Estimating Distributions of Counterfactuals with an Application ... - UCL

More documents

Recommendations

Info

370 CARNEIRO, HANSEN, AND HECKMAN θ. Conditional on θ and X, the potential outcomes are independent. If (12)–(15) accurately describe the data-generating process, we obtain the conditional independence assumptions used in matching (see, e.g., Cochrane and Rubin, 1973; Rosenbaum and Rubin, 1983). In matching it is assumed that Y s,a ⊥⊥ D s | X, Z,θ for all s. 14 From this assumption, we can identify ATE from the right-hand side of the following expression for continuous observed outcomes, which can be constructed if θ is observable, E(Y s,a − Y s ′ ,a | X, Z,θ) = E(Y s,a | X, Z,θ,D s = 1) − E(Y s ′ ,a | X, Z,θ,D s ′ = 1) In this case treatment on the treated, ATE, and MTE are the same parameter conditional on θ, X, and Z (Heckman, 2001; Aakvik et al., 2003). Our framework differs from matching by allowing the factors that generate the conditional independence that underlies matching to be unobserved by the analyst. In this sense, our approach is more robust than matching. The price for this robustness is the assumed independence between θ and (X, Z). Factor structure models are notorious for being identified by arbitrary normalization and exclusion restrictions. To reduce this arbitrariness and render greater interpretability to estimates obtained from our model, we adjoin a measurement system to choice equations (8) and outcome system (11). Various measurements can be interpreted as indicators of specific factors (e.g., test scores may proxy ability). Having measurements on the factors also facilitates identifiability under weaker assumptions as we demonstrate in Section 5. However, measurements are not strictly required for identification in our model. Outcome, measurement, and choice equations are interchangeable sources of identification in a sense that we make precise in Section 5. Consider a system of L measurements on the K factors, initially assumed to be for continuous outcome measures: (16) M 1 = µ 1 (X) + β 11 θ 1 +···+β 1K θ K + ε M 1 . M L = µ L (X) + β L1 θ 1 +···+β LK θ K + εL M ε M = (ε M 1 ,...,εM L ), E(εM ) = 0 and where we assume θ ⊥⊥ ε M ,θ ⊥⊥ ε s ,ε s ⊥⊥ (ε M ,ε Y ),ε M i ⊥⊥ ε M j ∀ i ≠ j, and i, j = 1,...,L. For interpretability, we assume θ i ⊥⊥ θ j , ∀ i ≠ j, i, j = 1,...,K. We develop the case with discrete measurements on latent continuous variables in Section 5. One can think of the outcome measures as an s-dependent measurement system. The measures (16) are the same across all s. Measurement system (16) allows for fallible measures of outcomes. Thus, in our schooling choice analysis we are not committed to the infallibility of test 14 Strictly speaking, matching models do not distinguish between X and Z. See Heckman and Navarro (2003).
EFFECTS OF UNCERTAINTY ON COLLEGE CHOICE 371 scores as measurements of ability. Measurement system (16) allows us to proxy unobservables accounting for measurement error and hence enables us to improve on the proxy procedure of Olley and Pakes (1996), which assumes no measurement error. 4.1. Choice Equations. Our analysis applies to both ordered discrete choice models and unordered choice models as analyzed by Cameron and Heckman (1998) and Hansen et al. (2001). In this article, we focus attention on a new ordered choice model. Other choice models can easily be accommodated in our framework and richer models are a source of additional identifying information. 15 For an ordered discrete choice model, let utility index I be written as (17) I = ϕ(Z) + ε W , ε W = γ ′ θ + ε I , σ 2 W = γ ′ ∑ θ γ + σ 2 I where E(ε 2 I ) = σ I 2, and ∑ θ is the covariance matrix of θ. A linear-in-parameters version that is the one developed in this article is written as ϕ(Z) = Zη. Choices are generated by index ϕ(Z) falling in various intervals. (18) D 1 = 1 if−∞of random utility model (8) in which states are ordered and pairwise contrasts possess a special structure. 16 We can parameterize the c s to be functions of state-specific regressors, e.g., c s = Q s ρ s where we restrict c s ≥ c s−1 . We could also follow a suggestion in Heckman et al. (1999) to incorporate one-sided shocks ν s and work with stochastic thresholds ˜c s in place of c s : ˜c s = c s + ν s , s = 1,...,S − 1, where ν s ≥ ν s−1 and ν s ≥ 0. 17 15 See Heckman and Navarro (2001) for a comparison among alternative models of completed schooling. Hansen et al. (2003) develop a parallel analysis for a one-factor multinomial choice model. 16 Specifically, it is assumed for (8) that µ s (Z ) is concave in s for each Z (Cameron and Heckman, 1998), that e s − e s−1 = τ s = 2,...,S with e 1 as an initial condition, that (∗) µ s (Z ) − µ s−1 (Z ) = ϕ(Z ) + c s−1 with µ 1 (Z ) as an initial condition and that c s ≥ c s−1 for all s = 1,..., ¯S. Changes in utilities across states are independent of s, except for an intercept. Then in (17) ε W = τ + e 1 . If we set all of the i.i.d. components of (9) to zero (the uniquenesses ε s ) we get the ordered probit model. As noted in the text, and developed in Heckman and Navarro (2001), we can generalize this model to allow e s − e s−1 = τ + χ s where χ s ≥ 0 is a one-sided random variable and still secure identification. The requirement (∗) precludes a strict random utility model because preferences are state specific. (The strict random utility model requires that µ s (Z ) not depend on s but Z can vary across s. See, e.g., Matzkin, 1993). 17 Write ν s = ∑ s j=2 ρ j , where ρ j ⊥ ρ j ′( j ≠ j ′ ),ρ j ⊥ ε W ,ρ j ⊥ (Z, Q),ρ j ≥ 0,ϕ(Z ) = Z ′ η. This model is identified under the assumptions in Cameron and Heckman (1998) even without any
Page 1 and 2: INTERNATIONAL ECONOMIC REVIEW May 2
Page 3 and 4: EFFECTS OF UNCERTAINTY ON COLLEGE C
Page 9: EFFECTS OF UNCERTAINTY ON COLLEGE C
Page 61 and 62:
EFFECTS OF UNCERTAINTY ON COLLEGE C
show all

Estimating Distributions of Counterfactuals with an Application ... - UCL

Create successful ePaper yourself

Delete template?

Save as template?