Transition Probabilities and Moment Restrictions in Dynamic Fixed Effects Logit Models

• Dynamic fixed-effects logit models help distinguish structural state dependence from serial correlation caused by unobserved heterogeneity in persistent discrete processes.The models feature lagged outcomes, strictly exogenous regressors, additive fixed effects, and logistic errors.Source: §1 · p. 2
• Conditional-likelihood methods can eliminate fixed effects but require covariate matching, ruling out time effects, producing N rates for continuous regressors, and limiting higher-order-lag extensions.These limitations motivate alternative moment-based methods.Source: §1 · pp. 2–3
• The paper introduces an analytic procedure for additive-fixed-effect models that uses logit transition probabilities and rational-fraction identities without numerical experimentation or symbolic computing.The method constructs moment restrictions for AR(p) models with arbitrary p ≥1, first-order panel vector autoregressions, and dynamic multinomial logit models.Source: §1 · p. 3
• The two-step method computes fixed-effect-free transition functions and differences distinct functions for the same transition probability, yielding all available binary-response moment equalities when T ≥ p + 2.Its period-by-period structure remains tractable as the number of periods and lag order increase.Source: §1 · p. 4
• The resulting moment functions also support identification results, including average marginal effects in arbitrary-order AR(p) models as differences of average transition functions.The approach solves repeated problems period by period rather than a single large full-likelihood system.Source: §1 · pp. 4–5
• The semiparametric efficiency bound for the AR(1) is presented for the base case of four waves of data.Source: §1 · p. 5

Extensions

• Extensions cover VAR(1) and the one-lag dynamic multinomial logit model, MAR(1).Source: §1 · p. 5

Empirical illustration and appendices

• The paper includes an empirical illustration of drug-consumption dynamics among young people, while simulations assess small-sample GMM performance and appendices provide proofs.The simulations are available in Appendix Section D.Source: §1 · p. 5

Model setting

• The models describe discrete outcomes using lagged outcomes, strictly exogenous regressors, additive individual fixed effects, and logistic errors.The framework includes dynamic binary-response specifications such as the canonical AR(1) model and treats the initial condition and covariates as observed data.Source: §2 · p. 5

Objective

• The analysis targets identification and estimation of the common parameter θ0 in short panels with fixed T.Average and counterfactual marginal effects are also among the quantities of interest.Source: §2 · p. 6
• The paper constructs moment functions free of the fixed effect, yielding conditional moment restrictions that can support identification and GMM estimation.The construction uses the law of iterated expectations after eliminating the fixed effect from suitable conditional functions.Source: §2 · p. 6

Contribution

• The proposed algebraic procedure replaces case-by-case numerical searches and provides a blueprint for valid moments in general dynamic fixed-effects logit models.It exploits the mechanics of the logistic assumption and is designed to remain tractable as model complexity increases.Source: §2 · p. 6
• Valid moment functions are characterized for arbitrary-lag AR(p) models with p > 1 and extended to multidimensional fixed-effects settings including VAR(1), network formation, and MAR(1) models.The authors describe the AR(p), p > 1 characterization as new to the literature and motivate multidimensional extensions where numerical searches are cumbersome or intractable.Source: §2 · p. 7

functions

• The method first computes transition functions for the model’s one-period-ahead transition probabilities.With p lags, the transition probability depends on the most recent p outcomes rather than the entire history.Source: functions · pp. 7–8
• At least T = p + 1 periods are required for analytical transition-function formulas accommodating arbitrary regressors and the initial condition.These functions are associated with transition probabilities in period t = p and are unique under the stated results.Source: functions · p. 8
• A further period, T ≥ p + 2, permits distinct transition functions for the same transition probabilities across periods t ∈ {p + 1, . . . , T − 1}.This additional variation is needed because the first transition-function construction alone does not immediately produce moment equality restrictions on θ0.Source: functions · p. 8
• Partial fraction decompositions adapted to the transition probabilities are the key algebraic ingredient in constructing the required transition functions.The procedure exploits rational-fraction structure rather than relying on numerical, model-by-model searches.Source: functions · p. 8
• The second step differences two transition functions associated with the same transition probability to obtain valid moment functions.The resulting procedure is then demonstrated in scalar and multidimensional fixed-effects models.Source: functions · p. 8

