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96e92f166e6212386862c24c05ff033a482fab84 | subsection | 39 | 48 | Biot pure shear stress | Then a^2 + b^2\gamma ^2 = b^2 = p, b^2\gamma = q and c^2=r, thus\widetilde{F}^T\widetilde{F} &= \begin{pmatrix} 1 & 0 & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix}\begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \begin{pmatrix} 1 & \g... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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25a01fc5b2925849474a6990b13c1afe40dae662 | subsection | 40 | 48 | Biot pure shear stress | We use Proposition and write \lambda _1=e^\alpha , \lambda _2=e^{-\alpha }, \lambda _3=1. | {
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} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
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e64e67aee67b9fbdcc304296ec4df9e37c8ed87a | subsection | 41 | 48 | Biot pure shear stress | Then\gamma &=\frac{\lambda _1^2-\lambda _2^2}{\lambda _1^2+\lambda _2^2} = \frac{e^{2\alpha }-e^{-2\alpha }}{e^{2\alpha }+e^{-2\alpha }} = \tanh (2\alpha )\,,\qquad a = \sqrt{\frac{\lambda _1^2+\lambda _2^2}{2}} = \sqrt{\frac{e^{2\alpha } + e^{-2\alpha }}{2}} = \sqrt{\cosh (2\alpha )}\,,\\
b &= \lambda _1\lambda _2\sqr... | {
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} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
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72e080ef35517b9fdaf6e3e49dc36ec96178bca9 | subsection | 42 | 48 | Biot pure shear stress | By lemformvonpkommutierenmitt, the tensors C and \widehat{T}^{\mathrm {Biot}}(C) commute if and only if C=U^2 is of the form (REF ), i.e. if and only ifU = {\cosh (\alpha )&\sinh (\alpha )&0\\\sinh (\alpha )&\cosh (\alpha )&0\\0&0&1}=\exp {0&\alpha &0\\\alpha &0&0\\0&0&0}\qquad \text{with $\alpha \in $}is a finite pure... | {
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} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
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"Robert J. Martin",
"Patrizio Neff"
] | [
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671e5b478a0af04230c2f4d23180f9e721ac2c69 | subsection | 43 | 48 | Biot pure shear stress | For example, the requirement w^{\prime }\left(\frac{1}{\lambda }\right) = -w^{\prime }(\lambda ) from Proposition is satisfied by the function w originally proposed by Valanis and Landel as a model for incompressible materials , which they defined via the equality w^{\prime }(\lambda )=2\mu (\lambda ), i.e. (assuming W... | {
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} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
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dfa76faf94735abe12fb7015f86833c7b5a9a668 | subsection | 44 | 48 | Linearization of finite simple shear | In order to verify that the notions of finite simple shear and finite pure shear stretch (as well as the rotation R given in (REF )) are compatible to the corresponding concepts of shear in linear elasticity via the identification \gamma =2, simply compare the linearizations&\frac{1}{\sqrt{\cosh (2)}}\, {1&\sinh (2)&0\... | {
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} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
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1379f1abfce66d222cd49235cb3082d51f238a4e | subsection | 45 | 48 | Shear monotonicity | Although simple shear is generally not a suitable concept for nonlinear elasticity if (Cauchy) pure shear stresses are concerned, it does occur in finite deformations under certain displacement boundary conditions in the context of so-called anti-plane shear deformations of the form \varphi (x_1,x_2,x_3)=(x_1\,,x_2\,,x... | {
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} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
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"Robert J. Martin",
"Patrizio Neff"
] | [
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fe34e36b1e337ff5305387cace75f85816df022c | subsection | 46 | 48 | Shear monotonicity | The Cauchy stress tensor \sigma corresponding to a simple shear deformation F_\gamma with the amount of shear \gamma \in is given by\sigma &=(\beta _0+\beta _1+\beta _{-1})+\left(\begin{matrix}\beta _1^2&(\beta _1-\beta _{-1})&0\\(\beta _1-\beta _{-1})&\beta _{-1}^2&0\\0&0&0\end{matrix}\right).Again, we can observe tha... | {
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} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
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631c9c454a2207ba8ce4584464df794b00b6519d | subsection | 47 | 48 | Shear monotonicity | In particular, APS-convexity is implied by rank-one convexity , thus every Cauchy stress response induced by a rank-one convex energy is shear monotone. | {
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} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
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] | [
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a6d795ad2ed8829fa123e71d27c590f8e18772a4 | abstract | 0 | 32 | Abstract | We present spine-local type inference, a partial type inference system for
inferring omitted type annotations for System F terms based on local type
inference. Local type inference relies on bidirectional inference rules to
propagate type information into and out of adjacent nodes of the AST and
restricts type-argument... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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cbe6870d39fb7102076e5b021721a7e37b565008 | subsection | 1 | 32 | Introduction | Local type inference is a simple yet effective partial
technique for inferring types for programs. In contrast to complete
methods of type inference such as the Damas-Milner system
which can type programs without any type annotations by restricting
the language of types, partial methods require the programmer
to provid... | {
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{
"arxiv_id": "",
"doi": "10.1145/345099.345100",
"end": 98,
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"raw": "Benjamin C. Pierce and David N. Turner. 2000. Local Type Inference. ACM Trans. Program. Lang. Syst. 22, 1 (Jan. 2000), 1–44. https://doi.org/10.11... | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
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81dbc7edf49375b76f2c16ee6f2ea3e127f38098 | subsection | 2 | 32 | Introduction | The inference systems presented in , will fail here
because the argument {\lambda }\, x.\, x does not synthesize a type.
The techniques proposed in the literature of local type inference for
dealing with cases similar to this include classifying and
avoiding such “hard-to-synthesize” terms and
utilizing the partial typ... | {
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{
"arxiv_id": "",
"doi": "10.1145/360204.360207",
"end": 121,
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"raw": "Martin Odersky, Christoph Zenger, and Matthias Zenger. 2001. Colored Local Type Inference. SIGPLAN Not. 36, 3 (Jan. 2001), 41–53. https://doi.org... | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
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6f784b744a229932ea8d6f714b665a6034494978 | subsection | 3 | 32 | Contributions | In this paper, we explore the design space of local type inference in
the setting of System F, by developing
spine-local type inference, an approach that both expands the
locality of type-argument inference to an application spine
and augments its effectiveness by using the contextual type
of the spine. In doing so, we... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0304-3975(86)90044-7",
"end": 305,
"openalex_id": "https://openalex.org/W2054969282",
"raw": "Jean-Yves Girard. 1986. The system F of variable types, fifteen years later. Theoretical Computer Science 45 (1986), 159 – 192. https://do... | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
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df5ad1670b85aa776cf8eb20b2e8154c7ab90ace | subsection | 4 | 32 | Internal and External Language | Type inference can be viewed as a relation between an
internal language of terms, where all needed typing
information is present, and an external language, in which
programmers work directly and where some of this information can be
omitted for their convenience. Under this view, type inference for the
external languag... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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18f7875e4ff8a340fc0f7982bd15c74e04932970 | subsection | 5 | 32 | Syntax | We take as our internal language explicitly typed System F (see
); we review its syntax below:\textbf {Types} && S,T,U,V ::=\ & X,Y,Z\ |\ S \rightarrow T\ |\ {\forall }\, X.\, T \\
\textbf {Contexts} && \Gamma ::=\ & \cdot \ |\ \Gamma ,X\ |\ \Gamma ,x\! : \! T \\
\textbf {Terms} && e,p ::=\ & x\ |\ {\lambda }\, x\! : \... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 258,
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"raw": "Jean-Yves Girard, Paul Taylor, and Yves Lafont. 1989. Proofs and Types. Cambridge University Press, New York, NY, USA.",
"source_ref_id": "cdb060a4fbf3f5f22f9418... | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
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751e4068a0e7cd88cc0d14f699f6cdb5283e583b | subsection | 6 | 32 | Terminology | In both the internal and external languages, we say that the
applicand of a term or type application is the term in the
function position. A head a is either a variable or
λ-abstraction (bare or annotated), and an application
spine (or just spine) is a view of an application
as consisting of some head (called the spine... | {
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"doi": "10.1093/logcom/13.5.639",
"end": 379,
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"source_ref_id": "ca37fde0bff0a... | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
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c99d79f01d1e2ea254a6847e78c28d156d37e506 | subsection | 7 | 32 | Type Inference Specification | The typing rules for our internal language are standard for explicitly
typed System F and are omitted (see Ch. 23 of for a
thorough discussion of these rules). We write {\Gamma \vdash e : T} to
indicate that under context \Gamma internal term e has type
