Temporally Local Maximum Likelihood with Application to SIS Model

Christian Gourieroux; Joann Jasiak

doi:10.1515/jtse-2022-0016

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Temporally Local Maximum Likelihood with Application to SIS Model

Christian Gourieroux und Joann Jasiak

Veröffentlicht/Copyright: 7. März 2023

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Aus der Zeitschrift Journal of Time Series Econometrics Band 15 Heft 2

Abstract

The parametric estimators applied by rolling are commonly used for the analysis of time series with nonlinear patterns, including 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 consider the TLML estimators of (a) constant parameters, (b) stochastic, stationary parameters and (c) parameters with the ultra-long run (ULR) dynamics bridging the gap between the constant and stochastic parameters. We show that the weights used in the TLML estimators have a strong impact on the inference. For illustration, we provide a simulation study of the epidemiological susceptible–infected–susceptible (SIS) model, which explores the finite sample performance of TLML estimators of a time varying contagion parameter.

Keywords: local maximum likelihood; rolling estimator; omitted heterogeneity; bias reduction; generalized linear model; SIS model; logistic growth; ultra long run

JEL Classification: C01; C13; C22

Corresponding author: Joann Jasiak, York University, Toronto, Canada, E-mail: jasiakj@yorku.ca

Funding source: Natural Sciences and Engineering Research Council of Canada

Research funding: The first author acknowledges financial support from the ACPR chair “Regulation and Systemic Risk”, the ERC DYSMOIA and the Agence Nationale de la Recherche (ANR-COVID) [Grant ANR-17-EURE-0010]. The second author gratefully acknowledges financial support of the Natural Sciences and Engineering Council of Canada (NSERC).
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Appendix A: Second-Order Expansion

Even though the TLML estimator is consistent, the finite sample bias adjustment formula is expected to be non-standard. We consider below the second-order expansion of the FOC [see Gosh and Subramanyan (1974), Efron (1975), Firth (1993), Kosmidis and Firth (2009)]. For expository purpose, we assume dim θ_t = 1. We define:

L 1 t θ ∞ * = ∂ ⁡ log ⁡ l ∂ θ y t | y t − 1 ; θ ∞ * , L 2 t θ ∞ * = ∂ 2 ⁡ log ⁡ l ∂ θ 2 y t | y t − 1 ; θ ∞ * , L 3 t θ ∞ * = ∂ 3 ⁡ log ⁡ l ∂ θ 3 y t | y t − 1 ; θ ∞ * .

The second-order expansion of the FOC in a neighbourhood of the pseudo-true value is:

∑ h = 0 T − 1 w ( h ) L 1 , T − h θ ∞ * + ∑ h = 0 T − 1 w ( h ) L 2 , T − h θ ∞ * ( θ ̂ T ( w ) − θ ∞ * ) + 1 2 ∑ h = 0 T − 1 w ( h ) L 3 , T − h θ ∞ * ( θ ̂ T ( w ) − θ ∞ * ) 2 = o P ,

where o_p is negligible in probability with respect to the left hand side of the equation, or equivalently,

(A.1) ∑ h = 0 T − 1 w ( h ) L 1 , T − h θ ∞ * + W T E 0 L 2 , t θ ∞ * ( θ ̂ T ( w ) − θ ∞ * ) + ∑ h = 0 T − 1 w ( h ) L 2 , T − h θ ∞ * − E 0 L 2 , t θ ∞ * ( θ ̂ T ( w ) − θ ̂ ∞ * ) + 1 2 W T E 0 L 3 , t θ ∞ * ( θ ̂ T ( w ) − θ ∞ * ) 2 ) = o P .

From (3.9), it follows that:

θ ̂ T ( w ) − θ ∞ * = I T w ; θ ∞ * 1 / 2 W T J θ ∞ * X T + o P ,

where X_T ∼ N(0, 1).

This expression can be plugged into the two last terms of the second-order expansion to get:

∑ h = 0 T − 1 w ( h ) L 1 , T − h θ ∞ * = W T J θ ∞ * ( θ ̂ T ( w ) − θ ∞ * ) + ∑ h = 0 T − 1 w ( h ) L 2 , T − h θ ∞ * + J θ ∞ * I T w ; θ ∞ * 1 / 2 W T J θ ∞ * X T + 1 2 W T E 0 L 3 , t θ ∞ * I T w ; θ ∞ * W T 2 J θ ∞ * 2 X T 2 = o P .