4.1.1 The number of moment restrictions in the AR(1)

• The AR(1) model’s transition probabilities form a basis for conditional-history probabilities, implying 2T − 2T linearly independent moment restrictions in general.The basis has 2T elements and spans the image of the conditional expectation operator; valid moments correspond to its kernel.Source: §4.1.1 · pp. 9–10
• A constructive partial-fraction procedure replaces impractical symbolic linear-system calculations and derives analytic valid moment functions for arbitrary T.The argument relies on distinct irreducible denominator terms, with sufficient conditions including γ0 ≠ 0 and a continuously distributed regressor with nonzero slope.Source: §4.1.1 · pp. 9, 11

4.1.2 Construction of valid moment functions for the pure model

• For the pure model without exogenous regressors, Lemma 1 constructs functions of three consecutive outcomes whose conditional expectations reproduce transition probabilities.The construction imposes a functional form that removes inadequate poles and a normalization matching the target transition probability, differing conceptually from conditional-likelihood approaches established by Cox (1958), Chamberlain (1985), and Magnac (2000).Source: §4.1.2 · pp. 11, 13
• The second construction step exploits time-invariant transition probabilities to generate valid moments across periods, requiring at least four total periods of observations.This requirement coincides with the conditional-likelihood requirement stated for the pure model.Source: §4.1.2 · p. 13
• For T = 3, the proposed moments account for all information in the pure AR(1) model, with the efficient moment corresponding to the conditional score.This extends the established efficiency result for the pure model to the proposed moment-restriction approach.Source: §4.1.2 · p. 14

4.1.3 Construction of valid moment functions with strictly exogenous regressors

• With strictly exogenous covariates, naive differences of transition functions generally fail, but partial-fraction constructions yield valid moments for T ≥ 3 and recursive extensions for T ≥ 4.The transition probabilities depend on covariates, while the constructed functions combine past outcomes and transitions using the logistic structure; numerical checks suggest linear independence and completeness of the available moment restrictions.Source: §4.1.3 · pp. 15–18

4.2 Semiparametric efficiency bound for the AR(1) with regressors

• For T = 3, under Assumption 1, the efficient score equals the efficient conditional moment and the finite semiparametric efficiency bound is V0(y0) = E[D(Xi, y0)Σ(Xi, y0)−1D(Xi, y0)′|Yi0 = y0]−1.The result fixes the initial condition y0 ∈ {0, 1} and requires the matrix in Assumption 1 to be nonsingular.Source: §4.2 · p. 19

4.3 Connections to other works on the AR(1) logit model

• The AR(1) transition functions can be algebraically re-expressed using δ = (eγ −1), linking the resulting moment conditions to Kitazawa (2022).The connection is obtained after algebraic manipulation of the transition functions from Lemma 2, or Lemma 1 without covariates.Source: §4.3 · p. 20
• For T = 3, the paper’s AR(1) moment functions coincide with those of Kitazawa (2022) and Honor´e and Weidner (2020), while T ≥4 yields additional identifying moments.The h-form and g-form connection provides a new perspective on Kitazawa (2022); the paper also notes that scalar methods are difficult to adapt to multivariate fixed-effect models because of computational barriers.Source: §4.3 · pp. 20–21

4.4 Moment restrictions for the AR(p) logit model, p > 1

• For AR(p) logit models, transition probabilities generate moment restrictions for each of the 2^p non-redundant transitions, extending the identification scheme to arbitrary lag order.The extension targets persistent stochastic processes and better empirical fit, while retaining the logistic index structure in common parameters, regressors, and the fixed effect.Source: §4.4 · pp. 21–22, §4.4.2 · p. 24

4.4.1 Impossibility results and number of moment restrictions when p ≥1

• Theorem 3 proves that no moment restrictions for common parameters exist when T ≤ p + 1, whereas beyond this threshold 2^T − (T − p + 1)2^p restrictions exist.Under weaker parameter or regressor conditions, additional moments may arise even when T ≤ p + 1.Source: §4.4.1 · pp. 22–23

4.4.2 Construction of transition probabilities with p > 1

• The construction starts from unique transition functions at t = p, shifts them across later periods, applies partial fraction decompositions, and differences equivalent functions to obtain linearly independent moments.The method requires T = p + 2 for identifying moments and generalizes the AR(1) blueprint because transition probabilities retain the same logistic functional form.Source: §4.4.2 · pp. 23–24

Appendix Section C.

