Identification of Multidimensional Hedonic Models. Lars Nesheim Centre for Microdata Methods and Practice (CeMMAP) UCL and IFS

Transcription

1 Identification of Multidimensional Hedonic Models Lars Nesheim Centre for Microdata Methods and Practice (CeMMAP) UCL and IFS October 2015

2 Abstract Nonparametric identification results for scalar nonseparable hedonic models have only recently been worked out (Heckman et al., 2010). This paper extends this work to multidimensional hedonic models. A fully nonseparable multidimensional hedonic model is point identified. Identification requires policy invariant normalizations, requires data from multiple markets with rich price variation, and requires observable multidimensional aggregate supply or demand shifters. Estimation is based on nonparametric estimation of the joint distribution of the data. When data from multiple markets, is not available, point identification can be obtained with additional restrictions. The paper shows that an additive specification results in a multidimensional transformation model and shows how to extend results for a scalar transformation model to the multidimensional case. These models are identified up to location and scale.

3 1 Introduction Nearly all empirical applications of hedonic models involve products with multidimensional characteristics. In fact, these methods were invented precisely to enable price comparison between differentiated products defined in multidimensional spaces (See Waugh, 1929; Griliches, 1961; Rosen, 1974). In hedonic housing market applications for example, housing quality is measured by a vector of characteristics including location within a city, size of dwelling, size of lot, age of dwelling, number of rooms, etc. The price of any house depends on all of these characteristics. Figure 1 shows that 2010 house prices in London vary in two geographic dimensions (controlling for other dwelling characteristics). Similarly, in labor market applications, wages depend on a vector of skills and occupational characteristics. Despite the importance of hedonic models in empirical work, a general theory of identification of structural utility and cost parameters in these models has not been fully developed. For one-dimensional hedonic models, the theory on identification of preferences and costs has only recently been worked out (See Ekeland et al., 2004; Heckman et al., 2010). In this paper, I extend this work to multidimensional hedonic models and derive nonparametric identification and estimation results for multidimensional hedonic models. I derive the following results. First, using data from multiple markets, a fully nonparametric multivariate hedonic model is point identified up to some normalizations. Choice of normalization does not affect counterfactual analysis of policies that change the distribution of observables (e.g. income through tax policy or housing supply). This identification result requires price variation across markets that is rich enough. That is, aggregate supply and demand shocks must vary enough. Given the identification results, estimation is straight forward. It is based on nonparametric estimation of the joint distribution of hedonic characteristics, prices and buyer and seller characteristics. Second, using data from only a single market, a full nonparametric multivariate hedonic model is not point identified. In this case, I present an additivity restriction that is sufficient for point identification. I show that the estimating equation in the additively separable model is a multivariate statistical transformation model. I show that the parameters of this multivariate transformation model are identified up to scale and location normalizations. This result generalizes existing results in the econometric literate on scalar transformation models (e.g. Ridder, 1990; Horowitz, 1998). Third, I provide conditions on utility and cost functions and on the distributions of buyers and sellers that are sufficient to prove that the 1

4 hedonic equilibrium price function is unique and twice-continuously differentiable. This result extends a result from the mathematical literature on optimal transportation to the analysis of hedonic equilibria. Finally, I provide some Monte Carlo experiments that illustrate how a sieve estimator performs in estimating parameters in the single market hedonic setting. The analysis is important for several reasons. The identification analysis provides insight into what data is required to identify structural utility and cost parameters in hedonic models. The analysis also describes straight forward methods to estimate these parameters. The identified structural functions can be used to estimate the both the mean and distribution of impacts of counterfactual policy changes, or changes in the economic environment. Section 2 describes the model, proves that equilibrium exists and is unique, and proves that the price function is differentiable. Section 3 analyses single-market identification. Section 4 analyses multi-market identification. Section 5 presents some results from a Monte Carlo analysis illustrating the performance of the single-market setting methods. Section 6 concludes. 2 Model Products (jobs, houses) have payoff relevant characteristics z Z R 2. They are traded in many markets m M. In market m they can be bought and sold at price p (z, m) where p : Z M R where M is a set of markets to be described below. The set Z is the set of feasible types of products; the same in all markets. Buyers and sellers cannot move between markets. There is a set of buyers with characteristics (x, ε) X E R nx R 2. The characteristics affect the buyers utility payoffs. While all agents in the economy observe (x, ε), outside observers have data only on x. The vector ε is unobservable. I make the following assumption about the distribution of buyer types. Assumption 1 The random variables (X, M, ε) are continuously distributed on their supports with ε (X, M). Their joint distribution is described by a pair of density functions (f XM, f ε ). Assume that (f XM, f ε ) C 2 (X M) C 2 (E). Further assume that 0 f XM < and 0 f ε <. One can allow for discrete components of X. However, this may result in the loss of point identification of the model parameters. I discuss this more in the section on identification. Next I describe the assumption on buyer utility. 2

