Partial Identification of Nonseparable Models using Binary Instruments

Size: px
Start display at page:

Download "Partial Identification of Nonseparable Models using Binary Instruments"

Transcription

1 Partial Identification of Nonseparable Models using Binary Instruments Takuya Ishihara October 13, 2017 arxiv: v2 [stat.me] 12 Oct 2017 Abstract In this study, we eplore the partial identification of nonseparable models with continuous endogenous and binary instrumental variables. We show that structural function is partially identified when it is monotone or concave in an eplanatory variable. D Haultfœuille and Février 2015) and Torgovitsky 2015) prove the point identification of the structural function under two key assumptions: a) the conditional distribution functions of an endogenous variable given the instruments have intersections and b) the structural function is strictly increasing in a scalar unobservable variable. However, we demonstrate that, even if the two assumptions do not hold, monotonicity or concavity provides an identifying power. Point identification is achieved when the structural function is flat or linear in eplanatory variables in a given interval. Keywords: Nonseparable models, partial identification, endogeneity, monotonicity, concavity, discrete outcomes, unobserved heterogeneity, instrumental variables. 1 Introduction In this study, we eamine the identification of a system of structural equations that takes the following form: Y = gx, ɛ) 1) X = hz, η), where Y R is a scalar response variable, X R is a continuous endogenous variable, Z {0, 1} is a binary instrument, and ɛ and η are unobservable scalar variables. For simplicity, we suppose that X is a scalar variable. It is straightforward to etend the results to a case in which X is a vector. This specification is nonseparable in unobservable variable ɛ and captures unobserved heterogeneity in the effect of X on Y. Such models have also been considered by, for eample, D Haultfœuille and Février 2015) and Torgovitsky 2015). University of Tokyo, Graduate School of Economics, ishihara-takuya143@g.ecc.u-tokyo.ac.jp. I gratefully acknowledge Katsumi Shimotsu and Hidehiko Ichimura for their guidance and support. I thank seminar participants of University of Tokyo, Otaru University of Commerce, Kanazawa University, Hiroshima University, and Shanghai Jiao Tong University. I am supported by Grant-in-Aid for JSPS Research Fellow 17J03043) from the JSPS. 1

2 D Haultfœuille and Février 2015) and Torgovitsky 2015) show that g is point identified when g, e) and hz, v) are strictly increasing in e and v and Z is independent of ɛ, η). Their results are important for empirical analyses, in which many instruments are binary or discrete, such as the intent-to-treat in a randomized controlled eperiment and quarter of birth used in Angrist and Keueger 1991). For nonseparable models, such as 1) with continuously distributed X, other nonparametric point identification results require Z to be continuously distributed. See, for eample, Newey and Powell 2003), Chernozhukov, Imbens, and Newey 2007), and Imbens and Newey 2009). D Haultfœuille and Février 2015) and Torgovitsky 2015) use two key assumptions when establishing point identification for g. First, F X Z 0) and F X Z 1) have intersections. Second, g, e) is strictly increasing in e. There are, however, many empirically important models that do not satisfy these assumptions. For eample, F X Z 0) and F X Z 1) do not have an intersection when Z has a strictly monotonic effect on X such as linear models X = β 0 + β 1 Z + η. Further, in many applications, instrumental variables have a strictly monotonic effect on endogenous variables such as LATE proposed by Imbens and Angrist 1994). For eample, as seen in Baird, McIntosh, and Özler 2011), cash transfer programs have been implemented in many countries. Then, if we use treatment indicator Z as the instrumental variable for income X, Z has a strictly monotonic effect on X, which violates the intersection assumption. Actually, in this eample, we can write X = βz + η, where β is transfer amount and η is initial household income. Hence, F X Z 0) and F X Z 1) never have an intersection in this eample. When Y is discrete or censored, g, e) is not strictly increasing in e. Many problems in economics involve dependent variables that are discrete or censored. For eample, development economists may want to analyze the effects of income changes on child education. If school attendance is used as a dependent variable, then Y is discrete. As another eample, suppose we want to analyze the effects of income changes on education ependiture. Then, education ependiture is censored at zero when children do not attend school. This study shows that when g, e) is monotone or concave in, we can partially identify g, even if F X Z 0) and F X Z 1) have no intersection and g, e) is not strictly increasing in e. g, e) is monotone or concave in in many economic models. For eample, the Engel function of a normal good is monotonically increasing in income, the demand function is ordinarily decreasing in price, and economic analyses of production often suppose that the production function is monotone and concave in inputs. Manski 1997) assumes these assumptions and shows that the average treatment response is partially identified. The partial identification approach using the concavity assumption introduced in this study is somewhat similar to that considered by D Haultfoeuille, Hoderlein, and Sasaki 2013). There is rich literature on the identification of nonseparable models with endogeneity. For eample, Chernozhukov and Hansen 2005), Chesher 2007), Chernozhukov et al. 2007), and Imbens and Newey 2009) consider the identification of nonseparable models with endogeneity. In particular, Imbens and Newey 2009) consider models similar to 1). Their study allows ɛ to be multivariate and shows that the quantile function of g, ɛ) is point identified, while in this analysis, ɛ is imposed to be scalar. Their result needed continuous instruments, whereas D Haultfœuille and Février 2015), Torgovitsky 2015), and the present study do not. This analysis adopts the control function approach, which is also used by, for eample, 2

3 Imbens and Newey 2009), D Haultfœuille and Février 2015), and Torgovitsky 2015). This approach requires the strict monotonicity of hz, v) in v and strong eogeneity: Z is independent of ɛ, η). The estimating equation approach followed by, for eample, Newey and Powell 2003), Chernozhukov and Hansen 2005), and Chernozhukov et al. 2007), does not require these assumptions. Chernozhukov and Hansen 2005) allow η to be multivariate and correlated with Z. However, they assume that independent and instrumental variables are discrete. By contrast, we suppose that the instrumental variable Z is binary. D Haultfœuille and Février 2015) consider the case in which the instrumental variable takes more than two values. In this case, they show point identification can be achieved using group and dynamical systems theories, even when F X Z z) and F X Z z ) have no intersection. The remainder of this study is organized as follows. Section 2 introduces the assumptions employed in this study. Sections 3 and 4 demonstrate the partial identification of g under the monotonicity and concavity assumption when conditional distributions have no intersections. Section 5 etends the result in Section 3 to a general case, where we allow Y to be discrete or censored. Section 6 concludes. 2 Model The following two assumptions are the same as those in D Haultfœuille and Février 2015) and Torgovitsky 2015). Assumption 1. The instrument is independent of the unobservable variables: Z ɛ, η). Assumption 2. i) Function g is continuous and g, e) is strictly increasing in e for all X. ii) For all z {0, 1}, hz, v) is continuous and strictly increasing in v. Assumption 1 is a strong eogeneity assumption and typically employed when using the control function approach. See, for eample, Imbens and Newey 2009), D Haultfœuille and Février 2015), and Torgovitsky 2015). Imbens and Newey 2009) also need Assumption 2ii) but allow ɛ to be multivariate. Chernozhukov and Hansen 2005) and Chernozhukov et al. 2007) employ Assumption 2i) but do not need Assumption 2ii). We rela a part of Assumption 2i) in Section 5, where we assume g, e) is nondecreasing in e. The net assumption regarding the conditional distributions of X conditional on Z differs from that imposed by D Haultfœuille and Février 2015) and Torgovitsky 2015). For any random variable U and W, let F U W denote the conditional distribution function of U conditional on W. Let X, X z, and Y,z denote interiors of the support of X, X Z = z, and Y X =, Z = z, respectively. Assumption 3. i) F X Z z) is continuous in for all z {0, 1} and F X Z 0) < F X Z 1) for all X 0 X 1. ii) X 0 = 0, 0 ), X 1 = 1, 1 ), and < 1 < 0 < 1 < 0 <. Conditions i) and ii) imply that F X Z z) is strictly increasing and continuous in conditional on X z. This assumption implies that F X Z 0) and F X Z 1) do not have any intersection on the support of X and X Z = 0 stochastically dominates X Z = 1. Thus, Z has a strictly monotonic effect on X. D Haultfœuille and Février 2015) and 3

4 Torgovitsky 2015) rule out this case because they assume F X Z 0) and F X Z 1) have intersections. Torgovitsky 2015) shows that the point identification of g holds when 1 = 0 = ξ, that is, F X Z 0) and F X Z 1) intersect at = ξ, and gξ, e) is strictly increasing in e. Even when X 0 = X 1 = X =, ), we can not identify g unless g, e) or g, e) is strictly increasing in e. Net, we impose restrictions on the conditional distributions of Y conditional on X and Z. Assumption 4. i) For all z,, y) {0, 1} X z Y,z, F Y X,Z y, z) is continuous in and y. ii) For all z, ) {0, 1} X z, we have Y,z = Y = y, y), where y < y. D Haultfœuille and Février 2015) and Torgovitsky 2015) also assume condition i) but do not assume condition ii). Conditions i) and ii) imply that F Y X,Z y, z) is strictly increasing and continuous in y conditional on Y. Hence, the conditional quantile function of Y conditional on X and Z is the inverse of F Y X,Z y, z). Condition ii) is not necessary for this study s result but without it, deriving the results can be cumbersome. We rela condition i) in Section 5 and allow Y to be discrete or censored. Finally, we impose the normalization assumption on unobservable variables and support condition of ɛ X =, Z = z. Assumption 5. i) ɛ U0, 1) and η U0, 1). ii) For all z, ) {0, 1} X z, the interior of the support of ɛ X =, Z = z is 0, 1). Condition i) is the usual normalization in a nonseparable model see, Matzkin 2003)). Torgovitsky 2015) does not use this normalization but D Haultfœuille and Février 2015) normalize ɛ to be uniformly distributed. D Haultfœuille and Février 2015) do not assume η U0, 1). The theorems in this study hold without this assumption, but the proof becomes more complicated. Condition ii) implies that g, e) y, y) = Y for all, e) X 0, 1). 3 Partial Identification through Monotonicity Let Y be the closure of Y. We establish the partial identification of g by showing that we can identify functions T U, y) : Y Y and T L,y) : Y Y and that they are i) strictly ) ) increasing in y ii) surjective, that is, T U, [y, y] = T L, [y, y] = [y, y], and iii) satisfy the following inequalities: g, e) T U, g, e)), 2) g, e) T L, g, e)). 3) T U, y) and T L, y) correspond to Q in D Haultfœuille and Février 2015). If T U,y) is identified for all, X X 0 X 1, we can obtain the lower bound of the structural function g, e) in the following manner. Here, we define G L u) F Y X= T U, u)) df X ). 4

