Mixture Matrix-valued Autoregressive Model

Fei Wu; Kung-Sik Chan

arxiv: 2312.06098 · v3 · submitted 2023-12-11 · 📊 stat.ME · math.ST· stat.TH

Mixture Matrix-valued Autoregressive Model

Fei Wu , Kung-Sik Chan This is my paper

Pith reviewed 2026-05-24 05:37 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH

keywords matrix time seriesautoregressive modelmixture modelregime switchingEM algorithmconsistencyasymptotic distribution

0 comments

The pith

The mixture matrix autoregressive model captures regime shifts in the dynamics between two sets of attributes in matrix time series.

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

Matrix time series track relationships between two groups of variables, such as countries and economic indicators, but linear matrix autoregressive models miss abrupt changes in those relationships. The paper introduces a mixture version that treats the observed series as switching among a small number of distinct linear regimes. An EM algorithm estimates the parameters, and the authors prove that the estimates are consistent and asymptotically normal. Simulations and real-data examples show the mixture recovers the switches that a single linear model cannot.

Core claim

The MMAR model represents matrix-valued time series as a finite mixture of linear MAR processes, each corresponding to a distinct regime; maximum-likelihood estimates obtained via the EM algorithm are consistent and asymptotically normal under standard regularity conditions.

What carries the argument

The mixture matrix autoregressive model, which decomposes the observed dynamics into a finite number of linear MAR components selected by latent regime indicators.

If this is right

The estimated regimes can be interpreted as distinct economic states such as expansion versus recession.
Forecasts can be formed by weighting the predictions of each component by its estimated probability.
The number of regimes can be selected by standard information criteria applied to the mixture likelihood.
Asymptotic normality supplies standard errors for testing whether a particular coefficient differs across regimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same mixture construction could be applied to tensor-valued series if an analogous linear tensor autoregressive model is available.
Regime probabilities produced by the fitted model might serve as leading indicators for policy changes.
If the number of regimes grows with sample size the consistency proof would need to be extended.

Load-bearing premise

The nonlinear patterns in the data are produced by switching among a small number of linear regimes rather than by some other form of nonlinearity.

What would settle it

A simulation in which the true process is a single linear MAR or a nonlinear process outside any finite mixture of MARs, yet the EM procedure still reports well-separated components with high likelihood.

Figures

Figures reproduced from arXiv: 2312.06098 by Fei Wu, Kung-Sik Chan.

**Figure 2.** Figure 2: Simulated data of Example 1. be made overall second-order stationary by making minor adjustments to the example’s parameters, while preserving the non-stationarity of the second component process. 4 Parameter Estimation Maximum likelihood estimation of the MMAR model can be implemented via an Expectation–Maximization (EM) algorithm (Dempster et al., 1977). Let Zt = (Zt,1, . . . , Zt,K) be the latent varia… view at source ↗

**Figure 3.** Figure 3: Clustered time series plot of the economic indicators, with phase 1 shaded yellow, [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗

**Figure 4.** Figure 4: One-step marginal predictive distribution for Q3 2021, with [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗

**Figure 5.** Figure 5: One-step marginal predictive distribution for Q4 2021, with [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗

read the original abstract

Time series of matrix-valued data are increasingly available in various areas including economics, finance, social science, among others. These data may shed light on the inter-dynamical relationships between two sets of attributes, for instance, countries and economic indices. The matrix autoregressive (MAR) model provides a parsimonious approach for analyzing such data. However, the MAR model, being a linear model with parametric constraints, cannot capture the nonlinear patterns in the data, such as regime shifts in the dynamics. We propose a mixture matrix autoregressive (MMAR) model for analyzing potential regime shifts in the dynamics between two attributes, for instance, due to recession versus expansion, or stable period versus pandemic. We propose an EM algorithm for maximum likelihood estimation. We derive some theoretical properties of the proposed method including consistency and asymptotic distribution, and illustrate its performance via simulations and real applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

MMAR adds a mixture layer to matrix AR for regime shifts with EM and claimed asymptotics, but the derivations need checking on identifiability.