Let us consider the variable:

Z T = ∑ h = 0 T − 1 ∑ k = 0 T − 1 w ( h ) w ( k ) I 2 h − k ; θ ∞ * − 1 / 2 ∑ h = 0 T − 1 w ( h ) ∂ 2 ⁡ log ⁡ l ∂ θ 2 y T − h | y T − h − 1 ; θ ∞ * ≡ I 2 , T w ; θ ∞ * − 1 / 2 ∑ h = 0 T − 1 ( w ( h ) ∂ 2 ⁡ log ⁡ l ∂ θ 2 y T − h | y T − h − 1 ; θ ∞ * ] ,

where I 2 h ; θ ∞ * = Cov 0 ∂ 2 ⁡ log ⁡ l ∂ θ 2 y t | y t − 1 ; θ ∞ * , ∂ ⁡ log ⁡ l ∂ θ 2 y t − h | y t − h − 1 ; θ ∞ * .

Then we get:

θ ̂ T ( w ) − θ ∞ * = I T w ; θ ∞ * 1 / 2 W T J θ ∞ * X T − I T w ; θ ∞ * 1 / 2 W T J θ ∞ * I 2 T w ; θ ∞ * 1 / 2 W T J θ ∞ * X T Z T − 1 2 E 0 L 3 , t θ ∞ * J θ ∞ * I T w ; θ ∞ * W T 2 J θ ∞ * 2 X T 2 + o P .

This expansion provides an approximation of the difference between the TLML estimator and the pseudo-true value as a quadratic function of the pair (X_T, Z_T), which is asymptotically normally distributed and has zero mean components with unitary variances and non-zero correlation, in general. Alternative bias adjustments could be based on the pseudo score [Lambrecht et al. (1997)].

Appendix B: Stationarity Condition

B.1 Condition for Ergodicity

Whenever the Markov chain is irreducible, we can use the sufficient conditions for ergodicity provided in Tweedie (1975). They are:

E [ p ̂ 2 ( t ) | p ̂ 2 ( t − 1 ) ] is bounded.
E [ p ̂ 2 ( t ) − p ̂ 2 ( t − 1 ) | p ̂ 2 ( t − 1 ) = x ] ≤ − ε , for x outside a compact set.

Condition (i) is satisfied since p ̂ 2 ( t ) < 1 .

Let us now consider condition (ii). We have:

E [ p ̂ 2 ( t ) − p ̂ 2 ( t − 1 ) | p ̂ 2 ( t − 1 ) = x ] = a x ( 1 − x ) − c x < − c x .

Therefore condition (ii) is satisfied with the compact set K = [ɛ/c, 1], and ɛ < c.

B.2 Condition for Irreducibility

The different types of behaviour discussed below Proposition 6 are related to the irreducibility properties of the Markov chain [see e.g. Rio (2017), Chapter 9, and Chotard and Auger (2019)]. This irreducibility follows from the condition 0 < 1 − c/a < 1, and replacing of B by B + (or P by P + ) in the definition of the chain.

Appendix C: Additional Figures

C.1 Quasi-Collinearity

To provide some insights on the quasi-collinearity, we compute the sample information matrix from the Hessian of the temporally local log-likelihood function and report its eigenvalues. As expected these eigenvalues are positive and one of them is small and close to 0.

Figure 12:

Eigenvalues of estimated information matrix: Constant a.

Figure 13:

Eigenvalues of estimated information matrix: Stochastic a.

C.2 Persistence of Estimates and Stochastic Parameters

Additional summary statistics provided below concern the joint dynamics of the stochastic parameters of interest and their local estimates. They are displayed in Figures 14 and 15 for the contagion parameter and reproductive number, respectively. For each value of w considered, the diagonal panels show the autocorrelations of the estimates and of the stochastic parameters, respectively. The off-diagonal panels show the cross-correlations between the estimates and stochastic parameters.

$Figure 14: Joint ACF of a t and a ̂ t ${\hat{a}}_{t}$ .$

Figure 14:

Joint ACF of a_t and a ̂ t .

$Figure 15: Joint ACF of R0,t and R ̂ 0 , t ${\hat{R}}_{0,t}$ .$

Figure 15:

Joint ACF of R_0,t and R ̂ 0 , t .