5 If T U, y) satisfying 2) is obtained for all, X, then we have G L g, e)) = F Y X= T U, g, e)) ) df X ) F Y X= g, e)) df X ) = P g, ɛ) g, e) X = )df X ) = P ɛ e X = )df X ) = e, 4) where the first inequality follows from 2) and the monotonicity of F Y X y ) in y and the third equality follows from the strict monotonicity of g, e) in e. Furthermore, G L u) is invertible because T U, y) is strictly increasing in y. Because T U,y) is surjective, we ) have G L [y, y] = [0, 1]. Hence, for all e 0, 1), we have g, e) G) L 1 e). 5) Similarly, define G U u) F Y X= T L, u)) df X ), and thus, we have g, e) G U ) 1 e). This idea is similar to that of D Haultfœuille and Février 2015). They use function T,y) that is strictly increasing in y and satisfies g, e) = T, g, e)). Define G u) FY X= T,u)) df X ). Then, similar to the above inequality, we have G g, e)) = e. Hence, if T,y) is strictly increasing in y, then G u) is invertible and we have g, e) = G ) 1 e). Net, we propose how to construct functions T U, y) and T L,y) that satisfy 2) and 3). As in Torgovitsky 2015), define π) : X 0 X 1 and π 1 ) : X 1 X 1 0 as π) Q X Z FX Z 0) 1 ), π 1 ) Q X Z FX Z 1) 0 ). 6) Here, for any random variables U and W, let Q U W τ w) denote the conditional τ-th quantile of U conditional on W = w, that is, Q U W τ w) inf{u : F U W u w) τ}. The following result is essentially proven by D Haultfœuille and Février 2015) Theorem 1). We state this result as a proposition because it plays a central role in the following and our assumptions somewhat differ from those in D Haultfœuille and Février 2015). Proposition 1. Then, under Assumptions 1 5, we have 1) y) Q Y X,Z FY X,Z y, 0) π), 1 ), 1) y) Q Y X,Z FY X,Z y, 1) π 1 ), 0 ). g π), e) = g π 1 ), e ) = 1) g, e)), 1) g, e)). 1 These functions correspond to s ij in D Haultfœuille and Février 2015). 5

6 Proof. Step.1 We show that, for all e 0, 1), P ɛ e X =, Z = 0) = P ɛ e X = π), Z = 1). 7) First, we eamine variable V F X Z X Z). This is called control variable in Imbens and Newey 2009). Let h 1 z, ) be the inverse function of hz, v) with respect to v. We have for all z, ) {0, 1} X z, F X Z z) = P hz, η) Z = z) = P η h 1 z, ) Z = z ) = P η h 1 z, ) ) = h 1 z, ), where the second equality follows from the strict monotonicity of h, v) in v and the third equality follows from Z ɛ, η). Therefore, we obtain V = h 1 Z, X) = η. Net, we show that the conditional distribution of ɛ conditional on X, Z) =, z) is the same as that of ɛ conditional on V = F X Z z). Because, z) F X Z z), z) is one-to-one and F X Z z) is continuous in, Z ɛ, η), and V = η, we have P ɛ e X =, Z = z) = P ɛ e V = F X Z z), Z = z ) = P ɛ e V = F X Z z) ). 8) Hence, the conditional distribution of ɛ conditional on X and Z solely depends on V = F X Z X Z). By definition, functions π) and π 1 ) satisfy that F X Z π) 1) = F X Z π 1 ) 0) = F X Z 0), F X Z 1). 9) Hence, events {X =, Z = 0} and {X = π), Z = 1} have the same V = F X Z X Z) and 7) follows from 8). 1) Step.2 We show that 7) implies g π), e) = g, e)). For all, e) X 0 0, 1), we have 1) g, e)) = Q Y X=π),Z=1 FY X=,Z=0 g, e)) ) = Q Y X=π),Z=1 P ɛ e X =, Z = 0)) = Q Y X=π),Z=1 P ɛ e X = π), Z = 1)) = Q Y X=π),Z=1 FY X=π),Z=1 gπ), e)) ) = gπ), e), where the third equality follows from 7). Similarly, we can prove that g π 1 ), e) = 1) 1) g, e)). 1) 1) By definition, y) and y) are strictly increasing, [y, y]) = [y, y], and 1) [y, y]) = [y, y]. Define π n ) π π) for n N if π n ) eists. Then, we 6

7 1) π n 1 ) obtain gπ n ), e) = for n N, and thus, we have 1) g, e)). Define n) y) 1) π n 1 ) 1) y) gπ n ), e) = n) g, e)). y) is strictly increasing in y and π n ) π 1 π 1 n) ) and y) have gπ n ), e) = In addition, n) n) y) is strictly increasing in y and 1) π n 1) ) n) [y, y]) = [y, y]. Similarly, define n) g, e)). 1) y) for n N; then, we n) [y, y]) = [y, y]. For all X, define 0) y) y. Here, we eamine the properties of π) and π 1 ). Because F X Z 0) < F X Z 1) conditional on X, we have π) = Q X Z FX Z 0) 1 ) < Q X Z FX Z 1) 1 ) =, π 1 ) = Q X Z FX Z 1) 0 ) > Q X Z FX Z 0) 0 ) =. Figure 1 illustrates this intuitively. Because X Z = 0 stochastically dominates X Z = 1 and functions π) and π 1 ) satisfy 9), these inequalities hold. To facilitate the illustration of our identification result, we first review the identification approach employed by D Haultfœuille and Février 2015) and Torgovitsky 2015) when 0 = 1 = ξ, even though Assumption 3 rules out the case of 0 = 1 = ξ. Pick an initial point 0 X i.e., 0 > ξ) and form a recursive sequence n+1 = π n ) for n > 0. Because 0 = 1 = ξ implies X 1 X 0, we have π) X 0 for all X and there eists a sequence {π n )} n=1. The sequence { n } is decreasing by 10) and n > ξ for all n 0 by the definition of π). Hence, sequence { n } converges to a limiting point. Because 9) implies F X Z n+1 1) = F X Z n 0) and F X Z z) is continuous in, we have F X Z lim n n 1) = F X Z lim n n 0). Because F X Z 0) < F X Z 1) is conditional on ξ, 0 ) and F X Z ξ 0) = F X Z ξ 1) = 0, the sequence { n } converges to ξ for any initial point 0 X. Figure 2 illustrates this ) n) intuitively. Define y) lim n y), which is strictly increasing and invertible in y. From the continuity of g, we obtain, for all X, A similar argument provides holds for any, and ) g, e)) = lim gπ n ), e) = gξ, e). n ) g, e)) = gξ, e). Therefore, g, e) = ) ) 1 ) ) g, e)). ) g, e)) = 10) ) g, e)) ) 1 ) ) ) Define T,y) y), and thus, T,y) is strictly increasing and satisfies g, e) = T, g, e)). Hence, as discussed above, g is point identified. 7

8 Under Assumption 3, this approach is not available because a convergent sequence {π n )} n=1 does not eist. When F X Z 0) and F X Z 1) have no intersections, π n ) lies in X 1 X0 c = 1, 0 ] when n is sufficiently large. If π n ) is in X 1 X0 c, then π n+1 ) does not eist. As seen in Lemma 1, for all X, {n : π n ) eists.} is a finite set under Assumption 3. For eample, in Figure 1, π), π 1 ), and π 2 ) eist but π 2 ) and π 3 ) do not. If we do not impose additional restrictions, then we cannot construct functions T U, y) and T L, y) that satisfy 2) and 3). First, we show that a set ΠM, defined below is nonempty and finite, even when F X Z 0) and F X Z 1) have no intersections. Net, we show that we can partially identify g, e) using Π M, when g, e) is nondecreasing in. For, ) X X, define Π M, as Π M, {n, m) : π n ) π m ), n, m Z, π n ) and π m ) eist.}, 11) where π 0 ). In Figure 1, Π M, = { 1, 2), 0, 2), 0, 1), 1, 2), 1, 1), 1, 0)}. The following lemma shows that Π M, is nonempty and finite even when F X Z 0) and F X Z 1) have no intersection. Lemma 1. Under Assumptions 1 5, Π M,, ) X X. defined by 11) is nonempty and finite for all Proof. Observe that if π n ) eists and π n ) X 0, then π n+1 ) also eists from 6). Suppose that n N {0} does not eist such that π n ) X 1 X0 c = 1, 0 ]. Then, there eists sequence { n } n=0 such that n = π n ). By 10), { n } n=0 is a decreasing sequence. Because n > 0, { n } n=0 converges to [ 0, 0 ). Because F X Z n+1 1) = F X Z n 0) holds from 9) and F X Z 1) = F X Z 0) from the continuity of F X Z. However, this equation violates Assumption 3. Hence, for all X, n N {0} eists such that π n ) X 1 X0 c. Consequently, π n ) does not eist for n > n. Similarly, for all X, n N {0} eists such that π n ) X 0 X1 c. Then, π n ) does not eist for n > n. Therefore, Π M, is finite for all, ) X X because a set {n, m) Z Z : π n ) and π m ) eist.} is finite. We proceed to show the nonemptyness of Π M,. For all, X, n, m) Z Z eists such that π n ) X 1 X0 c = 1, 0 ] and π m ) X 0 X1 c = [ 1, 0 ). It follows from Assumption 3ii) that π n ) < π m ). Under Assumptions 1 5, a set {n Z : π n ) eists.} is finite from Lemma 1. Hence, g cannot be point identified using the method proposed by D Haultfœuille and Février 2015) and Torgovitsky 2015)). We impose the following assumption: Assumption M Monotonicity). For all e 0, 1), g, e) is nondecreasing in. The monotonicity assumption holds in many economic models. For eample, the Engel function for a normal good is monotonically increasing in income, the demand function is ordinarily monotonically decreasing in price, and economic analyses of production often 8

9 suppose that the production function is monotonically increasing in input. Monotonicity assumptions of this type have been employed in many studies. Manski 1997) imposes a monotonicity assumption on a response function and shows that the average treatment response is partially identified. If n, m) Π M,, then Assumption M implies that Because n) n) n) g, e)) = gπ n ), e) gπ m ), e) = y) is strictly increasing in y and ). Hence, we have ) 1 m) g, e)) g, e) min n,m) Π M, n) m) g, e)). n) ) [y, y] = [y, y], we have g, e) ) 1 m) ) g, e)). Define T MU, y) T ML, y) min n,m) Π M, n) m) ma n,m) Π M, ) 1 m) ) y), ) 1 ) n) y). 12) Then, T MU, y) is strictly increasing and satisfies g, e) T MU, g, e)). 13) ) 1 ) T MU, y) achieves the tightest lower bound of g because function n) m) y) satisfies 2) for any n, m) Π M ML,. Similarly, T, y) is strictly increasing and satisfies g, e) T MU, g, e)). 14) Define G ML u) G MU u) B ML, e) sup y:y B MU, e) inf y:y F Y X= F Y X= { G ML y T MU, u) ) df X ), T ML, u) ) df X ), ) } 1 e), { G ) } MU 1 y e). G ML u) and G MU u) provide lower and upper bounds on g, e) on the basis of arguments 4) and 5). B ML, e) and B MU, e) strengthen these bounds. Theorem 1. Under Assumptions 1 5 and M, for all, e) X 0, 1), we have B ML, e) g, e) B MU, e). 9

