Identification of Impulse Response Functions for Nonlinear Dynamic Models

Christian Gourieroux; Quinlan Lee

REVIEW 2 major objections 8 minor 1 cited by

Identification of Impulse Response Functions for Nonlinear Dynamic Models

T0 review · 2 major / 8 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read In multivariate nonlinear dynamic models, structural shocks and impulse responses are only partially identified: the same transition density is compatible with every smooth volume-preserving transformation of the underlying shocks.

desk verdict A credible partial-identification result for nonlinear IRFs, with two overclaimed remedies (Props 5 and 7) that should be fixed before publication. read the letter →

arxiv 2506.13531 v2 pith:PQPQDAKG submitted 2025-06-16 econ.EM

Christian Gourieroux , Quinlan Lee This is my paper · ORCID

classification econ.EM MSC 62M1060J05

keywords NonlinearAutoregressiveModelGenerativeImpulseResponseFunctionsIndependentComponentAnalysisLocalProjectionsInnovationsPartialIdentificationRecurrentMarkovProcess

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether structural shocks and impulse response functions can be recovered uniquely in nonlinear multivariate dynamic models, and its answer is that in general they cannot. Any first-order Markov process with a positive transition density admits a nonlinear autoregressive representation $y_t = g(y_{t-1}; \varepsilon_t)$ with standard normal shocks $\varepsilon_t$, but once there are at least two series this representation is not unique: the shocks are identified only up to smooth invertible transformations $T$ of the uniform innovation with Jacobian determinant $+1$ or $-1$ everywhere. The nonlinear innovations and the impulse responses they generate are therefore partially identified, not point identified, which matters because impulse responses are the standard tool for saying what a shock does to the economy. The paper then catalogues conditions under which identification is restored — non-Gaussian independent sources, latent factors with different dynamics, and identifiable summary effects such as the pseudo impulse response — and it shows that the whole analysis depends on the universe of variables under study.

What carries the argument

The load-bearing object is the identified set described in Proposition 2: for $n \geq 2$, the collection of nonlinear autoregressive representations compatible with a given transition density is exactly the orbit of the true representation $(g, u_t)$ under smooth invertible transformations $T$ of the uniform shock with $|\det \partial T(u)/\partial u'| = 1$ everywhere. Existence of at least one representation is supplied by the recursive conditional-quantile inversion $\varepsilon_t = \Phi^{-1} \circ F(y_t \mid y_{t-1})$, which builds Gaussian innovations from the conditional cumulative distribution function of the transition. The multiplicity is generated by distribution-preserving maps, and in polar coordinates these include norm-dependent rotations of the Gaussian shock, $Q(\|\varepsilon_t\|^2)\varepsilon_t$; the polar decomposition of the Jacobian (a scale, a rotation, a unit-determinant symmetric part) organizes the group of admissible transformations. This mechanism converts the identification problem into a question about measure-preserving diffeomorphisms of the unit cube, which is what lets the paper import and use results from nonlinear independent component analysis.

What would settle it

Simulate a bivariate nonlinear Markov process from a known transition density, estimate that density nonparametrically, and fit two representations $g_0$ and $g_1$ that are related by a nontrivial smooth volume-preserving transformation $T$ (for instance a norm-dependent rotation with $|\det \partial T/\partial u'| = 1$). If both fits have the same likelihood while their horizon-2 impulse responses differ materially, Proposition 2's non-identification is confirmed in practice; if instead the transformed innovations fail a test of serial independence, the empirical identified set is smaller than the full orbit the paper describes.

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Extended reading notes

Core claim

The paper's central discovery is a characterization of the identified set for nonlinear dynamic models, stated as Proposition 2. Fix the transition density $f(y_t \mid y_{t-1})$ of a first-order Markov process: for a single series ($n=1$) the nonlinear autoregressive representation with Gaussian innovations is essentially unique, up to the sign change $\varepsilon \mapsto -\varepsilon$ and irregular non-smooth cases; for $n \geq 2$ the uniform shocks $u_t$ are identified only up to a continuously differentiable, invertible transformation $T$ of $[0,1]^n$ whose Jacobian determinant is identically $+1$ or identically $-1$. Any such volume-preserving transformation of the shocks produces a new representation with exactly the same likelihood, so no amount of data can choose among them. A concrete family is rotation of the Gaussian shock vector by an angle that depends on its norm, $\eta_t = Q(\|\varepsilon_t\|^2)\varepsilon_t$ with $Q$ a special orthogonal matrix function, the nonlinear analogue of the orthogonal-matrix indeterminacy of linear SVARs. Because the impulse response depends on which representative of this set is selected, the IRF is only partially identified; the identifiable objects are exactly the functionals of the transition density itself, such as the pseudo impulse response from a direct additive shock to $y_t$.

