arXiv subjects

Christian Gourieroux

Publications and source records attributed to Christian Gourieroux.

12 recordsLinked to original sources

arXiv2025

Nonlinear Impulse Response Functions and Local Projections

The goal of this paper is to extend the nonparametric estimation of Impulse Response Functions (IRF) by means of local projections in the nonlinear dynamic framework. We discuss the existence of a nonlinear autoregressive representation for Markov processes and explain how their IRFs are directly linked to the Nonlinear Local Projection (NLP), as in the case for the linear setting. We present a fully nonparametric LP estimator in the one dimensional nonlinear framework, compare its asymptotic properties to that of IRFs implied by the nonlinear autoregressive model and show that the two approaches are asymptotically equivalent. This extends the well-known result in the linear autoregressive model by Plagborg-Moller and Wolf (2017). We also consider extensions to the multivariate framework through the lens of semiparametric models, and demonstrate that the indirect approach by the NLP is less accurate than the direct estimation approach of the IRF.

econ.EM↗

arXiv2025

Identification of Impulse Response Functions for Nonlinear Dynamic Models

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.

econ.EM↗

arXiv2025

Nonlinear Fore(Back)casting and Innovation Filtering for Causal-Noncausal VAR Models

We show that the mixed causal-noncausal Vector Autoregressive (VAR) processes satisfy the Markov property in both calendar and reverse time. Based on that property, we introduce closed-form formulas of forward and backward predictive densities for point and interval forecasting and backcasting out-of-sample. The backcasting formula is used for adjusting the forecast interval to obtain a desired coverage level when the tail quantiles are difficult to estimate. A confidence set for the prediction interval is introduced for assessing the uncertainty due to estimation. We also define new nonlinear past-dependent innovations of mixed causal-noncausal VAR models for impulse response function analysis. Our approach is illustrated by simulations and an application to oil prices and real GDP growth rates.

econ.EM↗

arXiv2024

Forecast Relative Error Decomposition

We introduce a class of relative error decomposition measures that are well-suited for the analysis of shocks in nonlinear dynamic models. They include the Forecast Relative Error Decomposition (FRED), Forecast Error Kullback Decomposition (FEKD) and Forecast Error Laplace Decomposition (FELD). These measures are favourable over the traditional Forecast Error Variance Decomposition (FEVD) because they account for nonlinear dependence in both a serial and cross-sectional sense. This is illustrated by applications to dynamic models for qualitative data, count data, stochastic volatility and cyberrisk.

econ.EM↗

arXiv2022

Structural Modelling of Dynamic Networks and Identifying Maximum Likelihood

This paper considers nonlinear dynamic models where the main parameter of interest is a nonnegative matrix characterizing the network (contagion) effects. This network matrix is usually constrained either by assuming a limited number of nonzero elements (sparsity), or by considering a reduced rank approach for nonnegative matrix factorization (NMF). We follow the latter approach and develop a new probabilistic NMF method. We introduce a new Identifying Maximum Likelihood (IML) method for consistent estimation of the identified set of admissible NMF's and derive its asymptotic distribution. Moreover, we propose a maximum likelihood estimator of the parameter matrix for a given non-negative rank, derive its asymptotic distribution and the associated efficiency bound.

econ.EM↗

arXiv2022

Long Run Risk in Stationary Structural Vector Autoregressive Models

This paper introduces a local-to-unity/small sigma process for a stationary time series with strong persistence and non-negligible long run risk. This process represents the stationary long run component in an unobserved short- and long-run components model involving different time scales. More specifically, the short run component evolves in the calendar time and the long run component evolves in an ultra long time scale. We develop the methods of estimation and long run prediction for the univariate and multivariate Structural VAR (SVAR) models with unobserved components and reveal the impossibility to consistently estimate some of the long run parameters. The approach is illustrated by a Monte-Carlo study and an application to macroeconomic data.

