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 SusceptibleInfected-Susceptible (SIS) epidemiological model and illustrate their finite sample performance in a simulation study.

[1]  Zongwu Cai,et al.  Trending time-varying coefficient time series models with serially correlated errors , 2007 .

[2]  F. Hoppensteadt,et al.  Asymptotic Behavior of Solutions to a Population Equation , 1975 .

[3]  Changyong Xu,et al.  The threshold of a stochastic delayed SIRS epidemic model with temporary immunity and vaccination , 2018, Chaos, Solitons & Fractals.

[4]  A. Auger,et al.  Verifiable conditions for the irreducibility and aperiodicity of Markov chains by analyzing underlying deterministic models , 2015, Bernoulli.

[5]  J. Wallinga,et al.  Different Epidemic Curves for Severe Acute Respiratory Syndrome Reveal Similar Impacts of Control Measures , 2004, American journal of epidemiology.

[6]  C. Gouriéroux,et al.  Estimated reproduction ratios in the SIR model , 2021, The Canadian journal of statistics = Revue canadienne de statistique.

[7]  N. Reid,et al.  AN OVERVIEW OF COMPOSITE LIKELIHOOD METHODS , 2011 .

[8]  David Firth,et al.  Bias reduction in exponential family nonlinear models , 2009 .

[9]  B. Efron.  Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency) , 1975 .

[10]  Emmanuel Rio,et al.  Asymptotic Theory of Weakly Dependent Random Processes , 2017 .

[11]  Herbert W. Hethcote,et al.  An SIS epidemic model with variable population size and a delay , 1995, Journal of mathematical biology.

[12]  J. O. Irwin,et al.  MATHEMATICAL EPIDEMIOLOGY , 1958 .

[13]  R. Jennrich.  Asymptotic Properties of Non-Linear Least Squares Estimators , 1969 .

[14]  Chang‐Jin Kim,et al.  State Space Models with Regime Switching , 1999 .

[15]  How To Go Viral: A COVID-19 Model with Endogenously Time-Varying Parameters , 2020 .

[16]  Michael L. Stein,et al.  Local likelihood estimation for nonstationary random fields , 2011, J. Multivar. Anal..

[17]  Estimation for random coefficient autoregressive model , 2016 .

[18]  Jianqing Fan,et al.  Simultaneous Confidence Bands and Hypothesis Testing in Varying‐coefficient Models , 2000 .

[19]  Yasin Şimşek,et al.  Bridging the COVID-19 Data and the Epidemiological Model using Time Varying Parameter SIRD Model , 2020, 2301.13692.

[20]  James A. Yorke,et al.  Some equations modelling growth processes and gonorrhea epidemics , 1973 .

[21]  J. Ghosh,et al.  Second order efficiency of maximum likelihood estimators , 1974 .

[22]  Stephen E. Satchell,et al.  Approximating the Finite Sample Bias for Maximum Likelihood Estimators by Using the Score , 1996 .

[23]  Stanley R. Johnson,et al.  Varying Coefficient Models , 1984 .

[24]  James H. Matis,et al.  Stochastic Population Models: A Compartmental Perspective , 2000 .

[25]  R. Ahmad,et al.  Analysis and Prediction of COVID-19 Pandemic in Pakistan using Time-dependent SIR Model , 2020, 2005.02353.

[26]  C. Fraser,et al.  A New Framework and Software to Estimate Time-Varying Reproduction Numbers During Epidemics , 2013, American journal of epidemiology.

[27]  Zhou Zhou,et al.  Simultaneous inference of linear models with time varying coefficients , 2010 .

[28]  C. C. Heyde,et al.  Quasi-likelihood and Optimal Estimation , 2010 .

[29]  F. Karim,et al.  Are we there yet? An adaptive SIR model for continuous estimation of COVID-19 infection rate and reproduction number in the United States , 2020, medRxiv.

[30]  R. Tweedie.  Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space , 1975 .

[31]  Bruce E. Hansen,et al.  Convergence to Stochastic Integrals for Dependent Heterogeneous Processes , 1992, Econometric Theory.

[32]  D. Andrews.  CONSISTENCY IN NONLINEAR ECONOMETRIC MODELS: A GENERIC UNIFORM LAW OF LARGE NUMBERS , 1987 .

[33]  Christian Gourieroux,et al.  Analysis of Virus Propagation: A Transition Model Representation of Stochastic Epidemiological Models , 2020, 2006.10265.

[34]  Javier Rubio-Herrero,et al.  A Flexible Rolling Regression Framework for Time-Varying SIRD models: Application to COVID-19 , 2021, 2103.02048.

[35]  Jianqing Fan,et al.  Efficient Estimation and Inferences for Varying-Coefficient Models , 2000 .

[36]  Zhidong Teng,et al.  Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients , 2017 .

[37]  F. Canova,et al.  Estimating overidentified, nonrecursive, time‐varying coefficients structural vector autoregressions , 2015 .

[38]  Luke M. Froeb,et al.  Measuring and comparing smoothness in time series the production smoothing hypothesis , 1994 .

[39]  H. White.  A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity , 1980 .

[40]  D. Firth.  Bias reduction of maximum likelihood estimates , 1993 .

[41]  Giuseppe C. Calafiore,et al.  A Modified SIR Model for the COVID-19 Contagion in Italy , 2020, 2020 59th IEEE Conference on Decision and Control (CDC).

[42]  Giorgio E. Primiceri.  Time Varying Structural Vector Autoregressions and Monetary Policy , 2002 .

[43]  Debasis Mukherjee,et al.  Study of an S-I epidemic model with nonlinear incidence rate: Discrete and stochastic version , 2011, Appl. Math. Comput..

[44]  Zhiliang Ying,et al.  Semiparametric and Nonparametric Regression Analysis of Longitudinal Data , 2001 .

[45]  P. J. Huber.  The behavior of maximum likelihood estimates under nonstandard conditions , 1967 .

[46]  H. White.  Maximum Likelihood Estimation of Misspecified Models , 1982 .

[47]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[48]  Eric R. Ziegel,et al.  Generalized Linear Models , 2002, Technometrics.

[49]  Liangjian Hu,et al.  A Stochastic Differential Equation SIS Epidemic Model , 2011, SIAM J. Appl. Math..