T. For type inference in the external language, Figure REF shows
j... | {
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"arxiv_id": "",
"doi": "",
"end": 160,
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"raw": "Benjamin C. Pierce. 2002. Types and Programming Languages (1st ed.). The MIT Press.",
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"start": 111
}
]
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"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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67dde4d80797c07b231638960c79b77141b4c1d9 | subsection | 8 | 32 | Bidirectional Rules | We now consider more closely each judgment form and its rules starting
with \vdash _{\delta }, the point of entry for type inference. The two
modes for type inference, checking and synthesizing, are
indicated resp. by \vdash _{\Downarrow } (suggesting pushing a type down and
into a term) and \vdash _{\Uparrow } (sugges... | {
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"arxiv_id": "",
"doi": "10.1017/s0956796806006034",
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"Christopher Jenkins",
"Aaron Stump"
] | [
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5df5301188b3f4bd87ca097b19f6b6893b648708 | subsection | 9 | 32 | Bidirectional Rules | Shared by both
is the second premise of the (anonymous) rule introducing \vdash ^{\text{I}} that
\sigma solves precisely the meta-variables of the partially inferred
type T for application t\ t^{\prime }. | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
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ea9191eb1c5fe58e2c84e44e3d0e44b0a8c74f0d | subsection | 10 | 32 | Meta-variables | What are the “meta-variables” of elaborations and types? When t is
a term application with some type arguments omitted in its spine, its
partial elaboration p from spine-local type-argument inference under
context \Gamma fills in each missing type argument with either a
well-formed type or with a meta-variable (a type ... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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06808c9e4aaacd25135910fd96f46baacb25ffa3 | subsection | 11 | 32 | Meta-variables | We have for our partially
elaborated term that MV(\Gamma ,\texttt {pair}[X][ℕ]\ ({λ}\, x\! : \! ℕ.\, x)\ z) = \lbrace X\rbrace and also for our type that
{MV(\Gamma ,⟨X\times ℕ⟩)} = \lbrace X\rbrace . If we have a derivation of the
judgment above formed by \vdash ^{\text{I}} we can then derive with rule AppChk\Gamma \v... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
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] | 2,018 | en | Computer Science | [
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b925295c8fdef2b0209537e032d9a23c68bc8362 | subsection | 12 | 32 | Specification Rules | Judgment \vdash ^{\text{I}} serves as an interface to spine-local type-argument
inference. In Figure REF it is defined in terms of the
specification for contextual type-argument inference given by
judgments \vdash ^{\text{P}} and \vdash ^{\cdot }; we call it a “shim” judgment
because in Figure REF we give for it an alt... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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fee20b91a955832727bb992f37956c52db6d7959 | subsection | 13 | 32 | Specification Rules | The details of which S
to guess, or whether we should guess at all, are not present in this
specificational rule. In both cases, we elaborate the applicand to
p[X] of type T and check that it can be applied to t^{\prime } – we do
this even when we guess S for X to maintain the invariant that for
all elaborations p and ... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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a4e212be801724d56f63aa27e8f5533497dc62c4 | subsection | 14 | 32 | Specification Rules | Y \rightarrow ⟨X \times Y⟩,[ℕ → ℕ/X])\cdot z}and furthermore that \Gamma \vdash _{\Uparrow }z : ℕ. Then we have
instantiation [ℕ/Y] from synthetic type-argument inference and use
it to produce for the application the result type [ℕ/Y]\ ⟨X \times Y⟩
= ⟨X \times ℕ⟩ and the elaboration \texttt {pair}[X][ℕ]\ ({\lambda }\, ... | {
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{
"arxiv_id": "",
"doi": "",
"end": 816,
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"raw": "Luca Cardelli. 1997. An implementation of F<:. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.6158",
"source_ref_id": "df9d25bceaf823afd83f349b83dca... | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
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] | [
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d324dbd0a8a2f8e3cbe0c4f4d071060e484eb4c7 | subsection | 15 | 32 | Soundness, Weak Completeness, and Annotation Requirements | The inference rules in Figure REF for our external language are
sound with respect to the typing rules for our internal
language (i.e. explicitly typed System F), meaning that
elaborations of typeable external terms are typeable at the same
typeA complete list of proofs
for this paper can be found in the proof appendix... | {
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{
"arxiv_id": "",
"doi": "10.1016/s0168-0072(98)00047-5",
"end": 1271,
"openalex_id": "https://openalex.org/W1993209012",
"raw": "J. B. Wells. 1998. Typability and Type Checking in System F Are Equivalent and Undecidable. ANNALS OF PURE AND APPLIED LOGIC 98 (1998), ... | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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4bbdba505da5bd1b2cfa76566a5c882554774ffb | subsection | 16 | 32 | Soundness, Weak Completeness, and Annotation Requirements | They are defined
mutually recursively below:\lfloor {{\lambda }\, x\! : \! T.\, e} \rfloor
& = \lbrace {\lambda }\, x\! : \! T.\, t \mid t \in \lfloor {e} \rfloor \rbrace
\cup \lbrace {\lambda }\, x.\, t \mid t \in \lfloor {e} \rfloor \rbrace
\\ \lfloor {{\Lambda }\, X.\, e} \rfloor
& = \lbrace {\Lambda }\, X.\, t ... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
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d23492d03daa9edad37d76ce240d015db3ccacdf | subsection | 17 | 32 | Soundness, Weak Completeness, and Annotation Requirements | S.\, t^{\prime \prime } for some t^{\prime }
If e^{\prime } occurs as a maximal term application in e and if
\Gamma ^{\prime }
\vdash ^{\text{P}}t^{\prime } : T^{\prime \prime } \rightsquigarrow (p,\sigma _{id}) for some T and p, then
MV(\Gamma ,p)\!=\!\varnothing .
If e^{\prime } is a term application and t^{\prime ... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
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8a694bcdcfb12016e792a7631394a202e874262f | subsection | 18 | 32 | Soundness, Weak Completeness, and Annotation Requirements | However, doing so means type errors may now require
non-trivial reasoning from users to determine why some meta-variables
were introduced in the first place.Still, we find it somewhat inelegant that our characterization of
annotation requirements for type inference is not fully independent of
the inference system itsel... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
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568ef85792bf4991641b7f7cebd9c657c7c6e5b6 | subsection | 19 | 32 | Body | We conclude this section with some example programs for which the type
inference system in Figures REF and REF will and
will not be able to type. We start with the motivating example from
the introduction of checking that the expression \texttt {pair}\ ({\lambda }\, x.\, x)\ z has type ⟨(ℕ → ℕ) \times ℕ⟩, which is not ... | {
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"Christopher Jenkins",
"Aaron Stump"
] | [
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9f3b947f0067ad4c55a5bdb3b0a19f3b09cbd5ee | subsection | 20 | 32 | Body | Note that we elaborate the argument {\lambda }\, x.\, x of this
application to {\lambda }\, x\! : \! ℕ.\, x – we never pass down meta-variables to
term arguments, keeping type-argument inference local to the spine.In sub-derivation \mathcal {D}_2 we type (\texttt {pair}\ ({\lambda }\, x.\, x))\ z (parentheses added) wh... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
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ddf7f978f4231bd7ae92825248102cfedfe7b05b | subsection | 21 | 32 | Body | In this case the user can expect an error message
like the following:expected type: ?Xerror: We are not in checking mode, so boundvariable x must be annotatedwhere ?X indicates an unsolved meta-variable
corresponding to type variable X in the type of pair. The
situation above corresponds to condition (1) of Theorem REF... | {
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"Christopher Jenkins",
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ddf4018bd71edfce5c1495176bc9e2fdfb94157a | subsection | 22 | 32 | Body | In particular, with local type inference we would like to
avoid error messages like the following:synthesized type: →expected type: ?X := ℕ → ℕerror: type mismatchFrom this error message alone the programmer has no
indication of why the expected type is ℕ → ℕ! In our type inference
system we expand the distance inform... | {
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"Christopher Jenkins",
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] | [
"cs.PL"
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6ede30d975ea51548a5c47654998218218d8bab4 | subsection | 23 | 32 | Body | The reason for this is two-fold: first, a partial type
inference technique is needed as complete type inference for
F_{\le } is undecidable; second, global type inference
systems fail to infer principal types in F_{\le }
, , , whereas local type inference is able to
promise that it infers the “locally best” type argume... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/lics.1996.561306",
"end": 432,
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"raw": "J. Tiuryn and P. Urzyczyn. 1996. The subtyping problem for second-order types is undecidable. In Proceedings 11th Annual IEEE Symposium on Log... | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
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4618b9ac5b7a15a60fafbb56feca5c789ec35ddc | subsection | 24 | 32 | Body | First, our rules distinguish between
checking the argument of an application with a fully known expected
