Nonarametric estimation of Exact consumer surlus with endogeneity in rice Anne Vanhems February 7, 2009 Abstract This aer deals with nonarametric estimation of variation of exact consumer surlus with endogenous rices. The variation of exact consumer surlus is linked with the demand function via a non linear differential equation and the demand is estimated by nonarametric instrumental regression. We analyze two inverse roblems: smoothing the data set with endogenous variables and solving a differential equation deending on this data set. We rovide some nonarametric estimator, resent results on consistency and otimal choice of smoothing arameters, and comare the asymtotic roerties to some revious works. Keywords: Nonarametric regression, Instrumental variable, Inverse roblem JEL classifications: Primary C14; secondary C30 Toulouse Business School, 20 Boulevard Lascrosses, BP 7010, F-31068 Toulouse, France, a.vanhems@ esc-toulouse.fr
1 Introduction In structural econometrics, interest arameters are often defined imlicitly by a relation derived from the economic context and deending on the law of distribution of the data set. Such roblems require to exlicit the link between the arameter of interest and the law of data set and can be considered as inverse roblems. Deending on the regularity roerties of the relation to solve, they are either well-osed ie there exists a unique stable solution) or ill-osed. This work analyzes two mixed inverse roblems and is motivated by a articular economic relation, the link between the variation of exact consumer surlus associated to some rice variation and the observed demand function. Such a framework was studied in articular in Hausman and Newey 1995). Their objective is to measure the imact on the consumer welfare of a rice change for one good. One way to roceed is to calculate the variation of exact consumer surlus, which is a monetary way of measuring the change in welfare. To do so, consider one consumer, define y his income, q the demand in good and 1 the rice of a unique good. Assume that there exists a rice variation from to 1. The variation of exact consumer surlus for an income level y, denoted by S y, reresents the cost to ay to the consumer so that his welfare does not change for a rice change see Varian 1992)). The link between the interest arameter S y and the demand function q is given by the following nonlinear relation: { S y ) = q, y S y )) model. S y 1 ) = 0 1.1) The demand function q is not known and can be estimated using some econometric Consider Q, P, Y ) a random vector defining demand, rice and income, and a samle Q i, Y i, P i ) i=1,...,n of observations. The demand function q can be aroximated by the function g estimated by a nonarametric regression: { Q = g P, Y ) + U E U P, Y ) = 0 In their aer, Hausman and Newey 1995) analyze gasoline consumtion using data from the U.S. Deartment of Energy. They estimate semiarametrically the demand function, with a nonarametric estimation of g, and a arametric art including several exogenous variables like the year of survey, the city state of the household. They assume that the identification assumtion E U P, Y ) = 0 is fulfilled. The motivation for our work derives from the endogeneity of rice in the analysis of demand function. In this case, the identification condition E U P, Y ) = 0 is no more satisfied and the conditional mean does not identify the structural demand relationshi. To identify our interest arameter, we introduce some random variable W, called an instrument, 1