10 Proof. Step.1 First, we show that G ML ) 1 ) e) g, e) G MU 1 e). 15) Function T ML n), y) defined in 12) is strictly increasing in y because y) is strictly increasing in y for all n Z and X and Π M, is finite as per Lemma 1. Similarly, T MU n), y) is strictly increasing in y. Because [y, y]) = [y, y] for all n Z and X, we ) 1 ) ) 1 ) m) n) m) n) have [y, y]) = [y, y]. Because [y, y]) = [y, y] and Π M, ) ) is finite, it follows that G ML [y, y] = [0, 1]. Similarly, G MU [y, y] = [0, 1]. Therefore, inequalities 15) hold from 4) and 5). Step.2 Net, we show that B ML, e) g, e) B MU, e). 16) Because g, e) is nondecreasing in and g, e) ) G ML 1 e), we have g, e) gy, e) ) G ML 1 y e) for all y. Hence, it follows that { G ) } ML 1 g, e) sup y e). y:y Similarly, g, e) gy, e) ) G MU 1 y e) for all y and it follows that { G ) } MU 1 g, e) inf y e). y:y Therefore, inequalities 16) hold. In the first step, we show ) G ML 1 ) e) g, e) G MU 1 e). In the second step, we strengthen these bounds to B ML, e) g, e) B MU, e). Figure 3 illustrates this proof intuitively. This idea is similar to that in Manski 1997), who considers a case in which response function yt) is increasing, where yt) is a latent outcome with treatment t. He then uses the monotonicity of yt) to partially identify the average response function E[yt)] when outcome support is bounded. Our approach does not require information on the infimum and supremum of the support of outcomes. Theorem 1 shows that we can obtain nontrivial bounds of g by using the monotonicity assumption. However, without this assumption, we show that the sharp bounds of g become trivial bounds, that is y < g, e) < y. To see this, suppose that g is the true structural function. Let G be a set of functions that satisfy Assumption 2 i). Similarly to Torgovitsky 2015), we show that the identified set is G { g G : g 1 X, Y ), V ) Z, P Y gx, e)) = e for all e 0, 1).}. First, we show that if By the definition, 1) n) g, e)) = gπ n ), e) holds, then g, e) satisfies g 1 X, Y ), V ) Z. g, e)) = gπ), e) implies that F Y X,Z g, e), 0) = F Y X,Z gπ), e) π), 1) P g 1 X, Y ) e V = v, Z = 1) = P g 1 X, Y ) e V = v, Z = 0), 10

11 n) where v = F X Z 1). Therefore, if g, e)) = gπ n ), e) for all and e, then g, e) satisfies g 1 X, Y ), V ) Z. Define g φ, e) g, φ, e)), where φ : X 0, 1) n) 0, 1) is continuous and strictly increasing e. Because we have g, φ, e))) = gπ n ), φ, e)) by Proposition 1, if φ satisfies φ, e) = φπ n ), e) for all X and e 0, 1), 17) then g 1 φ X, Y ), V ) Z holds. Hence, if φ satisfies 17) and P Y g φ X, e)) = P ɛ φ, e) X = )df X ) for all e 0, 1), 18) then g φ G. Fi X and e 0, 1). Then, we can find the function φ such that φ, e ) is close to 0 or 1 and satisfies 17) and 18). Therefore, without the monotonicity assumption, the sharp bounds of g, e) become y and y. To illustrate Theorem 1, we consider the following eample: Y = X 1/3 ep α + βφ 1 ɛ) ) X = d + c1 Z) + η, where Φ ) is the standard normal distribution function and Z is a random Bernoulli variable with p = 0.5. Suppose that ɛ = ΦU) η = ΦV ) 0 U, V ) N 0 ) 1 ρ, ρ 1 )). Thus, ɛ U0, 1) and η U0, 1). In this eample, F X Z 1) = d for X 1 = d, 1 + d) and F X Z 0) = c d for X 0 = c + d, 1 + c + d). Hence, the conditional distribution functions F X Z 0) and F X Z 1) do not have any intersection. This eample satisfies the assumptions in Theorem 1. We calculate the bounds of g, 0.5) using Theorem 1 when α = 0.5, β = 0.5, c = 0.5, d = 0.05, and ρ = 0.3. These bounds are depicted in Figure 4. The bounds become tighter as the difference between g, e) and T U, g, e)) or T L, g, e))) reduces. The following theorem shows that, if g, e) is flat in in a given interval, inequalities 2) and 3) become equalities and structural function g is point identified. Theorem 2. Under Assumptions 1 5 and M, if there eists X such that g, e) is constant on [π ), π 1 )] for each e 0, 1), then g, e) is point identified for all, e) X 0, 1). This result holds even when interval [π ), π 1 )] is unknown. Proof. Step.1 First, we show that for all X, n Z eists such that π n ), π n+1 ) [π ), π 1 )]. Because π n ) is strictly increasing in, for all n Z, we have y π n ) π n y). 19) We consider the following four cases: i) π ) ii) π 1 ) iii) < π ), and iv) > π 1 ). In case i), it follows from 19) that π ) π 1 ) 11

12 π 1 ). In case ii), it follows from 19) that π ) π) π 1 ). In case iii), by the definition of π, we have π ) X 1. It follows from the proof of Lemma 1 that n Z eists such that π ) π n ) X 0 X1 c. Hence, n Z eists such that π n+2 ) π ) π n+1 ) and we can obtain π ) π n+1 ) π n ) π 1 ) from 19). In case iv), similar to case iii), there eists n Z such that π n ), π n+1 ) [π ), π 1 )]. Step.2 Net, we show that g is point identified. From step 1, for all, X, there eists n, m Z such that π n ), π n+1 ), π m ), π m+1 ) [π ), π 1 )]. Then, from 19), we have either π n+1 ) π m+1 ) π n ) π m ) or π m+1 ) π n+1 ) π m ) π n ). If π n+1 ) π m+1 ) π n ) π m ), then we have n + 1, m + 1), n, m) Π M, and m + 1, n) ΠM,. If πm+1 ) π n+1 ) π m ) π n ), then we have n + 1, m) Π M, and m + 1, n + 1), m, n) ΠM,. Because g, e) is constant in conditional on [π ), π 1 )], π n ), π m ) [π ), π 1 )] implies that n) g, e)) = m) g, e)). Therefore, g, e) = T MU, g, e)) and g, e) = T ML, g, e)). Hence, g, e) is identified because inequalities 13) and 14) become equalities. In the first step, we show that for all X, n Z eists such that π n ), π n+1 ) [π ), π 1 )]. In the second step, we show that g is point identified. Because g, e) is constant in conditional on [π ), π 1 )], we have g, e) = T MU, g, e)), and g, e) = T ML, g, e)) for all, X and e 0, 1). Hence, g, e) is point identified because inequalities 13) and 14) become equalities. Eample 1 Engel curve). We consider the case where we want to estimate the Engel function by using a cash transfer eperiment. Let Y be household ependiture on a normal good, X be household income, and Z be a treatment indicator of the cash transfer eperiment. Let β denote transfer amount. Then, we write the model 1) as the following form: Y = gx, ɛ) X = βz + η, where ɛ represents household preference and η is initial household income. If participants are randomly assigned to either the treatment or the control group, Z is independent of ɛ, η). In the proof of Proposition 1, we show that P ɛ e X =, Z = 0) = P ɛ e X = π), Z = 1). In this case, we can understand it intuitively. Observe that π) = + β. Then, households with X = and Z = 0 have the same initial household income η as households with X = π) = + β and Z = 1. Because Z is independent of household preference and initial household income, the household with X = and Z = 0 have the same conditional distribution of ɛ as the household with X = π) = + β and Z = 1. Hence, we obtain P ɛ e X =, Z = 0) = P ɛ e X = π), Z = 1). In this case, we cannot use the identification results of D Haultfœuille and Février 2015) and Torgovitsky 2015) because F X Z 0) and F X Z 1) do not have an intersection. However, because the Engel function gx, ɛ) is generally nondecreasing in household income X, we can partially identify g by using our identification result. 12

13 4 Partial Identification through Concavity In this section, we propose how to construct the lower and upper bounds of g, e) when g, e) is concave in. First, we show that a set Π C, defined below is nonempty and finite. Net, we show that we can partially identify g using Π C, when g, e) is concave in. For, ) X X, define Π C, as Π C, { n, m) : π n ) π m ) π n 1 ), n, m Z, π n ), π n 1 ) and π m ) eist. }. 20) In Figure 1, Π C, = {0, 1), 1, 0)}. The following lemma shows that ΠC, and finite, similar to Lemma 1. is nonempty Lemma 2. Under Assumptions 1 5, Π C,, ) X X. defined by 20) is nonempty and finite for all Proof. From the proof of Lemma 1, Π C, is finite. Hence, we show the nonemptyness of Π C,. From the proof of Lemma 1, for all, X, there eist n, m Z such that π m ), π n ) X 1 X0 c. Without loss of generality, we assume π n ) π m ). Then, π m 1 ) and π n 1 ) eist because π m ) X 1 and π n ) X 1. Because π m ) X0 c and pi n 1 ) X 1, we have π n ) π m ) π n 1 ) π m 1 ) from 19), and hence n, m) Π C,. Therefore, ΠC, is nonempty. Similar to Section 3, we impose the following assumption: Assumption C Concavity). For all e 0, 1), g, e) is concave in. The concavity assumption holds in many economic models. For eample, economic analyses of production often suppose that the production function is concave in inputs. Manski 1997) assumes a concavity assumption and shows that the average treatment response is partially identified. D Haultfoeuille et al. 2013) achieves the partial identification of the average treatment on treated effect using a locally concavity assumption. As seen in Section 3, if we identify functions T U, y) and T L,y) that are strictly increasing in y, surjective, and satisfy 2) and 3), we can obtain the lower and upper bounds of g, e). Hence, we consider how to construct functions T U, y) and T L, y) that are strictly increasing in y, surjective, and satisfy 2) and 3). If n, m) Π C,, from Assumption C, we have where n 1) n 1) have [ ] n) n 1) t,n, m) + 1 t,n, m)) g, e)) [ ] n) n 1) t,n, m) + 1 t,n, m)) y) = t,n, m) m) g, e)), n) y) + 1 t,n, m)) y) and t,n, m) = π n 1 ) π m )) / π n 1 ) π n )). Because y) and n) ) n 1) ) y) are strictly increasing in y, [y, y] = [y, y], and [y, y] = [y, y], we g, e) [ n) min t,n, m) n,m) Π C + 1 t,n, m)), 13 n 1) n) ] 1 ) m) g, e)).

