Instrumental Variables Estimation and Other Inverse Problems in Econometrics. February Jean-Pierre Florens (TSE)

Transcription

1 Instrumental Variables Estimation and Other Inverse Problems in Econometrics February 2011 Jean-Pierre Florens (TSE)

2 2 I - Introduction Econometric model: Relation between Y, Z and U Y, Z observable random elements U unobservable random noise The type of relations: - linear Y = Z, ϕ + U - non linear Y = ϕ(z) + U - non separable Y = ϕ(z, U) ϕ(z,.) bijective and distribution of U given. Assumption required on relation between U and Z.

3 3 Regression type models: - linear: E(UZ) = 0 - non linear: E(U Z) = 0 - non separable: U Z Exogeneity assumption

4 4 This hypothesis is not relevant in many cases because its does not characterized the parameter of interest. Two examples: i) Market model: demand function: D = a + bp + cw 1 + U (P price, W 1 some variable (revenue)) supply function : S = α + βp + γw 2 + V (W 2 some cost variable) equilibrium: We observe Q = S = D, P, W 1, W 2 The regression of Q on P, W 1, W 2 does not estimate neither the demand or the supply parameters. P is "endogenous". ii) Endogenous selection ζ is the level of a treatment Y = ϕ(ζ) + U outcome equation ζ not randomly selected. Z selection of ζ not independent of U Y = ϕ(z) + U but E(U Z) 0.

5 5 An example. Y = ϕ(z) + U Z = m(w ) + v U V bivariate normal distribution U and V non independent.

6 6 Graph 1

7 7 How to treat the endogeneity problem? Instrumental variables approach Ragnar Frisch (34) Confluence analysis Olaf Reiersol (41, 45) New variables W Independence relation between U and Z is replaced by an independence relation between U and W. Three models Y = Z, ϕ + U E(UW ) = 0 Y = ϕ(z) + U E(U W ) = 0 Y = ϕ(z, U) U W Other approach for endogeneity. The control function approach the example of model II. E(Y Z, W ) = ϕ(z) + E(U Z, W ) }{{} assumption h(v )

8 8 V function of Z, W "separated" form Z such that we may identified the additive model E(Y Z, W ) = ϕ(z) + h(v ) Ex : Z = m(w ) + V Separated : support condition such that ϕ(z) V = 0 A function of Z is almost surely equal to a function of V is almost surely constant.

9 9 Structure of the analysis of I.V. models. i) The independence relation between W and U generates a functional equation linking ϕ and F the distribution of (Y, Z, W ). ii) The existence of a solution is an assumption of the model (but may be tested in some cases) The unicity or local unicity (identification) need a dependence condition between Z and W. iii) The equation is usually ill-posed: then the plug-in method (replacing F by an estimator) does not lead to a consistent estimator. We need to introduce a regularization in the resolution. iv) The main characteristic of econometrics inverse problems is that the operator is not known and is estimated using the same data as the observed signal. The rate of convergence then depend as usual on the degree of ill-posedness (which depends on the dependence between Z and W and on the regularity of the function but also on the possibility of ignoring the estimation error of the operator. v) Usual problems of selection of the regularization parameters.

10 10 Other inverse problems in econometrics. - Some extensions of the I.V. models transformation models panel data semi parametric models - The game theoretic models - Dynamic models

11 11 Who? (in Econometrics) - R. Blundell, X.Chen, V. Chernozhukov, D. Christensen, P. Galardini, E. Gauthier, P. Hall, J. Horowitz, Y. Kitamura, W. Newey, D. Pouzo, J. Powell, O. Scaillet... - "Toulouse connection" M. Carrasco, S. Darolle, F. Fève, J. Johannes, E. Renault, G. Simon, A. Simoni, S. Sokullu, A. Vanhems.

12 12 II - Instrumental variables and integral equations Model 1: Y = Z, ϕ + U E(UW ) = 0 E( Z, ϕ W ) = E(Y W ) Cϕ = r C : covariance operator between Z and W (E(Y ) = 0 E(Z) = 0 for simplicity) Ex: Z, W L 2 [0,1] (uniform measure) Cϕ = 1 0 c(t, s)ϕ(s)ds c(t, s) = E(Z(s)W (t)) Hilbert Schmidt operator under minor assumption on c. Remark: Even if Z and W are equal (linear regression with functional covariates) the inverse problem remains.