such that E U Y, W ) = 0. The underlying function g is then defined through a second equation: E Q gp, Y ) Y, W ) = 0 1.2) Solutions of this second linear roblem have been extensively studied, in arametric as well as in nonarametric settings. The analysis of endogenous regressors, and more generally of simultaneity, has a great imact in structural econometrics. Since the earliest works of Amemiya 1974) and Hansen 1982), extensions to nonarametric and semiarametric models have been considered. Identification and estimation of g have been the subject of many recent economic studies Darolles, Florens, and Renault 2002), Newey and Powell 2003), Hall and Horowitz 2005), Gagliardini and Scaillet 2007), Blundell and Horowitz????), Blundell, Chen, and Kristensen 2007) to name but a few). In articular, the alication to Gasoline demand is studied in Blundell, Horowitz, and Parey 2008). In what follows, we use Hall and Horowitz 2005) methodology to estimate g. Our urose in this work is to mix both roblems 1.1) and 1.2) in a nonarametric setting. We lug some nonarametric instrumental regression estimator into the differential equation and study the asymtotic roerties of the associated estimated solution. We aly our rocedure to the gasoline consumtion database used in Hausman and Newey 1995). The aer roceeds in the following way. In the next section, we set the notations, the main equations to solve and the link with inverse roblems theory. We then resent our nonarametric estimator and recall the theoretical roerties of each inverse roblem. In section 4, we study the asymtotic behavior of our estimator. 2 Model Secification. In this section, we set the notations and link our model with inverse roblems theory. 2.1 The linear equation model. The objective of this art is to set the econometric model defining the demand function q. We follow the modelization of Hall and Horowitz 2005). Consider Q, P, Y, W, U) a random vector with all scalar random variables to fit with the emirical alication). We assume that P, Y and W are suorted on [0; 1] 1. Let Q i, P i, Y i, W i, U i ), for i 1, be indeendent and identically distributed as Q, P, Y, W, U). P and Y are endogenous and exogenous exlanatory variables, resectively. Data Q i, P i, Y i, W i ), for 1 i n, are observed. Let f P Y W denote the density distribution of P, Y, W ), and f Y the density of Y. Following Hall and Horowitz 2005) notations, we define for each y [0, 1] t y 1, 2 ) = 1 This assumtion is not very restrictive since we study solutions of differential equations that are defined locally 2
fp Y W 1, y, w)f P Y W 2, y, w)dw and the oerator T y on L 2 [0, 1] by T y ψ), y) = t y ξ, )ψξ, y)dξ. The solution g of equation 1.2) satisfies: T y g, y) = f Y y)e W Y {EQ Y = y, W )f P Y W, y, W ) Y = y} 2.1) where E W Y denotes the exectation oerator with resect to the distribution of W conditional on Y. Then, for each y for which Ty 1 exists, it may be roved that g, y) = f Y y)e W Y {EQ Y = y, W )Ty 1 f P Y W ), y, W ) Y = y}. 2.2 The nonlinear equation model. Our interest functional arameter S y is solution of the differential equation 1.1) deending on m, which can be rewritten: { S y ) = g, y S y )) S y 1 ) = 0 2.2) or equivalently: S y ) = 1 gt, y S y t))dt 2.3) The function S y is deending on g deending itself on the law of distribution of Q, P, Y, W ). These two roblems 2.1) and 2.3) can be considered as articular cases of inverses roblems. 2.3 Link with inverse roblems theory Studying our interest arameter S y is equivalent to solving both inverse roblems 2.1) and 2.3). Let start with the relation 2.3). The function S is defined by an imlicit nonlinear relation there is no restrictive assumtion on the form of the function g). Denote by A y the oerator defined by A y g, S) = S y + g., y S y ). Solving 2.3) is equivalent to inverting the oerator A y under the initial condition S y 1 ) = 0. Under regularity assumtions on g, following Vanhems 2006), there exists a unique solution: S y ) = Φ y [g]), where Φ y is continuous with resect to g. This