5 Assumption 2 Buyers have reservation utility u 0 (x, ε, m) = 0. A buyer (x, ε) who participates in market m and chooses product z obtains utility u (z, x, ε) p (z, m) with u C 5 (Z X E). There is a set of firms with characteristics (y, η) Y H R ny R 2. The characteristics affect the sellers costs. Outside observers have data only on y. Assumption 3 The random variables (Y, M, η) are continuously distributed on their supports with η (Y, M). Their joint distribution is described by a pair of density functions (f Y M, f η ). Assume that (f Y M, f η ) C 2 (Y M) C 2 (H). Also, assume that 0 f Y M < and 0 f η <. Further, (Y, η) (X, ε) M. Assumption 4 Sellers have reservation profit π 0 (y, η, m) = 0. Profits of a seller (y, η) who participates in market m and sells product z are p (z, m) c (z, y, η) with c C 4 (Z Y H). Given these assumptions, I can describe the buyers and sellers problems. Buyers have quasilinear utility given in Assumption 2. In market m, a buyer solves with FOC Firms in market m solve with FOC v (x, ε, m) = max {u (z, x, ε) p (z, m)} z Z D z u (z, x, ε) D z p (z, m) = 0. (1) π (y, η, m) = max {p (z, m) c (z, y, η)} z Z D z p (z, m) D z c (z, y, η) = 0. In market m an equilibrium is a price function p m and an assignment of buyers and sellers to products such that markets clear and all agents maximise their individual welfare. Following Chiappori et al. (2010), let α m be a non-negative measure on X E Y H Z. The support of α m (denoted spt(α)) is the smallest closed set of full mass. Pairs of buyers and sellers that lie outside the support of α are pairs that never match together. Thus, the measure α m represents an assignment of buyers and sellers to each other and to products. AN equilibrium is defined in terms of the price function p m and the assignment α m. Definition 1 The pair (α m, p m ) is a hedonic equilibrium if 3

6 1. markets clear. That is, if α m (A) = F XE M (A) A X E (2) α m (B) = F Y H M (B) B Y H (3) 2. all agents maximize individual welfare. That is, if v(x, ε, m) = u(z, x, ε) p(z, m) α m almost everywhere (4) π(y, η, m) = p(z, m) c(y, η, m) α m almost everywhere (5) The market clearing condition states that the projection of the equilibrium assignment measure onto the space of buyers equals the measure of buyers and that the projection onto the space of sellers equals the measure of sellers. The optimisation condition states that the set of agents who fail to maximise individual welfare has measure zero. Theorem 1 Under Assumptions 1-4, hedonic equilibrium exists. Proof. The stated assumptions are stronger versions of the assumptions stated in Chiappori et al. (2010). Thus, the conditions of Theorem 1 and Proposition 2 in that paper are satisfied. Hence, a hedonic equilibrium exists. See Chiappori et al. (2010) for details. Even though equilibrium exists it need not be unique and it need not be pure in the sense that almost all agents have pure strategies in equilibrium. To ensure uniqueness and purity and additional assumption is required. Assumption 5 For almost all (x, ε, y, η), 1. The problem s (x, ε, y, η) = max {u (z, x, ε) c (z, y, η)} z Z has a unique interior solution. 2. The matrix M + M T is positive definite where [ ] D 2 M = xy s Dxηs 2 Dεys 2 Dεηs 2 3. D zε > 0 and D zη < 0. Theorem 2 Equilibrium exists and equilibrium matching is pure. 4

