are definedinthetheorem. Given that ( ) 0 = ( ( ) ) 0 ( ) the rank condition imposed on ( ) is satisfiedifandonlyif ( ) ( 0 ) has full rank. This is a natural condition given identification arises from Assumption A.3. Also note the close similarity between the conditions and asymptotic distribution stated in Theorem 5 and the corresponding ones found in the asymptotic analysis of GMM estimators in the just-identified case. Next, we analyze the proposed estimator of : Theorem 6 Assume that A.1-A.5 hold together with the following rank condition, ( ) := ( ) 0 R has full rank. =()= Suppose furthermore that for =,. Then,ˆ ( ), asdefined by eq. (14), is consistent and satisfies where q + ( ) = ( ) ( ) { ˆ ( ) ( )} 0 1 =() Z Z Z K () R () R The estimator ( ) is first-order equivalent to ˆ ( ). +4 and + ( ) ( ) 1 ( ) K () R + K () 0 R 1

14 Proof. The proof of the first part of the theorem follows along the one of Theorem 5, except that we now have, with ˆ := ˆ ( ) and := ( ), 0= ˆ (ˆ ) = ˆ ( ) + ˆ ( ) 0 (ˆ ) where is situated on the line connecting ˆ and. Under the conditions on the bandwidths, it follows from Lemma 11 that q ( + ˆ ( ) ) ( ) (0 ( )) ˆ ( ) 0 ( ) The second estimator, ( ), is also analyzed using arguments for GMM estimators: Since ˆ ( ) ( ) (uniformly in ), it holds that sup kˆ ( ) k k ( ) k 0, where ( ) =argmin k ( ) k is unique. This shows consistency. To show asymptotic normality, we follow the usual steps: With := ( ), 0={ˆ ( ) } ˆ ( ) ˆ ( ) ˆ ( ) + 0 where = ( ). Here, q + {ˆ ( ) } ˆ ( µ ) 0 ( ) + {ˆ ( ) } ˆ ( ) 0 ( ) 1 ( ) ( ) 1 ( ) ( ) and ˆ ( ) ˆ ( ) 0 + {ˆ ( ) } ˆ ( ) 0 ( ) ( ) 0 It is now easily checked that ( ) has the same asymptotic distribution as ˆ ( ). 6 Nonparametric Estimation Using Rank Conditions The estimation method using rank conditions is based upon results analogous to those developed in Matzkin (010, 015). Let and be given. Let Z 0 denote a set that includes the values (1) (+1) satisfying the rank condition that guarantees identification and let denote a nonnegative continuous function on Z 0 that integrates to 1 on Z 0 and it is strictly positive at the values (1) (+1). If is continuously distributed, define and as Z log ( ) log ( ) = () Z 0 Z Z log ( ) + () Z 0 13 log ( ) Z 0 ()

15 and = Z log ( ) log ( ) () Z 0 Z Z log ( ) log ( ) () () Z 0 Z 0 If is discrete, = X log ( ) log ( ) () Z 0 + X log ( ) () X log ( ) () Z 0 Z 0 and = X log ( ) log ( ) () Z 0 X log ( ) () X log ( ) () Z 0 Z 0 Theorem 7 Let be given and let denote the value of epsilon satisfying = ( ). Let Z 0 denote a set included in the support of so that (1) (+1) Z 0,and() anonnegative function on Z 0 that is strictly positive at (1) (+1) Then, the derivatives of ( ) with respect to are identified as 1 = Proof. The identification result and the definition of () imply that is the unique minimizer of Z log 0 log Z The result is analogous for the case where is discrete. () 14