nonlinear inverse roblem is well-osed and defines a unique stable solution. The function g itself is solution of a second linear roblem 2.1). As recalled in introduction, this model is the foundation of many economic studies. Solving equation 2.1) is equivalent to trying to invert the oerator T y. Even when the robability distribution of P, Y, W ) is known, the calculation of a solution g from equation 2.1) is an ill-osed inverse roblem. However f P Y W is unknown in general and has to be estimated from an iid samle of P, Y, W ). Two stes are necessary in order to obtain an estimator of g. The 3
first ste is to stabilize equation 2.1), the second ste is to solve the stabilized equation where T y is relaced by its estimator. Under regularity assumtions on the function g and the oerator T y, there exists a unique solution g see Hall and Horowitz 2005) or Johannes, Van Bellegem, and Vanhems 2007) for a general overview). Remark 2.1. The best methodology would have been to try and solve both roblems in one ste and invert one oerator instead of two. Contrary to the oerator A y which is deterministic, T y also deends on the law of data set and has to be estimated. Therefore, it turns out to be imossible to write our model into a single inverse roblem to solve. We use a methodology in two stes to study our interest arameter S y. In the next section, we recall the estimation rocedure and theoretical roerties of both functions g and S y searately, before mixing both inverse roblems. 3 Estimation and identification In this section, we resent the nonarametric methodology used as well as the issues of identification and overidentification for both inverse roblems searately. We briefly recall the results in Hall and Horowitz 2005) and Vanhems 2006) in order to rove the asymtotic roerties of the final estimated functional arameter S y. 3.1 The linear inverse roblem We first consider the nonarametric instrumental regression defined in equation 2.1). It is a Fredholm equation of the first kind and generates an ill-osed inverse roblem. For the urose of estimation, we need to relace the inverse of T y by a regularized version. Indeed, it is well-known that the ill-osedness of this equation imlies that a consistent estimator of g is not found by a simle inversion of the estimated oerator T y. A modification of the inversion is always necessary and in what follows, we consider the Tikhonov regularization and relace T 1 y by T y + ai) 1 = T y + where I is the identity oerator and a > 0. 3.1.1 Estimation Consider K a kernel function of one dimension, centered and searable, h > 0 the bandwidth arameter and K h u) = 1/h)Ku/h). 2 To construct an estimator of g, y), let h, h y > 0 2 Note that we could have introduced some generalized kernel function to overcome edge effects, as in Hall and Horowitz 2005). It is not necessary in our context since we intent to estimate a local solution S y of the differential equation in the neighborhood of the initial condition. 4
two bandwidth arameters and define: f P Y W, y, w) = 1 n n K h P i )K hy y Y i )K h w W i ), i=1 f i) P Y W, y, w) = 1 n K h P i )K hy y Y i )K h w W i ), n 1) j=1,j i t y 1, 2 ) = f P Y W 1, y, w) f P Y W 2, y, w)dw, T y ψ), y, w) = t y ξ, )ψξ, y, w)dξ. The nonarametric estimator of g, y) is defined by: ĝ, y) = 1 n n T y + i=1 f i) P Y W ), y, W i)q i K hy y Y i ). 3.1) 3.1.2 Theoretical roerties In order to derive rates of convergence for Hall and Horowitz 2005) estimator, it is necessary to imose regularity conditions on the oerator T y. Assume that for each y [0, 1], T y is a linear comact oerator and note {φ y1, φ y2,...} the orthonormalized sequence of eigenvectors and λ y1 λ y2... > 0 the resective eigenvalues of T y. Assume that {φ yj } forms an orthonormal basis on L 2 [0, 1] and consider the following decomositions on this orthonormal basis: t y 1, 2 ) f P Y W, y, w) g, y) = j=1 λ yjφ yj 1 )φ yj 2 ), = j=1 k=1 d yjkφ yj )φ yk w), = j=1 b yjφ yj ). 