• Theorem 4 computes transition functions for arbitrary lag order by combining AR(p − 1) transition functions with partial fraction decompositions.For example, an AR(2) can be treated as an AR(1) with the second lag as an additional strictly exogenous regressor.Source: Appendix Section C · pp. 24–25
• For lag order p > 1, a recursive algorithm computes transition functions and yields 2T −(T + 1 −p)2p numerically linearly independent moment functions.Combined with Theorem 3, the two-step procedure recovers all moment equality conditions in the model.Source: Appendix Section C · pp. 25, 27
• The moment construction extends to known-link models with interaction effects and can accommodate misspecified individual unobserved heterogeneity when the fixed effect enters additively.The extension leverages rational-fraction identities and includes models such as random-effect AR(2) logit processes.Source: Appendix Section C · p. 27

4.5 Identification with more than one lag

• The proposed point-identification approach uses a continuous covariate component with unbounded support and a known-sign nonzero effect, alongside regularity conditions enabling identification at infinity.The limiting moment function resembles valid AR(1) moments, and monotonicity over ordered covariate regions identifies θ0.Source: §4.5 · pp. 28–29, Assumption 3. (i) θ0 = (γ01, γ02, β′ · p. 29

Assumption 3. (i) θ0 = (γ01, γ02, β′

• Theorem 5 establishes point identification of θ0 in higher-order dynamic logit models with T = 4 under Assumptions 2 and 3.The guarantee applies to short panels but requires knowledge of aspects of the data-generating process beyond the baseline setup.Source: Assumption 3. (i) θ0 = (γ01, γ02, β′ · p. 30
• For AR(p) logit models, θ0 is identified at infinity under Assumptions 2 and 3 when T ≥ 2 + p, with the construction generalizing to p ≥ 3.The scalar identification argument extends naturally to higher lag orders.Source: Assumption 3. (i) θ0 = (γ01, γ02, β′ · pp. 30–31

4.6 Average Marginal Effects in AR(p) logit models

• Average marginal effects are expectations of transition functions and are identified whenever θ0 is identified; for AR(1), T ≥ 3 with Xi3 − Xi2 near zero suffices.The result interprets AMEs as population-average causal effects of changing past outcomes from 0 to 1.Source: §4.6 · pp. 31–32
• For AR(p), multiperiod transition contrasts and survivor-function analogs are identified under conditions ensuring θ0 identification, including T ≥ p + 2.The construction accommodates counterfactuals over multiple future periods and survival in the same state.Source: §4.6 · pp. 32–33
• The multiperiod identification result follows by decomposing products of transition probabilities into linear combinations of transitions, then taking expectations.Partial fraction decomposition supplies the linear representation under appropriate regressor and common-parameter conditions.Source: §4.6 · p. 33

5 Multi-dimensional fixed effects models

• The scalar-case blueprint for valid moment functions carries over to VAR(1) and MAR(1) models with multidimensional fixed effects, without claiming exhaustiveness.The multidimensional extension is presented as a generalization while leaving broader possibilities for future work.Source: §5 · p. 34

5.1 Moment restrictions for the VAR(1) logit model

• In VAR(1) logit models, the approach avoids the strong regressor restriction used in prior identification work, though its first construction step differs from the scalar case.Only transition functions associated with πk|k appear to exist at this step.Source: §5.1 · p. 35
• The VAR(1) extension derives transition functions and valid moment restrictions that generalize the AR(1) structure, with additional functions available as T increases.The construction uses partial-fraction identities and extends sequentially beyond T = 4.Source: §5.1 · pp. 35–36

5.2 Moment restrictions for the dynamic multinomial logit model

• For dynamic multinomial logit models with fixed effects, the method provides an alternative estimation strategy and derives transition functions and moment functions for arbitrary numbers of alternatives.The baseline MAR(1) model has C + 1 alternatives, with transition functions initially constructed for staying in the same state and extended when T ≥ 3.Source: §5.2 · pp. 37–39