14 Define T CU,y) T CL,y) [ n) min t,n, m) n,m) Π C + 1 t,n, m)), n 1) ) 1 [ m) ma n) n,m) Π C t, n, m) + 1 t, n, m)), ] 1 ) m) y), ] ) n 1) y). 21) Then, T CU,y) defined in 21) is strictly increasing and satisfies g, e) T CU, g, e)). 22) Similarly, T CL,y) defined in 21) is strictly increasing and satisfies Define B CL, e) B CU, e) G CL u) G CU u) g, e) T CL, g, e)). 23) { y sup y,y :y<<y y y min F Y X= F Y X= ) G CL y [inf y,y :<y<y { y inf y,y :y <y< y y T CU,u) ) df X ), T CL,u) ) df X ), ) 1 e) + y y y ) B CL y, e) + { y y y ) B CL y, e) + y y y ) y G CU y y y ) } G ) CL 1 y e), ) G ) } CU 1 y e) ) 1 e) }]. G CL u) and G CU u) provide the lower and upper bounds of g, e) as per arguments 4) and 5). B CL, e) and B CU, e) strengthen these bounds. Theorem 3. Under Assumptions 1 5 and C, for all, e) X 0, 1), we have B CL, e) g, e) B CU, e). Proof. Similar to the proof of Theorem 1, we can obtain ) G CL 1 ) e) g, e) G CU 1 e). Because g, e) is concave in, if = ty + 1 t)y and t 0, 1), then we have g, e) tgy, e) + 1 t)gy, e) t ) G CL 1 ) y e) + 1 t) G CL 1 y e). Hence, we have { ) ) } y G ) CL 1 y G ) g, e) sup y,y :y<<y y y e) + CL 1 y y y e). y Because g, e) is concave in, if = ty + 1 t)y and t < 0, then we have g, e) tgy, e) + 1 t)gy, e). Because B CL, e) g, e) ) G CU 1 e), t < 0, and 1 t > 0, we have g, e) tb CL y, e) + 1 t) ) G CU 1 y e). Similarly, if = ty + 1 t)y and 14,

15 t > 1, then we have g, e) tgy, e) + 1 t)gy, e) t ) G CU 1 y e) + 1 t)b CL y, e). Hence, we have g, e) min [inf y,y :<y<y { y inf y,y :y <y< y y ) B CL y, e) + { y y y ) B CL y, e) + y y y ) y G CU y y y ) G ) } CU 1 y e) ) 1 e) }]., Similar to Theorem 1, we can show that ) G CL 1 ) e) g, e) G CU 1 e). We strengthen the bounds to B CL, e) g, e) B CU, e) using the concavity of g, e) in. Figure 5 illustrates this proof intuitively. A similar approach is used by Manski 1997). Manski 1997) uses the concavity of the response function to partially identify the average response function when the support of outcome is bounded. Our approach does not require information on the infimum and supremum of the support of outcomes. This identification approach is somewhat similar to that considered by D Haultfoeuille et al. 2013). They study the identification of nonseparable models with continuous, endogenous regressors, using repeated cross sections. They consider the following model: Y t = g t X t, A t ), t = 1,, T, where A t is an unobserved heterogeneous factor. They show that under the assumptions that A t V t F Xt X t ) = v A s V s F Xs X s ) = v and g t, a) = m t g, a)), the average treatment on treated effect AT T, ) E[g T, A T ) g T, A T ) X T = ] is identified when F XT ) = F Xt ). Under their assumption, AT T, ) is not identified if F XT ) F Xt ) for all t {1,, T 1}. However, they show that AT T, ) is partially identified if g, a) is locally concave. In several cases, such as production function, we may assume that both of Assumptions M and C hold. Then, it follows from Theorem 1 and 3 that ma{b ML, e), B CL, e)} g, e) min{b MU, e), B CU, e)}. 24) However, in this case, we can obtain the tighter bounds in the following manner. Define T MCU, y) min{t MU, y), T CU,y)}, T MCL, y) ma{t ML, y), T CL,y)}, G MCL u) F Y X= T MCU, u) ) df X ), G MCU u) F Y X= T MCL, u) ) df X ). Similarly to the above arguments, we have g, e) T MCU, g, e)), and hence we can obtain T MCL, g, e)) and g, e) Define G MCL ) 1 e) g, e) G MCU B MCL, e) sup y:y B MCU, e) inf y:y ) 1 e). { G ) } MCL 1 y e), { G ) } MCU 1 y e), 15

16 ˆB MCL, e) { y sup y,y :y<<y y y ) G MCL y ) 1 e) + y y y ) } G ) MCL 1 y e), ˆB MCU, e) min [inf y,y :<y<y { y inf y,y :y <y< y y) ˆBMCL y, e) + { ) y ˆBMCL y, e) + y y y y y ) y G MCU y y y ) G ) } MCU 1 y e) ) 1 e) }]., Then, from the above results, both of BMCU, e) and ˆB MCU, e) are upper bounds of g, e). Similarly, both of BMCL, e) and ˆB MCL, e) are also lower bounds of g, e). Therefore, we can obtain ma{ B MCL, e), ˆB MCL, e)} g, e) min{ B MCL, e), ˆB MCL, e)}. 25) Clearly, these bounds are tighter than 24). Similar to Theorem 2 in Section 3, the following theorem shows that if g, e) is linear in in a particular interval, then the inequalities 22) and 23) become equalities and structural function g is point identified. Theorem 4. Under Assumptions 1 5 and C, if X eists such that g, e) is linear in on [π ), π 1 )], then g, e) is point identified. This result holds even if interval [π ), π 1 )] is unknown. Proof. Similar to Theorem 2, for all, X, there eist n, m Z such that π n ), π n 1 ), and π m ) are in [π ), π 1 )], and π n ) π m ) π n 1 ). Because g, e) is linear in, we have [ ] n) n 1) t,n, m) + 1 t,n, m)) g m), e)) = g, e)). Similarly, for all, X, there eist n, m Z such that [ n) t, n, m) + 1 t, n, m)) m) g, e)) = ] n 1) g, e)). Hence, as described above, g, e) is point identified because inequalities 22) and 23) become equalities. Eample 2 quantile regression). Suppose gx, ɛ) = θ 0 ɛ) + θ 1 ɛ)x, where X is positive and θ 0 e) and θ 1 e) are strictly increasing in e. This model is a quantile regression model with endogeneity. The τ-th quantile function of g, ɛ) is θ 0 τ) + θ 1 τ). In this case, structural function g, e) = θ 0 e) + θ 1 e) is linear in. Hence, Theorem 4 shows that θ 0 e) and θ 1 e) are identified if binary instruments are available. 5 Etension: General Models In this section, we etend the results in Section 3 to more general models and allow Y to be discrete or censored. If outcomes are discrete or censored, then assumptions 2 and 4 are not satisfied. Hence, we replace these assumptions with the following ones: 16

17 Assumption 2. i) Function g, e) is nondecreasing in e for all X. ii) For all z {0, 1}, hz, v) is continuous and strictly increasing in v. Assumption 4. For all z, ) {0, 1} X z, we have Y,z = Y. Here, define y sup{y : y Y} and y inf{y : y Y}. Assumption 2 i) differs from Assumption 2i). Assumption 2i) imposes the strict monotonicity of g, e) in e, while Assumption 2 i) requires only the weak monotonicity of g, e) in e. For eample, if we consider g, e) = 1{e > 1 + epβ 0 + β 1 )) 1 }, then g, e) is not strictly increasing in e. Chesher 2010) also employs a weak monotonicity condition. Assumption 4 implies that Y is continuously distributed, but assumptions 2 and 4 allow outcomes that are discrete or censored. D Haultfœuille and Février 2015) and Torgovitsky 2015) do not consider the case in which outcomes are discrete or censored because they assume that g, e) is strictly increasing in e. Chesher 2010) considers instrumental variable models for discrete outcome. He shows that the structural function is partially identified using instruments. In this section, we show that g, e) is partially identified under assumptions 1, 2, 3, 4, 5, and M. Define F + Y X,Z y, z) P Y y X =, Z = z), F Y X,Z y, z) P Y < y X =, Z = z), Q + Y X,Z τ, z) sup{y : F Y X,Z y, z) τ} y, Q Y X,Z τ, z) inf{y : F + Y X,Z y, z) τ} y. F + Y X,Z y, z) and Q+ Y X,Z τ, z) are right continuous in y and τ. F Y X,Z y, z) and Q Y X,Z τ, z) are left continuous in y and τ. Under assumptions 2 and 4, Proposition 1 does not hold. Instead, we show the following proposition. Proposition 2. Define ˆT 1) y) Q Y X,Z Ť 1) y) Q + Y X,Z ˆT 1) y) Q Y X,Z Ť 1) y) Q + Y X,Z Then, under Assumptions 1, 2, 3, 4, and 5, we have ) F Y π), X,Z y, 0) 1, ) F +Y π), X,Z y, 0) 1, ) F π Y X,Z y, 1) 1 ), 0, ) F + π Y X,Z y, 1) 1 ), 0. 1) g π), e) ˆT g, e)), g π), e) Ť 1) g, e)), g π 1 ), e ) 1) ˆT g, e)), g π 1 ), e ) Ť 1) g, e)). 17

18 Proof. Because g, e) is nondecreasing in e, we have F Y X,Z g, e), 0) = P g, ɛ) < g, e) X =, Z = 0) P ɛ < e X =, Z = 0) = P ɛ e X = π), Z = 1) P g π), ɛ) g π), e) X = π), Z = 1) = F + Y X,Z gπ), e) π), 1), 26) where the first inequality follows from {ɛ : g, ɛ) < g, e)} {ɛ : ɛ < e} and the second inequality follows from {ɛ : ɛ e} {ɛ : g, ɛ) ) g, e)}. From the definition of Q Y X,Z τ, z), it follows that Q Y X,Z F +Y, X,Z y, z) z = inf{y : F + Y X,Z y, z) F + Y X,Z y, z)} y y for all y Y. Hence, inequality 26) implies that ) ˆT 1) g, e)) = Q Y X,Z F Y π), X,Z g, e), 0) 1 ) F +Y π), X,Z gπ), e) π), 1) 1 Q Y X,Z gπ), e). Similarly, because g, e) is nondecreasing in e, we have F + Y X,Z g, e), 0) F Y X,Z gπ), e) π), 1). ) Because Q + Y X,Z F Y, X,Z y, z) z = sup{y : F Y X,Z y, z) F Y X,Z y, z)} y y for all y Y, we have Ť 1) gπ), e) Ť 1) g, e)). Similarly, we have two inequalities, gπ 1 ), e) g, e)). 1) ˆT g, e)) and gπ 1 ), e) This approach is similar to the identification approaches in Athey and Imbens 2006) and Chesher 2010). Athey and Imbens 2006) show that the counterfactual distribution is partially identified using right and left continuous quantile functions when outcomes are discrete. Chesher 2010) uses the result in which the weak monotonicity of h, u) in u implies {u : h, u) h, τ)} {u : u τ} and {u : h, u) < h, τ)} {u : u < τ} and shows that structural function h is partially identified. ˆT n) Define y) and then, we have Similarly, define Ť 1) ˆT 1) π n 1 ) n) ˆT y) ˆT 1) y) and Ť n) y) Ť 1) π n 1 ) Ť 1) gπ n n) ), e) ˆT g, e)), gπ n ), e) Ť n) g, e)). ˆT 1) π n 1) ) y) for n N, and thus, we have ˆT 1) gπ n n) ), e) ˆT g, e)), gπ n ), e) Ť n) g, e)). 18 y) for n N, y) and Ť n) y) Ť 1) π n 1) )