The autocorrelation function (ACF) of a_t and R_0,t reveals the local-to-unity property of the ULR process. For w = 0.1, the weighted estimator is more global and appears almost uncorrelated with the underlying stochastic parameter. We observe that the cross-correlations, which often take small values, do not decay to zero with the lag. This is a consequence of the ULR a_t process. For larger w, the estimates display more persistence, although the autocorrelations of a ̂ t are smaller and decay faster than those of a_t.

C.3 Nonlinear Prediction Performance

Under the standard maximum likelihood approach, the values of the log-likelihood at the optimum can be used to perform tests based on the likelihood ratios, especially of the time varying hypotheses H_0T = {R_0T > 1}. It can be used to check the “joint” accuracy of a ̂ t , c ̂ t , as a global prediction measure. In this respect, Figure 16 displays the evolution of the locally weighted log-likelihood.

Figure 16:

Log-likelihoods, stochastic a, red dashed line: w = 0.1, black solid line: w = 0.5 and dotted green line: w = 0.9.

This evolution has to be compared with the trajectory of counts in Figure 6. As expected, the local nonlinear fit is improved during the episodes of counts evolving without sudden jumps or “local” trends, i.e. characterized by rather stable evolution. In contrast, the values of the log-likelihood are shown to decrease in the neighbourhoods of peaks or troughs.

Alternative measures of prediction performance could also be built by comparing at each date the observed number of infected individuals N₂(t) with its estimator-based prediction:

N ̂ 2 ( t ) = a ̂ t [ n − N 2 ( t − 1 ) ] N 2 ( t − 1 ) n + ( 1 − c ̂ t ) N 2 ( t − 1 ) .

The difference N 2 ( t ) − N ̂ 2 ( t ) depends on the theoretical prediction error (which is present when the true parameter values are used) and the estimation error due to replacing the true parameters by their estimates in the theoretical pointwise prediction formula.

References

Anderes, E., and M. Stein. 2011. “Local Likelihood Estimation for Nonstationary Random Fields.” Journal of Multivariate Analysis 102: 506–20. https://doi.org/10.1016/j.jmva.2010.10.010.Suche in Google Scholar

Andrews, D. 1987. “Consistency in Nonlinear Econometric Models; A Generic Uniform Law of Large Numbers.” Econometrica 55: 1465–71. https://doi.org/10.2307/1913568.Suche in Google Scholar

Brauer, F., L. Allen, P. Van den Driessche, and J. Wu. 2008. Mathematical Epidemiology In Lecture Notes in Mathematics, 1945, Mathematical Biosciences Subseries. Berlin Heidelberg: Springer.10.1007/978-3-540-78911-6Suche in Google Scholar

Breto, C., D. He, E. Ionides, and A. King. 2009. “Time Series Analysis via Mechanistic Models.” Annals of Applied Statistics 3: 319–48. https://doi.org/10.1214/08-aoas201.Suche in Google Scholar

Cai, Z. 2007. “Trending Time-Varying Coefficient Time Series Models with Serially Correlated Errors.” Journal of Econometrics 136: 163–88. https://doi.org/10.1016/j.jeconom.2005.08.004.Suche in Google Scholar

Cai, Z., J. Fan, and R. Li. 2000. “Efficient Estimation and Inferences for Varying-Coefficient Models.” Journal of the American Statistical Association 95: 838–902. https://doi.org/10.1080/01621459.2000.10474280.Suche in Google Scholar

Calafiore, G., C. Novara, and C. Possieri. 2020. “A Modified SIR Model for the COVID-19 Contagion in Italy.” ArXiv 2003.14301.10.1109/CDC42340.2020.9304142Suche in Google Scholar

Canova, F., and J. P. Forero. 2015. “Estimating Overidentified Nonrecursive Time-Varying Coefficients Structural Vector Autoregressions.” Quantitative Economics 6: 359–84. https://doi.org/10.3982/qe305.Suche in Google Scholar

Carpenter, J. 1999. “Test Inversion Bootstrap Confidence Intervals.” Journal of the Royal Statistical Society - Series B 61: 159–72. https://doi.org/10.1111/1467-9868.00169.Suche in Google Scholar