6 Empirical Illustration

• The empirical illustration revisits Deza (2015) to study drug-consumption dynamics among young adults, distinguishing causal state dependence within and between drugs from latent traits increasing general illicit-substance propensity.Source: §6 · p. 40
• Within-drug state dependence is positive for all substances, statistically significant for alcohol and marijuana but not hard drugs after accounting for unobserved heterogeneity.The absence of significance for hard drugs is based on the four-wave sample.Source: §6 · p. 43, Table 1 · p. 44
• Naive MLE estimates within-drug dependence at coefficients four to eight times larger than GMM, while random-effects estimates lie between the approaches.The naive estimator mistakes serial correlation from A_i for state dependence; the random-effects approach still overstates within-state dependence.Source: §6 · p. 43
• The model estimates a positive and statistically significant marijuana effect on hard-drug consumption, supporting a gateway interpretation consistent with Deza (2015).The other estimators also find a positive influence, although it is statistically insignificant for the random-effects estimator.Source: §6 · pp. 43–44
• Between-state-dependence estimates vary substantially across specifications, with disagreements between random-effects and GMM estimates for some cross-effects.Overall, the comparison indicates that flexible control of unobserved heterogeneity is important for accurately characterizing state dependence.Source: §6 · pp. 44–45

7 Conclusion

• The paper introduces a procedure for logistic dynamic discrete-choice models with potentially many lags while remaining agnostic about unobserved individual heterogeneity.The approach is motivated by concerns about misspecifying initial conditions and unit-specific effects.Source: §7 · p. 45
• For binary-response models, the paper also derives general expressions for average marginal effects, while noting that its model coverage is not exhaustive.The conclusion identifies extension to other settings with logit noise as an avenue for future work.Source: §7 · p. 45
• The framework permits estimation of dynamic ordered logit models with potentially arbitrary lag order.Source: §7 · p. 45

A Partial Fraction Decomposition

• The appendix establishes partial-fraction decomposition lemmas for arbitrary real parameters indexed by K and binary vectors indexed by M.Lemma 8 applies for K ≥1, while Lemma 9 fixes M ≥2 and Y = {0, 1}^M.Source: A Partial Fraction Decomposition · pp. 50–52

B Connection to Kitazawa and Honoré-Weidner

• For T = 3, the paper’s moment restrictions are identical to the moment functions of Honoré and Weidner (2020).The connection follows by rearranging the moment functions and using binary-outcome cancellation identities, including the initial condition Y_i0 = 0.Source: B Connection to Kitazawa and… · pp. 54–55

C The remaining steps for the AR(p) model with p > 1

• With at least T = p + 3 observed periods, additional transition functions can be constructed iteratively from those in Lemma 10.These extensions are summarized in Corollary 10.1 for periods after the initial condition.Source: C The remaining steps for the AR(p)… · p. 56
• The procedure generalizes valid moment functions from the AR(1) case to AR(p) models with p > 1.Proposition 4 provides one such generalized family, while Lemma 10 supplies the simplest functions when T ≥ p + 2.Source: C The remaining steps for the AR(p)… · pp. 55–56
• The constructed family contains precisely 2^T − (T + 1 − p)2^p distinct moment functions.The count aggregates choices across periods, conditioning states, and feasible ordered index sequences.Source: C The remaining steps for the AR(p)… · p. 57
• Numerical experiments for AR(1) and AR(2) suggest that Proposition 4's moment functions are linearly independent and therefore form a complete family under Theorem 3.The completeness claim is based on the reported numerical linear-independence evidence together with Theorem 3.Source: C The remaining steps for the AR(p)… · p. 57
• Functional differencing can be decomposed into equivalent period-by-period subproblems that recover all moment equality restrictions, which may help when the full likelihood is high-dimensional.The paper highlights this advantage for sophisticated models with a few lags.Source: C The remaining steps for the AR(p)… · pp. 57–58

D Simulation Experiments

• The section reports simulations assessing the finite-sample performance of GMM estimators based on the proposed moment conditions.The supplied passage identifies the simulation objective but does not provide numerical outcomes.Source: D Simulation Experiments · p. 58

D.1 Monte Carlo for an AR(3) logit model

• The AR(3) design uses T = 5, eight valid moment functions, four sample sizes, and 1000 replications to compare equally weighted and rescaled GMM estimators.The moment vector combines valid functions with initial conditions and covariates; the rescaling choice is motivated by improved finite-sample performance.Source: §D.1 · pp. 58–59
• For the AR(3), rescaled GMM nearly eliminates lag-parameter bias when N ≥8000, whereas the unnormalized estimator retains substantial bias in lag and slope coefficients.Both estimators show negative lag-parameter bias through N = 2000; the rescaled estimator is approximately centered at true values for larger samples.Source: §D.1 · p. 60
• In the AR(3), the slope coefficient is accurate even at N = 500, while precise transition-parameter estimation requires larger samples and becomes noisier toward earlier lags.The reported precision ranking places the first-lag coefficient as noisier than the second-lag coefficient, which is noisier than the third-lag coefficient.Source: §D.1 · pp. 60–61