19 0) Define ˆT y) = y and Ť 0) y) = y. If n, m) Π M,, then Assumption M implies that Define then we have Hence, we obtain ˆT n) g, e)) gπ n ), e) gπ m ), e) Ť m) g, e)). ˆT n) ) ˆT n) ) n) n) ˆT u) sup{y : ˆT g, e) Define T GU, y) min n,m) Π M, ) y) = sup{y n) : ˆT y ) min n,m) Π M, ˆT n) y) u} y, ) n) Ť ˆT m) g, e)) ). ) ) m) Ť y), then T GU, ˆT n) y)} y y for all y Y. y) satisfies g, e) T GU, g, e)). 27) ) ) ) Similarly, define T GL, y) ma m) n) m) n,m) Π Ť M, ˆT y) and Ť u) inf{y : Ť m) y) u} y, then T GL y) satisfies that, g, e) T GL, g, e)). 28) Define G GL u) B GU, e) inf y:y F + Y X T GU,u) ) df ), G GU u) F Y X T GL,u) ) df ), { B GL, e) sup inf{u : G GL y u) e} } y, y:y { sup{u : G GU y u) e} } y, where F + Y X y ) P Y y X = ) and F Y X y ) P Y < y X = ). GGL u) and G GU u) provide the lower and upper bounds on g, e) as per the argument similar to 4) and 5). B GL, e) and B GU, e) strengthen these bounds. Theorem 5. Under Assumptions 1, 2, 3, 4, 5, and M, for all, e) X 0, 1), we have B GL, e) g, e) B GU, e). Proof. First, we show that inf{u : G GL u) e} y g, e) sup{u : G GU u) e} y. 29) 19

20 Because T GU,y) satisfies 27), we have G GL g, e)) = = F + Y X= T GU,g, e)) ) df ) F + Y X= g, e)) df ) P g, ɛ) g, e) X = ) df ) P ɛ e X = ) df ) = e, where the second inequality follows from {ɛ : ɛ e} {ɛ : g, ɛ) g, e)}. Because g, e) y, we can obtain g, e) inf{u : G GL u) e} y. Similarly, because T GL, y) satisfies 28), we have G GU g, e)) F Y X= g, e)) df ) = P g, ɛ) < g, e) X = ) df ) P ɛ < e X = ) df ) = e, where the second inequality follows from {ɛ : g, ɛ) < g, e)} {ɛ : ɛ < e}. Hence, we can obtain g, e) sup{u : G GU u) e} y. Because g, e) is nondecreasing in and 29) holds, similar to Theorem 1, we have B GL, e) g, e) B GU, e). 6 Conclusions In this study, we consider the partial identification of nonseparable models using binary instruments. We show that partial identification can be achieved when g, e) is monotone or concave in, even if X is continuous and Z is binary. D Haultfœuille and Février 2015) and Torgovitsky 2015) show that g is point identified without monotonicity and concavity. They use two key assumptions to establish the point identification of g. First, F X Z 0) and F X Z 1) have intersections and second, g, e) is strictly increasing in a scalar unobservable. There are, however, many empirically important models that do not satisfy these assumptions. Thus, we provide bounds for structural functions without the use of these assumptions. 20

21 Figure 1: The left function is F X Z 1) and the right function is F X Z 0). Appendi: Figures 21

22 Figure 2: The left function is F X Z 1) and the right function is F X Z 0). Figure 3: The dashed lines denote ) G ML 1 ) e) and G MU 1 e). The solid lines represent B ML, e) and B MU, e). 22

23 Figure 4: The dashed line denotes g, 0.5). The solid lines are the bounds of g, 0.5), B ML, 0.5), and B MU, 0.5). α = 0.5, β = 0.5, c = 0.5, d = 0.05, andρ = 0.3. Figure 5: The dashed lines denote ) G CL 1 ) e) and G CU 1 e). The solid lines denote B CL, e) and B CU, e). 23

24 References Angrist, J. D. and A. B. Keueger 1991): Does Compulsory School Attendance Affect Schooling and Earnings? The Quarterly Journal of Economics, 106, Athey, S. and G. W. Imbens 2006): Identification and inference in nonlinear difference-in-differences models, Econometrica, 74, Baird, S., C. McIntosh, and B. Özler 2011): Cash or Condition? Evidence from a Cash Transfer Eperiment, The Quarterly Journal of Economics, 126, Chernozhukov, V. and C. Hansen 2005): An IV model of quantile treatment effects, Econometrica, 73, Chernozhukov, V., G. W. Imbens, and W. K. Newey 2007): Instrumental variable estimation of nonseparable models, Journal of Econometrics, 139, Chesher, A. 2007): Instrumental values, Journal of Econometrics, 139, ): Instrumental variable models for discrete outcomes, Econometrica, 78, D Haultfœuille, X. and P. Février 2015): Identification of nonseparable triangular models with discrete instruments, Econometrica, 83, D Haultfoeuille, X., S. Hoderlein, and Y. Sasaki 2013): Nonlinear differencein-differences in repeated cross sections with continuous treatments, Tech. rep., Boston College Department of Economics. Imbens, G. W. and J. D. Angrist 1994): Identification and estimation of local average treatment effects, Econometrica, 62, Imbens, G. W. and W. K. Newey 2009): Identification and estimation of triangular simultaneous equations models without additivity, Econometrica, 77, Manski, C. F. 1997): Monotone treatment response, Econometrica, Matzkin, R. L. 2003): Nonparametric estimation of nonadditive random functions, Econometrica, 71, Newey, W. K. and J. L. Powell 2003): Instrumental variable estimation of nonparametric models, Econometrica, 71, Torgovitsky, A. 2015): Identification of nonseparable models using instruments with small support, Econometrica, 83,

Unconditional Quantile Regression with Endogenous Regressors

Unconditional Quantile Regression with Endogenous Regressors Unconditional Quantile Regression with Endogenous Regressors Pallab Kumar Ghosh Department of Economics Syracuse University. Email: paghosh@syr.edu Abstract This paper proposes an extension of the Fortin,

More information

Control Functions in Nonseparable Simultaneous Equations Models 1

Control Functions in Nonseparable Simultaneous Equations Models 1 Control Functions in Nonseparable Simultaneous Equations Models 1 Richard Blundell 2 UCL & IFS and Rosa L. Matzkin 3 UCLA June 2013 Abstract The control function approach (Heckman and Robb (1985)) in a

More information

COUNTERFACTUAL MAPPING AND INDIVIDUAL TREATMENT EFFECTS IN NONSEPARABLE MODELS WITH DISCRETE ENDOGENEITY*

COUNTERFACTUAL MAPPING AND INDIVIDUAL TREATMENT EFFECTS IN NONSEPARABLE MODELS WITH DISCRETE ENDOGENEITY* COUNTERFACTUAL MAPPING AND INDIVIDUAL TREATMENT EFFECTS IN NONSEPARABLE MODELS WITH DISCRETE ENDOGENEITY* QUANG VUONG AND HAIQING XU ABSTRACT. This paper establishes nonparametric identification of individual

More information

Conditions for the Existence of Control Functions in Nonseparable Simultaneous Equations Models 1

Conditions for the Existence of Control Functions in Nonseparable Simultaneous Equations Models 1 Conditions for the Existence of Control Functions in Nonseparable Simultaneous Equations Models 1 Richard Blundell UCL and IFS and Rosa L. Matzkin UCLA First version: March 2008 This version: October 2010

More information

Identification in Triangular Systems using Control Functions

Identification in Triangular Systems using Control Functions Identification in Triangular Systems using Control Functions Maximilian Kasy Department of Economics, UC Berkeley Maximilian Kasy (UC Berkeley) Control Functions 1 / 19 Introduction Introduction There

More information

Additional Material for Estimating the Technology of Cognitive and Noncognitive Skill Formation (Cuttings from the Web Appendix)

Additional Material for Estimating the Technology of Cognitive and Noncognitive Skill Formation (Cuttings from the Web Appendix) Additional Material for Estimating the Technology of Cognitive and Noncognitive Skill Formation (Cuttings from the Web Appendix Flavio Cunha The University of Pennsylvania James Heckman The University

More information

IDENTIFICATION OF MARGINAL EFFECTS IN NONSEPARABLE MODELS WITHOUT MONOTONICITY

IDENTIFICATION OF MARGINAL EFFECTS IN NONSEPARABLE MODELS WITHOUT MONOTONICITY Econometrica, Vol. 75, No. 5 (September, 2007), 1513 1518 IDENTIFICATION OF MARGINAL EFFECTS IN NONSEPARABLE MODELS WITHOUT MONOTONICITY BY STEFAN HODERLEIN AND ENNO MAMMEN 1 Nonseparable models do not

More information

Flexible Estimation of Treatment Effect Parameters

Flexible Estimation of Treatment Effect Parameters Flexible Estimation of Treatment Effect Parameters Thomas MaCurdy a and Xiaohong Chen b and Han Hong c Introduction Many empirical studies of program evaluations are complicated by the presence of both

More information

Nonlinear Difference-in-Differences in Repeated Cross Sections with Continuous Treatments

Nonlinear Difference-in-Differences in Repeated Cross Sections with Continuous Treatments Nonlinear Difference-in-Differences in Repeated Cross Sections with Continuous Treatments Xavier D Haultfœuille Stefan Hoderlein Yuya Sasaki CREST Boston College Johns Hopkins March 29, 2015 Abstract This

More information

Nonadditive Models with Endogenous Regressors

Nonadditive Models with Endogenous Regressors Nonadditive Models with Endogenous Regressors Guido W. Imbens First Draft: July 2005 This Draft: February 2006 Abstract In the last fifteen years there has been much work on nonparametric identification

More information

How Revealing is Revealed Preference?

