Econ 2148, fall 2017 Instrumental variables II, continuous treatment

Transcription

1 Econ 2148, fall 2017 Instrumental variables II, continuous treatment Maximilian Kasy Department of Economics, Harvard University 1 / 35

2 Recall instrumental variables part I Origins of instrumental variables: Systems of linear structural equations Strong restriction: Constant causal effects. Modern perspective: Potential outcomes, allow for heterogeneity of causal effects Binary case: 1. Keep IV estimand, reinterpret it in more general setting: Local Average Treatment Effect (LATE) 2. Keep object of interest average treatment effect (ATE): Partial identification (Bounds) 2 / 35

3 Agenda instrumental variables part II Continuous treatment case: 1. Restricting heterogeneity in the structural equation: Nonparametric IV (conditional moment equalities) 2. Restricting heterogeneity in the first stage: Control functions 3. Linear IV: Continuous version of LATE 3 / 35

4 Takeaways for this part of class We can write linear IV in three numerically equivalent ways: 1. As ratio Cov(Z,Y )/Cov(Z,X). 2. As regression of Y on first stage predicted values X. 3. As regression of Y on X controlling for the first stage residual V. The literature on IV identification with continuous treatment generalizes these ideas to non-linear settings. 4 / 35

5 Takeaways continued 1. Moment restrictions: Assume one-dimensional additive heterogeneity in structural equation of interest nonparametric regression of Y on non-parametric prediction X. 2. Control functions: Assume one-dimensional heterogeneity in first stage relationship. X is independent of structural heterogeneity conditional on V = F X Z (X Z). 3. Continuous LATE: No restrictions on heterogeneity. Interpret linear IV coefficient as weighted average derivative. 5 / 35

6 Alternative ways of writing the linear IV estimand Linear triangular system: Y = β 0 + β 1 X + U X = γ 0 + γ 1 Z + V Exogeneity (randomization) conditions: Relevance condition: Under these conditions, Cov(Z,U) = 0, Cov(Z,V) = 0. Cov(Z,X) = γ 1 Var(Z) 0. β 1 = Cov(Z,Y ) Cov(Z,X). 6 / 35

7 Moment conditions Write Cov(Z,U) = 0 as Cov(Z,Y β 0 β 1 X) = 0 Let X be the predicted value from a first stage regression, Multiply Cov(Z,U) by γ 1, X = γ 0 + γ 1 Z. Cov( X,Y β0 β 1 X) = 0, and note Cov( X,X) = Var( X), to get two-stage least squares! β 1 = Cov( X,Y ) Var( X). 7 / 35

8 Conditional moment equalities Under the stronger mean independence restriction E[U Z] 0, for all z. 0 = E[(Y β 0 β 1 X) Z = z] Conditional moment equality Suggest 2 stage estimator: = E[Y Z = z] β 0 β 1 E[X Z = z] 1. Regress both Y and X (non-parametrically or linearly) on Z. 2. Then regress E[Y Z = z] or Y (linearly) on E[X Z = z]. two-stage least squares! 8 / 35

9 Control function perspective V is the residual of a first stage regression of X on Z. Consider a regression of Y on X and V, Y = δ 0 + δ 1 X + δ 2 V + W Partial regression formula: δ1 is the coefficient of a regression of Ỹ on X (or of Y on X), where Ỹ, X are the residuals of regressions on V. By construction: X = γ 0 + γ 1 Z = X Ỹ = β 0 + β 1 X + Ũ Cov(Z,U) = Cov(Z,V) = 0 implies Cov( X,Ũ) = 0, and thus δ 1 = β 1. 9 / 35

10 Recap Three numerically equivalent estimands: 1. The slope Cov(Z,Y )/Cov(Z,X). 2. The two-stage least squares slope from the regression Y = β 0 + β 1 X + Ũ, where Ũ = (β 1 V + U), and X is the first stage predicted value X = γ 0 + γ 1 Z. 3. The slope of the regression with control Y = δ 0 + δ 1 X + δ 2 V + W, where the control function V is given by the first stage residual, V = X γ 0 γ 1 Z. 10 / 35

11 Roadmap Nonparametric IV estimators generalize these approaches in different ways, dropping the linearity assumptions: 1. If heterogeneity in the structural equation is one-dimensional: conditional moment equalities 2. If heterogeneity in the first stage is one-dimensional: control functions 3. Without heterogeneity restrictions: continuous versions of the LATE result for the linear IV estimand Objects of interest: Average structural function (ASF) ḡ(x) = E[g(x, U)]. Quantile structural function (QSF) g τ (x) defined by P(g(x,U) < g τ (x)) = τ. Weighted averages of marginal causal effect, E[ω x g (x,u)]dx for weights ω x. 11 / 35

12 Moment restrictions Approach I: Conditional moment restrictions (nonparametric IV) Consider the following generalization of the linear model: Here the ASF ḡ equals g. Practice problem Y = g(x) + U X = h(z,v) Z (U,V) Under these assumptions, write out the conditional expectation E[Y Z = z] as an integral with respect to dp(x Z = z). Consider the special case where both X and Z have finite support of size n x and n z, and rewrite the integral as a matrix multiplication. 12 / 35