Load-bearing premise

The load-bearing premise — flagged by the paper itself in Sections 4.3 and 6 — is that the one-step-ahead conditional distribution of the full observed system is identified from the data and that the series really form a first-order Markov system in the variables observed; if the distribution shifts over time, the Markov order is wrong, or relevant variables are omitted, the described set of equally good shock structures no longer applies, and the characterization counts only smooth transformations, whereas the non-smooth cases in Appendix A.2 add still more multiplicity.

Editorial extensions

If this is right

With two or more series, two estimation exercises with different starting values or algorithms can converge to different functions $g$ inside the identified set and therefore to different impulse responses, all with the same asymptotic fit to the data.
Univariate nonlinear models are identified: for $n=1$ the Gaussian innovation and its IRF are unique up to the sign flip, so the failure of identification is a genuinely multivariate phenomenon.
Independence of non-Gaussian sources restores identification: if at most one source is Gaussian, the mixing structure and the sources are recovered up to scale and permutation (linear ICA).
Observed series that are transformations of independent latent factors with different dynamics — different autocovariance functions, or different transition-density derivatives — yield essentially unique factors and identifiable IRFs.
The pseudo impulse response, defined by shocking $y_t$ itself by $\Delta$, is always identifiable from the transition density, and Proposition 7 shows the structural IRF equals the PIRF plus a horizon-one correction; the unidentified part of the IRF is precisely this correction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The norm-dependent rotations the paper constructs double as a robustness diagnostic: compute the impulse response over a grid of such rotations and report the spread, which measures how much of the reported IRF is structural rather than an artifact of the chosen representative.
Because the identified set is an orbit of measure-preserving transformations, conventional identifying devices — sign restrictions, narrative records, Cholesky-type recursions — amount to selecting one element of the orbit; their identifying content could be audited by checking whether the selected shock survives adversarial volume-preserving perturbations.
The universe dependence implies that nonlinear impulse responses are not coherent under aggregation: a shock identified in a small system need not correspond to any shock in an augmented system, with direct consequences for factor-augmented and network models.
A testable extension: apply the covariance-based Markov tests of Section 6.1.1 to residuals from two representations in the identified set; if one passes and the other fails serial independence, the empirical identified set is strictly smaller than the full orbit of Proposition 2.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read the paper. The central result is real: in a multivariate Markov model, the nonlinear AR representation with Gaussian/uniform innovations is only identified up to measure-preserving diffeomorphisms, so nonlinear IRFs are partially identified rather than point identified. Proposition 2 is self-contained and correct under the stated smoothness assumptions. That is worth knowing for anyone doing nonlinear local projections or generative-model IRFs. The paper also does well to show the one-dimensional uniqueness, the polar-coordinate rotation examples, and the dependence on time unit and universe.

The soft spots are real but not fatal. Proposition 5 is the biggest. It presents the maximum IRF as sqrt(a'Φ^h DD'Φ'^h a) and calls it identifiable, but that formula only holds for the linear Gaussian VAR—it uses IRF(h,δ,a)=a'A^hDδ, which is linear in δ and history-invariant. In the nonlinear model (2.7) the IRF is nonlinear in δ, stochastic, and history-dependent, so the optimization in (4.1) does not produce that closed form. As written, the proposition is stated for the general framework and is false; at minimum it needs to be explicitly restricted to the linear illustration, and the identified-function argument for nonlinear models needs a separate derivation. Proposition 7 is also overclaimed: the expectation of the pseudo-IRF is identifiable from the transition density, but the full stochastic PIRF as a coupled counterfactual may depend on the chosen representation. The paper should either prove distributional identifiability or state the weaker conditional-expectation result. Minor issues: Proposition 4 leans on an external harmonic-function result and its eigenvalue argument is sketchy, and Appendix A.2 shows non-smooth transformations add more multiplicity, so Proposition 2's identified set is a smoothness-dependent description (the paper acknowledges this). The heavy self-citation is mostly legitimate—the cited results are prior and external in origin.