3.2) Under regularity conditions on the density f P Y W and the kernel K f P Y W has r continuous derivatives and K is of order r), on the function g, y), and on the rate of decrease of the coefficients b yj, λ yj and d yjk deending on constants α and β, it is roved that ĝ, y) 2β 1 τ converges to g, y) in mean square at the rate n 2β+α with τ = 2r 2r+1. In articular, the constants α and β are defined such that, for all j, b yj Cj β, j α Cλ yj and k 1 d yjk Cj α/2, C > 0, uniformly in y [0, 1]. 3.2 The nonlinear inverse roblem Consider now the second inverse roblem defined by equation 2.3). The function Ŝy) is defined as solution of the estimated system: {Ŝ y ) Ŝ y 1 ) = 0 = ĝ, y Ŝy )) 5
3.2.1 Estimation The estimated solution Ŝy is aroximated using numerical imlementation. Various classical algorithms can be used to calculate a solution, like Euler-Cauchy algorithm, Heun s method, Runge Kutta method. Hausman and Newey 1995) use a Buerlisch-Stoer algorithm from Numerical recies. Let briefly recall the general methodology. Consider a grid of equidistant oints 1,..., n where i+1 = i + h and 1 = 1. The differential equation 2.2) is transformed into a discretized version: {Ŝyi+1) Ŝ y0 = 0. = Ŝyi hĝ h i, y Ŝyi) 3.3) Where ĝ h is an aroximation of ĝ. In the articular case of Euler algorithm, ĝ h = ĝ. As recalled in Vanhems 2006), numerical aroximation of Ŝy does not imact the theoretical roerties of the estimator since they have a higher seed of convergence than nonarametric estimation methods. 3.2.2 Theoretical roerties It has been roved see Vanhems 2006)) that under some regularity assumtions on g, following Cauchy-Lischitz theorem, for each y [0, 1], there exists a unique solution S y defined in a neighborhood of the initial condition 1, 0). Again, under regularity conditions on ĝ, following Cauchy-Lischitz theorem, there exists a unique solution Ŝy defined on a neighborhood of the initial condition 1, 0). The stability of the inverse roblem 2.3) is fulfilled if the estimator Ŝy is consistent, that is if e 2 ĝ i.e. the derivative of ĝ with resect to the second variable) converges uniformly to e 2 g. Under the condition that e 2 ĝ e 2 g 0, the estimator Ŝy converges almost surely to S y and the nonlinear inverse roblem is well-osed. see Vanhems 2006) for more details). In order to derive rates of convergence, we need to exlicit the link between the solution S y and the function g. The main issue of this differential inverse roblem is its nonlinearity. The following reliminary result transforms the nonlinear equation into a linear roblem. The methodology used is closely related to functional delta method and close to result used in Hausman and Newey 1995) and Vanhems 2006). Denote I = [ 1 ε 1, 1 + ε 1 ], for ε 1 > 0 a closed neighborhood of 1 and D = {, y); I, y ε 2 }, for ε 2 > 0. Proosition 3.1. : i) Under the assumtion of consistency of Ŝy to S y, it can be roved that: I, Ŝy ) S y ) = where R 1,n ) = o P ĝ g ) and ĝ g = 1 t ]) gu,y S e ĝ g) t, y S y t)).e[ 2 yu))du dt+r 1,n, y) su ĝa, b) ga, b). a,b) D 6
ii) Assume moreover that g and K are at least continuously differentiable of order 2 ie r 2), then the revious decomosition can be transformed: t ]) I, Ŝy gu,y S e ) S y ) = ĝ g) t, y S y t)).e[ 2 yu))du dt+r 2,n, y) 1 where R 2,n ) = O P ĝ g 2) and ĝ g 2 = 1 D ĝ g) 2 a, b) dadb Introducing this exansion enables us to transform the nonlinear roblem into a linear one, u to a residual term. Hence the rate of convergence of Ŝy) towards S y ) can be deduced from both terms. the linear art. The rate of convergence of the estimated solution of the differential equation 2.2) is exected to be greater than the rate of convergence of the estimator of the function g since there is a gain in regularity. Moreover, we also exect a gain in dimension since we transform a function of two arguments into a function of one argument. the residual term, which is the counterart in the Taylor exansion. This term converges to zero by definition and we will neglect it in what follows. Rather we obtain an aroximation rate u to this remainder term, controlled in robability. 