How Revealing is Revealed Preference? How Revealing is Revealed Preference? Richard Blundell UCL and IFS April 2016 Lecture II, Boston University Richard Blundell () How Revealing is Revealed Preference? Lecture II, Boston University 1 / 55

More information

Control Variables, Discrete Instruments, and. Identification of Structural Functions

Control Variables, Discrete Instruments, and. Identification of Structural Functions Control Variables, Discrete Instruments, and arxiv:1809.05706v1 [econ.em] 15 Sep 2018 Identification of Structural Functions Whitney Newey and Sami Stouli September 18, 2018. Abstract Control variables

More information

Nonlinear Difference-in-Differences in Repeated Cross Sections with Continuous Treatments

Nonlinear Difference-in-Differences in Repeated Cross Sections with Continuous Treatments Nonlinear Difference-in-Differences in Repeated Cross Sections with Continuous Treatments Xavier D Haultfœuille Stefan Hoderlein Yuya Sasaki CREST Boston College Johns Hopkins August 13, 2013 Abstract

More information

Identifying Multiple Marginal Effects with a Single Instrument

Identifying Multiple Marginal Effects with a Single Instrument Identifying Multiple Marginal Effects with a Single Instrument Carolina Caetano 1 Juan Carlos Escanciano 2 1 Department of Economics, University of Rochester 2 Department of Economics, Indiana University

More information

The Econometric Evaluation of Policy Design: Part I: Heterogeneity in Program Impacts, Modeling Self-Selection, and Parameters of Interest

The Econometric Evaluation of Policy Design: Part I: Heterogeneity in Program Impacts, Modeling Self-Selection, and Parameters of Interest The Econometric Evaluation of Policy Design: Part I: Heterogeneity in Program Impacts, Modeling Self-Selection, and Parameters of Interest Edward Vytlacil, Yale University Renmin University, Department

More information

Stochastic Demand and Revealed Preference

Stochastic Demand and Revealed Preference Stochastic Demand and Revealed Preference Richard Blundell Dennis Kristensen Rosa Matzkin UCL & IFS, Columbia and UCLA November 2010 Blundell, Kristensen and Matzkin ( ) Stochastic Demand November 2010

More information

Generated Covariates in Nonparametric Estimation: A Short Review.

Generated Covariates in Nonparametric Estimation: A Short Review. Generated Covariates in Nonparametric Estimation: A Short Review. Enno Mammen, Christoph Rothe, and Melanie Schienle Abstract In many applications, covariates are not observed but have to be estimated

More information

Identification of Nonparametric Simultaneous Equations Models with a Residual Index Structure

Identification of Nonparametric Simultaneous Equations Models with a Residual Index Structure Identification of Nonparametric Simultaneous Equations Models with a Residual Index Structure Steven T. Berry Yale University Department of Economics Cowles Foundation and NBER Philip A. Haile Yale University

More information

Identification in Nonparametric Limited Dependent Variable Models with Simultaneity and Unobserved Heterogeneity

Identification in Nonparametric Limited Dependent Variable Models with Simultaneity and Unobserved Heterogeneity Identification in Nonparametric Limited Dependent Variable Models with Simultaneity and Unobserved Heterogeneity Rosa L. Matzkin 1 Department of Economics University of California, Los Angeles First version:

More information

Lectures on Identi cation 2

Lectures on Identi cation 2 Lectures on Identi cation 2 Andrew Chesher CeMMAP & UCL April 16th 2008 Andrew Chesher (CeMMAP & UCL) Identi cation 2 4/16/2008 1 / 28 Topics 1 Monday April 14th. Motivation, history, de nitions, types

More information

Quantile methods. Class Notes Manuel Arellano December 1, Let F (r) =Pr(Y r). Forτ (0, 1), theτth population quantile of Y is defined to be

Quantile methods. Class Notes Manuel Arellano December 1, Let F (r) =Pr(Y r). Forτ (0, 1), theτth population quantile of Y is defined to be Quantile methods Class Notes Manuel Arellano December 1, 2009 1 Unconditional quantiles Let F (r) =Pr(Y r). Forτ (0, 1), theτth population quantile of Y is defined to be Q τ (Y ) q τ F 1 (τ) =inf{r : F

More information

IDENTIFICATION IN NONPARAMETRIC SIMULTANEOUS EQUATIONS

IDENTIFICATION IN NONPARAMETRIC SIMULTANEOUS EQUATIONS IDENTIFICATION IN NONPARAMETRIC SIMULTANEOUS EQUATIONS Rosa L. Matzkin Department of Economics Northwestern University This version: January 2006 Abstract This paper considers identification in parametric

More information

Identification of Regression Models with Misclassified and Endogenous Binary Regressor

Identification of Regression Models with Misclassified and Endogenous Binary Regressor Identification of Regression Models with Misclassified and Endogenous Binary Regressor A Celebration of Peter Phillips Fourty Years at Yale Conference Hiroyuki Kasahara 1 Katsumi Shimotsu 2 1 Department

More information

Causal Inference Lecture Notes: Causal Inference with Repeated Measures in Observational Studies

Causal Inference Lecture Notes: Causal Inference with Repeated Measures in Observational Studies Causal Inference Lecture Notes: Causal Inference with Repeated Measures in Observational Studies Kosuke Imai Department of Politics Princeton University November 13, 2013 So far, we have essentially assumed

More information

Nonparametric Instrumental Variables Identification and Estimation of Nonseparable Panel Models

Nonparametric Instrumental Variables Identification and Estimation of Nonseparable Panel Models Nonparametric Instrumental Variables Identification and Estimation of Nonseparable Panel Models Bradley Setzler December 8, 2016 Abstract This paper considers identification and estimation of ceteris paribus

More information

Principles Underlying Evaluation Estimators

Principles Underlying Evaluation Estimators The Principles Underlying Evaluation Estimators James J. University of Chicago Econ 350, Winter 2019 The Basic Principles Underlying the Identification of the Main Econometric Evaluation Estimators Two

More information

INFERENCE APPROACHES FOR INSTRUMENTAL VARIABLE QUANTILE REGRESSION. 1. Introduction

INFERENCE APPROACHES FOR INSTRUMENTAL VARIABLE QUANTILE REGRESSION. 1. Introduction INFERENCE APPROACHES FOR INSTRUMENTAL VARIABLE QUANTILE REGRESSION VICTOR CHERNOZHUKOV CHRISTIAN HANSEN MICHAEL JANSSON Abstract. We consider asymptotic and finite-sample confidence bounds in instrumental

More information

IDENTIFICATION, ESTIMATION, AND SPECIFICATION TESTING OF NONPARAMETRIC TRANSFORMATION MODELS

IDENTIFICATION, ESTIMATION, AND SPECIFICATION TESTING OF NONPARAMETRIC TRANSFORMATION MODELS IDENTIFICATION, ESTIMATION, AND SPECIFICATION TESTING OF NONPARAMETRIC TRANSFORMATION MODELS PIERRE-ANDRE CHIAPPORI, IVANA KOMUNJER, AND DENNIS KRISTENSEN Abstract. This paper derives necessary and sufficient

More information

Correct Specification and Identification

Correct Specification and Identification THE MILTON FRIEDMAN INSTITUTE FOR RESEARCH IN ECONOMICS MFI Working Paper Series No. 2009-003 Correct Specification and Identification of Nonparametric Transformation Models Pierre-Andre Chiappori Columbia

More information

Limited Information Econometrics

Limited Information Econometrics Limited Information Econometrics Walras-Bowley Lecture NASM 2013 at USC Andrew Chesher CeMMAP & UCL June 14th 2013 AC (CeMMAP & UCL) LIE 6/14/13 1 / 32 Limited information econometrics Limited information

More information

Random Coefficients on Endogenous Variables in Simultaneous Equations Models

Random Coefficients on Endogenous Variables in Simultaneous Equations Models Review of Economic Studies (207) 00, 6 0034-6527/7/0000000$02.00 c 207 The Review of Economic Studies Limited Random Coefficients on Endogenous Variables in Simultaneous Equations Models MATTHEW A. MASTEN

More information

Missing dependent variables in panel data models

Missing dependent variables in panel data models Missing dependent variables in panel data models Jason Abrevaya Abstract This paper considers estimation of a fixed-effects model in which the dependent variable may be missing. For cross-sectional units

More information

ESTIMATION OF NONPARAMETRIC MODELS WITH SIMULTANEITY

ESTIMATION OF NONPARAMETRIC MODELS WITH SIMULTANEITY ESTIMATION OF NONPARAMETRIC MODELS WITH SIMULTANEITY Rosa L. Matzkin Department of Economics University of California, Los Angeles First version: May 200 This version: August 204 Abstract We introduce

More information

Econometrics of causal inference. Throughout, we consider the simplest case of a linear outcome equation, and homogeneous

Econometrics of causal inference. Throughout, we consider the simplest case of a linear outcome equation, and homogeneous Econometrics of causal inference Throughout, we consider the simplest case of a linear outcome equation, and homogeneous effects: y = βx + ɛ (1) where y is some outcome, x is an explanatory variable, and

More information

Counterfactual worlds

Counterfactual worlds Counterfactual worlds Andrew Chesher Adam Rosen The Institute for Fiscal Studies Department of Economics, UCL cemmap working paper CWP22/15 Counterfactual Worlds Andrew Chesher and Adam M. Rosen CeMMAP

More information

ON ILL-POSEDNESS OF NONPARAMETRIC INSTRUMENTAL VARIABLE REGRESSION WITH CONVEXITY CONSTRAINTS

ON ILL-POSEDNESS OF NONPARAMETRIC INSTRUMENTAL VARIABLE REGRESSION WITH CONVEXITY CONSTRAINTS ON ILL-POSEDNESS OF NONPARAMETRIC INSTRUMENTAL VARIABLE REGRESSION WITH CONVEXITY CONSTRAINTS Olivier Scaillet a * This draft: July 2016. Abstract This note shows that adding monotonicity or convexity

More information

Nonseparable multinomial choice models in cross-section and panel data

Nonseparable multinomial choice models in cross-section and panel data Nonseparable multinomial choice models in cross-section and panel data Victor Chernozhukov Iván Fernández-Val Whitney K. Newey The Institute for Fiscal Studies Department of Economics, UCL cemmap working

More information

On IV estimation of the dynamic binary panel data model with fixed effects

On IV estimation of the dynamic binary panel data model with fixed effects On IV estimation of the dynamic binary panel data model with fixed effects Andrew Adrian Yu Pua March 30, 2015 Abstract A big part of applied research still uses IV to estimate a dynamic linear probability

More information

Semiparametric Estimation of Structural Functions in Nonseparable Triangular Models

Semiparametric Estimation of Structural Functions in Nonseparable Triangular Models Semiparametric Estimation of Structural Functions in Nonseparable Triangular Models Victor Chernozhukov Iván Fernández-Val Whitney Newey Sami Stouli Francis Vella Discussion Paper 17 / 690 8 November 2017