13 13 Model 2: Y = ϕ(z) + U E(U Z) = 0 E(ϕ(Z) W ) = E(Y W ) ϕ(z)f(z w)dz = yf(y w)dy Kϕ = r K = Conditional expectation operator from L 2 Z into L2 W Hilbert Schmidt also under minor assumption on f(z, w) ( ) 2 f(z, w) f(z)f(w)dzdw < f(z)f(w)

14 14 Model 3: U W P rob(u u, Z = z W = w)dz = P rob(u u) = F 0 (u) F 0 is given. P rob(ϕ(z, U) ϕ(z, u), Z = z W = w)dz = F 0 (u) F (ϕ(z, u), z w)dz = F 0 (u) where F (y, z w) = P rob(y y, Z = z W = w) Non linear integral equation. T (ϕ) = F 0 ϕ(., u) L 2 z u

15 15 Identification: Unicity of the solution In models 1 and 2 : identification C or K one to one. Comments on the conditional expectation operator: K one to one: "strong identification" or "completness". Z W jointly normal K one to one Rank [Cov(Z, W )] = Rank [V ar(z)] SVD decomposition of K: (λ j, ϕ j, ψ j ) j=1,... ϕ j L 2 Z ψ j L 2 W λ 1 = 1 λ λ 2 j: eigen values of K K K one to one λ j 0 Link with the density : j. f(z, w) f(z)f(w) = 1 + j=2 λ j ϕ j (z)ψ j (w) K one to one if all the λ j corresponding to ϕ j 0 are non zero.

16 16 In model 3: local unicity of the solution: Frechet derivative of the operator: T ϕ ( ϕ) = ϕ(z, u)f(ϕ(z, u), z w)dz where f(y, z w) density of Y, Z given W. T ϕ one to one if Z is strongly identified by W given U (E(a(Z, U) W, U) = 0 a.s. a = 0 a.s.)

17 17 III - Estimation Linear system: Cϕ = r or Kϕ = r Computation of a regularized solition. e.g. Tikhonov. ϕ α = (αi + K K) 1 K r ϕ α = arg min( Kϕ r 2 + α ϕ 2 ) Estimation of K and r by parametric or non parametric methods. Ex : non linear separable model : K : L 2 Z L2 W (w.r.t. true densities) Kϕ = E(ϕ(Z) W ) K ψ = E(ψ(W ) Z)

18 18 We have to solve: αϕ(z) + = ϕ(ζ)dζ ydy ˆf(ζ w) ˆf(w z)dw ˆf(y w) ˆf(w z)dw ˆf: usual non parametric estimator. This equation reduces to a system of linear equations with unknown elements. Ex of linear models: Ĉϕ = 1 n z i, ϕ w i Ĉ = 1 n i w i, ϕ z i i ˆr = 1 n yi w i

19 19 - Many extensions: - Other regularizations: Landweber Fredman iterations Iterated Tikhonov Spectral cut off... - Other penalizations: Sobolev norm Hilbert scale norms - Bayesian methods - Non linear systems : Difficult to implement numerically. ex : Non linear Tiknonov: min T (ϕ) r 2 + α ϕ 2 F OC αϕ + T ϕ (T (ϕ)) = T ϕ r and T, T ϕ and r estimated.

20 20 Better approach: iterated methods. Ex : - Choice of regularization parameters. min 1 α ˆK ˆϕ α ˆr 2 or 1 α 2 ˆK ˆK ˆϕ α ˆK ˆr 2 ˆϕ α : Iterated Tiknonov estimation. Bayesian approach: α treated as an hyper parameter.

21 21 - Some pictures: Graph 1

22 22 - Some pictures: Graph 2

23 23 Graph 3

24 24 Graph 4

25 25 Graph 5

26 26 Graph 6

27 27 IV - Some questions about the rate of convergence of the estimators 1) simple problem: linear model Regularization by ordinary Tikhonov method. 2) ˆϕ α = (αi + ˆK ˆK) 1 ˆK ˆr ˆϕ α ϕ = (αi + ˆK ˆK) 1 ( ˆK ˆr ˆK ˆKϕ) I [ + (αi + ˆK ˆK) ] 1 ˆK ˆK (αi + K K) 1 K K ϕ II + (αi + K K) 1 K Kϕ ϕ III

28 28 3) Usual regularization bias: ϕ α ϕ 2 0 if α 0 III = α(αi + K K) 1 ϕ = ϕ α ϕ Regularity condition under ϕ implies ϕ α ϕ 2 O (function (α)) e.g. α β Ex : ϕ R(K K) β 2 (β 2) ϕ, ϕ j 2 < j=1 λ 2β j

29 29 Usual approach: L a differential operator Kϕ L a ϕ a = degree if ill-posedness. ϕ D(L b ϕ) β = b. a