7 Proof. With this additional condition, the Twisted Buyer condition in Chiappori et al. (2010) is satisfied. Thus, Theorem 2 in that paper applies and so equilibrium is unique and is pure. The paper Chiappori et al. (2010) does not provide conditions to ensure that the equilibrium price function is continuously differentiable. Since the identification analysis to follow and since most empirical applications assume that hedonic price functions are differentiable, it is useful to have some conditions that ensure that this is the case. That is why the differentiability conditions above are included. That is also why the second condition in Assumption 5 is added. In addition, to ensure differentiability two additional conditions are required. First, the 4th derivatives of the surplus function cannot be too big. Second, the optimal map from buyers to sellers cannot hit the boundary of the spaces. Assumption 6 There exists a constant c 0 > 0 such that for any w X E, z Y H, and any ξ X E, ψ Y H (s p,q s ij,p s q,rs s ij,rs ) s r,k s s,l ξ i ξ j ψ k ψ l c 0 ξ 2 ψ 2,k,l,p,q,r,s Assumption 7 The spaces of buyers and sellers are s convex. That is, 1. For all (y, η) Y H, the set (D yη s) 1 (X E, y, η) is convex. 2. For all (x, ɛ) X E, the set (D xɛ s) 1 (x, ε, Y H) is convex. These conditions enable me to state the following theorem. Theorem 3 Under Assumptions 1-7, the equilibrium price function is twice continuously differentiable. Proof. Under the stated conditions, the envelope theorem implies that s(x, ε, y, η) C 4. Thus, Theorem 2.1 of Ma et al. (2005) implies that v(x, ε, m) C 3 and that π(y, η, m) C 3. Next recall that the price function satisfies p(z, m) = u(x, z, ε) v(x, ε). Differentiating this equation and using the envelope theorem D z p = D z u(z, x, ε) + (D x u D x v) D z x = D z u Lemma 4 If s (x, ε, y, η) 0 almost surely, everyone enters the market almost surely. Next I turn to a discussion of identification. 5

8 3 Identification The hedonic identification problem is the following. Given data on (x i, z i, p i ) N i=1 where p = p (z) + ξ and ξ is measurement error, one wants to estimate (p, u, F ε ). The demand function z = d (x, ε) satisfies D z p (z) = D z u (z, x, ε). The data provide direct estimates of p and F Z X. 3.1 Failure of point identification in a single market Even if F ε is known, multiple models for d are consistent with data. Suppose the elements of ε are independent and identically distributed as uniform random variables. Then, by construction, a demand model consistent with data is d A 1 (x, ε 1 ) = ( 1 F Z1 X) (x, ε1 ) (6) d A 2 (x, ε 1, ε 2 ) = ( ) [ 1 F Z2 Z 1 X x, ( ) ] 1 F Z1 X (x, ε1 ), ε 2. An observationally equivalent demand model is d B 1 (x, ε 1, ε 2 ) = ( ) [ 1 F Z1 Z 2 X x, ( ) ] 1 F Z2 X (x, ε2 ), ε 1 (7) d B 2 (x, ε 2 ) = ( F Z2 X) 1 (x, ε2 ). The second demand model can be derived from the first through a measure preserving transformation T : X E E. This leads to the following lemma. Lemma 5 Suppose F ε is multivariate independent uniform. Let T : X E E be a measure preserving mapping a.s. X and let T be the set of all such measure preserving transformations. Then the set of demand models consistent with the data is { d : d = d A T for some T T } where d A is defined in equation (6). Two elements in the identified set are given in (6) and (7). Another simple example could be obtained by first mapping ε A into the real line, rotating it, and then mapping back into the unit square. None of these alternative models is structural. All can be used to predict distribution of demand if price held fixed but not otherwise. For 6