Chotard, A., and A. Auger. 2019. “Verifiable Conditions for the Irreducibility and Aperiodicity of Markov Chains by Analyzing Underlying Deterministic Models.” Bernoulli 25: 112–47. https://doi.org/10.3150/17-bej970.Suche in Google Scholar

Cooke, K., and J. Yorke. 1973. “Some Equations Modelling Growth Processes and Gonorrhea Epidemics.” Mathematical Biosciences 16: 75–101. https://doi.org/10.1016/0025-5564(73)90046-1.Suche in Google Scholar

Cori, A., N. Ferguson, C. Fraser, and S. Cauchemez. 2013. “A New Framework and Software to Estimate Time Varying Reproduction Numbers during Epidemics.” American Journal of Epidemiology 178: 1505–12. https://doi.org/10.1093/aje/kwt133.Suche in Google Scholar PubMed PubMed Central

Dahlhaus, R. 2012. “Locally Stationary Processes.” In Handbook of Statistics, 30, 351–412. North Holland, Amsterdam: North Holland Publishing Company.10.1016/B978-0-444-53858-1.00013-2Suche in Google Scholar

Dahlhaus, R., S. Richter, and W. Wu. 2019. “Towards a General Theory of Nonlinear Locally Stationary Processes.” Bernoulli 25: 1013–44.10.3150/17-BEJ1011Suche in Google Scholar

Das, P., D. Mukherjee, and A. Sarkar. 2011. “Study of an SI Epidemic Model with Nonlinear Incidence Rate: Discrete and Stochastic Version.” Applied Mathematics and Computation 218: 2509–15. https://doi.org/10.1016/j.amc.2011.07.065.Suche in Google Scholar

Dureau, J., K. Kalogeropoulos, and M. Buguelin. 2013. “Capturing the Time Drivers of an Epidemic Using Stochastic Dynamical Systems.” Biostatistics 14: 541–55. https://doi.org/10.1093/biostatistics/kxs052.Suche in Google Scholar PubMed

Efron, B. 1975. “Defining the Curvature of a Statistical Problem (With Applications to Second-Order Efficiency).” Annals of Statistics 3: 1189–242. https://doi.org/10.1214/aos/1176343282.Suche in Google Scholar

Elliot, S., and C. Gourieroux. 2021. “Estimated Reproduction Ratios in the SIR Model.” Canadian Journal of Statistics 49: 992–1017. https://doi.org/10.1002/cjs.11663.Suche in Google Scholar PubMed PubMed Central

Fan, J., and I. Gijbels. 1996. Local Polynomial Modelling and its Applications. New York: Chapman and Hill.Suche in Google Scholar

Fan, J., and W. Zhang. 2000. “Simultaneous Confidence Bands and Hypothesis Testing in Varying Coefficient Models.” Scandinavian Journal of Statistics 27: 715–31. https://doi.org/10.1111/1467-9469.00218.Suche in Google Scholar

Fahrmeir, L., and H. Kauffmann. 1985. “Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in the Generalized Linear Models.” Annals of Statistics 13: 342–68.10.1214/aos/1176346597Suche in Google Scholar

Firth, D. 1993. “Bias Reduction of Maximum Likelihood Estimates.” Biometrika 80: 27–38. https://doi.org/10.1093/biomet/80.1.27.Suche in Google Scholar

Francq, C., and A. Gautier. 2004. “Estimation of Time Varying ARMA Models with Markovian Changes in Regimes.” Statistics & Probability Letters 70: 243–51. https://doi.org/10.1016/j.spl.2004.10.009.Suche in Google Scholar

Froeb, L., and R. Koyak. 1994. “Measuring and Comparing Smoothness in Time Series: The Production Smoothing Hypothesis.” Journal of Econometrics 64: 97–122. https://doi.org/10.1016/0304-4076(94)90059-0.Suche in Google Scholar

Godambe, V., and C. Heyde. 1987. “Quasi-Likelihood and Optimal Estimation.” International Statistical Review 55: 231–44.10.2307/1403403Suche in Google Scholar

Gosh, J., and K. Subramanyan. 1974. “Second-Order Efficiency of Maximum Likelihood Estimators.” Sankhya, A. 36: 325–58.Suche in Google Scholar

Gourieroux, C., and J. Jasiak. 2020. “Analysis of Virus Transmission: A Transition Model Representation of Stochastic Epidemiological Models.” Annals of Economics and Statistics 140: 1–26. https://doi.org/10.15609/annaeconstat2009.140.0001.Suche in Google Scholar

