Maximum likelihood estimation of time series models: the Kalman filter and beyond

Abstract

The purpose of this chapter is to provide a comprehensive treatment of likelihood inference for state space models. These are a class of time series models relating an observable time series to quantities called states, which are characterized by a simple temporal dependence structure, typically a first order Markov process.The states have sometimes substantial interpretation. Key estimation problems in economics concern latent variables, such as the output gap, potential output, the non-accelerating-inflation rate of unemployment, or NAIRU, core inflation, and so forth. Time-varying volatility, which is quintessential to finance, is an important feature also in macroeconomics. In the multivariate framework relevant features can be common to different series, meaning that the driving forces of a particular feature and/or the transmission mechanism are the same.The objective of this chapter is reviewing this algorithm and discussing maximum likelihood inference, starting from the linear Gaussian case and discussing the extensions to a nonlinear and non Gaussian framework.

A state-space approach for longitudinal outcomes: An application to neuropsychological outcomes

“…While this step is useful in constructing the joint likelihood of a series, the state estimates obtained via this method are sub-optimal. 28,29,18 After all, it does seem intuitively appealing to utilize all data up until the last time point, j, to infer on a latent variable or a parameter of interest. Letting boldD i , j = boldP i , j | j boldG i , j + 1 ′ + boldP i , j + 1 | j − 1, the Kalman smoother can be derived via the law of iterated expectation in this fashion 16 : and …”

Kredilerin GSYİHye Oranı Açığı ve Sistemik Banka Krizleri Arasındaki İlişki: Türkiye Örneği

“…Its sensitivity to the length of the observation period and the possibility of uncertainty in end-point estimates of time-series are much lower compared to alternative methods. (Özbek & Özlale, 2005;Proietti & Luati, 2012;Johnson, 2013).…”

Automatic Identification and Forecasting of Structural Unobserved Components Models with UComp

“…Maximum likelihood is the usual method applied to these systems. There are many issues related to the identification and estimation of UC models that are beyond the scope of this paper and are addressed in many excellent references (see, for example, Harvey 1989;Young et al 1999;De Jong 1991;Durbin and Koopman 2012;Proietti and Luati 2013;Pelagatti 2015;Casals et al 2016;Villegas and Pedregal 2018). Only those issues specifically relevant to this paper will be mentioned below.…”

“…UComp is a library that implements comprehensive procedures for the automatic identification, estimation and forecasting of univariate unobserved components models (UC). All the implemented models belong to the class of structural UC proposed in a seminal work of Harvey (1989) and extended in many ways by a long and rich literature stream (see, e.g., De Jong 1991;Young, Pedregal, and Tych 1999;Durbin and Koopman 2012;Proietti and Luati 2013;Pelagatti 2015, etc. ).…”