Applications of state space models in finance

Used abbreviations and symbols xxiii
¯F T +l
f t−1
minimum MSE variance matrix for a l-period ahead forecast
vector of lagged factor realizations
ft
variance of one-step ahead prediction errors (univariate case)
fk,t
¯f
˜fk,t
realizations of the k-th risk factor
steady-state value of ft
mean zero realizations of the k-th risk factor
fibt
FIBOR rate
F Xt
exchange rate factor
Ht
variance-covariance matrix of observation disturbances
h (i) i-th draw for h obtained from an importance density
ht
conditional variance in a GARCH model;
log-volatility process in a SV model
h unconditional variance of observation errors
i∗ T
the most probable state at time T in the Viterbi algorithm
J Φ
index matrix
Kt
¯K
Kalman gain matrix
steady-state value of the Kalman gain matrix
Kɛ
kurtosis for the unconditional distribution of ɛt
Kz
kurtosis for the unconditional distribution of zt
k number of explanatory variables
ku kurtosis of a series
Lt
element of the covariance matrix of the state vector
l order of test statistics;
lead time
lev correlation between R2 i,t and Ri,t−1
M number of draws from a simulation
m dimension of the state vector;
number of components in a mixture model
mt+1
stochastic discount factor
N t
variance matrix of the weighted sum of future innovations
Nt
number of pairs of sector ranks
N dimension of a time series vector
OILt
oil factor
P 1
covariance matrix of the initial state vector
P ∗, P ∞ elements of the covariance matrix of the initial state vector
P t
covariance matrix of the state vector
¯P steady-state value of P t
p ∗ likelihood of the most probable path in the Viterbi algorithm
p number of lagged conditional variance terms of a GARCH model
pt
price of an asset at time t
Qt variance-covariance matrix of state errors;
variance-covariance matrix of standardized residuals
Q0 initial variance-covariance matrix of state errors
variance-covariance matrix of state errors (extended state vector)
Q †
t