Lectures on Identi cation 2

Transcription

1 Lectures on Identi cation 2 Andrew Chesher CeMMAP & UCL April 16th 2008 Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

2 Topics 1 Monday April 14th. Motivation, history, de nitions, types of model, parametric and nonparametric IV models. 2 Today. 1 Quantiles and a non-additive IV model. 2 Triangular models and control function methods. 3 Models with excess heterogeneity and index restrictions. 3 Monday April 21st. Discrete endogenous variables, set identi cation and control function methods. Set identi cation in IV models for binary data. 4 Wednesday April 23rd. Seminar on Endogeneity and Discrete Outcomes - set identi cation in IV models for discrete data. Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

3 Quantiles 1 Random variable A with distribution function: has quantile function When A is continuously distributed F A (a) Pr[A a] Q A (p) inffa : F A (a) pg p = F A (Q A (p)) a = Q A (F A (a)) but not when A is discrete. For example: binary A has Pr[A = 0] = p 0 2 (0, 1) p0, a < 1 0, p p0 F A (a) = Q 1, a = 1 A (p) = 1, p > p 0 So e.g. if p 0 = 0.5, Generally F A (Q A (0.3)) = 0.5 F A (Q A (0.7)) = 1 F A (Q A (p)) p Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

4 Quantiles 2 Random variable A has distribution and quantile function: F A (a) Pr[A a] Q A (p) inffa : F A (a) pg De nition: U 2 [0, 1] has a uniform distribution, Unif (0, 1) if F U (u) = u then Q U (p) = p For continuous A, B = F A (A) Unif (0, 1) This because Pr[A α] F A (α) fa : a αg = fa : F A (a) F A (α)g Pr[F A (A) F A (α)] = F A (α) So Pr[B b] = b Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

5 Quantiles 3 Random variable A has distribution and quantile function: F A (a) Pr[A a] Q A (p) inffa : F A (a) pg De nition: U 2 [0, 1] has a uniform distribution, Unif (0, 1) if F U (u) = u then Q U (p) = p For continuous A, For continuous A it follows from B = F A (A) Unif (0, 1). F A (A) = U ) Q A (U) = A For discrete and continuous A, Q A (U) has the same distribution as A. Conditioning on X = x Q A (U) = A Q AjX (Ujx) has the distribution A has when X = x Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

6 Normalization Continuous Y, scalar continuous U, h strictly monotonic (increasing): Y = h(x, U) Can normalize U Unif (0, 1) under monotonicity - free choice of units of measurement. Consider for continuous W Y = h (X, W ) = h (X, Q W (F W (W ))) = h(x, U) where and U F W (W ) h(x, ) h (X, Q W ()) Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

7 The IV model nonadditive U Chernozhukov and Hansen (2005), Chernozhukov, Imbens, Newey (2007): Y = h(x, U) U k Z h strictly increasing in scalar U Unif (0, 1). De ne a(τ, x, z) Pr[U τjx = x, Z = z] Independence U k Z requires for all τ, z. E X jz [a(τ, X, z)jz = z] = Pr[U τjz = z] = τ h strictly increasing with respect to U implies for each x (and z) fu : u τg = fu : h(x, u) h(x, τ)g So Pr[U τjx = x, Z = z] = Pr[h(X, U) h(x, τ)jx = x, Z = z] a(τ, x, z) = Pr[Y h(x, τ)jx = x, Z = z] So for all τ, z Pr[Y h(x, τ)jz = z] = τ Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

8 Identification of non-additive structural functions 5 Figure 1: Contours of a joint density function of Y 1 and Y 2 conditional on X at 3 values of X. The line marked hh is the structural function h(y 2 ; X; q ), not varying across X 2 fx 1 ; x 2 ; x 3 g, drawn with q = 0:5. h X = x 3 Y 1 X = x 2 X = x 1 h Y 2

9 The IV model nonadditive U h monotonic (normalized increasing) in scalar U. Y = h(x, U) U k Z For all τ, z Pr[Y h(x, τ)jz = z] = τ identi cation of h if restrictions on h and F UXZ guarantee a unique solution. rich support for Z ; all or nothing identi cation. set identi cation is a possibility. local independence: could just have Q U jz (τjz) = τ then identify h(x, τ) Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