13 Moment restrictions Solution Using additivity of structural equation, and independence, k(z) = E[Y Z = z] = E[g(X) Z = z] + E[U Z = z] = E[g(X) Z = z] = g(x)dp(x = x Z = z). In the finite support case, let k = (k(z 1 ),...,k(z nz )), g = (g(x 1 ),...,g(x nx )), and let P be the n z n x matrix with entries P(X = x Z = z). Then the integral equation can be written as k = P g. 13 / 35

14 Moment restrictions Completeness The function k(z) = E[Y Z = z] and the conditional distribution P X Z are identified. In the finite-support case, the equation k = P g implies that g is identified if the matrix P has full column rank n x. The analogue of the full rank condition for the continuous case (integral equation) is called completeness. Completeness requires that variation in Z induces enough variation in X, like the instrument relevance condition in the linear case. Completeness is a feature of the observable distribution P X Z, in contrast to the conditions of exogeneity / exclusion, or restrictions on heterogeneity. 14 / 35

15 Moment restrictions Ill posed inverse problem Even if completeness holds, estimation in the continuous case is complicated by the ill posed inverse problem. Consider the discrete case. The vector g is identified from g = (P P) 1 P k Suppose that P P has eigenvalues close to zero. Then g is very sensitive to minor changes in P k. 15 / 35

16 Moment restrictions Continuous analog: notation k(z) = E[Y Z = z]f Z (z) (Pg)(z) = g(x)f X,Z (x,z)dx (P k)(x) = k(z)f X,Z (x,z)dz T = P P Thus the moment conditions can be rewritten as k = Pg or P k = T g, Therefore g = T 1 P k, if the inverse of T exists which is equivalent to completeness. 16 / 35

17 Moment restrictions T is a linear, self-adjoint ( symmetric) positive definite operator on L 2. Functional analysis: If f X,Z (x,z) 2 fxdz, then 0 is the unique accumulation point of the eigenvalues of T, and the eigenvectors form an orthonormal basis of L 2. Implication: g is not a continuous function of P k in L 2. Minor estimation errors for k can translate into arbitrarily large estimation errors for g. Takeaway: Estimation needs to use regularization, convergence rates are slow. 17 / 35

18 Moment restrictions Estimation using series Implementation is surprisingly simple. Use series approximation g(x) k j=1 β j φ j (x). Then we get and thus E[φ j (Z)Y ] k j=1 β j E[φ j (Z)φ j (X)] β (E[φ j (Z)φ j (X)]) 1 j,j (E[φ j (Z)Y ]) j. Sample analog: Two stage least squares, where the regressors φ j (X) are instrumented by the instruments φ j (Z). 18 / 35

19 Moment restrictions Additive one-dimensional hetereogeneity is crucial for conditional moment equality Consider the following non-additive example: Average structural function: Y = X 2 U X = Z + V ( ( )) (U,V) N 0, ḡ(x) = E[x 2 U] = 0. Conditional moment equality is solved by g(x) = x: E[Y g(x) Z = z] = E[(Z + V) 2 U Z = z] z = 2zE[VU] + E[V 2 U] z = / 35

20 Moment restrictions Non-additive heterogeneity Consider now the slightly more general model Y = g(x,u) X = h(z,v) Z (U,V) where dim(u) = 1 and g is strictly monotonic in U. We can assume w.l.o.g. U Uniform([0, 1]). Here the QSF g τ (x) equals g(x,τ). Practice problem Under these assumptions, show that the conditional probability P(Y g(x,τ) Z = z) equals τ. Propose an estimator for g(, τ). 20 / 35

21 Moment restrictions Solution Conditional probability: This implies P(Y g(x,τ) Z = z) = P(g(X,U) g(x,τ) Z = z) g(,τ) argmin g( ) = P(U τ Z = z) = P(U τ) = τ E [ (E[1(Y g(x)) Z] τ) 2]. This suggests a series minimum distance estimator: ĝ( ) = argmin g:g(x)= β j φ j (x) i ) 2 (Ê[1(Y g(x)) Z = Zi ] τ, with Ê given in turn by series regression. 21 / 35

22 Moment restrictions One-dimensional hetereogeneity is crucial for conditional quantile restriction Consider the following example where heterogeneity U is multidimensional: Y = U 1 X + U 2 X = Z + V (U 1,U 2,V) N(0,Σ) Without proof: In this case, for generic Σ, P(Y g τ (X) Z = z) τ, where g τ is the quantile structural function. 22 / 35

23 Control functions Approach II: Control functions Consider now the alternative model Y = g(x,u) X = h(z,v) Z (U,V) where dim(v) = 1 and h is strictly monotonic in V. We can assume w.l.o.g. V Uniform([0, 1]). 23 / 35