9 example, if the distribution of x were held fixed, but a single individual s value of x changed and nothing else, then these models could be used to predict how that individual would respond (partial equilibrium). This is useful for some partial equilibrium predictions. However, it doesn t allow one to much about welfare nor is it very useful for policies that change the distribution of x or the distribution of supply. Such policies change the price and so also change the demand function. Even if d (x, ε) were identified, still u is not. The first-order condition is D z p (d (x, ε)) = D z u (d (x, ε), x, ε). The function u is a function of n x + 4 variables, but, in a single market, we only observe its values on a n x + 2 dimensional manifold. Without further structure, it is impossible to estimate D z u (d (x, ε) + z, x, ε). That is, what is the marginal utility for a counterfactual product that was not purchased. To infer marginal utility for choices that are not observed in the current equilibrium, one needs either data from multiple markets or further restrictions on u. Without further restrictions, the best one can do is bound the utility values using revealed preference. The argument is easiest to describe in terms of discrete changes in z. If consumer (x, ε) chose to purchase z, then they must have preferred that choice to the alternative z + z. This implies that u(z + z, x, ε) u(zz, x, ε) p(z + z) p(z). (8) That is, the utility benefit from moving to z + z must have been less than the cost. 3.2 Single market identification One restriction that is sufficient for identification is an additivity restriction. Recall the buyers first-order condition from equation (1). This FOC is D z u (z, x, ε) = D z p (z). Suppose utility takes the additive form D z u (z, x, ε) = u 0 (z) + v (x) + ε and define D z T (z) = D z p (z) u 0 (z) 7

10 The first-order conditions become D z T (z) = v (x) + ε (9) with D zz T > 0. This is a bivariate transformation model with the restriction that T (z) is a convex function and is one-to-one and onto. Using the change of variables formula, this model implies that the density of z given x satisfies Defining and f Z X (z, x) = f ε (D z T (z) v (x)) det D zz T (z). (10) A = {z Z : Z 1 z 1, Z 2 z 2 } D z T (A) v (x) = {ε : ε = T (z) v (x) for some z Z}. and integrating over A, this implies that the cumulative distribution function of the data is F Z X (z 1, z 2, x) = z 1 z 2 f ε (D z T (s) v (x)) D zz T (s) ds (11) = f ε (ε) dε. D zt (A) v(x) The identifying power of the additive model stems in from the additivity that defines the boundary of the region of integration. Differentiating with respect to z and x provides linearly independent information about the parameters. We can now show identification by differentiating with respect to z and x. The derivative of (11) w.r.t. z i is F Z X (z 1, z 2, x) z i = z i = = D zt (A(z)) v(x) D zt (A(z)) v(x) D zt (A(z)) v(x) = D zt (A(z)) v(x) f ε (ε) dε ( [ ]) 2 T (z) f ε (ε) dε 2 2 T (z) dε 1 z 1 z i z 2 z i ( 2 T D ε1 f ε (ε) + D ε2 f ε (ε) z 1 z i D ε f ε dε 8 T (D z T ) z i. 2 T z 2 z i ) dε 1 dε 2

11 The second line is a multivariate version of Leibniz rule for differentiating under an integral (see for example Flanders, 1973). The third line is Green s Theorem. The final line follows since the derivative of T does not depend on ε. Thus, the derivative with respect to z equals an unknown integral times a function that depends only on z. In a similar manner, one can compute the derivative of (11) with respect to x j as F Z X (z 1, z 2, x) x j = x j = = D zt (A(z)) v(x) D zt (A(z)) v(x) D zt (A(z)) v(x) f ε (ε) dε f ε (ε) D ε f ε dε [ v1 dε 2 v ] 2 dε 1 x j x j T D x v. The derivative of the CDF equals an unknown integral multiplied by a function that depends only on x. The restrictions imposed by separability can be summarized by D z F Z X = (D zz T ) D ε f ε dε (12) D x F Z X = (D x v) D ε f ε dε. (13) This is a system of linear partial differential equations in the unknown functions (T, v, F ε ). Normalize D x v (x 0 ) = I and define FZ X 0 = F Z X (z, x 0 ) Fε 0 = D ε f ε (ε) dε so that D zt (z) v(x 0 ) D z F 0 Z X = (D zz T ) F 0 ε = (D zz T ) D x F 0 Z X This is a linear second order elliptic partial differential equation in the unknown T (z). Appealing to the literature on partial differential equations (See for example Evans et al., 2000) this system of equations 9

