Applications of state space models in finance

xxii Used abbreviations and symbols
θ (i) i-th draw of θ
ϑ coefficient related to EGARCH or GJR-GARCH component
ρ correlation coefficient
ρS t
Spearman’s rank correlation coefficient
Σ matrix of combined state vector and initial covariance matrix
σ2 unconditional standard deviation;
prediction error variance
σt
conditional standard deviation
˜σ 2 approximated p.e.v. in terms of unstandardized GLS residuals
positive scaling factor to which the variance is proportional
σ2 ∗
˜σ 2 ∗(ψ∗) ML estimator of σ2 ∗ for given ψ∗ without explanatory variables
˜σ +2 p.e.v. for models with fixed regression coefficients
˜σ +2
∗
ML estimator of σ2 ∗ that depends on GLS residuals
ςt
state disturbances
τ time index
υt
combined vector of state and observation disturbances
ξt ¯ξ
state vector
mean state vector
ξ ∗
mean-corrected state vector
ξ †
t
ˆξ t
extended state vector
smoothed state vector
Ψ quadratic matrix used in the context of GLS
ψ vector of unknown parameters
ψ∗ concentrated vector of unknown parameters
ζt state disturbances
Other letters
A selection matrix
at
mean state vector
a1
mean vector of the initial state vector
a constant term as part of the initial state vector
at
location parameter
B matrix of unconditional factor loadings
bt
scaling parameter
BMRt benchmark residual factor
ct
system vector in state equation
ct
consumption at time t
d number of diffuse elements in the state vector
Dt
matrix of conditional standard deviations
Di,t
rank difference
dt
system vector in observation equation
F t
variance matrix of the one-step ahead prediction error