Who is this for? Applied macroeconomists using local projections or machine-learning reduced forms, and time-series econometricians thinking about nonlinear shocks. The negative result is the contribution; the remedies are less developed. I would send it to a serious referee, with the expectation of major revision. The central finding deserves to be in the literature; the overclaims need to be scrubbed.

Referee Report

2 major / 8 minor

Summary. The paper studies identification of impulse response functions (IRFs) in nonlinear multivariate dynamic models. It introduces a Gaussian (or uniform) nonlinear autoregressive representation of a Markov process and shows that, in dimension n≥2, this representation is not unique: the identified set consists of transformations that leave the standard normal or uniform distribution invariant. Proposition 2 characterizes this set as smooth invertible transformations with Jacobian determinant ±1, and Proposition 3 provides examples based on radius-dependent rotations. The paper then discusses remedies: non-Gaussian innovations and independent component analysis (Proposition 6), factor models with sources having different dynamics (Propositions 8 and 9), pseudo-IRFs (Proposition 7), and the role of the universe of variables (Section 6). The overall message is that nonlinear innovations and IRFs are generally partially identified, not point identified, unless additional restrictions are imposed.

Significance. If the results hold, the paper makes a useful conceptual contribution by formalizing the extent of underidentification of nonlinear IRFs. The proof of Proposition 2 in Appendix A.3 is self-contained, and the rotation examples in Proposition 3 are correct. The discussion of the dependence of Markov properties, innovations, and IRFs on the universe of variables (Section 6) is a valuable reminder. The paper also usefully connects the identification problem to nonlinear ICA and to dynamic factor models. Its contribution is primarily formal and conceptual rather than an estimation theory; no consistency or inference results are claimed. The central negative result—that nonlinear IRFs are partially identified in general multivariate settings—is credible and internally consistent.

major comments (2)

[Section 4.3.3, Proposition 7] The statement 'The function PIRFt is identifiable' is not proved and, as stated, is too strong. The random variable PIRFt(h,Δ,yt) in (4.10) is constructed by feeding the same future innovations ε_{t+1},... into the baseline and perturbed paths, so its full distribution depends on the particular function g selected from the identified set, not only on the transition density. What the text actually establishes in (4.11) is that E[PIRFt(h,Δ,yt)|yt] = E(yt+h|yt+Δ) − E(yt+h|yt), which is a functional of the transition density and hence identifiable. Because Proposition 7 is presented as one of the paper's remedies for partial identification, the proposition should be restated and proved for this conditional expectation (or for another identifiable summary), and the non-identifiability of the stochastic PIRF should be acknowledged.
[Section 3.2.2 and Appendix A.4, Proposition 4] The dimension count in Proposition 4 and Lemma 4 appears incorrect. The conditions (A.4) are the Cauchy-Riemann equations for the map b1 + i b2, so the space of harmonic pairs of total degree at most m has real dimension 2(m+1), not 2m; for m=1 the paper's own parametrization leaves the four parameters b1,00, b2,00, c00, and d11 free. Moreover, the set {b1,h,k, b2,h,k : h+k ≥ m} in the proposition is infinite-dimensional, so the statement as written is not well defined. This does not undermine the paper's qualitative underidentification conclusion, but the proposition and Lemma 4 need to be corrected or replaced by a precise statement.

minor comments (8)