type and synthesizing its argument when incomplete information is
available to keep meta-variables spine-local, whereas
in their approach meta-variables and typing constraints are passed
downwards to check term argu... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/237721.237729",
"end": 902,
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"raw": "Martin Odersky and Konstantin Läufer. 1996. Putting Type Annotations to Work. In Proceedings of the 23rd ACM SIGPLAN-SIGACT Symposium on Principl... | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
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15b8081f2c6ef782e563140e58214a62266bc607 | subsection | 25 | 32 | Algorithmic Inference Rules | The type inference system presented in Section do not
constitute an algorithm. Though the rules forming
judgment \vdash ^{\cdot } indicate where and how we use
contextually-inferred type arguments, they do not specify
what their instantiations are or even whether this
information is available to use, and it is not obvi... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
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913f12b9d1ace0eb31bd009b0d18c023b0cb7e33 | subsection | 26 | 32 | Prototype Matching | Figure REF lists the rules for the prototype matching
algorithm. We read the judgment {\overline{X}} \Vdash ^\text{:=}T := P \Rightarrow (\sigma ,W) as: “solving for meta-variables {\overline{X}}, we match type
T to prototype P and generate solution \sigma and decorated
type W,” and we maintain the invariant that dom(\... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
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4238bf227b72e7a433236014cee0454a3a282323 | subsection | 27 | 32 | Prototype Matching | Stuck decorations occur when the expected arity of a
spine head (as tracked by a given prototype) is greater than the arity
of the type of the head and are the mechanism by which we propagate a
contextual type to a head that is “over-applied” – a not-uncommon
occurrence in languages with curried applications!Turning to... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
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e23893e57d3d34adfe6a488158678cee7dcf4f10 | subsection | 28 | 32 | Decorated Type Inference | We now discuss the rules in Figures REF and
REF which implement contextual type-argument inference
(as specified by Figures REF and REF ) by
using the prototype matching algorithm. We begin by giving a reading
for judgments \Vdash ^{\text{?}} – read {\Gamma ;P \Vdash ^{\text{?}}t : W \rightsquigarrow (p,\sigma )} as: “... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0... | |
e3c639acb5cd7fe84e79434e27c49421ba461e21 | subsection | 29 | 32 | Decorated Type Inference | We are justified in requiring that matching T to ?\!\rightarrow P
generates empty solution \sigma _{id} since we have in general that the
meta-variables solved by our prototype matching judgment are a subset
of the meta-variables it was asked to solve:If {\overline{X}} \Vdash ^\text{:=}T := P \Rightarrow (\sigma ,W) th... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
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0.046761635690927505,
-0.008477835915982723,
0.004586120136082172,
0.... | |
d9835d9d4f50cd31847135077c0fdee6113f49d4 | subsection | 30 | 32 | Decorated Type Inference | W,\sigma ) \cdot t^{\prime } : W^{\prime }
\rightsquigarrow (p^{\prime },W^{\prime }) as: “under \Gamma , elaborated applicand p of
decorated type W together with solution \sigma can be
applied to t^{\prime }; the application has decorated type W^{\prime } and
elaborates p^{\prime } with solution \sigma ^{\prime }.” Th... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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c6bbae2655b1bc68df0732bfa5e793d5557c34fa | subsection | 31 | 32 | Decorated Type Inference | Indeed, the judgment \vdash ^{\cdot } provides
more flexibility in reasoning about type inference than does
\Vdash ^{\cdot }, as in rule PForall we may freely decline to
guess a contextual type argument even when this would be justified
and instead try to learn it synthetically. In contrast, algorithmic
rule ?Forall re... | {
"cite_spans": []
} | 1805.10383 | Spine-local Type Inference | [
"Christopher Jenkins",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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9e34b562a57ce287c06b2f8d74ead788b07540d9 | abstract | 0 | 55 | Abstract | Unobserved heterogeneous treatment effects have been emphasized in the recent
policy evaluation literature (see e.g., Heckman and Vytlacil, 2005). This paper
proposes a nonparametric test for unobserved heterogeneous treatment effects in
a treatment effect model with a binary treatment assignment, allowing for
individu... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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-... | |
c6d11516f9d0077f93c3b7b8e80aaa254b922217 | subsection | 1 | 55 | Body | 11em11emcolorlinks = true,
urlcolor = black,
citecolor = black,
linkcolor = blackassumptionassumptionassumptions
assumption213Testing for unobserved heterogeneous treatment effects in a nonseparable model with endogenous selection
[]Yu-Chin HsuInstitute of Economics, Academia Sinica. Email: ychsu@econ.sinica.edu.
[]Ta-... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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... | |
f421290b36a11b43e38071146f4eba100a88fb5f | subsection | 2 | 55 | Introduction | Heterogeneous treatment effects due to unobserved latent variables has been emphasized in the policy evaluation literature. See e.g.
,
, ,
,
,
,
,
, , ,
,
,
,
,
,
and among many others. The interpretation and credibility of the instrumental variable (IV) approach relies on the hypothesis that treatment effects are hom... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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0... | |
a388bcb7521699b2d588e4e0335fac36b1c038da | subsection | 3 | 55 | Introduction | We distinguish the cases whether covariates include continuous variables.
In sec:simulations, we conduct Monte Carlo experiments to study the finite-sample performance of the proposed test.
sec:empirical illustrates our testing approach by two empirical applications. sec:extensions extends our approach to the Regressio... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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0... | |
dae01a246e9595ac42042b83ef00b87247936248 | subsection | 4 | 55 | Model and Testable Restrictions | We consider the following nonseparable treatment effect model:Y=g(D,X,\epsilon )where Y \in R is outcome variable, D{0,1} denotes treatment status, XdX are covariates, is an unobserved random disturbance of general form (e.g. without invoking any restriction on the dimensionality of ), and g is an unknown but smooth fu... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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22b712cf68bd25fb529a0c67cec2937416f4bcb5 | subsection | 5 | 55 | Model and Testable Restrictions | Throughout, we maintain {ass_a}.
\end{}Moreover, let (x,z) = E(Y|X=x,Z=z). Under H0 and {ass_a}, we have
\mu (x,z) = \left[g(0,x,\epsilon )|X=x\right] + \delta _0(x) p(x,z), \ \ \text{for }\ z=0,1.
In the above equation system, we treat [g(0,X,)|X=x] and 0(x) as two unknowns. Solve the equations, then we identify 0(x... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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29a5464e7667f640dfc6c892509fd02c09b5353d | subsection | 6 | 55 | Model and Testable Restrictions | Moreover, shows that such a monotonicity condition is observationally equivalent to the weak monotonicity of (REF ) in the error term \eta . point out assc can be relaxed to the strict monotonicity of (Y\le y;D=1|X,Z=1)-(Y\le y;D=1|X,Z=0) in y\in S^\circ _{Y|X,D=1}, the interior region of S_{Y|X,D=1}.Note that the seco... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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0... | |
07b9ddb3602b73f25b8d75e88720298b98835d31 | subsection | 7 | 55 | Model and Testable Restrictions | Then H0 holds if and only if WZ|X.From now on, we maintain assa,assc,assb,assd. By thm1, testing the null hypothesis _0 is equivalent to testing the conditional independence condition W\protect {\protect {\perp }Z|X. It is worth pointing out that {thm1} is related to \cite {LW_2014_JoE}, who show that H}_0 holds if and... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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... | |
a0b5aa4819f0244e09f9c8b72a9c43c420966616 | subsection | 8 | 55 | Case 1: discrete covariates | We first discuss the case where X takes only a finite number of values. Let \lbrace (Y_i,D_i,X^{\prime }_i, Z_i)^{\prime }: i\le n\rbrace be a random sample of (Y,D,X^{\prime },Z)^{\prime }. By thm1, we test _0 via the following model restrictions:F_{W|XZ}(\cdot \ |x,0) = F_{W|XZ}(\cdot \ |x,1), \ \forall \ x\in S_X,wh... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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... | |
bb056a17c0505d9fd50fd1dc4c26a6c8f680426b | subsection | 9 | 55 | Case 1: discrete covariates | Moreover, the probability distribution of Y given (D,X,Z) admits a uniformly continuous density function f_{Y|DXZ} and (Y^2)<\infty .Theorem 3.2
Suppose assa,assc,assb,assd,ass5 hold. Then, under _0,\hat{\mathcal {T}}_n \overset{d}{\rightarrow } \ \sup _{w \in R}; \ x\in S_X|Z(w,x)|,whereZ(,x) is a mean--zero Gaussian... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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-0.011079638265073... | |
15aabadb0ea1ad37e370fcecdb037b5ffd696a90 | subsection | 10 | 55 | Case 1: discrete covariates | Namely,
\begin{align*}
&\hat{\psi }_{wx,i} = \left[1\right.(\hat{W}_i \le w) - \frac{\sum _{j=1}^n 1}{(}\hat{W}_j\le w; X_j=x)\end{align*}{\sum _{j=1}^n 1}(X_j=x) ] [ 1(Xi=x,Zi=0)P(X=x,Z=0) - 1(Xi=x,Zi=1) P(X=x,Z=1) ];wx,i = (w,x) [ Wi - j=1n Wj 1(Xj=x)j=1n 1(Xj=x)] [ 1(Xi=x,Zi=0) P(X=x,Z=0) - 1(Xi=x,Zi=0)P(X=x,Z=1) ],... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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-0.007105281576514244... | |
5cb712d9bc50c58dee2873956d757934a1297ebd | subsection | 11 | 55 | Case 2: Continuous Covariates | We now consider the case where X\in is continuously distributed with a finite support.