16 Employing techniques closely related to those in Matzkin (015), it can be shown that, under some regularity conditions, the asymptotic distribution if are continuous and is discrete is given, for some finite matrix by +3 ( ( ) ( )) (0) 7 Nonparametric Estimation Using a Best-Fit Condition and Averaging over z To improve the estimator based on the best-fit condition, one can average over all the values of Such method is appropriate even when the rank conditions for identification of the derivatives are not satisfied. To detail the method, we note that by our assumptions on the relationship between and, for any given 0 there exists a unique value, 0 of corresponding to it. Hence, for any given 0 and any and there is a unique value of corresponding to 0 Such value satisfies Denote and ( ) ( 0 )=argmin R + =0 µ µ ( ) ( ) ( ) = 0 0 ( ) = ³ ( ) ³ 0 ( ) 0 By combining the arguments in the identification sections, we can establish the following result. Proposition 8 Fix, and 0 Assume and are continuously distributed, and that the best-fit condition for identification is satisfied. 1 Then, ( 0 ) is the unique solution to the minimization over of the function ( 0 ) = ( 0 ) Z µ log ( 0 ) + ( 0 ) log ( 0) µ log ( 0 ) + + log ( 0) () where () is an arbitrary nonnegative function over whichintegratesto1. 15

17 The above result leads to a natural estimator by replacing unknown functions with their nonparametric estimators. 8 Extensions The above estimators assumed that we had already identified a set of covariates that satisfy A.3 (or A.3 ). However, in some situations, one may have more s available that potentially satisfy A.3. Or one may worry that the s in the data set do not directly satisfy A.3. We here discuss two modifications of the estimators that can address these two situations Availability of "over-identifying" external covariates Suppose we have available multiple sets of s, each of which satisfies Assumption A.3. In this case, one can obtain more precise estimates through the following generalization of the proposed estimator. Let 1 be 1 sets of co-variates, each of which satisfies A.-A.3: A.* For =1 : is distributed independently of conditional on and ( )= has a continuous distribution characterized by a density ( ) which is twice continuously differentiable. A.3* For =1 : The conditional density ( ) satisfies the following restriction: ( ) =0 = Recall that we interpreted the best-fit restriction as a type of moment condition which lead to just identification of. Assumptions A.*-A.3* can therefore be thought of generating overidentifying moment restrictions. Similar to GMM-estimators, these can then be combined to obtain amoreefficient estimator. We here focus on the estimation of ; the analysis of the corresponding estimator of follows along the same lines. Given the conditional kernel density estimators ˆ ( ), =1,wecollectthe "moment conditions" in ˆ ( ) =(ˆ 1 ( ) 0 ˆ ( ) 0 ) 0 R where ˆ ( ) := ˆ ( ) = For a given choice of ( ), we then propose to estimate ( ) by ˆ ( ) =argmin R + ˆ ( ) 0 ˆ ( ) ˆ ( ) (17) 16

18 for some weighting matrix ˆ ( ) R. To state the limiting distribution of the estimator, we define ( ) =( 1 ( ) 0 ( ) 0 ) 0 R where ( ) := ( ) The following theorem generalizes Theorem 6, where, for simplicity, we assume that the same bandwidths is used across the density estimates: = Theorem 9 Assume that A.1, A.*-A.3* and A.4-A.5 hold, ˆ,andthematrix ( ) := ( ) 0 ( ) ( ) R has full rank, where ( ) :=( ) ( 0 ) =() R. Suppose furthermore that for =, +4 and + +.Then,ˆ ( ), as defined by eq. (14), is consistent and satisfies q { ˆ ( ) ( )} (0 Ω ( )) + where Ω ( ) = 1 ( ) ( ) 0 ( ) ( ) ( ) ( ) 1 ( ) and ( ) =[ ( )] =1 with ( ) = ( ) ( ) =() Z Z Z K () R () R K () R K () 0 R Proof. Consistency follows along the same lines as in the proof of Theorem 6. To show asymptotic normality of the estimator, firstnotethattheestimatorˆ = ˆ ( ) satisfies the following first-order condition, ˆ (ˆ ) 0 ˆ ( ) ˆ(ˆ ) =0w.p.1, where ˆ (ˆ) = ˆ ( ) () and dependence on ( ) is suppressed since these are kept fixed. A Taylor expansion of ˆ(ˆ) around = ( ) yields ˆ(ˆ) = ˆ()+ ˆ( )(ˆ ), where is situated on the line segment connecting ˆ and. Substituting this into the first-order condition, and rearranging the terms yields q h i 1 q + (ˆ ) = ˆ(ˆ) 0 ˆ ˆ( ) ˆ(ˆ) 0 ˆ + ˆ() where h i 1 q ˆ(ˆ) 0 ˆ ˆ( ) ˆ(ˆ) 0 ˆ [ 0 ] 1 0 and + ˆ() (0). Inthecasewhere, wecanusea-test to test for whether the chosen s indeed are valid co-variates satisfying A.*-A.3. 17