More information

Quantile and Average Effects in Nonseparable Panel Models 1

Quantile and Average Effects in Nonseparable Panel Models 1 Quantile and Average Effects in Nonseparable Panel Models 1 V. Chernozhukov, I. Fernandez-Val, W. Newey MIT, Boston University, MIT First Version, July 2009 Revised, October 2009 1 We thank seminar participants

More information

A Test for Rank Similarity and Partial Identification of the Distribution of Treatment Effects Preliminary and incomplete

A Test for Rank Similarity and Partial Identification of the Distribution of Treatment Effects Preliminary and incomplete A Test for Rank Similarity and Partial Identification of the Distribution of Treatment Effects Preliminary and incomplete Brigham R. Frandsen Lars J. Lefgren August 1, 2015 Abstract We introduce a test

More information

Testing Rank Similarity

Testing Rank Similarity Testing Rank Similarity Brigham R. Frandsen Lars J. Lefgren November 13, 2015 Abstract We introduce a test of the rank invariance or rank similarity assumption common in treatment effects and instrumental

More information

A Discontinuity Test for Identification in Nonparametric Models with Endogeneity

A Discontinuity Test for Identification in Nonparametric Models with Endogeneity A Discontinuity Test for Identification in Nonparametric Models with Endogeneity Carolina Caetano 1 Christoph Rothe 2 Nese Yildiz 1 1 Department of Economics 2 Department of Economics University of Rochester

More information

Econ 2148, fall 2017 Instrumental variables II, continuous treatment

Econ 2148, fall 2017 Instrumental variables II, continuous treatment Econ 2148, fall 2017 Instrumental variables II, continuous treatment Maximilian Kasy Department of Economics, Harvard University 1 / 35 Recall instrumental variables part I Origins of instrumental variables:

More information

New Developments in Econometrics Lecture 16: Quantile Estimation

New Developments in Econometrics Lecture 16: Quantile Estimation New Developments in Econometrics Lecture 16: Quantile Estimation Jeff Wooldridge Cemmap Lectures, UCL, June 2009 1. Review of Means, Medians, and Quantiles 2. Some Useful Asymptotic Results 3. Quantile

More information

Weak Stochastic Increasingness, Rank Exchangeability, and Partial Identification of The Distribution of Treatment Effects

Weak Stochastic Increasingness, Rank Exchangeability, and Partial Identification of The Distribution of Treatment Effects Weak Stochastic Increasingness, Rank Exchangeability, and Partial Identification of The Distribution of Treatment Effects Brigham R. Frandsen Lars J. Lefgren December 16, 2015 Abstract This article develops

More information

leebounds: Lee s (2009) treatment effects bounds for non-random sample selection for Stata

leebounds: Lee s (2009) treatment effects bounds for non-random sample selection for Stata leebounds: Lee s (2009) treatment effects bounds for non-random sample selection for Stata Harald Tauchmann (RWI & CINCH) Rheinisch-Westfälisches Institut für Wirtschaftsforschung (RWI) & CINCH Health

More information

Lecture 9: Quantile Methods 2

Lecture 9: Quantile Methods 2 Lecture 9: Quantile Methods 2 1. Equivariance. 2. GMM for Quantiles. 3. Endogenous Models 4. Empirical Examples 1 1. Equivariance to Monotone Transformations. Theorem (Equivariance of Quantiles under Monotone

More information

The properties of L p -GMM estimators

The properties of L p -GMM estimators The properties of L p -GMM estimators Robert de Jong and Chirok Han Michigan State University February 2000 Abstract This paper considers Generalized Method of Moment-type estimators for which a criterion

More information

The changes-in-changes model with covariates

The changes-in-changes model with covariates The changes-in-changes model with covariates Blaise Melly, Giulia Santangelo Bern University, European Commission - JRC First version: January 2015 Last changes: January 23, 2015 Preliminary! Abstract:

More information

Identifying Effects of Multivalued Treatments

Identifying Effects of Multivalued Treatments Identifying Effects of Multivalued Treatments Sokbae Lee Bernard Salanie The Institute for Fiscal Studies Department of Economics, UCL cemmap working paper CWP72/15 Identifying Effects of Multivalued Treatments

More information

NBER WORKING PAPER SERIES

NBER WORKING PAPER SERIES NBER WORKING PAPER SERIES IDENTIFICATION OF TREATMENT EFFECTS USING CONTROL FUNCTIONS IN MODELS WITH CONTINUOUS, ENDOGENOUS TREATMENT AND HETEROGENEOUS EFFECTS Jean-Pierre Florens James J. Heckman Costas

More information

Identification of Instrumental Variable. Correlated Random Coefficients Models

Identification of Instrumental Variable. Correlated Random Coefficients Models Identification of Instrumental Variable Correlated Random Coefficients Models Matthew A. Masten Alexander Torgovitsky January 18, 2016 Abstract We study identification and estimation of the average partial

More information

A Simple Way to Calculate Confidence Intervals for Partially Identified Parameters. By Tiemen Woutersen. Draft, September

A Simple Way to Calculate Confidence Intervals for Partially Identified Parameters. By Tiemen Woutersen. Draft, September A Simple Way to Calculate Confidence Intervals for Partially Identified Parameters By Tiemen Woutersen Draft, September 006 Abstract. This note proposes a new way to calculate confidence intervals of partially

More information

Identi cation of Positive Treatment E ects in. Randomized Experiments with Non-Compliance

Identi cation of Positive Treatment E ects in. Randomized Experiments with Non-Compliance Identi cation of Positive Treatment E ects in Randomized Experiments with Non-Compliance Aleksey Tetenov y February 18, 2012 Abstract I derive sharp nonparametric lower bounds on some parameters of the

More information

Fuzzy Differences-in-Differences

Fuzzy Differences-in-Differences Review of Economic Studies (2017) 01, 1 30 0034-6527/17/00000001$02.00 c 2017 The Review of Economic Studies Limited Fuzzy Differences-in-Differences C. DE CHAISEMARTIN University of California at Santa

More information

Midterm 1. Every element of the set of functions is continuous

Midterm 1. Every element of the set of functions is continuous Econ 200 Mathematics for Economists Midterm Question.- Consider the set of functions F C(0, ) dened by { } F = f C(0, ) f(x) = ax b, a A R and b B R That is, F is a subset of the set of continuous functions

More information

Semiparametric Identification in Panel Data Discrete Response Models

Semiparametric Identification in Panel Data Discrete Response Models Semiparametric Identification in Panel Data Discrete Response Models Eleni Aristodemou UCL March 8, 2016 Please click here for the latest version. Abstract This paper studies partial identification in

More information

Data Models. Dalia A. Ghanem. May 8, Abstract. Recent work on nonparametric identification of average partial effects (APEs) from panel

Data Models. Dalia A. Ghanem. May 8, Abstract. Recent work on nonparametric identification of average partial effects (APEs) from panel Testing Identifying Assumptions in Nonseparable Panel Data Models Dalia A. Ghanem May 8, 2015 Abstract Recent work on nonparametric identification of average partial effects (APEs) from panel data require

More information

Characterizations of identified sets delivered by structural econometric models

Characterizations of identified sets delivered by structural econometric models Characterizations of identified sets delivered by structural econometric models Andrew Chesher Adam M. Rosen The Institute for Fiscal Studies Department of Economics, UCL cemmap working paper CWP44/16

More information

Identification Analysis for Randomized Experiments with Noncompliance and Truncation-by-Death

Identification Analysis for Randomized Experiments with Noncompliance and Truncation-by-Death Identification Analysis for Randomized Experiments with Noncompliance and Truncation-by-Death Kosuke Imai First Draft: January 19, 2007 This Draft: August 24, 2007 Abstract Zhang and Rubin 2003) derives

More information

Sensitivity checks for the local average treatment effect

Sensitivity checks for the local average treatment effect Sensitivity checks for the local average treatment effect Martin Huber March 13, 2014 University of St. Gallen, Dept. of Economics Abstract: The nonparametric identification of the local average treatment

More information

Semiparametric Estimation of Invertible Models

Semiparametric Estimation of Invertible Models Semiparametric Estimation of Invertible Models Andres Santos Department of Economics University of California, San Diego e-mail: a2santos@ucsd.edu July, 2011 Abstract This paper proposes a simple estimator

More information

1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty.

1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty. 1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty. Let E be a subset of R. We say that E is bounded above if there exists a real number U such that x U for

More information

Semi and Nonparametric Models in Econometrics

Semi and Nonparametric Models in Econometrics Semi and Nonparametric Models in Econometrics Part 4: partial identification Xavier d Haultfoeuille CREST-INSEE Outline Introduction First examples: missing data Second example: incomplete models Inference

More information

QUANTILE MODELS WITH ENDOGENEITY

QUANTILE MODELS WITH ENDOGENEITY QUANTILE MODELS WITH ENDOGENEITY V. CHERNOZHUKOV AND C. HANSEN Abstract. In this article, we review quantile models with endogeneity. We focus on models that achieve identification through the use of instrumental

More information

Job Displacement of Older Workers during the Great Recession: Tight Bounds on Distributional Treatment Effect Parameters

Job Displacement of Older Workers during the Great Recession: Tight Bounds on Distributional Treatment Effect Parameters Job Displacement of Older Workers during the Great Recession: Tight Bounds on Distributional Treatment Effect Parameters using Panel Data Brantly Callaway October 24, 205 Job Market Paper Abstract Older

More information

CALIFORNIA INSTITUTE OF TECHNOLOGY

CALIFORNIA INSTITUTE OF TECHNOLOGY DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 IDENTIFYING TREATMENT EFFECTS UNDER DATA COMBINATION Yanqin Fan University of Washington Robert

More information

Comments on: Panel Data Analysis Advantages and Challenges. Manuel Arellano CEMFI, Madrid November 2006

Comments on: Panel Data Analysis Advantages and Challenges. Manuel Arellano CEMFI, Madrid November 2006 Comments on: Panel Data Analysis Advantages and Challenges Manuel Arellano CEMFI, Madrid November 2006 This paper provides an impressive, yet compact and easily accessible review of the econometric literature

More information

Continuous Treatments

Continuous Treatments Continuous Treatments Stefan Hoderlein Boston College Yuya Sasaki Brown University First Draft: July 15, 2009 This Draft: January 19, 2011 Abstract This paper explores the relationship between nonseparable

More information

Math 104: Homework 7 solutions

Math 104: Homework 7 solutions Math 04: Homework 7 solutions. (a) The derivative of f () = is f () = 2 which is unbounded as 0. Since f () is continuous on [0, ], it is uniformly continous on this interval by Theorem 9.2. Hence for

More information

Comparison of inferential methods in partially identified models in terms of error in coverage probability

Comparison of inferential methods in partially identified models in terms of error in coverage probability Comparison of inferential methods in partially identified models in terms of error in coverage probability Federico A. Bugni Department of Economics Duke University federico.bugni@duke.edu. September 22,

More information

Risk Aversion over Incomes and Risk Aversion over Commodities. By Juan E. Martinez-Legaz and John K.-H. Quah 1

Risk Aversion over Incomes and Risk Aversion over Commodities. By Juan E. Martinez-Legaz and John K.-H. Quah 1 Risk Aversion over Incomes and Risk Aversion over Commodities By Juan E. Martinez-Legaz and John K.-H. Quah 1 Abstract: This note determines the precise connection between an agent s attitude towards income

More information

Dealing with a Technological Bias: The Difference-in-Difference Approach

Dealing with a Technological Bias: The Difference-in-Difference Approach Dealing with a Technological Bias: The Difference-in-Difference Approach Dmitry Arkhangelsky Current version November 9, 2017 The most recent version is here Abstract I construct a nonlinear model for

More information

Identification and Estimation of a Triangular model with Multiple Endogenous Variables and Insufficiently Many Instrumental Variables

Identification and Estimation of a Triangular model with Multiple Endogenous Variables and Insufficiently Many Instrumental Variables Identification and Estimation of a Triangular model with Multiple Endogenous Variables and Insufficiently Many Instrumental Variables Liquan Huang Umair Khalil Neşe Yıldız First draft: April 2013. This

More information

NONPARAMETRIC IDENTIFICATION AND ESTIMATION OF TRANSFORMATION MODELS. [Preliminary and Incomplete. Please do not circulate.]

