*J. Durbin and S.J. Koopman*

- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199641178
- eISBN:
- 9780191774881
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641178.003.0004
- Subject:
- Mathematics, Probability / Statistics

This chapter begins with a set of four lemmas from elementary multivariate regression which provides the essentials of the theory for the general linear state space model from both a classical and a ...
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This chapter begins with a set of four lemmas from elementary multivariate regression which provides the essentials of the theory for the general linear state space model from both a classical and a Bayesian standpoint. The four lemmas lead to derivations of the Kalman filter and smoothing recursions for the estimation of the state vector and its conditional variance matrix given the data. The chapter also derives recursions for estimating the observation and state disturbances, and derives the simulation smoother, which is an important tool in the simulation methods employed later in the book. It shows that allowance for missing observations and forecasting are easily dealt with in the state space framework.Less

This chapter begins with a set of four lemmas from elementary multivariate regression which provides the essentials of the theory for the general linear state space model from both a classical and a Bayesian standpoint. The four lemmas lead to derivations of the Kalman filter and smoothing recursions for the estimation of the state vector and its conditional variance matrix given the data. The chapter also derives recursions for estimating the observation and state disturbances, and derives the simulation smoother, which is an important tool in the simulation methods employed later in the book. It shows that allowance for missing observations and forecasting are easily dealt with in the state space framework.

*J. Durbin and S.J. Koopman*

- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199641178
- eISBN:
- 9780191774881
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641178.003.0005
- Subject:
- Mathematics, Probability / Statistics

Computational algorithms in state space analyses are mainly based on recursions, that is, formulae in which the value at time t + 1 is calculated from earlier values for t, t − 1, …, 1. This chapter ...
More

Computational algorithms in state space analyses are mainly based on recursions, that is, formulae in which the value at time t + 1 is calculated from earlier values for t, t − 1, …, 1. This chapter deals with the question of how these recursions are started up at the beginning of the series, a process called initialisation. It provides a general treatment in which some elements of the initial state vector have known distributions while others are diffuse, that is, treated as random variables with infinite variance, or are treated as unknown constants to be estimated by maximum likelihood. The discussions cover the exact initial Kalman filter; exact initial state smoothing; exact initial disturbance smoothing; exact initial simulation smoothing; examples of initial conditions for some models; and augmented Kalman filter and smoother.Less

Computational algorithms in state space analyses are mainly based on recursions, that is, formulae in which the value at time *t* + 1 is calculated from earlier values for *t*, *t* − 1, …, 1. This chapter deals with the question of how these recursions are started up at the beginning of the series, a process called initialisation. It provides a general treatment in which some elements of the initial state vector have known distributions while others are diffuse, that is, treated as random variables with infinite variance, or are treated as unknown constants to be estimated by maximum likelihood. The discussions cover the exact initial Kalman filter; exact initial state smoothing; exact initial disturbance smoothing; exact initial simulation smoothing; examples of initial conditions for some models; and augmented Kalman filter and smoother.