19 As is standard for GMM-type estimators, an efficient estimator arises by choosing ˆ to be a consistent estimator of = 1 ( ) in which case the asymptotic variance of ˆ ( ) takes the form Ω ( ) = ( ) 0 1 ( ) ( ) 1. Check whether we can allow for overlapping s Identifying an index Another scenario is that we have covariates, =( 1 ) R where.wethenwish to construct a set of co-variates 0 R from this candidate set which satisfies A.-A.3. This can be done in a number of ways. One natural way is to try to construct 0 as an index based on the set of initial covariates, 0 = for some matrix R. This matrix should be chosen in order to maximize how well the resulting explains the variation in. This can be done by choosing ˆ =arg max R X log ˆ ( ) =1 This estimator was originally proposed in Fan et al (009) as a dimension reduction device and they show that ˆ is -consistent and asymptotically normally distributed. Thus, the first-step estimation of the index ˆ = ˆ, willnotaffect the asymptotic properties of the final nonparametric estimators of ( ) as derived earlier. 8.1 Shape Constraints Another way of improving on the nonparametric estimators of and is to impose shape constraints on and ( ). Recallthat log ( ) =log ( ( ) )+log ( ) (18) If we assume that, for any given, 7 log ( ) is concave, then, for any given (), 7 log ( ) is also concave. This in particular implies that Assumption A.3 holds and so eq. (5) is valid. Moreover, by imposing shape constraints, the restricted estimator will in general enjoy considerably finite-sample improvements over the corresponding unconstrained estimator. Thus, imposing log-concavity c shape constraints can be used to improve on the nonparametric estimation of, and thereby ( ). 18

20 There is a large literature on shape-constrained nonparametric estimation. Cule and Samworth (010a,b) have developed estimators of multivariate log-concave densities. However, this work focuses on unconditional densities on so does not directly apply to our setting where the object of interest is a conditional density. Matzkin (1999) develop shape-constrained estimators in a very general set-up, which includes ours and so her general numerical algorithm can be applied to our problem. Alternatively, for computational purposes, for any given, one can treat the kernel density estimator ˆ ( ) as a kernel regression estimator where the dependent variable is ( ) and so the concavity constraints on ˆ ( ) w.r.t. can be thought of as a constraint on a nonparametric kernel regression function. This allows us to import existing shape-constrained estimators developed in a regression framework. One example is Du, Parmeter and Racine (013) who show how shape constraints can be imposed by solving a constrained least-squares problem; something than can be readily done in standard software packages. 9 Semiparametric Estimation Using the Best-Fit Condition The nonparametric estimators developed in the previous section suffer from the usual curse of dimensionality. In the case of moderate-size of high-dimensional demand systems, this means that very large sample sizes are needed to get reasonably precise estimates of. If such samples are not available, semiparametric constraints are needed. We here consider two different sets of semiparametric constraints. 9.1 Parametric Demand Model Suppose that the demand function ( ) belongs to a known parametric family { ( ) : Θ} where 0 Θ is the unknown finite-dimensional parameter that generated data, ( ) = 0 ( ). The distribution of is on the other hand not known to us and is left as a nonparametric component. In this case, the following semiparametric profile MLE can be used to estimate 0, and thereby ( ): ˆ =argmax Θ X =1 n log ˆ o ( ( ) )+log ( ) where P =1 ˆ ( ) = K ( ( ) ) K ( ) P =1 K ( ) The analysis of this estimator should follow along the same lines as in XXXX. 19