NONPARAMETRIC IDENTIFICATION AND ESTIMATION OF TRANSFORMATION MODELS. [Preliminary and Incomplete. Please do not circulate.] NONPARAMETRIC IDENTIFICATION AND ESTIMATION OF TRANSFORMATION MODELS PIERRE-ANDRE CHIAPPORI, IVANA KOMUNJER, AND DENNIS KRISTENSEN Abstract. This paper derives sufficient conditions for nonparametric transformation

More information

Program Evaluation with High-Dimensional Data

Program Evaluation with High-Dimensional Data Program Evaluation with High-Dimensional Data Alexandre Belloni Duke Victor Chernozhukov MIT Iván Fernández-Val BU Christian Hansen Booth ESWC 215 August 17, 215 Introduction Goal is to perform inference

More information

PRICES VERSUS PREFERENCES: TASTE CHANGE AND TOBACCO CONSUMPTION

PRICES VERSUS PREFERENCES: TASTE CHANGE AND TOBACCO CONSUMPTION PRICES VERSUS PREFERENCES: TASTE CHANGE AND TOBACCO CONSUMPTION AEA SESSION: REVEALED PREFERENCE THEORY AND APPLICATIONS: RECENT DEVELOPMENTS Abigail Adams (IFS & Oxford) Martin Browning (Oxford & IFS)

More information

Instrumental variable models for discrete outcomes

Instrumental variable models for discrete outcomes Instrumental variable models for discrete outcomes Andrew Chesher The Institute for Fiscal Studies Department of Economics, UCL cemmap working paper CWP30/08 Instrumental Variable Models for Discrete Outcomes

More information

CAEPR Working Paper # Identifying Multiple Marginal Effects with a Single Binary Instrument or by Regression Discontinuity

CAEPR Working Paper # Identifying Multiple Marginal Effects with a Single Binary Instrument or by Regression Discontinuity CAEPR Working Paper #205-009 Identifying Multiple Marginal Effects with a Single Binary Instrument or by Regression Discontinuity Carolina Caetano University of Rochester Juan Carlos Escanciano Indiana

More information

Solution of the 7 th Homework

Solution of the 7 th Homework Solution of the 7 th Homework Sangchul Lee December 3, 2014 1 Preliminary In this section we deal with some facts that are relevant to our problems but can be coped with only previous materials. 1.1 Maximum

More information

Chapter 6 Stochastic Regressors

Chapter 6 Stochastic Regressors Chapter 6 Stochastic Regressors 6. Stochastic regressors in non-longitudinal settings 6.2 Stochastic regressors in longitudinal settings 6.3 Longitudinal data models with heterogeneity terms and sequentially

More information

Identification of Panel Data Models with Endogenous Censoring

Identification of Panel Data Models with Endogenous Censoring Identification of Panel Data Models with Endogenous Censoring S. Khan Department of Economics Duke University Durham, NC - USA shakeebk@duke.edu M. Ponomareva Department of Economics Northern Illinois

More information

What s New in Econometrics? Lecture 14 Quantile Methods

What s New in Econometrics? Lecture 14 Quantile Methods What s New in Econometrics? Lecture 14 Quantile Methods Jeff Wooldridge NBER Summer Institute, 2007 1. Reminders About Means, Medians, and Quantiles 2. Some Useful Asymptotic Results 3. Quantile Regression

More information

Generalized instrumental variable models, methods, and applications

Generalized instrumental variable models, methods, and applications Generalized instrumental variable models, methods, and applications Andrew Chesher Adam M. Rosen The Institute for Fiscal Studies Department of Economics, UCL cemmap working paper CWP43/18 Generalized

More information

Counterfactuals via Deep IV. Matt Taddy (Chicago + MSR) Greg Lewis (MSR) Jason Hartford (UBC) Kevin Leyton-Brown (UBC)

Counterfactuals via Deep IV. Matt Taddy (Chicago + MSR) Greg Lewis (MSR) Jason Hartford (UBC) Kevin Leyton-Brown (UBC) Counterfactuals via Deep IV Matt Taddy (Chicago + MSR) Greg Lewis (MSR) Jason Hartford (UBC) Kevin Leyton-Brown (UBC) Endogenous Errors y = g p, x + e and E[ p e ] 0 If you estimate this using naïve ML,

More information

A Course in Applied Econometrics. Lecture 5. Instrumental Variables with Treatment Effect. Heterogeneity: Local Average Treatment Effects.

A Course in Applied Econometrics. Lecture 5. Instrumental Variables with Treatment Effect. Heterogeneity: Local Average Treatment Effects. A Course in Applied Econometrics Lecture 5 Outline. Introduction 2. Basics Instrumental Variables with Treatment Effect Heterogeneity: Local Average Treatment Effects 3. Local Average Treatment Effects

More information

Using Instrumental Variables for Inference about Policy Relevant Treatment Parameters

Using Instrumental Variables for Inference about Policy Relevant Treatment Parameters Using Instrumental Variables for Inference about Policy Relevant Treatment Parameters Magne Mogstad Andres Santos Alexander Torgovitsky July 22, 2016 Abstract We propose a method for using instrumental

More information

Nonparametric Instrumental Variable Estimation Under Monotonicity

Nonparametric Instrumental Variable Estimation Under Monotonicity Nonparametric Instrumental Variable Estimation Under Monotonicity Denis Chetverikov Daniel Wilhelm arxiv:157.527v1 [stat.ap] 19 Jul 215 Abstract The ill-posedness of the inverse problem of recovering a

More information

A Test for Rank Similarity and Partial Identification of the Distribution of Treatment Effects Preliminary and incomplete

A Test for Rank Similarity and Partial Identification of the Distribution of Treatment Effects Preliminary and incomplete A Test for Rank Similarity and Partial Identification of the Distribution of Treatment Effects Preliminary and incomplete Brigham R. Frandsen Lars J. Lefgren April 30, 2015 Abstract We introduce a test

More information

Closest Moment Estimation under General Conditions

Closest Moment Estimation under General Conditions Closest Moment Estimation under General Conditions Chirok Han and Robert de Jong January 28, 2002 Abstract This paper considers Closest Moment (CM) estimation with a general distance function, and avoids

More information

Endogeneity and Discrete Outcomes. Andrew Chesher Centre for Microdata Methods and Practice, UCL

Endogeneity and Discrete Outcomes. Andrew Chesher Centre for Microdata Methods and Practice, UCL Endogeneity and Discrete Outcomes Andrew Chesher Centre for Microdata Methods and Practice, UCL July 5th 2007 Accompanies the presentation Identi cation and Discrete Measurement CeMMAP Launch Conference,

More information

Identification with Latent Choice Sets: The Case of the Head Start Impact Study

Identification with Latent Choice Sets: The Case of the Head Start Impact Study Identification with Latent Choice Sets: The Case of the Head Start Impact Study Vishal Kamat University of Chicago 22 September 2018 Quantity of interest Common goal is to analyze ATE of program participation:

More information

Supplementary material to: Tolerating deance? Local average treatment eects without monotonicity.

Supplementary material to: Tolerating deance? Local average treatment eects without monotonicity. Supplementary material to: Tolerating deance? Local average treatment eects without monotonicity. Clément de Chaisemartin September 1, 2016 Abstract This paper gathers the supplementary material to de

More information

d(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N

d(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x

More information

HETEROGENEOUS CHOICE

HETEROGENEOUS CHOICE HETEROGENEOUS CHOICE Rosa L. Matzkin Department of Economics Northwestern University June 2006 This paper was presented as an invited lecture at the World Congress of the Econometric Society, London, August

More information

Groupe de lecture. Instrumental Variables Estimates of the Effect of Subsidized Training on the Quantiles of Trainee Earnings. Abadie, Angrist, Imbens

Groupe de lecture. Instrumental Variables Estimates of the Effect of Subsidized Training on the Quantiles of Trainee Earnings. Abadie, Angrist, Imbens Groupe de lecture Instrumental Variables Estimates of the Effect of Subsidized Training on the Quantiles of Trainee Earnings Abadie, Angrist, Imbens Econometrica (2002) 02 décembre 2010 Objectives Using

More information

Closest Moment Estimation under General Conditions

Closest Moment Estimation under General Conditions Closest Moment Estimation under General Conditions Chirok Han Victoria University of Wellington New Zealand Robert de Jong Ohio State University U.S.A October, 2003 Abstract This paper considers Closest

More information

NUCLEAR NORM PENALIZED ESTIMATION OF INTERACTIVE FIXED EFFECT MODELS. Incomplete and Work in Progress. 1. Introduction

NUCLEAR NORM PENALIZED ESTIMATION OF INTERACTIVE FIXED EFFECT MODELS. Incomplete and Work in Progress. 1. Introduction NUCLEAR NORM PENALIZED ESTIMATION OF IERACTIVE FIXED EFFECT MODELS HYUNGSIK ROGER MOON AND MARTIN WEIDNER Incomplete and Work in Progress. Introduction Interactive fixed effects panel regression models

More information

Decomposing Real Wage Changes in the United States

Decomposing Real Wage Changes in the United States Decomposing Real Wage Changes in the United States Iván Fernández-Val Aico van Vuuren Francis Vella October 31, 2018 Abstract We employ CPS data to analyze the sources of hourly real wage changes in the

More information