30 30 - I Usual "variance term" (αi + ˆK ˆK) 1 ˆK (ˆr ˆKϕ) 2 min { 1 α ˆr ˆKϕ 2, 1 } α ˆK (ˆr ˆKϕ) 2 2 ˆr ˆKϕ 2 O ( ) 1 nh + q h2ρ ˆK (ˆr ˆKϕ 2 O ( ) 1 n + h2ρ dimw = q For ex : ρ regularity of E(y w) ( ) 1 I O αnh + q h2ρ ( ) 1 2ρ O 2ρ+q h"optimal" α n Question : link between ρ and β? Under some regularity assumptions ρ = a + b E(Y W ) = E(ϕ(Z) W )

31 31 -II Estimation of the operator: II = α [(αi + ˆK ] ˆK) 1 (αi + K K) 1 ϕ = α(αi + ˆK ˆK) 1 (K K ˆK ˆK)(αIK K)ϕ II = (αi + ˆK 1 ˆK) ˆK ( ˆK K)[α(αI + K K) 1 ϕ] + (αi + ˆK ˆK) 1 ( ˆK K )[αk(αi + K K) 1 ϕ] II 2 1 α ˆK K 2 α β α ˆK ˆK 2 α (β+1) 2 2 ˆK K 2 ˆK K 2 O ( ) 1 + h2ρ nhp+q dimz = p 0 faster then the other terms under "good" conditions.

32 32 - Notion of weak or strong instruments: very weak instruments: we cannot neglect the estimation of K. else: trade off between 1 2ρ n 2ρ+q α and α β. optimal α : ] [n 2ρ 1 β+1 2ρ+q rate of convergence: estimate ϕ. ] [n 2ρ β β+1 2ρ+q If ρ = a + b rate n 2b 2(a+b)ρ+q β measure the strength of the instruments to

33 33 V - Extensions of Instrumental Variables Estimation and Other inverse problems in Econometrics Direct extensions of I.V. regression. Y = ϕ(z) + U E(U W ) = 0 ϕ additive form ϕ 1 (Z 1 ) + ϕ 2 (Z z ) Semi parametric ϕ 1 (Z 1 ) + Z 2β Single index ϕ(z) = ϕ 1 (Z β) Transformation model ϕ(y ) = Z + X β + U E(U W ) = 0 W = Z, X more general ϕ(y ) = ψ(z) + X β + U E(U W ) = 0

34 34 Local I.V. models: E(Y W ) W E(ψ(Z) W ) = E(Z W ) (W W E(λ(Z) W ) = E( E(Y Z, W ) W ) W E(Z W ) W IR, Z IR). Natural question: ϕ Z = ψ or ϕ Z = λ? More general problem: Two inverse problems Kϕ = r Lψ = S Test of hypothesis Aϕ = ψ A given operator.

35 Expenditure share, food, with kids data φ0 φ1 ψ1 ψ2 E(Y Z) 0.4 share log total expenditure

36 Expenditure share, catering, kids data φ0 φ1 ψ1 ψ2 E(Y Z) 0.25 share log total expenditure

37 Expenditure share, food, with kids data φ0 φ1 ψ1 ψ2 E(Y Z) 0.4 share log total expenditure

38 38 - Dynamic I.V. models: additive form problem: Y t stochastic process Z t, W t two filtrations. Y t : Λ t + M t Λ t : Predictable w.r.t. Z t M t : Martingale type process E(M t M s W s ) = 0 leads to h t = E(λ t W t ) h t stochastic intensity of Y t w.r.t. W λ t structural intensity (derivative of Λ t ).

39 39 Change of time models Hyp: ϕ t increasing sequence of stopping time adapted to Z t such that Y ϕt = U t U t : know process independent of W t. leads to a sequence of non linear integral equations: dz ϕt 0 h(s Z s, W s )g s (z W s )ds = H u (t) h(s Z s, W s ): stochastic intensity of Y t given Z t and W t g s (z W s )ds: density on Z s given W s H u (t): compensator of U t w.r.t. its own history.

40 40 - Game theoretic models. Ex: auction models (first price, private values). N + 1 bidders. (ξ i ) i=1,...,n+1 private values iid F ξ i [0, 1] x i = bid for bidder i. Nash equilibrium: x i = ξ i c.d.f. of X = G ξi 0 F N (u)du F N (ξ i ) = σ F (ξ i ) }{{} F (.)increasing. strategy function (given). F Goσ F = 0 Non linear inverse problem.

41 41 Frechet derivative is usually an integral operator. In the auction case : T F ( F F (u) )(u) = ξ F 0 N (u)du many other game models of the same kind. ξ 0 F N 1 (u)du

42 42 Conclusion - Need for applications - Asymptotic normality of the estimators - Many questions: selection of the regularization parameters numerical methods for non linear problems resolution under constraints bootstrap tests