21 10 Application to Demand Prediction The proposed identification and estimation strategy can be used in the context of demand prediction when only finite price variation is available in data. Suppose we have observed a repeated cross section of demands and incomes across a finite set of price regimes, ( () () ()), =1 and =1, where the number of price regimes is small or moderate relative to the sample size in each regime. In price regime {1}, data has been generated by the following relationship () = ( () () ()) so that the observed consumers all face the common price () which we treat as observed. Using the techniques developed in the previous section, we can obtain an estimator of ( ) := ( () ) as a function of ( ). This can in turn then be employed to construct bounds for counterfactual demands for a consumer characterized by a particular value E who face a new set of prices 0 { () ( )} and income level 0 by using the general methodology developed in Blundell, Kristensen and Matzkin (014), BKM14 henceforth: First, we estimate the so-called intersection income levels ˆ =(ˆ (1) ˆ ( )) as the solutions to 0 0 ˆ( ˆ () )= 0 =1 (19) where ˆ ( ) is our estimator of the demand in price regime with ˆ ( ) =( 1: () 0 ˆ1: ( )) +1 (). These are in turn then used to construct the following support set estimator containing the set of estimated demands for the particular consumer, Ŝ 0 0 = { B 0 0 ˆ P 0} where P [ (1) ( )] and B 0 0 = n R = 0 o (0) is the budget set. The large sample properties of this support estimator are provided in BKM14 under high-level conditions on the initial demand estimators. These conditions are straightforward to verify for our particular estimator and so the asymptotic theory of BKM14applies. In most applications though, the value is not directly observed and is instead estimated from data in the following way: We choose a particular consumer in our sample characterized by his 0

22 income level, say,, and corresponding observed consumption decision in,say,priceregime. We know that there is a unique solving = ( ) but since is unknown, we cannot directly infer it. Instead it can be estimated as the implicit solution to = ˆ ( ˆ) Given ˆ, we can now proceed as above with the only change being that the intersection income levels are now computed as 0 0 ˆ( ˆ () ˆ) = 0 =1 (1) The inference developed in BKM14 has to be adjusted to take into account the first-step estimation error in ˆ. This can be done by formulating ˆ as a two-step GMM estimator: Define ˆ 1 ( ) = 0 ˆ( 0 ) 0 = 01: () 1: () ˆ1:( )+ 0+1 () 0 for =1, and collect these moment conditions in ˆ 1 ( ) ={ ˆ 1 ( )} =1 ˆ =argmin X ˆ 1 ( ˆ) so that The first-step estimation of is added as an additional moment condition, ˆ ( ˆ) = ˆ 1: ( ˆ) so that, with =( 1 ), (ˆ ˆ) =arg min ˆ ( ) X 0 E 0 We then replace BKM14 s Lemma 4 by the following modified version: Lemma 10 Assume that XXXX are satisfied. Then q " # + ˆ (0 Σ) ˆ where where = Σ = 1 1 ( ) ( ) = 0 1