4 Asymtotic behavior of the estimated solution In this section, we aim at giving the asymtotic behavior of the solution of the differential equation obtained after estimating the regression function observed in an endogenous setting. Note first that all the asymtotic results will be given using the L 2 norm which will be written.. The different other norms will be clearly secified. 4.1 Assumtions Here are the assumtions required for the consistency and mean square convergence. In articular we rovide rates of decay for the generalized fourier coefficients defined in equations 3.2). We also introduce the following decomosition: ] gu,y S e 2 yu))du t m y, t) = 1 [,]t).e[ 1 = c yjk φ yj )φ yk t) j=1 k=1 We then make the following assumtions, mostly adated from Hall and Horowitz 2005) and Vanhems 2006). 7
[A1] The data Q i, P i, Y i, W i ) are indeendent and identically distributed as Q, P, Y, W ), where P, Y, W are suorted on [0, 1]. [A2] The distribution of P, Y, W ) has a density f P Y W with r 2 derivatives, each derivative bounded in absolute value by C > 0, uniformly in and y. The functions EQ 2 Y = y, W = w) and EQ 2 P =, Y = y, W = w) are bounded uniformly by C. [A3] The constants α, β, ν satisfy β > 0, ν > 0, β + ν > 1/2, α > 1 2ν, and β + ν) 1/2 α < 2β + ν). Moreover, b yj Cj β, j α Cλ yj, k 1 d yjk Cj α/2 and k 1 c yjk Cj ν uniformly in y, for all j 1. [A4] The arameters a, h, h y satisfy a n ατ/2β+α), h n γ, h y n 1/2r+1) as n goes to infinity, where τ = 2r/2r + 1). [A5] The kernel function K is a bounded and Lebesgue integrable function defined on [0, 1]. Ku)du = 1 and K is of order r 2. Moreover, K is continuously differentiable of order r with derivatives in L 2 [0, 1]). [A6] For each y [0, 1], the function φ yj form an orthonormal basis for L 2 [0, 1] and su su y max j φ yj ) <. [A7] su,y e 2 ĝ, y) e 2 g, y) converges in robability to 0. Remark on assumtion [A3] that allows to control the regularity of T y, g and m y. 4.2 Consistency Proosition 4.1. Under assumtions [A1] [A7], our estimator Ŝy) is consistent and converges in robability to S y ). Then we can aly the result of Proosition 3.1 and write: Ŝ y ) S y ) = ĝ g) t, y S y t)).m y, t)dt + R 2,n, y) = I, y) + R 2,n, y) 4.3 Asymtotic mean square roerties Theorem 4.2. Consider assumtions [A1] [A7] and the following roerty: su y [0,1] E{I, y)} 2 d su y [0,1] E{ ĝ g)t, y)m y, t)dt} 2 d 4.1) Then, we can rove that: su y [0,1] E I., y) 2 2β+ν) 1 τ ) = On 2β+α ) 4.2) 8
Remark 4.1. Note that the rate of convergence deends on the arameter ν which can be interreted as the regularity induced by solving the differential equation. It is faster than n τ 2β 1 2β+α, which is the rate obtained by Hall and Horowitz 2005). The condition 4.1) is quite natural in economics, it means that we neglect the comensated income in the surlus equation. A Proofs Proof of Proosition 3.1. Proof. This roof is directly taken from Vanhems 2006). Under the assumtions of consistency, there exists a unique solution to 2.2) S y ) = Φ y [g] ). The objective is to try and characterize the functional Φ y that is the exact deendence between S y and m. To rove this result, it is not necessary to imose the strong assumtions required later in the aer. Indeed, consider the oerator A y defined on the following saces: { C 1 D) Cb,0 1 