read the letter

The paper's main move is to let a matrix autoregressive process switch between a small number of linear regimes via a mixture, with latent indicators and an EM algorithm for fitting. This targets matrix time series that show nonlinear behavior, such as shifts between recession and expansion periods in economic data. The extension keeps the matrix structure instead of vectorizing, which is a direct and sensible step from existing MAR work. They also run simulations and a couple of real applications to illustrate recovery of regimes and parameter estimates. The claim of consistency and asymptotic normality for the estimator is stated up front, which at least signals they tried to ground the method formally. The soft spot is that mixture identifiability is rarely automatic, especially with autoregressive parameters and matrix constraints. Without seeing the exact regularity conditions or how they rule out label switching or component collapse, it is hard to judge whether the asymptotics hold under the kinds of separation one sees in practice. The abstract does not flag any special handling for the matrix case, so that part will need referee attention. This is aimed at time-series people in economics, finance, or social science who already use MAR models and want a regime-switching version. A reader who needs a tool for detecting shifts in inter-attribute dynamics would find the setup useful if the theory checks out. I would send it to referees rather than desk reject, mainly to get the derivations and simulation design examined.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a mixture matrix autoregressive (MMAR) model to capture regime shifts in the dynamics of matrix-valued time series data. It introduces an EM algorithm for maximum likelihood estimation of the model parameters and derives consistency and asymptotic distribution results for the estimator. Performance is assessed through simulation studies and illustrated with real-data applications.

Significance. If the theoretical claims hold under appropriate regularity conditions, the MMAR framework provides a useful extension of the linear MAR model to handle nonlinear regime-dependent dynamics in matrix time series, with direct relevance to applications in economics and finance. The explicit derivation of consistency and asymptotic normality is a methodological strength that supports reliable inference, distinguishing the contribution from purely algorithmic proposals.

major comments (2)

[Theoretical Properties section] Theoretical Properties section: the claim of consistency and asymptotic distribution for the EM estimator requires explicit statement of the identifiability conditions for the mixture components (including label-switching resolution and separation of the regime-specific MAR coefficient matrices); without these, the asymptotic normality result cannot be verified and is load-bearing for the central theoretical contribution.
[Assumptions section (preceding the EM algorithm)] Assumptions section (preceding the EM algorithm): the weakest modeling assumption—that observed nonlinearity arises exactly from a finite mixture of linear MAR processes—needs a concrete discussion of how violations (e.g., continuous regime variation) would affect the estimator's consistency; this directly impacts whether the derived asymptotics apply to the intended data-generating processes.

minor comments (2)

[Simulation section] Simulation section: report the specific matrix dimensions (p,q) and the true mixing proportions used in the Monte Carlo experiments to allow reproducibility of the reported finite-sample performance.
[Notation] Notation: ensure the matrix autoregressive lag operator is defined uniformly (e.g., via Kronecker or vec notation) before its first use in the likelihood derivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments, which help clarify the theoretical foundations of the MMAR model. We address each major comment below and outline the corresponding revisions.

read point-by-point responses

Referee: [Theoretical Properties section] Theoretical Properties section: the claim of consistency and asymptotic distribution for the EM estimator requires explicit statement of the identifiability conditions for the mixture components (including label-switching resolution and separation of the regime-specific MAR coefficient matrices); without these, the asymptotic normality result cannot be verified and is load-bearing for the central theoretical contribution.

Authors: We agree that the asymptotic results require explicit identifiability conditions. In the revised manuscript, we will add a new subsection to the Theoretical Properties section that states the necessary identifiability assumptions. These will include standard constraints to resolve label-switching (e.g., ordering of the mixture proportions or the Frobenius norms of the regime-specific coefficient matrices) and conditions ensuring sufficient separation between the distinct MAR regimes (e.g., a minimum distance between the parameter vectors of different components). With these additions, the consistency and asymptotic normality theorems will be stated under the complete set of regularity conditions. revision: yes
Referee: [Assumptions section (preceding the EM algorithm)] Assumptions section (preceding the EM algorithm): the weakest modeling assumption—that observed nonlinearity arises exactly from a finite mixture of linear MAR processes—needs a concrete discussion of how violations (e.g., continuous regime variation) would affect the estimator's consistency; this directly impacts whether the derived asymptotics apply to the intended data-generating processes.