To extend the empirical process argument used in the proof of thm:testnoX to this case, we propose a modified Kolmogorov–Smirnov test statistic. Such a modification allows the generated variable W to be constructed from the unknown f... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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... | |
d2ae44d9c3e38768ff635eda34c31f7df46a56e4 | subsection | 12 | 55 | Case 2: Continuous Covariates | Then, we estimate \delta (X_i) by\hat{\delta }(X_i)=\frac{\sum _{j\ne i} Y_j Z_j K\left( \frac{X_j-X_i}{h} \right)\times \sum _{j\ne i} K\left( \frac{X_j-X_i}{h} \right) -\sum _{j\ne i} Y_j K\left( \frac{X_j-X_i}{h} \right)\times \sum _{j\ne i} Z_jK\left( \frac{X_j-X_i}{h} \right)}{\sum _{j\ne i} D_j Z_j K\left( \frac{... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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407c088a0537aea3e780669129fcaad302177417 | subsection | 13 | 55 | Case 2: Continuous Covariates | For z=0,1, \sup _{(x,z)\in S_{XZ}} f_{X|Z}(x|z)\le \overline{f}<+\infty and \inf _{x\in S_X} |f_{XZ}(x,1)-f_{XZ}(x,0)|>0.Assumption E
For z\in \lbrace 0,1\rbrace , functions f_{X|Z}(x|z), p(x,z) and \mu (x,z) are continuous in x.Assumption F
The support of K is a convex (possibly unbounded) subset of with nonempty in... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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0.... | |
e4ffb912102d54e058df18d9f6ac6fcc1bd9bc61 | subsection | 14 | 55 | Case 2: Continuous Covariates | Note that in the definition of \tilde{G}(w,x,z), there contains no nonparametric elements estimated in the indicate function.To establish asymptotic properties for inference, we make the following assumption.Assumption I
\sup _{x\in S_X}\left| [\hat{\delta }(x)]-\delta (x)\right|=o_p(n^{-\frac{1}{2}}) and \sup _{xz\in... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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... | |
6b18213558be71248e1087c5d305ab991386e3b6 | subsection | 15 | 55 | Case 2: Continuous Covariates | Then, under H0,
\begin{align*}
& \hat{\mathcal {T}}^c_n\overset{d}{\rightarrow }\sup _{w\in ; \ x\in S_X} |\mathcal {Z}^c(w,x)|
\end{align*}
where Zc(,) is a mean--zero Gaussian process with the following covariance kernel
\text{Cov} \left[ \mathcal {Z}^c(w,x), \mathcal {Z}^c(w^{\prime },x^{\prime }) \right] = \left[ ... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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341f5f8858f652cb1ac82d3805d1d45823e05d35 | subsection | 16 | 55 | Monte Carlo Simulations | In this section, we investigate the finite sample performance of our tests with a simulation study. The data are simulated as follows:&Y = D + X + [\gamma +(1-\gamma ) D]\times \epsilon ; \\
& D = 1[ () 0.5 Z ],
where (\epsilon ,\eta ) conforms to a joint normal distribution with zero mean, unit variance and correlati... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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... | |
28a76d5936917bbcc5d67347564caef77bd0a26c | subsection | 17 | 55 | Monte Carlo Simulations | To compute the suprema, we calculate the test statistic by using n/20 grid points in the support [\min _{i=1}^n(\hat{W}_i),\max _{i=1}^n(\hat{W}_i)], as well as in the support [\min _{i=1}^n(X_i),\max _{i=1}^n(X_i)].
table:tab2 reports the size and power properties of our test, which are qualitatively similar to the re... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
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0.04402856156229973,
0.012769198045134544,
0.017666345462203026,
0.040916357189416885,
-0.018780028447508812,
0.05187010020017624,
-0.007422001101076603,
-0.... | |
6bf423b341635669f33da0e047cbea37b97693e8 | subsection | 18 | 55 | Monte Carlo Simulations | In particular, when \gamma is closer to 1, it is more difficult to detect such a “local” alternative. Therefore, we obtain relatively small power even when sample size reaches n=2000 in Panel B. For relatively “small” sample size, e.g., n=1000, our results show that our test performs better with a larger bandwidth choi... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.07331324368715286,
0.03201731666922569,
-0.04550793021917343,
-0.03044544719159603,
-0.028018968179821968,
-0.06800246238708496,
0.04590471461415291,
0.025394096970558167,
0.010659721679985523,
0.03165105730295181,
-0.00851556845009327,
0.05963950231671333,
-0.016237571835517883,
-0.010... | |
fc14ce1c016d34ccc61453d7a7ed974062326d9e | subsection | 19 | 55 | The Effect of Job Training Program on Earnings | We now apply our tests to study the effects of the job training program on earnings, i.e., the National Job Training Partnership Act (JTPA), commissioned by the Department of Labor. This program began funding training from 1983 to late 1990's to increase employment and earnings for participants.
The major component of ... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.035223495215177536,
0.00838618353009224,
-0.027836939319968224,
0.007565878797322512,
-0.03485722094774246,
-0.00268029747530818,
0.03885572776198387,
0.04105338081717491,
-0.014139761216938496,
0.018451130017638206,
-0.00449832109734416,
-0.000663397426251322,
-0.05207216739654541,
-0.... | |
4100c9cab9a2b819963ac1a23cf85433f3e50a1b | subsection | 20 | 55 | The Effect of Job Training Program on Earnings | The effects of JTPA training programs on earnings have also been studied by e.g. , under a general framework allowing for unobserved heterogeneous treatment effects.The data is publicly available at http://upjohn.org/services/resources/employment-research-data-center/national-jtpa-study.Our sample consists of 11,204 ob... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.03785121440887451,
0.004834424704313278,
-0.0350429005920887,
-0.0023218211717903614,
-0.02638901397585869,
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0.03357769176363945,
0.021092895418405533,
-0.00028235785430297256,
0.034401871263980865,
0.0015243508387356997,
0.014171312563121319,
-0.035073425620794296,
... | |
768a42c0ca49860b46db5fccce51c630621af85f | subsection | 21 | 55 | The Impact of Fertility on Family Income | The second empirical illustration considers the heterogeneous impacts of children on parents' labor supply and income. Recently, have studied the heterogeneous effects of fertility on family income within the general LATE framework.
To deal with the endogeneity of fertility decisions, , , , , among many others, suggest... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.05114130303263664,
0.011122317984700203,
-0.07073123008012772,
0.0422312431037426,
-0.04058349132537842,
-0.031459834426641464,
0.020383287221193314,
0.025479109957814217,
0.026318242773413658,
0.060051362961530685,
-0.026272471994161606,
0.03362632170319557,
-0.011404572054743767,
-0.0... | |
3c53f8102d513804cecb1dc1cf0f3a1f1974f2f9 | subsection | 22 | 55 | The Impact of Fertility on Family Income | Some covariates, i.e., age, years in education, and working hours per week, are treated as continuous variables.Similar to the previous empirical illustration, we use the second kernel Gaussian kernel with various bandwidth choices for robustness check.