Gourieroux, C., and J. Jasiak. 2022a. “Long Run Risk in Stationary Structural Vector Autoregressive Models.” ArXiv 2202.09473.Suche in Google Scholar

Gourieroux, C., and J. Jasiak. 2022b. “Long Run Predictions.” Annals of Economics and Statistics 145: 75–90. https://doi.org/10.2307/48655902.Suche in Google Scholar

Gourieroux, C., J. Kim, and N. Meddahi. 2022. Stationary Ultra Long Run Component. CREST DP.Suche in Google Scholar

Gourieroux, C., and Y. Lu. 2020. SIR Model wtih Stochastic Transmission. CREST DP.10.2139/ssrn.3730349Suche in Google Scholar

Gray, A., D. Greenhalgh, L. Hu, X. Mao, and J. Pan. 2011. “A Stochastic Differential Equation SIS Epidemic Model.” SIAM Journal on Applied Mathematics 71: 876–902. https://doi.org/10.1137/10081856x.Suche in Google Scholar

Greenberg, J., and F. Hoppensteadt. 1975. “Asymptotic Behaviour of Solutions to a Population Equation.” SIAM Journal on Applied Mathematics 28: 662–74. https://doi.org/10.1137/0128055.Suche in Google Scholar

Hansen, B. 1992. “Convergence to Stochastic Integrals for Dependent Heterogeneous Processes.” Econometric Theory 8: 489–500. https://doi.org/10.1017/s0266466600013189.Suche in Google Scholar

Hastie, T., and R. Tibshirani. 1993. “Varying Coefficient Models.” Journal of the Royal Statistical Society: Series B 55: 757–96. https://doi.org/10.1111/j.2517-6161.1993.tb01939.x.Suche in Google Scholar

Hethcote, H., and P. Van den Driessche. 1995. “An SIS Epidemic Model with Variable Population Size and a Delay.” Journal of Mathematical Biology 34: 177–94. https://doi.org/10.1007/bf00178772.Suche in Google Scholar

Hethcote, H., and J. Yorke. 1994. Gonorrhea Transmission Dynamics and Control In Lecture Notes in Biomathematics, vol. 56. Berlin: Springer-Verlag.Suche in Google Scholar

Ho, P., T. Lubik, and C. Matthes. 2023. “How to Go Viral: A COVID-19 Model with Endogenously Time Varying Parameters.” Journal of Econometrics 232: 70–86. https://doi.org/10.1016/j.jeconom.2021.01.001.Suche in Google Scholar PubMed PubMed Central

Huber, P. 1967. “The Behaviour of Maximum Likelihood Estimates under Nonstandard Conditions.” In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 221–33.Suche in Google Scholar

Jennrich, R. 1969. “Asymptotic Properties of Nonlinear Least Squares Estimators.” The Annals of Mathematical Statistics 40: 633–43. https://doi.org/10.1214/aoms/1177697731.Suche in Google Scholar

Kim, C., and C. Nelson. 1999. State Space Models with Regime Switching. Cambridge, MA: Cambridge MIT Press.10.7551/mitpress/6444.001.0001Suche in Google Scholar

Kosmidis, I., and D. Firth. 2009. “Bias Reduction in Exponential Family Nonlinear Models.” Biometrika 96: 793–804. https://doi.org/10.1093/biomet/asp055.Suche in Google Scholar

Lambrecht, B., W. Perraudin, and S. Satchell. 1997. “Approximating the Finite Sample Bias for Maximum Likelihood Estimators Using the Score.” Econometric Theory 13: 310–2. https://doi.org/10.1017/s0266466600005806.Suche in Google Scholar

Lin, D., and Z. Ying. 2001. “Semiparametric and Nonparametric Regression Analysis of Longitudinal Data (With Discussion).” Journal of the American Statistical Association 96: 103–26. https://doi.org/10.1198/016214501750333018.Suche in Google Scholar

Matis, J., and T. Kiffe. 2000. Stochastic Population Models: A Compartmental Perspective. New-York: Springer.10.1007/978-1-4612-1244-7Suche in Google Scholar

McCullagh, P., and J. Nelder. 1989. Generalized Linear Models, 2nd ed. Chapman & Hall.10.1007/978-1-4899-3242-6Suche in Google Scholar