Authors: We acknowledge that a discussion of potential misspecification is warranted. In the revised Assumptions section, we will add a paragraph addressing violations of the finite-mixture assumption. Specifically, we will note that if the true process exhibits continuous regime variation rather than discrete switches, the MMAR estimator converges to the parameter vector that minimizes the Kullback-Leibler divergence to the true data-generating distribution. We will also briefly discuss the resulting bias in the estimated regime-specific matrices and the implications for the validity of the derived asymptotic normality under such departures. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard EM and asymptotic theory for mixtures

full rationale

The paper proposes the MMAR model as an extension of the linear MAR model to capture regime shifts via a finite mixture of linear processes, then applies the standard EM algorithm for MLE and states that consistency and asymptotic normality are derived under regularity conditions. No equations or steps in the provided abstract or description reduce a claimed result to a fitted input by construction, nor do they rely on self-citations for load-bearing uniqueness or ansatz smuggling. The theoretical claims follow the conventional route for mixture autoregressive models without redefining inputs as outputs or renaming known patterns as novel derivations. The derivation chain is therefore self-contained against external statistical benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 1 invented entities

The model relies on standard mixture model assumptions and introduces mixture components as the key new element, with parameters estimated from data.

free parameters (3)

number of mixture components
Chosen or estimated to represent different regimes.
regime-specific MAR coefficients
Parameters for each component's autoregression.
mixing proportions
Probabilities of each regime.

axioms (1)

domain assumption The time series is generated from a finite mixture of matrix autoregressive processes
Fundamental assumption for the MMAR model to apply to regime shifts.

invented entities (1)

latent regime indicators no independent evidence
purpose: To indicate which MAR component is active at each time
Standard in mixture models but applied here to matrix data.

pith-pipeline@v0.9.0 · 5669 in / 1293 out tokens · 59879 ms · 2026-05-24T05:37:44.731984+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a mixture matrix autoregressive (MMAR) model... EM algorithm for maximum likelihood estimation... consistency and asymptotic distribution
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

top-Lyapunov exponent γ = lim (1/t) E(log ∥Dt...D1∥)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

and Michailidis, G

Basu, S. and Michailidis, G. (2015). Regularized estimation in sparse high-dimensional time series models. The Annals of Statistics , 43(4):1535–1567. Chan, K.-S. (1993). Asymptotic behavior of the gibbs sampler. Journal of the American Statistical Association, 88(421):320–326. Chan, K.-S. and Tong, H. (1990). On likelihood ratio tests for threshold autor...

work page 2015
[2]

Mixture Matrix-valued Autoregressive Model

Springer. Fong, P. W., Li, W. K., Yau, C., and Wong, C. S. (2007). On a mixture vector autoregressive model. Canadian Journal of Statistics , 35(1):135–150. Gallaugher, M. P. and McNicholas, P. D. (2018). Finite mixtures of skewed matrix variate distributions. Pattern Recognition, 80:83–93. Gao, X., Shen, W., Zhang, L., Hu, J., Fortin, N. J., Frostig, R. ...

work page arXiv 2007
[3]

Assume that, E log+ ∥D1∥ < ∞ and E log+ ∥η0∥ < ∞

In model (13), let {(Dt, ηt)} be a sequence of strictly stationary and ergodic sequence. Assume that, E log+ ∥D1∥ < ∞ and E log+ ∥η0∥ < ∞. Also assume that its top-Lyapunov exponent γ, defined in (15), is strictly negative. Then, ˜Xt = ∞X j=0 tY i=t−j+1 Di ! ηt−j (S.1.1) is the unique strictly stationary solution to equation (13). Theorem S.1.2 (Theorem 4...

work page 2014
[4]

First notice that {Xt} is a time homogeneous Markov chain, as it is strictly stationary and its unique stationary solution is given by (17). Define the transition kernel by, P(Xt, ·) = Pr(Xt+1 ∈ ·|X t), The 1-step transition density is, f(Xt+1|Xt; θ) = f(yt+1|Xt; θ) = KX k=1 ft+1(yt+1|Xt; θk), indicating that f(Xt+1|Xt; θ) > 0 for all Xt+1 and Xt. Therefo...

work page 1993
[5]

(2007) and this proposition is proved

Since the MMAR model is a special case of the mixture VAR model with parameter restrictions, let Bk ⊗ Ak play the role of Θ k1 in Theorem 1 of Fong et al. (2007) and this proposition is proved. Proof of Proposition

work page 2007
[6]