12 has a unique solution up to some boundary conditions. For example, one could impose the boundary conditions that for all z Z D z1 T (z) = Φ 1 (z 1 ) D z2 T (z) = Φ 1 (z 2 ) where Φ is the standard normal CDF. Once T is known, one can use T to eliminate the unknowns in equation (13) and recover D x v. Then once both T and v are known, the distribution of f ε can be compute directly from the residuals Estimation The identification proof makes use of a partial differential equation. One could base an estimator on those equations. However, it iseasier to base estimation on constrained sieve maximum likelihood estimation where the constraints impose the normalizations defined in the previous section. The likelihood function is defined in equation (10). Rewrite the transformation model 9 as T 1 (z) = v 1 (x) + Φ 1 (u 1 ) (14) T 2 (z) = v 2 (x) + Φ 1 (u 2 ). With this change of variable, the likelihood function can be written as ln f Z X (z, x) = ln f u (u 1, u 2 ) 0.5u u 2 2 (15) + log det (D zz T (z)) u 1 = Φ (T 1 (z) v 1 (x)) u 2 = Φ (T 2 (z) v 2 (x)) Then, without loss of generality one can transform each dimension of z and x so that they all variables lie in the unit interval. Then to impose the boundary condition, one can define the sieve T (z 1, z 2 ) = z 1 α ij B i (z 1 )B j (z 2 ) + Φ 1 (s)ds + 0 Φ 1 (s)ds (16) 10

13 and impose the constraints that, for all (z 1, z 2 ), α ij B i (0) z 1 B j (z 2 ) = 0 (17) α ij B i (1) z 1 B j (z 2 ) = 0 α ij B i (z 1 ) z 1 B j (0) = 0 α ij B i (z 1 ) z 1 B j (1) = 0 and α ij B i (0) B j(z 2 ) z 2 = 0 (18) α ij B i (1) B j(z 2 ) z 2 = 0 α ij B i (z 1 ) B j(0) z 2 = 0 α ij B i (z 1 ) B j(1) z 2 = 0 Next, to impose that D zz T (z) is positive definite, define J(z) = β ijb i (z 1 )B j (z 2 ) and impose the constraint that log J(z 1, z 2 ) = log det (D zz T (z 1, z 2 )) (19) Next approximate the unknown functions (v 1, v 2 ) as v 1 (x 1, x 2 ) = v 2 (x 1, x 2 ) = γ 1,ij B i (x 1 )B j (x 2 ) (20) γ 2,ij B i (x 1 )B j (x 2 ) subject to v 1 (0, 0) = 0 and v 2 (0, 0) = 0. Finally, define ln f u = δ ij B i (u 1 )B j (u 2 ) (21) subject to f u is a density. 11

14 Then the estimation problem is { } max ln f u (u 1h, u 2h ) 0.5u 2 1h 0.5u 2 2h + ln J(z h ) θ h (22) subject to exp (ln J(z)) ( T 11 (z)t 22 (z) T 12 (z) 2) = 0 (23) B i (0) α ij B j (z 2 ) = 0 z 1 (24) α ij B i (1) z 1 B j (z 2 ) = 0 (25) α ij B i (z 1 ) z 1 B j (0) = 0 (26) α ij B i (z 1 ) z 1 B j (1) = 0 (27) α ij B i (0) B j(z 2 ) z 2 = 0 (28) α ij B i (1) B j(z 2 ) z 2 = 0 (29) α ij B i (z 1 ) B j(0) z 2 = 0 (30) α ij B i (z 1 ) B j(1) z 2 = 0 (31) v 1 (0, 0) = 0 (32) v 2 (0, 0) = 0 (33) f u (u)du = 1 (34) exp (B u α u ) du = 1 (35) 4 Multimarket identification Recall the the first-order conditions D z p (z, m) = D z u (z, x, ε). 12