econ.EM↗

arXiv2021

Temporally Local Maximum Likelihood with Application to SIS Model

The parametric estimators applied by rolling are commonly used in the analysis of time series with nonlinear features, such as structural change due to time varying parameters and local trends. This paper examines the properties of rolling estimators in the class of Temporally Local Maximum Likelihood (TLML) estimators. We study the TLML estimators of constant parameters, stochastic and stationary parameters and parameters with the Ultra Long Run (ULR) dynamics bridging the gap between the constant and stochastic parameters. Moreover, we explore the properties of TLML estimators in an application to the Susceptible-Infected-Susceptible (SIS) epidemiological model and illustrate their finite sample performance in a simulation study.

stat.ME↗

arXiv2021

Generalized Covariance Estimator

We consider a class of semi-parametric dynamic models with strong white noise errors. This class of processes includes the standard Vector Autoregressive (VAR) model, the nonfundamental structural VAR, the mixed causal-noncausal models, as well as nonlinear dynamic models such as the (multivariate) ARCH-M model. For estimation of processes in this class, we propose the Generalized Covariance (GCov) estimator, which is obtained by minimizing a residual-based multivariate portmanteau statistic as an alternative to the Generalized Method of Moments. We derive the asymptotic properties of the GCov estimator and of the associated residual-based portmanteau statistic. Moreover, we show that the GCov estimators are semi-parametrically efficient and the residual-based portmanteau statistics are asymptotically chi-square distributed. The finite sample performance of the GCov estimator is illustrated in a simulation study. The estimator is also applied to a dynamic model of cryptocurrency prices.

econ.EM↗

arXiv2020

Uncertainty on the Reproduction Ratio in the SIR Model

The aim of this paper is to understand the extreme variability on the estimated reproduction ratio $R_0$ observed in practice. For expository purpose we consider a discrete time stochastic version of the Susceptible-Infected-Recovered (SIR) model, and introduce different approximate maximum likelihood (AML) estimators of $R_0$. We carefully discuss the properties of these estimators and illustrate by a Monte-Carlo study the width of confidence intervals on $R_0$.

stat.ME↗

arXiv2020

SIR Model with Stochastic Transmission

The Susceptible-Infected-Recovered (SIR) model is the cornerstone of epidemiological models. However, this specification depends on two parameters only, which implies a lack of flexibility and the difficulty to replicate the volatile reproduction numbers observed in practice. We extend the classic SIR model by introducing nonlinear stochastic transmission, to get a stochastic SIR model. We derive its exact solution and discuss the condition for herd immunity. The stochastic SIR model corresponds to a population of infinite size. When the population size is finite, there is also sampling uncertainty. We propose a state-space framework under which we analyze the relative magnitudes of the observational and stochastic epidemiological uncertainties during the evolution of the epidemic. We also emphasize the lack of robustness of the notion of herd immunity when the SIR model is time discretized.

q-bio.PE↗

arXiv2020

Analysis of Virus Propagation: A Transition Model Representation of Stochastic Epidemiological Models

The growing literature on the propagation of COVID-19 relies on various dynamic SIR-type models (Susceptible-Infected-Recovered) which yield model-dependent results. For transparency and ease of comparing the results, we introduce a common representation of the SIR-type stochastic epidemiological models. This representation is a discrete time transition model, which allows us to classify the epidemiological models with respect to the number of states (compartments) and their interpretation. Additionally, the transition model eliminates several limitations of the deterministic continuous time epidemiological models which are pointed out in the paper. We also show that all SIR-type models have a nonlinear (pseudo) state space representation and are easily estimable from an extended Kalman filter.

q-bio.PE↗

arXiv2020

Time Varying Markov Process with Partially Observed Aggregate Data; An Application to Coronavirus

A major difficulty in the analysis of propagation of the coronavirus is that many infected individuals show no symptoms of Covid-19. This implies a lack of information on the total counts of infected individuals and of recovered and immunized individuals. In this paper, we consider parametric time varying Markov processes of Coronavirus propagation and show how to estimate the model parameters and approximate the unobserved counts from daily numbers of infected and detected individuals and total daily death counts. This model-based approach is illustrated in an application to French data.

stat.ME↗