(2007) and this Proposition is proved

Similar to the proof of Proposition 3, let Bk ⊗ Ak play the role of Θk1 in Theorem 3 of Fong et al. (2007) and this Proposition is proved. 39 Proof of Proposition

work page 2007
[7]

By Theorem S.1.2, the results holds

Under condition (14), it follows that, E log+ ∥D1∥ = KX k=1 αk log+(∥Φk∥) < ∞, and E log+ ∥η0∥ < ∞ because of normality. By Theorem S.1.2, the results holds. S.2 Preliminaries for the Proofs of Theorem 1 and 2 We begin with some notations and properties of matrices. Let M be an m × n matrix and M(g, h) be the (g, h)-th entry of M. There exists a commutati...

work page 1979
[8]

Since it is more convenient to take partial derivatives of the log-likelihood function w.r.t

= s 1 − X g≥h [V −1 i (g, h)]2. Since it is more convenient to take partial derivatives of the log-likelihood function w.r.t. vech U −1 k and vech V −1 k , our idea is to first derive the Fisher information matrix w.r.t. γ , and then use the delta method to derive the Fisher information matrix w.r.t. θ. Also, observe that vech U −1 k and vech −1 V −1 k ar...

work page 1962
[9]

(S.3.2) The proofs follows the ideas in Kalliovirta et al. (2016). First notice that {lt(θ)} is also a strictly stationary and ergodic process. By Theorem S.2.1, it suffices to show that E(supθ∈Θ |lt(θ)|) < ∞. Since the parameter space Θ is compact and Uk and Vk are positive definite, we have c1 ≤ det(Vk ⊗ Uk) ≤ C1 and c2 ≤ αk ≤ C2 for each k ∈ {1, . . . ...

work page 2016
[10]

We use the results in Sweeting (1980) to prove asymptotic normality

Let ˙LT (θ) = ∂ ∂θ LT (θ) and IT (θ) = −¨LT (θ) = − ∂2 ∂θ∂θT LT (θ). We use the results in Sweeting (1980) to prove asymptotic normality. Let Γ be the matrix (θ(1), . . . ,θ(dim(Θ))), where θ(i) ∈ Θ, i = 1, 2, . . . ,dim(Θ). Define IT (Γ) to be IT with ith row evaluated at θ(i). It suffices to show that, sup θ∈Θ IT (θ) T − pmax a.s. → E −¨lt(θ) , (S.4.8) ...

work page 1980
[11]

Since sup√T −pmax∥θ−θ0∥F ≤c ∥θ − θ0∥F = op(1), condition (S.4.9) holds if ∥∂vec(IT (θ))/∂θT∥F (T −pmax) = Op(1) uniformly for θ ⊂ Θ

By mean value inequality, sup√T −pmax∥θ−θ0∥F ≤c ∥IT (θ) − IT (θ0)∥F (T − pmax) ≤ sup√T −pmax∥θ−θ0∥F ≤c ∥θ−θ0∥F sup√T −pmax∥θ−θ0∥F ≤c ∥∂vec(IT (θ))/∂θT∥F T − pmax . Since sup√T −pmax∥θ−θ0∥F ≤c ∥θ − θ0∥F = op(1), condition (S.4.9) holds if ∥∂vec(IT (θ))/∂θT∥F (T −pmax) = Op(1) uniformly for θ ⊂ Θ. By Theorem S.2.1, it suffices to show that E sup θ∈Θ ∥∂vec ¨...

work page 1980
[12]

Therefore, Pr hTSθ = 0 = 1 ⇔ Pr hT ∂γT ∂θ Sγ = 0 = 1 (S.6.2) Let m be a dim( γ)-vector

(S.6.1) 49 By the chain rule, Sθ = ∂lt(θ) ∂θ = ∂γT ∂θ ∂lt(θ) ∂γ ≜ ∂γT ∂θ Sγ. Therefore, Pr hTSθ = 0 = 1 ⇔ Pr hT ∂γT ∂θ Sγ = 0 = 1 (S.6.2) Let m be a dim( γ)-vector. Consider the conditions when Pr mTSγ = 0 = 1, (S.6.3) Recall that, ∂lt(γi) ∂γi = αift(yt|Ft−1; γi)PK k=1 αkft(yt|Ft−1; γk) ∂ log(ft(yt|Ft−1; γi)) ∂γi , ∂lt(γi) ∂αi = ft(yt|Ft−1; γi) − ft(yt|Ft...