23 with being a block-diagonal matrix with its th element taking the form () = 1 ( ) ( ) 1 ( ) =1 and defined in the proof. Here, the right-hand side matrices in the definition of () are given in Theorem 6 with = 1:. Proof. Note that ˆ 1 ( ) = 0 0 ˆ( ) 0 = for =1,sothat ˆ 1 ( ) = 1 ˆ( )+1 () where ˆ( h ) = ˆ1: (1) 0 ˆ i 0, 1: ( ) 0 1 = diag 1 () = 01: () 1: () ˆ1:( )+ 0+1 () 0 01: 0+1 (1) 1: (1) 01: (1) ( ) 0 ( ) 1: ( ) Similarly, with = [0 0 ] and () =, ˆ ( ˆ) = ˆ( ) + (). In total, ( ) = ˆ ( )+ (), where q + { ˆ ( ) ( )} (0) with defined in the theorem. The result now follows from standard arguments for GMM estimators. We can now obtain that a 1 confidence set for S 0 0 is given by where S = S X 1 { B 0 0 P 0} X 1 = :(ˆ ) 0 Σ 1 (ˆ ) (1 ) + ª Here, (1 ) is the (1 )th quantile of the distribution, and Σ R is the component of Σ defined in Lemma Practical Implementation We here discuss how the nonparametric (and semiparametric?) estimators can be implemented in a given sample.

24 11.1 Bandwidth Selection Various bandwidth selection methods for kernel estimators of (conditional) densities have been proposed in the literature; see Bashtannyk and Hyndman (001), Fan and Yim (004), and Hall, Racine and Li (004) and references therein. A completely data-driven bandwidth choice can be obtained through cross-validation, as in Fan and Yim (004) and Hall et al (004), or using plug-in rules, as in Bashtannyk and Hyndman (001). All the above methods are designed so as to obtain an optimal estimate of the conditional density ( ). However, in our application, the object of interest is ( ) (). The above bandwidth selectors are not optimal for this object since they will undersmooth relative to the optimal bandwidth for estimation of the derivative. A simple solution is to compute either of the above bandwidths and then scale it up according to the optimal rate stated the above papers. However, bandwidth selectors targeting derivatives have recently been developed: Li and Racine (014) consider the case of derivatives of quantile regressions which proves to be related to bandwidth selection for conditional density estimators; Henderson et al (014) consider the case of derivatives of regression functions and, as pointed out earlier, for a given, one can think of the kernel density estimator of as a nonparametric regression of ( ) onto ( ). Härdle, Marron and Wand (1990) develop cross-validation methods for kernel estimators of density derivatives. 11. Confidence Intervals A simple approach to construct confidence intervals for ˆ ( ) is to use the asymptotic approximation of its distribution as derived in Section 4. These may, however, be imprecise in small and moderate samples. Instead one could employ bootstrapping which in general will provide more accurate coverage; nonparametric bootstrap methods include, amongst others, Hall and Horowitz (013) and Hall and Kang (001). 1 Finite-sample performance We here investigate the performance of the estimator through simulations. We consider two simulation designs (DGP s), a random coefficient model and a simultaneous-equations model. describe each of the DGP s and the corresponding results in the following two subsections. We 3

25 1.1 Random-coefficient model The first model is a bivariate ( =) random coefficient model where = and = + for =1. In total, = + We assume,, and are mutually independent with Ω.Thus, + Ω and + Ω. As such its density is given by ( ) = where Σ () = Ω.Inparticular, ½ 1 q exp 1 ¾ () ( )0 Σ 1 ()( ) Σ () ˆ ( ) :=argmax ( ) = which is the inverse ( ) = of the structural relation = ( ) =. For given values of ( ), we implement the estimator of () defined as ˆ( ) = arg max ˆ ( ) where ˆ ( ) is the kernel estimator of the conditional density using a matrix of bandwidths,. The bandwidth matrix are chosen using the multivariate version of Silverman s Rule-of-Thumb, = 1(+1) ˆΣ1 where ˆΣ isthesamplecovariancematrixof( ). The results for the estimator ˆ ( ) =(ˆ 1 ( ) ˆ ( )) are reported in Figures 1-4. In each figure we fix at a particular value, say,, and then plot the estimates of the function 7 1 ( ) and 7 ( ). The results show that the kernel-based estimator works quite well, with small biases and not too big variances. 1. Simultaneous equations model The second DGP is a bivariate simultaneous equations model where 1 = 1 () 1 + () 1 = () () 4