10 Example: triangular Gaussian model Suppose Y and X are generated by a triangular Gaussian model: U V Y = α 0 + α 1 X + U X = g(z ) + V 0 0 jz = z N σuu σ, UV σ UV σ VV Note that it could be σ UV (z) and still U k Z. The distribution of (Y a 0 a 1 X ) given Z = z is: N (α 0 a 0 ) + (α 1 a 1 ) g(z), σ UU + 2(α 1 a 1 )σ UV + (α 1 a 1 ) 2 σ VV With τ = 0.5, we seek a 0 and a 1 such that P[Y a 0 + a 1 X jz] = 0.5 for all z where! (α P[Y a 0 + a 1 X jz] = Φ 0 a 0 ) (α 1 a 1 ) g(z) (σ UU + 2(α 1 a 1 )σ UV + (α 1 a 1 ) 2 σ VV ) 1/2 Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

11 Triangular models Consider complete models for outcomes X and Y with triangular structure A: Y = h(x, U) X = g(z, V ) and incomplete models B: Y = h(x, U) Consider the impact of discrete vs continuous variation in (Y, X, Z ) on the nature of identi cation. We nd: Y discrete Y continuous X discrete A:set B: set A: set B: point X continuous A: point B: set A:point B: point Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

12 Triangular Gaussian model Y and X are generated by a triangular Gaussian model: Y = α 0 + α 1 X + U X = g(z ) + V 0 σuu, 0 U σ N uv V σ uv σ vv Conditional distribution of U given V and Z σuv N V, σ uu σ vv The expected value of Y given V and Z. σ 2 uv σ vv U k Z E [Y jv = v, Z = z] = α 0 + α 1 (g(z) + v) + σ uv σ vv v Conditioning on V = v, Z = z is the same as conditioning on X = x g(z) + v, Z = z so E [Y jx = x, Z = z] = α 0 + α 1 x + σ uv σ vv (x g(z)) Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

13 Additive latent variable model X and Y are generated by Y = h(x ) + U X = g(z ) + V E [UjV = v, Z = z] = c(v) Does not imply E [UjZ = z] is free of z. distinct from the IV model If c() is linear then E [UjZ = z] is free of z. If V k Z then E [UjZ = z] is free of z. E [V jz = z] = d Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

14 Additive latent variable model X and Y are generated by Y = h(x ) + U X = g(z ) + V E [UjV = v, Z = z] = c(v) Condition on V = v, Z = z. E [Y jv = v, Z = z] = h(g(z) + v) + c(v) Conditioning on V = v, Z = z is the same as conditioning on X = x g(z) + v, Z = z so E [Y jx = x, Z = z] = h(x) + c(x g(z)) If e.g. E [V jz = z] = d then E [X jz = z] = g(z) + d and g(z) is identi ed up to an additive constant. Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

15 Additive latent variable model: partial di erences X and Y are generated by Y = h(x ) + U X = g(z ) + V E [UjV = v, Z = z] = c(v) Condition on X = x, Z = z. E [Y jx = x, Z = z] = h(x) + c(x Consider two values of Z: z 0 and z 00 and let Then x 0 g(z 0 ) + d = E [X jz = z 0 ] E [V jz = z] = d x 00 g(z 00 ) + d = E [X jz = z 00 ] g(z)) E [Y jx = x 0, Z = z 0 ] = h(x 0 ) + c(d) E [Y jx = x 00, Z = z 00 ] = h(x 00 ) + c(d) E [Y jx = x 0, Z = z 0 ] E [Y jx = x 00, Z = z 00 ] = h(x 0 ) x(x 00 ) Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