For the critical value, we use 5,000 bootstrapped samples and sear... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.06925767660140991,
0.011853131465613842,
-0.01890094019472599,
0.00248465733602643,
-0.0034838595893234015,
-0.037161167711019516,
0.042805518954992294,
0.018717879429459572,
0.0165516696870327,
0.04253092780709267,
-0.021112913265824318,
0.02463681809604168,
-0.019923023879528046,
-0.0... | |
db67dc97b11f911de64174bc8643e0598b7cdd24 | subsection | 23 | 55 | Extensions | Our analysis naturally extends to the Fuzzy Regression Discontinuity (FRD) design, which has recently become a popular tool to address causal inference questions in empirical studies , , .Consider a nonparametric FRD design: LetY=Y(0)\times (1-D)+Y(1)\times D,where Y is the observed outcome variable, (Y(0),Y(1))\in ^2 ... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.10107546299695969,
0.020614877343177795,
-0.036377400159835815,
0.017074795439839363,
-0.0027447084430605173,
-0.016784874722361565,
0.030090700834989548,
0.030365362763404846,
0.028168071061372757,
0.04370170831680298,
-0.06390459090471268,
-0.002403288846835494,
0.02592500112950802,
0... | |
32a71001dd25c6a4876d5e190670009f31e5b1c3 | subsection | 24 | 55 | Extensions | The assignment of the treatment is given byD = 1[(X,R)],
where R is a continuous running variable, and \theta (\cdot ,\cdot ) is monotone in R, and \eta \in R is an unobserved error term. Moreover, let R=0 be the cutoff point of the running variable, and we assume the probability of receiving the treatment is a contin... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.046755965799093246,
0.007023313548415899,
-0.05185272917151451,
-0.002205038210377097,
-0.005394332110881805,
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0.03283904492855072,
0.04654232785105705,
0.021058496087789536,
0.026689354330301285,
-0.029893908649683,
-0.00950684305280447,
0.022965967655181885,
0.020... | |
477795ed21a710b0bc6a7fdd8a5d7a4d820c818e | subsection | 25 | 55 | Proof of prop1 | For the “if” part, under (REF ), we haveg(1,x,\epsilon ) - g(0,x,\epsilon ) = m(1,x) - m(0,x) \equiv \delta _0(x), \ \ \forall x \in {S}_X.For the “only if” part, () impliesg(d,x,\epsilon ) = d \times [ g(1,x,\epsilon ) - g(0,x,\epsilon ) ] + g(0,x,\epsilon ) = d \times \delta _0(x) + g(0,x,\epsilon ).Therefore, (REF )... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.028139935806393623,
0.03528174012899399,
-0.03164979815483093,
-0.024416429921984673,
-0.013398515991866589,
-0.020113034173846245,
0.024813197553157806,
0.027849990874528885,
0.011696996167302132,
0.003620498813688755,
0.009362175129354,
-0.03851692005991936,
-0.020570842549204826,
0.0... | |
a986cb641d28c87c477413911159560e26257b63 | subsection | 26 | 55 | Proof of thm1 | Because prop1 provides the only if part, then it suffices to show the if part. Suppose W \protect {\protect {\perp }Z \ \vert \ X. By the definition of W, we have: for any y\in ,
\begin{multline*}
P\end{multline*}(Y\le y, D=1 | X, Z = 1 ) + P}(Y + \delta (X) \le y, D=0 | X, Z = 1 ) \\
=P(Yy, D=1 | X, Z = 0 ) + P(Y + (X... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.025460083037614822,
0.004423861391842365,
-0.010510483756661415,
-0.0334077812731266,
0.010075725615024567,
-0.03566547483205795,
0.04039442911744118,
0.04283517971634865,
0.027733033522963524,
0.04619121178984642,
-0.017725953832268715,
-0.04439115896821022,
-0.007879049517214298,
-0.0... | |
09bf89bfeb254e0391a00e28fe8196ee2b87bee2 | subsection | 27 | 55 | Proof of thm1 | Suppose W \protect {\protect {\perp }Z \ \vert \ X. By the definition of W, we have: for any y\in ,
\begin{multline*}
P\end{multline*}(Y\le y, D=1 | X, Z = 1 ) + P}(Y + \delta (X) \le y, D=0 | X, Z = 1 ) \\
=P(Yy, D=1 | X, Z = 0 ) + P(Y + (X) y, D=0 | X, Z = 0 ).
It follows thatP(Y y, D=1 | X, Z = 1 ) - P(Y y, D=1 | X... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.005940803326666355,
-0.014964566566050053,
-0.04209309071302414,
-0.014285399578511715,
0.010095931589603424,
-0.007085466757416725,
0.0512809231877327,
0.0435887835919857,
0.04154365137219429,
0.037789154797792435,
-0.004799955524504185,
-0.03333260118961334,
0.007066389080137014,
-0.0... | |
5aab8eacdf9270eb26668405e805962d145c9e4e | subsection | 28 | 55 | Proof of thm:testnoX | Let 1WXZ(w,x,z) = 1(W w) 1XZ(x,z) and 1WXZ(w,x,z) = 1(W w) 1XZ(x,z). Let further 1W()XZ(w,x,z)=1(W() w) 1XZ(x,z), where W()=Y+(1-D)(X), be a function indexed by ()SX.
By definition, 1W()XZ(w,x,z)=1WXZ(w,x,z) and 1W()XZ(w,x,z)=1WXZ(w,x,z).
We first derive the asymptotics of \sqrt{n}[\hat{F}_{W|XZ}(w|x,z)-F_{W|XZ}(w|x,z)... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.03470538184046745,
0.021854927763342857,
-0.05390475317835808,
0.0014212570386007428,
-0.000017706097423797473,
-0.011415841989219189,
0.021061312407255173,
-0.01658959873020649,
-0.028112273663282394,
-0.004025305155664682,
0.0038555175997316837,
-0.018741516396403313,
0.0242662951350212... | |
ab64fe2b86e89782a9a579eb73f86266137cf5b6 | subsection | 29 | 55 | Proof of thm:testnoX | Therefore, we have
\begin{multline*}
\sqrt{n}\ \left\lbrace [ 1\right._{W(\hat{\delta })XZ} (\cdot , x, z)] -F_{W|XZ}(w|x,z) \\
+\sqrt{n} \left\lbrace _n [ 1\right._{WXZ} (\cdot , x, z)] - [1\end{multline*}_{WXZ} (\cdot , x, z)]
-f_{WDXZ}(w,0,x,z) \times \sqrt{n} [\hat{\delta }(x) - \delta (x)]+o_p(1).
Moreover, _n [1... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
0.001484801759943366,
0.02238360047340393,
-0.02279556915163994,
-0.020598405972123146,
-0.011634284630417824,
0.03022625297307968,
0.008132555522024632,
0.0033834788482636213,
-0.010367863811552525,
0.0019415904534980655,
0.00504279462620616,
-0.0049893916584551334,
-0.022231020033359528,
... | |
1cc8ec61f5c6b0236cf78544a9a37995f8245e6a | subsection | 30 | 55 | Proof of thm:testnoX | Moreover, applying lemmaB1, we have&&\sqrt{n} \left[\hat{F}_{W|XZ}(w|x,1) - \hat{F}_{W|XZ}(w |x,0)\right]-\sqrt{n} \left[ F_{W|XZ}(w|x,1)- F_{W|XZ}(w |x,0)\right]\\
&=&\sqrt{n} _n \left\lbrace [1\right.(W \le w) - F_{W|XZ}(w|x,1)] \times \frac{1}{_}{XZ}(x,1)(X=x,Z=1) }-n n {[1(W w) - FW|XZ(w|x,0)] 1XZ(x,0) (X=x,Z=0) }+... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.01265918742865324,
0.04972107708454132,
-0.03818358853459358,
-0.021503068506717682,
-0.0020202049054205418,
-0.04431860148906708,
0.012621034868061543,
-0.019763289019465446,
-0.016894178465008736,
0.010896516032516956,
-0.012704971246421337,
-0.03201805427670479,
0.0038858898915350437,
... | |
e06416b70e31fd20c5f09b0d41f8237e1484d7ec | subsection | 31 | 55 | Proof of thm:testnoX | By definition,F_{W|XZ}(w|x,z)=\frac{ [1}{_}{WXZ} (w,x,z)][1XZ(x,z)] and FW|XZ(w|x,z) = n [1WXZ(w,x,z)]n [1XZ(x,z)].In the expectation[ 1W()XZ (, x, z )] discussed below, we treat as an index rather than a random object. Note that \begin{multline*}
_n [1\end{multline*}_{\hat{W}XZ} (\cdot , x, z)]
= _n [ 1WXZ (, x, z)] -... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.014606758952140808,
0.004151555243879557,
-0.04508711397647858,
-0.0002496943052392453,
-0.005502337124198675,
0.03412822633981705,
0.008471004664897919,
-0.004456816706806421,
-0.015782015398144722,
-0.004804051481187344,
-0.010516256093978882,
-0.0094936303794384,
0.017033588141202927,
... | |
7351286a23b6e47b10ed154be8bbac2f7d038977 | subsection | 32 | 55 | Proof of thm:testnoX | Thus, by Slutsky�s theorem, we have
\begin{align*}
&\sqrt{n} \left[\hat{F}_{W|XZ}(w|x,1)-\hat{F}_{W|XZ}(w |x,0)\right]-\sqrt{n} \left[ F_{W|XZ}(w|x,1)- F_{W|XZ}(w |x,0)\right]\\
&= \frac{\sqrt{n} \big \lbrace _n [1}{_}{WXZ} (w,x,1)] - [1\end{align*}_{WXZ} (w,x,1)] \big \rbrace - f_{WDXZ}(w,0,x,1) \times \sqrt{n} [\hat{... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
0.004841023590415716,
0.0476396381855011,
-0.011055386625230312,
-0.0341503880918026,
-0.012749172747135162,
0.011375832371413708,
0.010109308175742626,
-0.014923627488315105,
-0.023346779868006706,
0.021439362317323685,
0.016510598361492157,
-0.00408187136054039,
-0.02655123919248581,
-0.... | |
799c2652818af744b2a30746b937fad0736ca77b | subsection | 33 | 55 | Proof of thm:testnoX | Moreover, applying lemmaB1, we have&&\sqrt{n} \left[\hat{F}_{W|XZ}(w|x,1) - \hat{F}_{W|XZ}(w |x,0)\right]-\sqrt{n} \left[ F_{W|XZ}(w|x,1)- F_{W|XZ}(w |x,0)\right]\\
&=&\sqrt{n} _n \left\lbrace [1\right.(W \le w) - F_{W|XZ}(w|x,1)] \times \frac{1}{_}{XZ}(x,1)(X=x,Z=1) }-n n {[1(W w) - FW|XZ(w|x,0)] 1XZ(x,0) (X=x,Z=0) }+... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
0.0005975770764052868,
0.04120563343167305,
-0.039557408541440964,
-0.01365890447050333,
-0.008874473161995411,
-0.038214411586523056,
0.0025810750667005777,
-0.01822204701602459,
-0.02180846408009529,
0.005555130075663328,
-0.021182747557759285,
-0.03299503028392792,
0.010927123948931694,
... | |
3cf177dfb40875b1384727dcbb2f9e84b8499c1c | subsection | 34 | 55 | Proof of lem:approximation | Fix X=x and w.l.o.g., let z=1.