26 Figure 1: Estimation of 1 ( ) with = 1 fixed. Figure : Estimation of 1 ( ) with = fixed. 5

27 Figure 3: Estimation of ( ) with = 1 fixed. Figure 4: Estimation of ( ) with = fixed. 6

28 With =( 1 ) 0 and =( 1 ) 0, this model can be written as = () where " () = 1 () () () 1 () # Note that () = 1 () (). Thus,if 1 () 6= () () the model is invertible with = 1 () =: ( ) One particular choice of that satisfies eq. () is 1 () = () () = () for some function () 6= 0. The estimator of ( ) is implemented as in the first simulation study. The results for the estimator ˆ are reported in Figures 5-8 below. As before, the estimator performs well and is able to capture the curvature of the inverse demand function with good precision. 13 Empirical Application In this empirical application we use a sample couples with and without children from the Family Expenditure Survey in the UK. The sample consists of 479 families from the 000 survey. We focus on two commodities described in Table 1, food share () andservicesshare () which we write as functions of log(total expenditure) and two unobserved tastes: 1 comprising family size calculated using equivalence scales, and comprising the average birth cohort of two adults in each household. 7

29 Figure 5: Estimation of 1 ( ) with = 1 fixed. Figure 6: Estimation of 1 ( ) with = fixed. 8

30 Figure 7: Estimation of ( ) with = 1 fixed. Figure 8: Estimation of ( ) with = fixed. 9

31 Table 1: Descriptive Statistics () [To be completed] 14 Conclusions In this paper we have developed conditions for identification and estimation of individual demands in the multiple good case with nonadditive/nonseparable heterogeneity. The focus has been on the case of discrete prices (finite markets) and many heterogeneous consumers. Identification has been shown when one of two conditions is satisfied: Either the conditional density of the vector of unobserved heterogeneity given the excluded variables is maximized at a unique value of the excluded variable; or a rank condition holds on this conditional density. We derived indirect and average derivative methods that identify the value of the vector of unobserved tastes of each consumer and the demand function of each consumer. This work has shown that nonparametric demand systems with nonseparable heterogeneity can be identified and that identification can be achieved when prices are either continuous or discrete. The methods presented employ shape restrictions on the demand and/or distribution of the unobservable heterogeneity. The identification methods are constructive. We have also shown how asymptotically normal estimators can be constructed for the values and derivatives of the functions and distributions. The usefulness of the estimators is illustrated through an empirical application using UK household consumer data. References Ai, C. (1997) A Semiparametric Maximum Likelihood Estimator, Econometrica 65,

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34 Richter, M.K. (1966): Revealed Preference Theory, Econometrica, 34, Stoye, J., and S. Hoderlein (013): Testing Stochastic Rationality and Predicting Stochastic Demand: The Case of Two Goods, Manuscript. Varian, H. (198): The Nonparametric Approach to Demand Analysis, Econometrica, 50, A Lemmas Lemma 11 Suppose that Assumptions A.6-A.7 hold. 1. As for =, and +, q ( + ˆ ( ) ) ( ) (0 ( )) where ( ) = Z Z Z ( ) K () K () K () () ( ) R R R 0 R. As 0 and As 0 and + +, ˆ ( ) 0, ( ) 0 ˆ ( ) 0 ( ) 0 Proof. We have ˆ ( ) 1 = ˆ ( ) ˆ ( ) + ˆ ( ) ˆ ( ) By standard arguments for kernel estimators (see, e.g. XXXX), q + ˆ ( ) ( ˆ ) ( ) ( ) ³ 0 ( ) where, and are the bias components and Z Z Z ( ) = ( ) K () K () K () () R R R 0 R Similarly, Ãs! ˆ ( ) = ( ) Ãs! ˆ ( ) = ( ) ˆ Ãs! ( ) = ( )