16 Additive latent variable model: partial derivatives X and Y are generated by Y = h(x ) + U X = g(z ) + V E [UjV = v, Z = z] = c(v) Condition on X = x, Z = z. E [Y jx = x, Z = z] = h(x) + c(x E [X jz = z] = g(z) + d Consider partial derivatives with respect to x and z E [V jz = z] = d r x E [Y jx = x, Z = z] = r x h(x) + c 0 (x g(z)) g(z)) r z E [Y jx = x, Z = z] = r z h(x) c 0 (x g(z))r z (g(z) r z E [X jz = z] = r z g(z) Since r z h(x) = 0 (exclusion restriction), if r z g(z) 6= 0 r x E [Y jx = x, Z = z] + r z E [Y jx = x, Z = z] r z E [X jz = z] = r x h(x) Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

17 Non-additive latent variable model Structural functions: X and Y are generated by Y = h(x, U) X = g(z, V ) h weakly monotonic (increasing) in U - allows discrete Y g strictly monotonic (increasing) in V - requires continuous X. Unobservables: (U, V ) independent of Z. Can normalize V Unif (0, 1) - then g(z, v) = Q X jz (vjz) Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

18 Non-additive latent variable model Structural functions: X and Y are generated by Y = h(x, U) X = g(z, V ) (U, V ) k Z h weakly monotonic (increasing) in U - g strictly monotonic (increasing) in V. Express Y in terms of V and Z Y = h(g(z, V ), U) Q Y jvz (τjv, z) = h(g(z, v), Q U jv (τjv)) Since (V = v \ Z = z), ( X = x g(z, v) \ Z = z): Q Y jxz (τjx, z) = h(x, Q U jv (τjv)) x g(z, v) = Q X jz (vjz) so Q Y jxz (τjq X jz (vjz), z) = h(x, Q U jv (τjv)) Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

19 Non-additive latent variable model: partial di erences Structural functions: X and Y are generated by Y = h(x, U) X = g(z, V ) (U, V ) k Z h weakly monotonic (increasing) in U - g strictly monotonic (increasing) in V. Q Y jxz (τjq X jz (vjz), z) = h(x, Q U jv (τjv)) Consider Z 2 fz 0, z 00 g de ne x 0 Q X jz (vjz 0 ) x 00 Q X jz (vjz 00 ) Then is identi ed by h(x 0, Q U jv (τjv)) h(x 00, Q U jv (τjv)) Q Y jxz (τjq X jz (vjz 0 ), z 0 ) Q Y jxz (τjq X jz (vjz 00 ), z 00 ) Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

20 Angrist-Krueger QJE (1991) cohort: W : log wage, S: years of schooling, B: quarter of birth W = h(s, U) S = g(b, V ) Quantiles of years of schooling (S) by quarter of birth Q S jb (vjb) b 2 f1, 2, 3, 4g v = b = b = b = b = Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

21 Angrist-Krueger QJE (1991) Estimated returns to schooling for median earner Q W jsb (.5jQ S jb (vjb 0 ), b 0 ) Q W jsb (.5jQ S jb (vjb 00 ), b 00 ) Q S jb (vjb 0 ) Q S jb (vjb 00 ) v = b 0 = 1 b 00 = (.121).012 (.090).508 (.393).210 (.218).176 (.228) b 0 = 1 b 00 = (.030).045 (.014).048 (.095).064 (.106).060 (.111) b 0 = b 00. = 4 (.022) (.011) (.079) (.083) (.089) Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

22 Non-additive latent variable model: partial derivatives Structural functions: X and Y are generated by Y = h(x, U) X = g(z, V ) (U, V ) k Z Q Y jxz (τjq X jz (vjz), z) = h(x, Q U jv (τjv)) where x g(z, v) Di erentiate left and right with respect to z r x Q Y jxz (τjx, z)j x =QX jz (v jz )r z Q X jz (vjz) + r z Q Y jxz (τjx, z)j x =QX jz (v jz ) r x h(x, Q U jv (τjv))j x =g (z,v ) r z g(z, v) Since g(z, v) and r z g(z, v) are identi ed by Q X jz (vjz) and r z Q X jz (vjz) r x Q Y jxz (τjx, z)j x =QX jz (v jz ) + r z Q Y jxz (τjx, z)j x =QX jz (v jz ) r z Q X jz (vjz) = r x h(x, Q U jv (τjv)) Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