15 Let z m = d (x, ε, m). In a single market, it is impossible to learn the precise value of D z u (z m + z, x, ε) because outcome z = z m+ z is never observed. However, with multiple markets, one can learn more about this object by finding a market m with price p (z, m ) such that z = d (x, ε, m ) = z m + z. Identification can be achieved by fixing (z, x), imposing a normalisation on the distribution F ε and tracing out D z (u, x, ε) by looking at the distribution of v = D z p (z, m) across markets. Equilibrium implies that the marginal price should vary across markets due to aggregate cost or demand shifters. Since I have assumed that only the distributions of observables vary across market, then the price will vary across markets in response to variations in F Y M and F X M. Define the random variable v = D z p (z, m) and rewrite the first order condition v = D z u (z, x, ε). The continuum of economies (F XM (x, m), F Y M (y, m), F ε ) generates data on F V ZX. I will use this data to construct an estimate of D z u (z, x, ε) for all (z, x, ε). Let Range(D z u) be the range of D z u. Assumption 5 implies that D z u (z, x, ε) : E V is one-to-one and onto. We need to ensure that the range of D z u does not depend on (z, x). Assumption 8 Assume that Range(D z u) is independent of (z, x). Define the random variable v = D z p(z, m). This random variable must have full support. That is the must be enough price variation to identify the gradient of marginal utility. Assumption 9 Suppose that (V, Z, X) has support on V Z X. 13

16 4.1 Normalization 1: Choice of F ε The vector ε is not observed and has no natural units. These variables enter model in non-additive fashion. One natural normalization is to define each element of ε as a percentile rank of consumers in some dimension of marginal utility space. This is a normalization as long as the distribution of ε is not changed by changes in (Y, X, M) or by policies that change these distributions. It is not a normalization otherwise. For example, ε 1 might measure the percentile rank of D z1 u conditional on (z, x) and ε 2 might measure the percentile rank of D z2 u conditional on (z, x) and ε 1. Of course, other normalizations are possible. However, this normalization chooses the units of measurement so that elements of ε are independent uniform random variables. 4.2 The identified set There is a family of demand functions, distribution functions F ε and utility functions that are consistent with the data. Let [ ] Q d(x, m, u) = Z1 XM(u 1, x, m). (36) Q Z2 Z 1 XM(u 2, Q Z1 XM(u 1, x, m), x, m) The identified set of demand functions includes all functions of the form d(x, m, ε) = d (x, m, R [F ε (ε), x]) (37) where F ε is a distribution function and R is any invertible measure preserving transformation satisfying det ( R u ) = 1. Choice between d and d is a normalization. Both predict the same distribution of demand in each market m. I can then use d and p to identify D z u. To estimate D z u(z, x, ε), the equation is D z u(z, x, ε) = D z p (d(x, m, ε), m ) (38) where m solves z = d(x, m, ε). 4.3 Estimation An estimator based on this normalization is as follows. Let utility be u (z 1, z 2, x, ε 1, ε 2 ) = u 1 (z 1, z 2, x, ε 1 ) + u 2 (z 2, x, ε 1, ε 2 ). Recall that (ε 1, ε 2 ) x, ε 1 ε 2, and ε i U [0, 1]. Interpretation: ε 1 is the rank in the population in terms of marginal utility of z 1. ε 2 is the rank of marginal utility of z 2 conditional on z 1. 14

17 The FOC become D z1 u 1 (z 1, z 2, x, ε 1 ) = D z1 p (z 1, z 2, m) D z2 u 1 (z 1, z 2, x, ε 1 ) + D z2 u 2 (z 2, x, ε 1, ε 2 ) = D z2 p (z 1, z 2, m). Solve for (ε 1, ε 2 ) to obtain ε 1 = w1 1 (v 1, z 1, z 2, x) ( [ ε 2 = w2 1 v2 D z2 u 1 z1, z 2, x, w1 1 (v 1, z 1, z 2, x) ], z 2, x, w1 1 [v 1, z 1, z 2, x] ). Therefore, w 1 1 (v 1, z, x) = F V1 ZX (v 1, z, x) w 1 2 (v 1, v 2, z, x) = F V2 V 1 ZX (v 2, v 1, z, x) where w 1 1 is the inverse of D z1 u 1 with respect to ε 1 and w 1 2 is the inverse of D z2 u 2 with respect to ε 2. Estimation is based on any nonparametric estimator of F V ZX. 5 Monte Carlo results for the single-market case This section is not yet complete. 6 Conclusion Multimarket data are required for point identification of an unrestricted hedonic model. When sufficient multimarket data are available, estimation can be based on nonparametric estimation of a multidimensional distribution function. This estimation problem is computationally simple but highly demanding of data. Using single-market data, point identification can still be achieved if restrictions such as additive separability are imposed A Figures 15

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