Note that&&\hat{G}(w,x,1)-\tilde{G}(w,x,1) \\
&=& _n \left\lbrace 1\right.^{*}_{XZ}(x,1) \hat{f}_{XZ}(X,0)(w-\hat{W}) \left[ 1\right.(\hat{W} \le w) - 1( W w)] }= n { 1*XZ(x,1) fXZ(X,0)(w-W) [ 1(W w) - 1( W w) ] 1 (|W-w|n-r) }+ n { 1*XZ(x,1) fXZ(X,0)(w-W) [ 1(W w)-1( W w)] 1 (|W-w|> n-r) ... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.032321639358997345,
0.04422479122877121,
-0.008606894873082638,
0.0004740278236567974,
-0.007340277545154095,
-0.010926484130322933,
0.05472398176789284,
0.01310109905898571,
-0.03644195944070816,
-0.017656343057751656,
-0.01345208939164877,
-0.04178312048316002,
-0.002088774461299181,
... | |
e755ca23342959634d1babced5c4cb18ba9b4c4d | subsection | 35 | 55 | Proof of lem:approximation | Because W is a bounded random variable and w belongs to a compact set, then \sqrt{ \hat{W}^2-2w\cdot (\hat{W})+ w^2}=O(1). Moreover, by lem:smalldistance1, | _2|\le o(n^{-k}) for any k>0. Hence, T2 = op(n-12).Fix X=x and w.l.o.g., let z=1.
Note that&&\hat{G}(w,x,1)-\tilde{G}(w,x,1) \\
&=& _n \left\lbrace 1\right.^{*}_{... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.06219850108027458,
0.03573819622397423,
-0.03119080886244774,
-0.003505914006382227,
0.010742822661995888,
-0.01970026083290577,
0.004574092570692301,
-0.012619764544069767,
-0.028474584221839905,
0.00462368642911315,
-0.03247262164950371,
-0.0003450120857451111,
-0.004982289392501116,
... | |
61117be76818866c6631780c606023b1c593749a | subsection | 36 | 55 | Proof of lem:approximation | Then, we have _1 = o_p(n^{-\frac{1}{2}}).For term _2, note that| _2 |\le \frac{\overline{K}}{h}\times \sqrt{ (w-\hat{W})^2}\times \sqrt{ \left(|\hat{W}-W|>n^{-r}\right)}\\
\le \frac{\overline{K}}{h}\times \sqrt{ \hat{W}^2-2w\cdot (\hat{W})+ w^2} \times \sqrt{ \left[|\hat{\delta }(X)-\delta (X)|>n^{-r}\right]},where \ov... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.04122425988316536,
0.0230074692517519,
-0.01382279023528099,
0.016370700672268867,
0.00233049807138741,
0.000963571306783706,
0.0015218800399452448,
-0.014913661405444145,
-0.007632285822182894,
0.006171433255076408,
-0.022763358429074287,
0.006919023580849171,
-0.007132621016353369,
-0... | |
bebb08b7591986604850f1aca190b7693d565f0f | subsection | 37 | 55 | Proof of thm:testcontX | By lem:approximation, we have\hat{\mathcal {T}}^c_n=\sqrt{n} \left| \tilde{G}(w, x, 1) - \tilde{G}(w, x, 0) \right|+o_p(1).Let 1*WXZ(w,x,z)1 (Ww,Xx,Z=z).
Note that
\tilde{G}(w,x,z) = _{1}(w,x,z)+_{2}(w,x,z)+o_p(n^{-1/2})
where
\begin{align*}
&_{1}(w,x,z)\equiv \frac{1}{n}\sum _{i=1}^n 1\end{align*}^{*}_{W_iX_iZ_i}(w,... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.017210884019732475,
0.053616177290678024,
-0.01855357736349106,
-0.04220328480005264,
0.00830028485506773,
-0.01196217443794012,
-0.009047920815646648,
-0.016463248059153557,
-0.00008171307854354382,
0.02551116794347763,
-0.01396095659583807,
-0.024046411737799644,
-0.01743975281715393,
... | |
86dcfb889220ff3934aecd899e4c789e5237d078 | subsection | 38 | 55 | Proof of thm:testcontX | Therefore,_{2}(w,x,z)= \frac{1}{n(n-1)}\sum _{i=1}^n\sum _{j\ne i}\zeta ^*_{n,ij}(w,x,z),which is a \mathcal {U}-process indexed by (w,x,z_\ell ). By and ,&&_{2}(w,x, z)-_2(w,x,z)\\
&=& \frac{2}{n}\sum _{i=1}^n\left\lbrace [\zeta ^*_{n,ij}(w,x,z)|Y_i,D_i,X_i,Z_i]- [\zeta ^*_{n,ij}(w,x,z)]\right\rbrace + o_p(n^{-1/2}).w... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.03822972625494003,
0.04198252409696579,
-0.024271145462989807,
0.012379657477140427,
-0.008260732516646385,
0.02219642885029316,
-0.011929626576602459,
0.04027393087744713,
-0.019969157874584198,
0.015293415635824203,
-0.012379657477140427,
0.0027936226688325405,
0.019969157874584198,
-... | |
66b828a65164841b85bad4c5bf2507c94bacbf10 | subsection | 39 | 55 | Proof of thm:testcontX | Thus,
\begin{multline*}
_2(w, x, 1) - _2(w, x, 0)-\left[ _2(w,x,1)- _2(w,x,0)\right] \\
= _n\left\lbrace \Big [\frac{1}{^}*_{XZ}(x,1)\right.{f_{XZ}(X,1)}-\frac{1}{^}*_{XZ}(x,0)\end{multline*}{f_{XZ}(X,0)}\Big ] f_{XZ}(X,0)f_{XZ}(X,1)\big [ \lambda (W-w)- (\lambda (W-w)|X)\big ] +o_p(n^{-\frac{1}{2}}).We now turn to _1(... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.0033602616749703884,
0.014081822708249092,
-0.030208026990294456,
0.005225378554314375,
-0.02692786231637001,
0.027034658938646317,
0.003011267399415374,
0.032374460250139236,
0.0019814481493085623,
-0.013189313001930714,
-0.021679598838090897,
-0.03542577847838402,
0.0028491662815213203,... | |
6c9038745ab28b2ddf3233b97c63321b0c26fcc0 | subsection | 40 | 55 | Proof of thm:testcontX | \end{multline*}
Note that [n,ij(w,x,z)|Yi,Di,Xi,Zi]=0 and
\begin{align*}
& [\xi _{n,ji}(w,x,z)|Y_i,D_i,X_i,Z_i] = \left\lbrace [\xi _{n,ji}(w,x,z)|X_j,Z_j,Y_i,D_i,X_i,Z_i]\big |Y_i,D_i,X_i,Z_i\right\rbrace \\
&= \Bigg \lbrace 1\end{align*}^{*}_{X_jZ_j}(x,z) f_{XZ}(X_j,z^{\prime }) P(Ww;D=0|Xj,Zj)[Wi-(W|Xj)]1hK(Xi-Xjh) ... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.0205812007188797,
0.011404335498809814,
-0.0326492004096508,
-0.0010145662818104029,
-0.024502156302332878,
0.00026794467703439295,
0.03393075615167618,
-0.004199388902634382,
0.011038175784051418,
-0.0003156216407660395,
-0.015447343699634075,
-0.006499326787889004,
-0.000546378258150070... | |
a5b7514c6194c1cc00b36d94048cd8eaaacfd1b3 | subsection | 41 | 55 | Proof of thm:testcontX | Therefore, under _0,&&\sqrt{n} \left[ \tilde{G}(w, x, 1) - \tilde{G}(w, x, 0) \right]\\
&=&\sqrt{n} \left\lbrace _1(w, x, 1) - _1(w, x, 0)-\left[ _1(w,x,1)- _1(w,x,0)\right]\right\rbrace \\
&+&\sqrt{n} \left\lbrace _2(w, x, 1) - _2(w, x, 0)-\left[ _2(w,x,1)- _2(w,x,0)\right]\right\rbrace +o_p(1)\\
&=& \sqrt{n} \times _... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.02625875361263752,
0.03713759779930115,
-0.01509763952344656,
-0.021269438788294792,
0.0028341449797153473,
-0.013983816839754581,
-0.005439422558993101,
-0.004001370165497065,
-0.015311249531805515,
0.03967040032148361,
-0.00014602235751226544,
-0.04470549151301384,
0.013991445302963257,... | |
4d4aae928d0765c788df45503b0949ed3c0b0efd | subsection | 42 | 55 | Proof of thm:testcontX | By definition,_2(w,x,z)
&=&\frac{1}{n(n-1)}\sum _{i=1}^n\sum _{j\ne i} \lbrace 1*XiZi(x,z) (Wi-w) 1h K(Xj-Xih)1(Zj=z') }= 1n(n-1)i=1nji n,ij(w,x,z)
where \zeta _{n,ij}(w,x,z)=1*XiZi(x,z) (Wi-w) 1h K(Xj-Xih)1(Zj=z').