work page 2021
[13]

AIC BIC HQ GIC T = 200 36.60% 93.60% 78.60% 99.80% T = 400 15.80% 97.80% 82.20% 99.80% T = 800 16.20% 98.40% 86.40% 100.00% Table S.7.1: Percentage of correctly selecting K = 2 in Scenario 3 with the AR orders given. AIC BIC HQ GIC T = 200 67.20% 66.80% 70.20% 61.20% T = 400 62.00% 95.60% 92.60% 95.60% T = 800 45.40% 99.20% 97.60% 99.20% Table S.7.2: Perc...

work page 2022

[1] [1]

and Michailidis, G

Basu, S. and Michailidis, G. (2015). Regularized estimation in sparse high-dimensional time series models. The Annals of Statistics , 43(4):1535–1567. Chan, K.-S. (1993). Asymptotic behavior of the gibbs sampler. Journal of the American Statistical Association, 88(421):320–326. Chan, K.-S. and Tong, H. (1990). On likelihood ratio tests for threshold autor...

work page 2015

[2] [2]

Mixture Matrix-valued Autoregressive Model

Springer. Fong, P. W., Li, W. K., Yau, C., and Wong, C. S. (2007). On a mixture vector autoregressive model. Canadian Journal of Statistics , 35(1):135–150. Gallaugher, M. P. and McNicholas, P. D. (2018). Finite mixtures of skewed matrix variate distributions. Pattern Recognition, 80:83–93. Gao, X., Shen, W., Zhang, L., Hu, J., Fortin, N. J., Frostig, R. ...

work page arXiv 2007

[3] [3]

Assume that, E log+ ∥D1∥ < ∞ and E log+ ∥η0∥ < ∞

In model (13), let {(Dt, ηt)} be a sequence of strictly stationary and ergodic sequence. Assume that, E log+ ∥D1∥ < ∞ and E log+ ∥η0∥ < ∞. Also assume that its top-Lyapunov exponent γ, defined in (15), is strictly negative. Then, ˜Xt = ∞X j=0 tY i=t−j+1 Di ! ηt−j (S.1.1) is the unique strictly stationary solution to equation (13). Theorem S.1.2 (Theorem 4...

work page 2014

[4] [4]

First notice that {Xt} is a time homogeneous Markov chain, as it is strictly stationary and its unique stationary solution is given by (17). Define the transition kernel by, P(Xt, ·) = Pr(Xt+1 ∈ ·|X t), The 1-step transition density is, f(Xt+1|Xt; θ) = f(yt+1|Xt; θ) = KX k=1 ft+1(yt+1|Xt; θk), indicating that f(Xt+1|Xt; θ) > 0 for all Xt+1 and Xt. Therefo...

work page 1993

[5] [5]

(2007) and this proposition is proved

Since the MMAR model is a special case of the mixture VAR model with parameter restrictions, let Bk ⊗ Ak play the role of Θ k1 in Theorem 1 of Fong et al. (2007) and this proposition is proved. Proof of Proposition

work page 2007

[6] [6]

(2007) and this Proposition is proved

Similar to the proof of Proposition 3, let Bk ⊗ Ak play the role of Θk1 in Theorem 3 of Fong et al. (2007) and this Proposition is proved. 39 Proof of Proposition

work page 2007

[7] [7]

By Theorem S.1.2, the results holds

Under condition (14), it follows that, E log+ ∥D1∥ = KX k=1 αk log+(∥Φk∥) < ∞, and E log+ ∥η0∥ < ∞ because of normality. By Theorem S.1.2, the results holds. S.2 Preliminaries for the Proofs of Theorem 1 and 2 We begin with some notations and properties of matrices. Let M be an m × n matrix and M(g, h) be the (g, h)-th entry of M. There exists a commutati...

work page 1979

[8] [8]

Since it is more convenient to take partial derivatives of the log-likelihood function w.r.t