I) CI) A y : u, v) A y u, v) where C 1 D) = {u CD) and continuously differentiable} and C 1 ε 2,0I) = { v C b,0 I), continuously differentiable and v < ε 2 /ε 1 } where D = {u, v) ; x ε 1, y ε 2 }. C 1 D),. ) and CI),. ) are Banach saces. Moreover we define the following norm:. = max v, v ) ) on Cb,0 C 1 I). We can easily see that b,0 1 I),. is a Banach sace. As a matter of fact, to rove it, we have to use the uniform convergence of functions and its alication to differentiability. The use of such a norm allows us to have the continuity and linearity of the following function: ) Cb,0 1 D : I),. CI),. ) y y So, we have: x I, A y u, v)x) = v x) + ux, y vx)). Define an oen subset O of C 1 D) C 1 b,0 I) and g, S y) O. A y is continuous on O it is a sum of continuous alications) and A y g, S y ) = 0. Let us check the hyothesis of the imlicit function theorem. A y is in fact continuously differentiable thanks to the same argument) so we can take its derivative with the second variable d 2 A y g, S y ). Moreover, we have: 9
h Cb,0 1 I), I, d 2A y g, S y )h)) = h ) + g, y S y )).h) e 2 We have to rove that d 2 A y g, S y ) is a bijection. Let us show first the surjectivity: v CI),?h C 1 b,0 I); I, h ) + e 2 g, y S y )).hx) = v) This is a linear differential equation, so we can solve it and find that: I, h) = 1 vs).e[ s ]) gt,y S e 2 yt))dt ds Therefore, d 2 A y g, y S y ) is surjective. Let us now demonstrate the injectivity, that is Ker d 2 A y g, y S y )) = {0} We are going to solve d 2 A y g, y S y )h = 0, h Cb,0 1 I). We find again a linear differential equation we can solve and find: I, h) = ce 1 gt,y S e 2 yt))dt and h 1 ) = 0 Therefore, we get c = 0. Thus, we have demonstrated that d 2 A y g, S y ) is bijective. Let us now demonstrate the bi-continuity of d 2 A y g, S y ). In the usual imlicit function theorem, this assumtion is not required, but here we consider infinite dimension saces that is why we need a more general theorem with further assumtions to satisfy. The continuity of d 2 A y g, S y ) has already been roved since A y is continuously differentiable. The continuity of the reversible function is given by an alication of Baire Theorem: if an alication is linear continuous and bijective on two Banach saces, the reversible alication is continuous. Therefore, we can aly the imlicit function theorem: U an oen subset around g and V an oen subset around S y such as: u U, A y u, v) = 0 has a unique solution in V Let us note: v = Φ y [u] this unique solution for u U. Now we are going to differentiate the relation: A y u, Φ [u]) = 0, u U and aly it in g, S y = Φ y [g]). Let us first differentiate A y : h C 1 D) C 1 b,0 I), da y g, S y )h)) = d 1 A y g, S y )dgh)) + d 2 A y g, S y )ds y h)) = dgh), y S y )) + ds y h)) ) + e 2 g, y S y ))dsh)) 10
The differential of A y leads to a linear differential equation in ds y h) that we can solve. Now we aly it with dgh) = ĝ g and ds y h) = dφ y [g] ĝ g) in order to find: dφ y [g] ĝ g) ) = e 2 g, y Φ y [g] ).dĝ g)) ĝ g), y Φ y [g] )) Solving it leads us to: dφ y [g] ĝ g)) = = = 1 1 ĝ g) t, y Φ y [g] t)).e[ s ĝ g) t, y S y [g] t)).e[ s 1 ĝ g) t, y S y [g] t)).v, t)) dt ]) gu,y Φ e 2 y[g]u))du dt ]) gu,y S e 2 y[g]u))du dt So the statement is roved. Proof of Proosition 4.1. Proof. The roof is based on the same roerties as in Vanhems 2006). To rove the consistency of Ŝy), we need to rove there exists a unique solution to each differential system 2.2) and??). Following Cauchy-Lischitz theorem, g and ĝ satisfy the Lischitz condition: g, y 2 ) g, y 1 ) k y 2 y 1, for all, y 1, y 2 ), ĝ, y 2 ) ĝ, y 1 ) k y 2 y 1, for all, y 1, y 2 ). Under the assumtion that K and f P Y W are continuously differentiable of order r 2, both conditions are satisfied. Moreover, to guarantee the stability of the inverse roblem, we need to imose that the estimated Lischitz factor k converges in robability to k of or in other words that e 2 ĝ converges uniformly in robability to e 2 g. This condition is fulfilled under assumtion [A7]. Proof of Theorem 4.2: Proof. We analyze the following term: ĝ g)t, y)dt. The objective is to rove that: 1 su y [0,1] E{ ĝ g)t, y)dt} 2 d = On τ 2β+ν) 1 2β+α ) 1 11