D.2 Monte Carlo for a VAR(1) logit model

• Estimator θ̂a is negatively biased for transition and slope parameters across the three sample sizes, while rescaled and iterated GMM demonstrate better accuracy, especially for γ12 and γ21.For the iterated GMM, convergence is attained in approximately 91% of replications for N = 2000 and approximately 100% for N = 8000 and 16000.Source: §D.2 · pp. 62–63
• In the bivariate VAR(1) design, estimator ˆθc slightly outperforms ˆθb for transition parameters at N = 2000, 8000, and 16000, especially for γ12 and γ21.These cross-equation transition parameters are identified as especially relevant for the empirical application.Source: §D.2 · p. 63
• In a trivariate extension, the rescaled estimator ˆθb performs very poorly, possibly because large rescaling factors create optimization problems, whereas ˆθc and ˆθa perform better.The paper treats this behavior as potentially design-specific and leaves fuller analysis for future work.Source: §D.2 · p. 63

E Proofs of Theorem 1 and Theorem 3

• The proof generalizes the AR(1) argument to higher lag orders and time indices, identifying where the higher-order structure changes the decomposition.The text explicitly compares the AR(1) case with the generalized result for t and k ranges.Source: E Proofs of Theorem 1 and Theorem 3 · pp. 64, 66, 67, 68
• The argument represents transition-probability terms as polynomials in e^a and uses distinct irreducible factors to invoke partial-fraction decompositions.The proof notes that the relevant numerator and denominator objects are polynomials in e^a, with dependence on covariates.Source: E Proofs of Theorem 1 and Theorem 3 · p. 65
• The proof strategy decomposes conditional choice probabilities into components that depend on the initial condition and components that do not.The decomposition separates terms denoted T0(a), T1(a), and T2(a), with the latter independent of the initial condition.Source: E Proofs of Theorem 1 and Theorem 3 · p. 65
• The proof constructs unique decompositions for history-indexed terms and shows that the associated functions form a basis of the relevant image space.It applies basis arguments to obtain rank and dimension implications, including the stated cardinality formula when T ≥ p + 2.Source: E Proofs of Theorem 1 and Theorem 3 · pp. 64, 67, 68

F Proofs of Propositions 1, 2, 4

• Propositions 1, 2, and 4 are proved using the same strategy based on the law of iterated expectations.The text develops Proposition 1 explicitly and states that the remaining cases follow by analogous arguments.Source: F Proofs of Propositions 1, 2, 4 · p. 69

G Proofs of Lemma 1 and Lemma 2

• The proposed functional is constructed to be zero whenever the current response satisfies Y_it ≠ 0.This restriction is the first component of the proof’s treatment of the conditional moments.Source: G Proofs of Lemma 1 and Lemma 2 · p. 69
• For current outcomes 0 and 1, the proof imposes separate relations involving next-period covariates, fixed effects, and transition probabilities.The displayed terms distinguish the 0-to- and 1-to-transition expressions through ϕ_0|0 and ϕ_1|1.Source: G Proofs of Lemma 1 and Lemma 2 · pp. 69–70

H Proofs of Lemmas 3,10 and Corollaries 3.1, 10.1

• The construction uses partial-fraction expansion to relate transition-probability expressions across cases, with analogous proof steps left implicit for the remaining cases.The proof explicitly develops Lemma 3 and states that the other cases follow the same logic.Source: H Proofs of Lemmas 3,10 and… · pp. 70–71
• The proofs rely on a rational-fraction identity, measurability, iterated expectations, and induction to establish the lemmas and Theorem 4.Theorem 4 is proved by finite induction, with separate treatment of the p = 2 edge case.Source: H Proofs of Lemmas 3,10 and… · pp. 70–71, Proof of Theorem 4 · pp. 74, 77

J.1 Identification for T = 3 with variability in the initial condition

• For T = 3, the moment function under Yi0 = 1, Yi−1 = 0 is strictly monotone in γ2, identifying γ2 once γ1 and β are identified.The corresponding moment function varies monotonically with γ1 and with βk according to the sign of Xi3k − Xi1k.Source: §J.1 · p. 80
• Under Yi0 = 0, Yi−1 = 1, strict monotonicity in γ2 likewise identifies γ2 uniquely after γ1 and β are identified.This moment is associated with m(0,1) in Honoré and Weidner (2020).Source: §J.1 · p. 81
• Some matched initial-condition and regressor configurations produce moment functions that are identically zero and contain no parameter information.The lack of information is stated for cases requiring Yi0 and Xi3 = Xi2 to match specified conditions.Source: §J.1 · pp. 79, 81

