BasicStateSpace.jl

WORK IN PROGRESS

A package for basic operations with discrete time state space operations

State Space model

• Transition equation:

$$ s_t = c + A s_{t-1} + B u_t $$

• Measurement equation:

$$ y_t = d + C s_{t} + e_{t} $$

with $E(u) = E(e) = 0$.

• Variance of shocks:

$$ Var(u) = Q$$

• Variance of measurement errors:

$$ Var(e) = H$$

Computations

• unconditional mean:

$$s = (I - A)^{-1}d$$

$$y = d + C s$$

• unconditional variance:

$$ Var(s) = A Var(s) A' + B Q B'$$

$$ Var(y) = C Var(s) C' + H$$

• mean forecast:

$$ s^f_{T+k} = c + A^k s_T$$$

$$ y^f_{T+k} = d + C s^f_{T+k}$$

• variance of forecast error:

$$ Var(s^f_{T+k} - s_{T+k}) = \sum_{i=1:k} A^{i-1}BQB'{A^{i-1}}'$$

$$ Var(y^f_{T+k} - y_{T+l}) = C Var(s^f_{T+k}) C' + H$$

• impulse response funtion (IRF)to shock on $u_{i1}$:

$$ s^r_1 = c + A s^r_0 + Bu_{i1} $$

$$ s^r_t = c + A s^r_{t-1} 1 < t <= T$$

$$ y^r_t = d + Cs^r_t$$

• variance of IRF taking into account future shocks: identical to variance of forecast error