To rove the result, we follow the demonstration in Hall and Horowitz 2005) and define: D ny ) = gx, y)f P Y W x, y, w)t y + f P Y W f P Y W )t, y, w)dxdw}dt A n1y ) = 1 n A n2y ) = 1 n A n3y ) = 1 n A n4y ) = 1 n 1 { n i=1 n i=1 n i=1 n i=1 1 T + y f P Y W )t, y, W i )Q i K hy y Y i )dt, 1 {T + y 1 { T + y 1 { T + y f i) P Y W f P Y W )}t, y, W i )Q i K hy y Y i )dt D ny ), T + y )f P Y W }t, y, W i )Q i K hy y Y i )dt + D ny ), T y + i) ) f P Y W f P Y W )}t, y, W i )Q i K hy y Y i )dt. Then ĝt, y) = A 1 n1y ) + A n2y ) + A n3y ) + A n4y ) and the theorem will follow if we rove that: E A n1y. 1 gt, y)dt 2 = On τ 2β+ν) 1 2β+α ), A.1) E A njy 2 2β+ν) 1 τ = On 2β+α ), forj = 2, 3, 4. A.2) To derive A.1), note that EA n1y ) gt, y)dt 2 2 I 1 1 2 + I 2 2 ) with I 2 2 = Oh 2r y a 2 ) and I 1 = a b yj c yjk λ j + a) 1 φ yk ). k j Therefore, I 1 2 = k a j b yj c yjk λ j + a) 1 2 C 2 a j b yj j ν λ j + a) 1 2 We then divide the series u to the sum over j J a 1/α and the comlementary art. Following Hall and Horowitz 2005), we bound the right-hand side by a 2 j J b yjj ν /λ j ) 2 + j>j b yjj ν ) 2. Under assumtions[a3] and [A4], we rove that: EA n1y ) 1 gt, y)dt 2 = On τ 2β+ν) 1 2β+α ). A.3) Using [A2], we deduce that [ ) ] 2 nvar{a n1y )} const.e Kh 2 y y Y ) T y + f P Y W )t, y, W )dt. 1 12
Then we rove, from an exansion of T + y f P Y W and 1 [ 1,] in their generalized Fourier series, that 1 var{a n1y )}d const. nh y jklq j d jk d l c jq c lq λ j + a)λ l + a) 1 λ j j 2γ const. nh y λ j + a) 2 Using the same series decomosition as reviously, we rove that E A n1y EA n1y 2 = var{a n1y )}d = O nh y ) 1 a α+1 2γ)/α) ) 2β+ν) 1 τ = O n 2β+α Result A.1) is imlied by this bound and A.3). References Amemiya, T. 1974): Multivariate regression and simltaneous equation models when the deendent variables are truncated normal, Econometrica, 42, 999 1012. Blundell, R., X. Chen, and D. Kristensen 2007): Semi-Nonarametric IV Estimation of Shae-Invariant Engel Curves, Econometrica,. 1613 1669. Blundell, R., and J. Horowitz????): A Nonarametric Test of Exogeneity, Rev. Econ. Stud., 74, 1035 1058. Blundell, R., J. Horowitz, and M. Parey 2008): Measuring the rice resonsiveness of Gasoline demand., Discussion Paer. Darolles, S., J.-P. Florens, and E. Renault 2002): Nonarametric Instrumental Regression, Working Paer # 228, IDEI, Université de Toulouse I. Gagliardini, P., and O. Scaillet 2007): A Secification Test for Nonarametric Instrumental Variable Regression, Swiss Finance Institute Research Paer No. 07-13. Hall, P., and J. L. Horowitz 2005): Nonarametric methods for inference in the resence of instrumental variables, Ann. Statist., 33, 2904 2929. Hansen, L. 1982): Large samle roerties of generalized method of moment estimators, Econometrica, 50, 1029 1054. Hausman, J., and W. Newey 1995): Nonarametric estimation of exact consumer surlus and deadweight loss, Econometrica, 636), 1445 1476. 13
Johannes, J., S. Van Bellegem, and A. Vanhems 2007): Projection estimation in nonarametric instrumental regression, Discussion Paer, Institut de statistique, Université catholique de Louvain. Newey, W. K., and J. L. Powell 2003): Instrumental variable estimation of nonarametric models, Econometrica, 71, 1565 1578. Vanhems, A. 2006): Nonarametric study of solutions of differential equations, Econometric Theory, 221), 127 157. Varian, H. 1992): Microeconomic Analysis. W.W. Norton, New York. 14