J.2 Proof of Theorem 5

• The proof obtains the limiting moments by sending W_i2 to infinity and extending the moment function continuously at that limit.Assumptions on the regressors and dominated convergence justify the limiting operations.Source: §J.2 · pp. 82–83
• Dominating functions and Leibniz’s integral rule provide the technical control needed for differentiation and passage to the limit.The proof applies the Dominated Convergence Theorem repeatedly before using the integral rule.Source: §J.2 · pp. 83–84
• The limiting 2Kx-equation system has at most one solution and identifies (γ02, β0) under the theorem’s assumptions.The argument uses dominated convergence and the validity of the original moment for arbitrary Xi.Source: §J.2 · pp. 83, 85
• Given (γ02, β0), strict monotonicity of the moment for other initial conditions point-identifies γ01.The relevant initial conditions are y0 ∈ {(0,1), (1,0), (1,1)}.Source: §J.2 · p. 85

K Proof of Proposition 3

• When s < p, polynomial degree comparisons establish the required decomposition, with strict inequality unless the future-state sequence equals the reference sequence.The proof compares numerator and denominator degrees for the transition-probability representation.Source: K Proof of Proposition 3 · p. 86
• Partial-fraction decomposition yields a unique set of known coefficients that rewrites the relevant rational expression in terms of transition probabilities.The proof then combines this decomposition with Theorem 4.Source: K Proof of Proposition 3 · pp. 86–87
• The resulting expression is identified when θ0 is identified by assumption.This conclusion is stated for both the general decomposition and the s ≥ p case.Source: K Proof of Proposition 3 · p. 87, Case 2: s ≥p · p. 88

Case 2: s ≥p

• When s ≥ p, the proof invokes the same degree-based argument to obtain a unique coefficient set and the corresponding identification result.The case is treated as an extension of the preceding argument.Source: Case 2: s ≥p · pp. 87–88

L Proof of Lemma 4

• Lemma 4 is verified by direct calculation.Source: L Proof of Lemma 4 · pp. 88–89

N Dynamic network formation with transitivity

• Graham (2013) identifies the network parameters for T = 3 using conditional likelihood, but the model excludes other covariates such as time-specific effects.Source: N Dynamic network formation with… · p. 90
• The network model contains state dependence, transitivity, and unrestricted dyad-level fixed effects for undirected links among three agents.Source: N Dynamic network formation with… · p. 90
• The VAR(1) results can be repurposed for the three-agent dynamic network model.Source: N Dynamic network formation with… · p. 91
• For T = 3, adaptations of Lemmas 4 and 5 produce eight transition functions and eight moment functions meaningfully depending on model parameters.Source: N Dynamic network formation with… · p. 91
• The moment functions are valid because they satisfy equation (1) through the law of iterated expectations.Source: N Dynamic network formation with… · p. 91

E) Conclusion

• The arguments established for initial condition Y0 = 0 also hold for Y0 = 1.Source: Q Proof of Theorem 2 · p. 94, E) Conclusion · p. 106

A) Preliminary calculations

• The derivation begins with the conditional density and likelihood of the three-period history given the initial condition and regressors.Source: A) Preliminary calculations · pp. 94–96, B) Scores and nonparametric tangent set · p. 96
• The candidate efficient score is formed from an efficient moment using D(X), Σ(X), and Ω(X)=D(X)'Σ(X)^−1.Source: A) Preliminary calculations · p. 95

B) Scores and nonparametric tangent set

• The nonparametric tangent set is characterized through nuisance scores for the conditional heterogeneity distribution, following Hahn (2001).Source: B) Scores and nonparametric tangent set · pp. 96–97

C) Verification of condition i) ψeff

• The proof verifies orthogonality conditions by characterizing the tangent-set orthocomplement and decomposing slope-score differences into conditional-expectation identities.Source: C) Verification of condition i) ψeff · pp. 97–98, D) Verification of condition ii) Sθ… · p. 105

E) Conclusion

• The appendix concludes that the proposed score is the efficient score of the AR(1) model and derives its semiparametric efficiency bound.Source: E) Conclusion · pp. 105–106