Let \zeta ^*_{n,ij}(w,x,z)=\frac{1}{2}\left[ \zeta _{n,ij}(w,x,z)+ \zeta _{n,ji}(w,x,z)\right].
Then, \... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.03093106672167778,
0.04443575069308281,
-0.031175218522548676,
0.009666912257671356,
-0.011139456182718277,
0.026795733720064163,
-0.012413627468049526,
0.03090054728090763,
-0.00011534058285178617,
0.021302303299307823,
-0.0064395214430987835,
-0.0031014992855489254,
0.003715695347636938... | |
d2eb79fbf767928f1de684444a3029a0d4880012 | subsection | 43 | 55 | Proof of thm:testcontX | Moreover, by \cite {powell1989semiparametric},
\begin{multline*}
\frac{2}{\sqrt{n}}\sum _{i=1}^n\left\lbrace [\zeta ^*_{n,ij}(w,x,z)|Y_i,D_i,X_i]- [\zeta ^*_{n,ij}(w,x,z)]\right\rbrace \\
= _n\left\lbrace 1\right.^*_{XZ}(x,z) f_{XZ}(X,z^{\prime }) \lambda (W-w)-u^e(w,x) \\
+_n\left\lbrace 1\right.^*_{XZ}(x,z^{\prime })... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.026433555409312248,
0.02328961156308651,
-0.017444316297769547,
-0.013728050515055656,
-0.01691015064716339,
0.021656591445207596,
0.015338177792727947,
0.02269439771771431,
-0.02077140286564827,
0.021290306001901627,
-0.04584665223956108,
0.0032622243743389845,
-0.0013335057301446795,
... | |
3d4d5e44fe4734346db807a16399073f3206a747 | subsection | 44 | 55 | Proof of thm:testcontX | By a similar decomposition argument on \hat{\delta }(X)-\delta (X) in lem:smalldistance1, we have_1(w,x,z)=-\frac{1}{n(n-1)}\sum _{i=1}^n\sum _{j\ne i} \xi _{n,ij}(w,x,z)+o_p(n^{-1/2})where \xi _{n,ij}(w,x,z)= 1*WiXiZi(w,x,z) fXZ(Xi,z') (1-Di) [Wj- (Wj|Xi)]1hK(Xj-Xih) p(Xi,1)-p(Xi,0)[ 1(Zj=1)fXZ(Xi,1)
- 1(Zj=0)fXZ(Xi,0... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.024167466908693314,
0.022855345159769058,
-0.03506113588809967,
-0.012198163196444511,
-0.013800173066556454,
0.044978342950344086,
-0.007365432567894459,
0.01527249626815319,
-0.008429625071585178,
-0.014601178467273712,
-0.010382551699876785,
-0.016111643984913826,
-0.004592428915202618... | |
2ad5b8e39c2c1b86ca7bc71be5268d4036586d7e | subsection | 45 | 55 | Proof of thm:testcontX | \end{multline*}
Note that [n,ij(w,x,z)|Yi,Di,Xi,Zi]=0 and
\begin{align*}
& [\xi _{n,ji}(w,x,z)|Y_i,D_i,X_i,Z_i] = \left\lbrace [\xi _{n,ji}(w,x,z)|X_j,Z_j,Y_i,D_i,X_i,Z_i]\big |Y_i,D_i,X_i,Z_i\right\rbrace \\
&= \Bigg \lbrace 1\end{align*}^{*}_{X_jZ_j}(x,z) f_{XZ}(X_j,z^{\prime }) P(Ww;D=0|Xj,Zj)[Wi-(W|Xj)]1hK(Xi-Xjh) ... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.0205812007188797,
0.011404335498809814,
-0.0326492004096508,
-0.0010145662818104029,
-0.024502156302332878,
0.00026794467703439295,
0.03393075615167618,
-0.004199388902634382,
0.011038175784051418,
-0.0003156216407660395,
-0.015447343699634075,
-0.006499326787889004,
-0.000546378258150070... | |
3fc92c656af9626461ee0ce5d51988b643bc59ed | subsection | 46 | 55 | Proof of thm:testcontX | Therefore, under _0,&&\sqrt{n} \left[ \tilde{G}(w, x, 1) - \tilde{G}(w, x, 0) \right]\\
&=&\sqrt{n} \left\lbrace _1(w, x, 1) - _1(w, x, 0)-\left[ _1(w,x,1)- _1(w,x,0)\right]\right\rbrace \\
&+&\sqrt{n} \left\lbrace _2(w, x, 1) - _2(w, x, 0)-\left[ _2(w,x,1)- _2(w,x,0)\right]\right\rbrace +o_p(1)\\
&=& \sqrt{n} \times _... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.03811001032590866,
0.027552712708711624,
-0.017437845468521118,
-0.026133885607123375,
0.007498426362872124,
-0.022136759012937546,
-0.016644522547721863,
-0.0016791364178061485,
0.0029158429242670536,
0.03454005718231201,
0.0027461175341159105,
-0.03627926483750343,
0.020717930048704147,... | |
ec5c1a25fdc0f420a2a63e04d0e12c256debe722 | subsection | 47 | 55 | Technical Lemmas | Let \Delta p(x)\equiv p(x,1) - p(x,0), which is strictly positive by assa.Lemma 8.1
Suppose assa,ass5 hold. Then, we have\sqrt{n}[\hat{\delta }(x)-\delta (x)]= \frac{1}{\Delta p(x)}\times \sqrt{n}_n \left\lbrace \big [W - (W|X=x,Z=0)\big ]\times \frac{1}{_}{XZ}(x,1)\right.{ (X=x,Z=1)} \\
-\frac{1}{\Delta p(x)}\times \... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.02358529344201088,
0.01215115562081337,
-0.03524063900113106,
0.002929092152044177,
-0.013379238545894623,
0.002454258967190981,
0.0256295558065176,
0.014637832529842854,
0.01471411157399416,
0.009237319231033325,
-0.014691228047013283,
-0.029046829789876938,
-0.014523414894938469,
-0.0... | |
a946d8d8dfcf92a805e010cd123e6fbe66a05ec7 | subsection | 48 | 55 | Technical Lemmas | Therefore,&=&\frac{\left[_{n}(1)-(1)\right] (0) + (1) \left[_n(0)-(0)\right]}{(1) (0)-(0)(1)}\\
&-& \frac{\left[_{n}(0)-(0)\right] (1)+(0) \left[ _n(1)-(1)\right]}{(1) (0)-(0)(1)}+o_p(n^{-1/2})\\
&=&\frac{_{n}(1) (0) -(0) _n(1)-_{n}(0) (1) + (1) _n(0)}{(1) (0)-(0)(1)}\\
&+& \frac{2\left[(0)(1)-(1)(0)\right]}{(1) (0)-(0... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.016491323709487915,
0.04872644692659378,
-0.023249562829732895,
-0.0032189355697482824,
-0.00565220694988966,
-0.0032475399784743786,
0.02399708889424801,
0.02913823351264,
0.0035011647269129753,
0.02004588395357132,
-0.007330324966460466,
-0.009504251182079315,
-0.019145801663398743,
0... | |
54076b6b4db8c021d7625d07ef86c12ea930b0a4 | subsection | 49 | 55 | Technical Lemmas | Then for any k>0 and r\in (\frac{1}{4},\iota ),\sup _{x\in S_X}n^k \times P[|(x)-(x)|> n-r]0.First, by a similar decomposition of \hat{\delta }(x)-\delta (x) as that in the proof of thm:testnoX, it suffices to show&\sup _x n^k \times P{|an(x,z) - a(x,z)|>an-r}0;xnk P{|bn(x,z) - b(x,z)|>bn-r}0;xnk P{|qn(x,z) - q(x,z)|>q... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.007064434699714184,
0.08111130446195602,
-0.010047364979982376,
0.008140120655298233,
-0.021651042625308037,
0.022108780220150948,
0.043332599103450775,
0.005565340165048838,
-0.028303511440753937,
0.034849174320697784,
0.008941163308918476,
-0.0008763789664953947,
-0.03143139183521271,
... | |
13d68c14bc40b645db2e1c3ded18d2ed443de459 | subsection | 50 | 55 | Technical Lemmas | It follows that
P{1n|=1n( Txzj-E Txzj)|>nxz} 2(-a4nhn-2r2C +23K a n-r).For sufficiently large, we have 23K a n-r1. Therefore, for sufficiently large n,