23 Non-additive latent variable model: remarks Structural functions: X and Y are generated by Y = h(x, U) X = g(z, V ) (U, V ) k Z h weakly monotonic (increasing) in U - g strictly monotonic (increasing) in V. could proceed with U k Z jv which implies Q U jvz (τjv, z) is free of z. but must be able to identify the function of X and Z that delivers V. could have local independence, e.g. Q U jvz (0.5jv, z) is free of z. Chesher (2003), Ma and Koenker (2006), Lee (2007), Imbens and Newey (2003). restrictions on number of latent variates. Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

24 Non-additive latent variable model: excess heterogeneity Scalar outcome W is determined by Y = h(θ(x, Z 1 ), U) where U is a latent vector random variable, θ(x, Z 1 ) is a scalar valued function of endogenous scalar X and vector Z 1. The continuous endogenous X is determined by X = g(z, V ) where h is strictly increasing in scalar continuous latent variate V and Z = (Z 1, Z 2 ). There is the covariation restriction: (U, V ) independent of Z. Study identi cation and estimation of index relative sensitivity: θ(x, z 1 )/ x θ(x, z 1 )/ z 1j. Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

25 Non-additive latent variable model: excess heterogeneity Outcome Y is determined by Y = h(θ(x, Z 1 ), U) Continuous endogenous Y given by X = g(z, V ) V = r(x, Z ) where r is the inverse function such that X = g(z, r(x, Z )). (U, V ) k Z. Normalize V Unif (0, 1). Then r(x, z) = F X jz (xjz). The conditional distribution function F Y jxz (yjx, z) P[Y yjx = x, Z = z]: Z Z F Y jxz (yjx, z) = df U jv (ujr(x, z)) s(θ(x, z 1 ), r(x, z), y) h(θ(x,z 1 ),u)y Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

26 Non-additive latent variable model: excess heterogeneity F Y jxz (yjx, z) = Z Z h(θ(x,z 1 ),u)y df U jv (ujr(x, z)) s(θ(x, z 1 ), r(x, z), y) To study identi cation of r x θ/r z1 θ consider derivatives of F Y jxz. r x F Y jxz = r θ s r x θ + r r s r x r r z1 F Y jxz = r θ s r z1 θ + r r s r z1 r r z2 F Y jxz = r r s r z2 r r x F X jz = r x r r z1 F X jz = r z1 r r z2 F X jz = r z2 r There is identi cation of r x θ/r z1 θ if r z2 F X jz 6= 0 because r θ s r x θ = r x F Y jxz r x F X jz r z 2 F Y jxz r z2 F X jz r θ s r z1 θ = r z1 F Y jxz r z1 F X jz r z 2 F Y jxz r z2 F X jz Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28

27 Non-additive latent variable model: excess heterogeneity With continuous Y r x F X jz (xjz) = 1 r v Q X jz (vjz) r zi F X jz (xjz) = where v F X jz (xjz) There is identi cation of r x θ/r z1 θ if r z2 F X jz 6= 0 because r θ s r x θ = r x F Y jxz r x F X jz r z 2 F Y jxz r z2 F X jz r θ s r z1 θ = r z1 F Y jxz r z1 F X jz r z 2 F Y jxz r z2 F X jz In terms of quantile derivatives: r θ s r x θ = r θ s r z1 θ = 1 r τ Q Y jxz r x Q Y jxz + r z 2 Q Y jxz r z2 Q X jz r z i Q X jz (vjz) r v Q X jz (vjz) 1 r τ Q Y jxz r z1 Q Y jxz r z1 Q X jz r z2 Q Y jxz r z2 Q X jz Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28!!

28 Types of restrictions The triangular model results require X continuous A: Y = h(x, U) X = g(z, V ) The IV model results require Y continuous B: Y = h(x, U) What is the nature of identi cation when these conditions fail? We nd: Y discrete Y continuous X discrete A:set B: set A: set B: point X continuous A: point B: set A:point B: point Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/ / 28