= s 1 − X g≥h [V −1 i (g, h)]2. Since it is more convenient to take partial derivatives of the log-likelihood function w.r.t. vech U −1 k and vech V −1 k , our idea is to first derive the Fisher information matrix w.r.t. γ , and then use the delta method to derive the Fisher information matrix w.r.t. θ. Also, observe that vech U −1 k and vech −1 V −1 k ar...

work page 1962

[9] [9]

(S.3.2) The proofs follows the ideas in Kalliovirta et al. (2016). First notice that {lt(θ)} is also a strictly stationary and ergodic process. By Theorem S.2.1, it suffices to show that E(supθ∈Θ |lt(θ)|) < ∞. Since the parameter space Θ is compact and Uk and Vk are positive definite, we have c1 ≤ det(Vk ⊗ Uk) ≤ C1 and c2 ≤ αk ≤ C2 for each k ∈ {1, . . . ...

work page 2016

[10] [10]

We use the results in Sweeting (1980) to prove asymptotic normality

Let ˙LT (θ) = ∂ ∂θ LT (θ) and IT (θ) = −¨LT (θ) = − ∂2 ∂θ∂θT LT (θ). We use the results in Sweeting (1980) to prove asymptotic normality. Let Γ be the matrix (θ(1), . . . ,θ(dim(Θ))), where θ(i) ∈ Θ, i = 1, 2, . . . ,dim(Θ). Define IT (Γ) to be IT with ith row evaluated at θ(i). It suffices to show that, sup θ∈Θ IT (θ) T − pmax a.s. → E −¨lt(θ) , (S.4.8) ...

work page 1980

[11] [11]

Since sup√T −pmax∥θ−θ0∥F ≤c ∥θ − θ0∥F = op(1), condition (S.4.9) holds if ∥∂vec(IT (θ))/∂θT∥F (T −pmax) = Op(1) uniformly for θ ⊂ Θ

By mean value inequality, sup√T −pmax∥θ−θ0∥F ≤c ∥IT (θ) − IT (θ0)∥F (T − pmax) ≤ sup√T −pmax∥θ−θ0∥F ≤c ∥θ−θ0∥F sup√T −pmax∥θ−θ0∥F ≤c ∥∂vec(IT (θ))/∂θT∥F T − pmax . Since sup√T −pmax∥θ−θ0∥F ≤c ∥θ − θ0∥F = op(1), condition (S.4.9) holds if ∥∂vec(IT (θ))/∂θT∥F (T −pmax) = Op(1) uniformly for θ ⊂ Θ. By Theorem S.2.1, it suffices to show that E sup θ∈Θ ∥∂vec ¨...

work page 1980

[12] [12]

Therefore, Pr hTSθ = 0 = 1 ⇔ Pr hT ∂γT ∂θ Sγ = 0 = 1 (S.6.2) Let m be a dim( γ)-vector

(S.6.1) 49 By the chain rule, Sθ = ∂lt(θ) ∂θ = ∂γT ∂θ ∂lt(θ) ∂γ ≜ ∂γT ∂θ Sγ. Therefore, Pr hTSθ = 0 = 1 ⇔ Pr hT ∂γT ∂θ Sγ = 0 = 1 (S.6.2) Let m be a dim( γ)-vector. Consider the conditions when Pr mTSγ = 0 = 1, (S.6.3) Recall that, ∂lt(γi) ∂γi = αift(yt|Ft−1; γi)PK k=1 αkft(yt|Ft−1; γk) ∂ log(ft(yt|Ft−1; γi)) ∂γi , ∂lt(γi) ∂αi = ft(yt|Ft−1; γi) − ft(yt|Ft...

work page 2021

[13] [13]

AIC BIC HQ GIC T = 200 36.60% 93.60% 78.60% 99.80% T = 400 15.80% 97.80% 82.20% 99.80% T = 800 16.20% 98.40% 86.40% 100.00% Table S.7.1: Percentage of correctly selecting K = 2 in Scenario 3 with the AR orders given. AIC BIC HQ GIC T = 200 67.20% 66.80% 70.20% 61.20% T = 400 62.00% 95.60% 92.60% 95.60% T = 800 45.40% 99.20% 97.60% 99.20% Table S.7.2: Perc...

work page 2022