P{1n|=1n(Txzj-ETxzj)|>nxz} 2 (- n2-2r2C +1) =o(n-k)where the inequality comes from {ass_G}. Note that the upper bound does not depend onor z. Therefore,
\sup _{x,z}P... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.01548522338271141,
0.016309067606925964,
-0.029307501390576363,
-0.00102599139790982,
-0.004687522072345018,
0.02651558443903923,
0.01624804176390171,
-0.010336196050047874,
-0.03893427550792694,
0.012891639024019241,
-0.024394948035478592,
0.0005411246092990041,
-0.03896478936076164,
0... | |
20495bdef5dfb7ffeb438cce71bd097935c611a6 | subsection | 51 | 55 | Technical Lemmas | Therefore,&=&\frac{\left[_{n}(1)-(1)\right] (0) + (1) \left[_n(0)-(0)\right]}{(1) (0)-(0)(1)}\\
&-& \frac{\left[_{n}(0)-(0)\right] (1)+(0) \left[ _n(1)-(1)\right]}{(1) (0)-(0)(1)}+o_p(n^{-1/2})\\
&=&\frac{_{n}(1) (0) -(0) _n(1)-_{n}(0) (1) + (1) _n(0)}{(1) (0)-(0)(1)}\\
&+& \frac{2\left[(0)(1)-(1)(0)\right]}{(1) (0)-(0... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.016491323709487915,
0.04872644692659378,
-0.023249562829732895,
-0.0032189355697482824,
-0.00565220694988966,
-0.0032475399784743786,
0.02399708889424801,
0.02913823351264,
0.0035011647269129753,
0.02004588395357132,
-0.007330324966460466,
-0.009504251182079315,
-0.019145801663398743,
0... | |
cbacfd1e7d40e8d5f80d3b61850840a92e263aa2 | subsection | 52 | 55 | Technical Lemmas | Then for any k>0 and r\in (\frac{1}{4},\iota ),\sup _{x\in S_X}n^k \times P[|(x)-(x)|> n-r]0.First, by a similar decomposition of \hat{\delta }(x)-\delta (x) as that in the proof of thm:testnoX, it suffices to show&\sup _x n^k \times P{|an(x,z) - a(x,z)|>an-r}0;xnk P{|bn(x,z) - b(x,z)|>bn-r}0;xnk P{|qn(x,z) - q(x,z)|>q... | {
"cite_spans": []
} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
] | [
"econ.EM"
] | 2,018 | en | Economics | [
-0.007064434699714184,
0.08111130446195602,
-0.010047364979982376,
0.008140120655298233,
-0.021651042625308037,
0.022108780220150948,
0.043332599103450775,
0.005565340165048838,
-0.028303511440753937,
0.034849174320697784,
0.008941163308918476,
-0.0008763789664953947,
-0.03143139183521271,
... | |
c0e3b88d90d7d2fbba26d3cf7f0c20ba8f912514 | subsection | 53 | 55 | Technical Lemmas | It follows that
P{1n|=1n( Txzj-E Txzj)|>nxz} 2(-a4nhn-2r2C +23K a n-r).For sufficiently large, we have 23K a n-r1. Therefore, for sufficiently large n,
P{1n|=1n(Txzj-ETxzj)|>nxz} 2 (- n2-2r2C +1) =o(n-k)where the inequality comes from {ass_G}. Note that the upper bound does not depend onor z. Therefore,
\sup _{x,z}P... | {
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} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
Observational Data | [
"Yu-Chin Hsu",
"Ta-Cheng Huang",
"Haiqing Xu"
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5ac116999c9f18744e22d7db8070b78bcbeda009 | subsection | 54 | 55 | Tables | Descriptive Statistics for the 1999 and 2000 Censuses
[Table: NO_CAPTION]Note: Data from the 1\% and 5\% PUMS in 1990 and 2000. Own calculations using the PUMS sample weights. The sample consists of married mother between 21 and 35 years of age with at least one child.Descriptive Statistics for the 1999 and 2000 Census... | {
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} | 1803.07514 | Testing for Unobserved Heterogeneous Treatment Effects with
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"Yu-Chin Hsu",
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db2b263d7c6dd8808ebc05b0003fbff7f722340a | abstract | 0 | 40 | Abstract | Yang-Baxter string sigma-models provide a systematic way to deform coset
geometries, such as $AdS_p \times S^p$, while retaining the $\sigma$-model
integrability. It has been shown that the Yang-Baxter deformation in target
space is simply an open-closed string map that can be defined for any geometry,
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} | 10.1007/JHEP06(2018)161 | 1803.07498 | Yang-Baxter Deformations Beyond Coset Spaces (a slick way to do TsT) | [
"I. Bakhmatov",
"E. Ó Colgáin",
"M. M. Sheikh-Jabbari",
"H. Yavartanoo"
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7af548e251b836cb0986bd1bd80187171b1437c7 | subsection | 1 | 40 | Introduction | Klimcik's pioneering work on integrable deformations of \sigma -models , paved the way for their application to string \sigma -models and AdS/CFT geometries , . Thanks to this breakthrough, we now understand noncommutative , , and marginal deformations , of AdS/CFT geometries in a new light: they are part of a larger f... | {
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"raw": "C. Klimcik, “Yang-Baxter sigma models and dS/AdS T duality,” JHEP 0212, 051 (2002) [hep-th/0210095].",
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... | 10.1007/JHEP06(2018)161 | 1803.07498 | Yang-Baxter Deformations Beyond Coset Spaces (a slick way to do TsT) | [
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6021a8aaca52ed5d2e0a2aa3e390361d0b1da5b3 | subsection | 2 | 40 | Introduction | Building on this observation, a general prescription for transforming the dilaton, RR sector, as well as introducing a Killing vector I, was presented in , where the method was applied to explicit coset and non-coset geometries alike In a series of papers , , , the same map has been embedded in DFT, where \Theta = \bet... | {
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"I. Bakhmatov",
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02df8c35d8cbaebbc03d786dfeae563ff316d7c5 | subsection | 3 | 40 | Introduction | While it is easy to invert () for explicit solutions, such as AdS_2 \times S^2 and the Schwarzschild black hole , for arbitrary G and \Theta extracting g and B, so that one can check the EOMs, is challenging.To overcome this difficulty, we work perturbatively in the deformation parameter \Theta about an arbitrary backg... | {
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"I. Bakhmatov",
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"M. M. Sheikh-Jabbari",
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