[Section 2.3, Eq. (2.9) and Remark 4] Equation (2.9) and the corresponding PIRF formula in Remark 4 give the cumulative response (Id + Φ + ... + Φ^h), whereas the definition in (2.7) and the formula used in Proposition 5 are the horizon-h responses Φ^h Dδ and Φ^h Δ. Please correct these displays or label them explicitly as cumulative IRFs.
[Section 4.3.3] The sentence referring to simulations in 'Section 4.2.2' should point to Section 4.3.2, which is where the simulation consistency argument appears.
[Section 4.3.3, Remark 4] The cross-reference to 'Proposition 6' in Remark 4 appears to be a typo; the relation IRF = PIRF(...) is the content of Proposition 7.
[Section 3.1, Proposition 2] The codomain of T is written as (0,1]^n, and the Jacobian condition is stated on mixed open/closed domains; please make the domain and codomain of T consistent (presumably [0,1]^n or (0,1)^n).
[Appendix A.3, proof of Proposition 2] The proof shows that any two representations yield a transformation T with |det| = 1, but it does not explicitly state the converse: every such T defines an observationally equivalent representation via g̃(yt−1, T(ut)) = g(yt−1, ut). Adding a sentence would make the 'identified up to' characterization complete.
[Section 4.2, Eq. (4.5)] In display (4.5), the components of the inverse c.d.f. vector are all written as F_1^{-1}(u_i,t); they should be F_i^{-1}(u_i,t) for i=1,...,n to match the statement that Fi is the c.d.f. of wi,t.
[Section 4.1] It would help to state explicitly that Proposition 5 is a linear-model illustration and that no nonlinear analogue of the 'maximum IRF' identification is claimed; the current transition from the nonlinear underidentification problem to the SVAR(1) formula could be misread as proposing a nonlinear remedy.
[Section 6.1.1] The word 'pormanteau' should be 'portmanteau'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: central identified-set characterization is proven self-contained; self-citations are auxiliary only.

full rationale

The paper's load-bearing negative result—that multivariate Gaussian/uniform nonlinear innovations and IRFs are only partially identified, with the identified set given by C^1 measure-preserving transformations of [0,1]^n—is established in Proposition 2 via the Jacobian formula in Appendix A.3. The proof starts from two representations y_t = h(y_{t-1};u_t) = \tilde h(y_{t-1};\tilde u_t), writes \tilde u_t = T(u_t) under the independence assumption, and obtains |det ∂T/∂u'| = 1 from equality of uniform densities. It does not import the target result from the cited literature. Proposition 1 is the standard Rosenblatt multivariate inverse-CDF construction, and Proposition 4's harmonic-function dimension count is proved in Appendix A.4. Self-citations (Gourieroux–Jasiak 2022; Gourieroux–Monfort–Renne 2017; Gourieroux–Jasiak 2023b) are used for extensions or as lemmas whose original sources are external (Comon, Darmois, Rosenblatt, Hall); none of these citations is the sole support for a central claim. The skeptical concern about Proposition 5 is a substantive correctness issue—the closed-form maximization uses the linear IRF formula a'Φ^h Dδ, whose nonlinear generalization is not established—but it is not circularity, since the optimized value is a function of the identifiable reduced-form covariance DD'. No fitted parameter is relabelled as a prediction, and no result is forced by definition or by an author-imported uniqueness theorem. The acknowledged caveats (smoothness of transformations in Appendix A.2; identifiability of the transition density in Section 4.3) weaken the exact description of the identified set without making the derivation circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No new latent physical entities are postulated. The nonlinear innovations and pseudo IRF are definitions built from the transition density, not free-floating additions. The load-bearing assumptions are the Markov structure, identification of the transition density, smoothness of representations, and the external identifying conditions imported from ICA and factor-model literatures.

assumptions (7)

domain assumption The observed process (yt) is Markov of order 1 on R^n with strictly positive transition density f(yt|yt-1) > 0.
Proposition 1 and Corollary 1 rely on this for the existence of the nonlinear autoregressive representation via Rosenblatt inversion. The positivity assumption excludes truncated supports and absorbing states.
domain assumption The transition density f(yt|yt-1) is identified from observations, typically via stationary ergodicity.
Section 4.3 states that 'usually the transition density is identifiable from a sequence of observations' under stationary ergodicity. The entire identified-set analysis is relative to this density.
domain assumption Candidate structural innovations must be independent of yt-1 and mutually independent, with Gaussian or uniform normalization.
Section 2.3 imposes these assumptions to define shocks. Proposition 2 derives the identified set by requiring both representations to have innovations satisfying this independence.
domain assumption The representation function g is one-to-one and continuously differentiable in the innovation with strictly positive Jacobian.
The Jacobian formula in Proposition 2 requires this smoothness. Appendix A.2 shows that non-differentiable transformations enlarge the multiplicity, so this is a substantive restriction.
standard math Standard mathematical tools: Rosenblatt transform, Jacobian formula, polar decomposition of matrices, harmonic function series expansions.
These tools are used in the proofs in Appendices A and B without further derivation.
domain assumption For factor-model identification, the sources have different dynamics, e.g., linearly independent autocovariance functions (Proposition 8) or linearly independent score-transition sequences (Proposition 9).
These sufficient conditions must be checked in applications; they are not guaranteed by the data.
domain assumption At most one of the independent sources is Gaussian for linear ICA identification.
Proposition 6 cites Comon (1994) and related work; the number of Gaussian sources controls the degree of partial identification.

