Computing exact score vectors for linear Gaussian state space models

Abstract

A recursive formula for computing the exact value of score vectors is proposed for a general form of the linear Gaussian state space model, which is more desirable than approximate values in some statistical analyses. Unlike most extant methods, our formula calculates all components of the score vector simultaneously. This approach significantly simplifies its programing, in particular, with some matrix-oriented programing languages, such as MATLAB. We also consider a way of handling initial conditions that depend on unknown parameters. This issue has not yet been explicitly addressed in the existing literature in the context of exact score computing for a general case, such as the one that we consider in this paper. It is also shown that our formula is especially useful for calculating score tests with an outer product of gradient asymptotic covariance matrix estimator.

Keywords:

• Recursive formula
• State space model
• Score vector
• Score test
• Initial condition

MSC CODES:

• 62H15
• 15A99
• 60G35

Notes

1 See also Nagakura (2013), which derived a vector version of the formula of Koopman and Shephard (1992) by using the results in matrix calculus derived in this paper.

2 Segal and Weinstein (1989) discuss how to reconstruct an observationally equivalent state space model when ${\mathbf{S}}_t$ or ${\mathbf{R}}_t{\mathbf{Q}}_t{\mathbf{R}}_t^{\prime }$ is singular for applying their formula.

3 For example, this holds when c_ij and c_ji are the same function of $\mathit{\theta }$, which implies that C is symmetric at any value of $\mathit{\theta }.$

4 Our notation of the commutation matrix ${\mathbf{K}}_{m,n}$ follows the notation presented by Magnus and Neudecker (1999), which differs slightly from the one proposed by Neudecker and Wansbeek (1983). The matrix ${\mathbf{K}}_{m,n}$ here corresponds to the matrix ${\mathbf{P}}_{n,m}$ in Neudecker and Wansbeek (1983). Note, in particular, that the order of the subscripts is opposite.

5 The element- and component-wise versions of equations in (6) and (7), that calculate $\partial {\mathbf{a}}_{t+1|t}/\partial {\theta }_i$ and $\partial {\mathbf{P}}_{t+1|t}/\partial {\theta }_i,$ respectively, are known as the filter sensitivity equations and Riccati-type sensitivity equations, respectively in the engineering literature. In addition, calculating the exact score with those formulas is referred to as the forward filter evaluation of the score (see Astrom (1981) and Sandell and Yared (1978)).

6 This is not an issue for nonstationary components in the state vector because the diffuse initial setting does not rely on the parameter vector.