Nicholls, D., and B. Quinn. 1980. “The Estimation of Random Coefficient Autoregressive Model I.” Journal of Time Series Analysis 1: 37–46. https://doi.org/10.1111/j.1467-9892.1980.tb00299.x.Suche in Google Scholar

Primiceri, G. 2005. “Time Varying Structural Vector Autoregressions and Monetary Policy.” The Review of Economic Studies 72: 821–52. https://doi.org/10.1111/j.1467-937x.2005.00353.x.Suche in Google Scholar

Public Health Ontario (PHO). 2021. Enhanced Epidemiological Summary: Trends of COVID-19 Incidence in Ontario. 2021-05-23.Suche in Google Scholar

Rifhat, A., L. Wang, and Z. Teng. 2017. “Dynamics for a Class of Stochastic SIS Epidemic Models with Nonlinear Incidence and Periodic Coefficients.” Physica A 480: 176–90. https://doi.org/10.1016/j.physa.2017.04.016.Suche in Google Scholar

Rio, E. 2017. Asymptotic Theory of Weakly Dependent Random Processes In Probability Theory and Stochastic Modelling, vol. 80. Berlin Heidelberg: Springer.10.1007/978-3-662-54323-8Suche in Google Scholar

Rubio-Herrero, J., and Y. Wang. 2021. “A Flexible Rolling Regression Framework for Time-Varying SIRD Model: Application to COVID-19.” ArXiv 2103.02048.Suche in Google Scholar

Shapiro, M., F. Karim, G. Murcioni, and A. Augustine. 2020. Are We There Yet? An Adaptive SIR Model for Continuous Estimation of COViD-19 Infections Rate and Reproduction Number in the United States. DP Anthem, Inc.10.1101/2020.09.13.20193896Suche in Google Scholar

Stroock, D., and S. Varadhan. 1979. Multidimensional Diffusion Processes. Berlin Heidelberg: Springer.Suche in Google Scholar

Tweedie, R. 1975. “Sufficient Conditions for Ergodicity and Recurrence of Markov Chains on a General State Space.” Stochastic Processes and their Applications 3: 385–403. https://doi.org/10.1016/0304-4149(75)90033-2.Suche in Google Scholar

Varin, C., N. Reid, and D. Firth. 2011. “An Overview of Composite Likelihood Methods.” Statistica Sinica 21: 5–42.Suche in Google Scholar

Waqas, M., M. Farooq, R. Ahmed and A. Ahmed. 2020. “Analysis and Prediction of COVID-19 Pandemic in Pakistan Using Time Dependent SIR Model.” ArXiv: 2005.02353.Suche in Google Scholar

Wallinga, J., and P. Teunis. 2004. “Different Epidemic Curves for Severe Acute Respiratory Syndrome Reveal Similar Impacts of Control Measures.” American Journal of Epidemiology 160: 509–16. https://doi.org/10.1093/aje/kwh255.Suche in Google Scholar PubMed PubMed Central

White, H. 1982. “Maximum Likelihood Estimation of Misspecified Models.” Econometrica 50: 1–25. https://doi.org/10.2307/1912526.Suche in Google Scholar

Xu, C., and X. Li. 2018. “The Threshold of a Stochastic Delayed SIRS Epidemic Model with Temporary Immunity and Vaccination.” Chaos, Solitons & Fractals 111: 227–34. https://doi.org/10.1016/j.chaos.2017.12.027.Suche in Google Scholar

Yu, Z., L. Liu, D. Bravata, L. Williams, and R. Tepper. 2013. “A Semi-parametric Recurrent Event Model with Time-Varying Coefficients.” Statistics in Medicine 32: 1016–26. https://doi.org/10.1002/sim.5575.Suche in Google Scholar PubMed PubMed Central

Zhou, Z., and W. Wu. 2010. “Simultaneous Inference of Linear Models with Time Varying Coefficients.” Journal of the Royal Statistical Society - Series B 72: 513–31. https://doi.org/10.1111/j.1467-9868.2010.00743.x.Suche in Google Scholar

Received: 2022-05-30

Revised: 2022-10-31

Accepted: 2023-02-07

Published Online: 2023-03-07

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local maximum likelihood; rolling estimator; omitted heterogeneity; bias reduction; generalized linear model; SIS model; logistic growth; ultra long run