24 Control functions Practice problem Write V as a function of X and Z. Show that X U V. Derive an expression for E[Y X,V]. Write the average structural function (ASF) E[g(x, U)] in terms of observable distributions. Propose an estimator for the ASF. 24 / 35

25 Control functions Solution V as a function of X and Z : Let x = h(z,v). Then F X Z (x z) = P(h(Z,V) x Z = z) = P(h(z,V) h(z,v)) = P(V v) = v, and thus V = F X Z (X Z). Conditional independence: Write X U V as h(z,v) U V = v, which follows immediately from Z (U,V). 25 / 35

26 Control functions Solution continued Conditional expectation: E[Y X = x,v = v] = E[g(x,U) X = x,v = v] = E[g(x,U) V = v] Since V Uniform([0, 1]) by assumption, the law of iterated expectations gives E[g(x,U)] = E[E[g(x,U) V]] = 1 0 E[Y X = x,v = v]dv. 26 / 35

27 Control functions Possible estimator Estimate F X Z using kernel regression: F X Z (x z) = i for some kernel function K. Impute V i : Flexibly regress Y i on X i and Vi. K (Z i z)1(x i x) / K (Z i z) i V i = FX Z (X i Z i ). Integrate predicted values for x, v over uniform distribution for v. 27 / 35

28 Control functions One-dimensional hetereogeneity in the first stage is crucial for control function Consider the following example where heterogeneity V is multidimensional: Average structural function: Y = X + U X = V 1 Z + V 2 (U,V 1,V 2 ) N(µ,Σ) Control function Ṽ = F X Z (X Z). g(x) = E[x + U] = x. Conditional independence U X Ṽ is violated, since U Z Ṽ does not hold: Σ E[U Z,Ṽ] = µ U + Φ 1 V2,U + ZΣ Vq,U (Ṽ) ΣV2,V 2 + 2ZΣ V1,V 2 + Z 2 Σ V1,V 1 28 / 35

29 Continuous LATE Approach III: Continuous LATE Consider the model without restrictions on heterogeneity: Y = g(x,u) X = h(z,v) Z (U,V) Assume first that X R, Z {0,1}. Potential outcome notation: X z = h(z,v). Assume X 0 X 1 (for non-negative weights). 29 / 35

30 Continuous LATE LATE for binary instrument Linear IV slope: As in part I of class, Denominator: Numerator: β := Cov(Z,Y ) E[Y Z = 1] E[Y Z = 0] = Cov(Z,X) E[X Z = 1] E[X Z = 0]. E[X Z = 1] E[X Z = 0] = E[X 1 X 0 ]. E[Y Z = 1] E[Y Z = 0] = E[g(X 1,U) g(x 0,U)] [ X 1 ] = E g (x,u)dx X 0 = E[g (x,u)1(x 0 x X 1 )]dx 30 / 35

31 Continuous LATE Taking rations yields: β = E[g (x,u) ω]dx where ω = 1(X 0 x X 1 ) E[1(X 0 x X 1 )dx. Linear IV gives a weighted average of the slopes (marginal causal effects) g (x,u). 31 / 35

32 Continuous LATE General instrument Now drop restriction that Z {0,1}, but assume that X 0. Then Y = g(h(z,v),u) = g(0,u) + Thus [ Cov(Z,Y ) = E = 0 g (x,u)1(x h(z,v))dx. 0 (Z E[Z]) E[g (x,u) ϖ]dx g (x,u)1(x h(z,v))dx 0 where ϖ(x) = E[1(x h(z,v)) (Z E[Z]) V]. ] 32 / 35

33 Continuous LATE If h is increasing in Z, then ϖ 0. Taking ratios as before yields β = Cov(Z,Y ) Cov(Z,X) = E[g (x,u) ω]dx 0 where ω = ϖ(x) 0 E[ϖ(x)]dx. As before, linear IV is a weighted average of marginal causal effects g (x,u). 33 / 35

34 References References Nonparametric IV: Newey, W. K. and Powell, J. L. (2003). Instrumental Variable Estimation of Nonparametric Models. Econometrica, 71(5): Horowitz, J. L. (2011). Applied Nonparametric Instrumental Variables Estimation. Econometrica, 79(2): Chernozhukov, V., Imbens, G. W., and Newey, W. K. (2007). Instrumental variable estimation of nonseparable models. Journal of Econometrics, 139(1):4 14. Hahn, J. and Ridder, G. (2011). Conditional moment restrictions and triangular simultaneous equations. The Review of Economics and Statistics, 93(2): / 35

35 References Control functions: Imbens, G. W. and Newey, W. (2009). Identification and Estimation of Triangular Simultaneous Equations Models Without Additivity. Econometrica, 77: Kasy, M. (2011). Identification in triangular systems using control functions. Econometric Theory, 27(03): Continuous LATE: Angrist, J. D., Graddy, K., and Imbens, G. W. (2000). The interpretation of instrumental variables estimators in simultaneous equations models with an application to the demand for fish. The Review of Economic Studies, 67(3): / 35