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Pith. "Pith review of Identification of Impulse Response Functions for Nonlinear Dynamic Models." pith.science (2026). https://pith.science/paper/PQPQDAKG

@misc{pith2026250613531,
  author       = {Pith},
  title        = {Pith review of: Identification of Impulse Response Functions for Nonlinear Dynamic Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQPQDAKG}},
  note         = {Machine review of arXiv:2506.13531}
}

read the original abstract

We explore the issues of identification for nonlinear Impulse Response Functions in nonlinear dynamic models and discuss the settings in which the problem can be mitigated. In particular, we introduce the nonlinear autoregressive representation with Gaussian innovations and characterize the identified set. This set arises from the multiplicity of nonlinear innovations and transformations which leave invariant the standard normal density. We then discuss possible identifying restrictions, such as non-Gaussianity of independent sources, or identifiable parameters by means of learning algorithms, and the possibility of identification in nonlinear dynamic factor models when the underlying latent factors have different dynamics. We also explain how these identification results depend ultimately on the set of series under consideration.

Figures

Figures reproduced from arXiv: 2506.13531 by the authors.

**Figure 1.** Piecewise Diffeomorphisms in [0, 1]2 in red and Diffeomorphisms in R 2 in blue. In each set of plots, the first two correspond to a(ρ) = 1, and the last two a(ρ) = 0.2ρ. can restrict the functions on [0, 1]2 to the harmonic functions that admit a series expansion as: b1(u) = X∞ h=0 X∞ k=0 (b1,h,ku h 1u k 2 ), b2(u) = X∞ h=0 X∞ k=0 (b2,h,ku h 1u k 2 ), (3.5) and analyze the identification issues by means of the coeff… view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Bandwidth-Free Inference for Recursive Nonlinear Impulse Response Functions
econ.EM 2026-08 conditional novelty 6.0 of 10

The paper provides first-order, bootstrap, and finite-simulation inference for recursive nonlinear impulse responses computed from empirical residual quantiles, without density estimation or quantile-smoothing bandwidths.

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Works this paper leans on

14 extracted references · 14 canonical work pages · cited by 1 Pith paper

[1]

Let us consider a Gaussian AR(1) model: yt = ρyt−1 +εt, where the εt’s are i.i.d. N(0, 1). It is easily seen that y2τ =ρ2y2(τ−1) +ε2τ +ρε2(τ−1). In this simple case, the function ˜g is still affine in y2(τ−1) with an autoregressive coefficient ρ2 and we have ˜ε2τ =ε2τ +ρε2(τ−1)

work page
[2]

In the nonlinear dynamic framework, the nonlinear innovation ˜ε2τ is no longer a func- tion of ε2τ and ε2(τ−1) only, but depends also on y2(τ−1) in general. Therefore, the definition of nonlinear innovation depends in a complicated way on the selected time unit, with important consequences when considering the definition of shocks and the computation of I...

work page
[3]

These shocks cannot change the past, which explains the need for the independence betweenyt−1 and εt

work page
[4]

This explains the assumption of cross sectional independence between the components of εt

The shock on ε1,t, say, has to be performed without an effect on ε2,t. This explains the assumption of cross sectional independence between the components of εt

work page
[5]

This is done by imposing the conditions of identical distribution, where any level of shock corresponds to a given quantile of the common distribution

It will be useful to fix the level of shocks δ1,δ 2 in a coherent way. This is done by imposing the conditions of identical distribution, where any level of shock corresponds to a given quantile of the common distribution. These common quantiles are what is used to define small as well as extreme shocks. Remark 1: The literature has suggested an alternati...

work page 1989
[6]

It is completely deterministic and not stochastic

work page
[7]

It is history invariant, that is, it does not depend on the current environment yt−1

work page
[8]

Indeed, the Gaussian VAR(1) is a very special case of dynamic models and such properties are in general, not indicative of what arises in a nonlinear dynamic framework

It is a linear function with respect to the magnitude δ. Indeed, the Gaussian VAR(1) is a very special case of dynamic models and such properties are in general, not indicative of what arises in a nonlinear dynamic framework. 2.4 Examples Example 1: Conditionally Gaussian Model The model can be written as: yt =m(yt−1) +D(yt−1)εt, (2.10) where εt∼ IIN (0,I...

work page 1996

Show all 14 references

[9]

dimension

The Ω symmetric positive definite matrix with det Ω = 1 can be written as Ω = ExpV , whereV is a symmetric traceless matrix (with Tr V = 0). Note that the “dimension” of the group generated by the different components are 1 for λ, n(n−1) 2 for Q (or B), n(n+1) 2 − 1 for Ω (or ...
[10]

structural form

a(ρ) = 0.2ρ, For each case, we represent in u1 u2 ∈ [0, 1]2, the curves transformed of the segments (u1 = k 75, 0≤u2 < 1),k = 0, 1,..., 75 (resp. 0≤u1≤ 1,u 2 = k 75),k = 0, 1,..., 75. They correspond to homotopic deformations of the square [Smale (1959)]. These transfor- matio...

1959
[11]

In particular, some tuning parameters for accelerating the algorithm are supposed to be defined unambiguously

The learning algorithm provides for each starting value and each iteration step a unique value ˆgp,T . In particular, some tuning parameters for accelerating the algorithm are supposed to be defined unambiguously
[12]

Can we use such an estimate in practice and for which purpose? 4.3.2 Simulations Let us first consider the simulation issues

ˆgpT,T converges to the identified set when T , pT tend to infinity. Can we use such an estimate in practice and for which purpose? 4.3.2 Simulations Let us first consider the simulation issues. Indeed, simulated trajectories are usually based on the nonlinear autoregressive s...

1993
[13]

IRFt(h,δ,y t−1) =PIRFt(h,g (yt−1,εt +δ)−g(yt−1,ε 0),g (yt−1,ε 0))
[14]

the diagonal elements of matrix A are equal to 1

The function PIRFt is identifiable. Therefore, the identification issue on the IRF is just due to the different effects at horizon 1 when applying a change on yt instead of a change on εt. The PIRF is in general stochastic and its expectation conditional on yt is equal to: E [...

1991 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Let us consider a Gaussian AR(1) model: yt = ρyt−1 +εt, where the εt’s are i.i.d. N(0, 1). It is easily seen that y2τ =ρ2y2(τ−1) +ε2τ +ρε2(τ−1). In this simple case, the function ˜g is still affine in y2(τ−1) with an autoregressive coefficient ρ2 and we have ˜ε2τ =ε2τ +ρε2(τ−1)

work page

[2] [2]

In the nonlinear dynamic framework, the nonlinear innovation ˜ε2τ is no longer a func- tion of ε2τ and ε2(τ−1) only, but depends also on y2(τ−1) in general. Therefore, the definition of nonlinear innovation depends in a complicated way on the selected time unit, with important consequences when considering the definition of shocks and the computation of I...

work page

[3] [3]

These shocks cannot change the past, which explains the need for the independence betweenyt−1 and εt

work page

[4] [4]

This explains the assumption of cross sectional independence between the components of εt

The shock on ε1,t, say, has to be performed without an effect on ε2,t. This explains the assumption of cross sectional independence between the components of εt

work page

[5] [5]

This is done by imposing the conditions of identical distribution, where any level of shock corresponds to a given quantile of the common distribution

It will be useful to fix the level of shocks δ1,δ 2 in a coherent way. This is done by imposing the conditions of identical distribution, where any level of shock corresponds to a given quantile of the common distribution. These common quantiles are what is used to define small as well as extreme shocks. Remark 1: The literature has suggested an alternati...

work page 1989

[6] [6]

It is completely deterministic and not stochastic

work page

[7] [7]

It is history invariant, that is, it does not depend on the current environment yt−1

work page

[8] [8]

Indeed, the Gaussian VAR(1) is a very special case of dynamic models and such properties are in general, not indicative of what arises in a nonlinear dynamic framework

It is a linear function with respect to the magnitude δ. Indeed, the Gaussian VAR(1) is a very special case of dynamic models and such properties are in general, not indicative of what arises in a nonlinear dynamic framework. 2.4 Examples Example 1: Conditionally Gaussian Model The model can be written as: yt =m(yt−1) +D(yt−1)εt, (2.10) where εt∼ IIN (0,I...

work page 1996