This package implements some common routines for state-space models. Provided algorithms include:
- Kalman filter (
kalman_filter) - Chandrasekhar recursions (
chand_recursion): "Using the "Chandrasekhar Recursions" for Likelihood Evaluation of DSGE Models" (2012) - Tempered particle filter (
tempered_particle_filter): "Tempered Particle Filtering" (2016) - Kalman smoothers:
hamilton_smoother: James Hamilton, Time Series Analysis (1994)koopman_smoother: S.J. Koopman, "Disturbance Smoother for State Space Models" (Biometrika, 1993)
- Simulation smoothers:
carter_kohn_smoother: C.K. Carter and R. Kohn, "On Gibbs Sampling for State Space Models" (Biometrika, 1994)durbin_koopman_smoother: J. Durbin and S.J. Koopman, "A Simple and Efficient Simulation Smoother for State Space Time Series Analysis" (Biometrika, 2002)
StateSpaceRoutines.jl is a registered Julia package in the General registry. To install, open your Julia REPL, type ] (enter package manager), and run
pkg> add StateSpaceRoutinesStateSpaceRoutines.jl is currently compatible with Julia 1.x.
To use StateSpaceRoutines.jl with Julia v0.7, please check out tag 0.2.0. To do this, click on the drop-down menu that reads branch: main on the left-hand side of the page. Select tags, then v0.2.0. If you've already cloned the repo, you can simply run git checkout v0.2.0.
The StateSpaceRoutines.jl package is not precompiled by default because when running code in parallel, we want to re-compile
the copy of StateSpaceRoutines.jl on each processor to guarantee the right version of the code is being used. If users do not
anticipate using parallelism, then users ought to change the first line of src/StateSpaceRoutines.jl from
isdefined(Base, :__precompile__) && __precompile__(false)
to
isdefined(Base, :__precompile__) && __precompile__(true)
s_{t+1} = C + T*s_t + R*ϵ_t (transition equation)
y_t = D + Z*s_t + u_t (measurement equation)
ϵ_t ∼ N(0, Q)
u_t ∼ N(0, E)
Cov(ϵ_t, u_t) = 0
kalman_filter(y, T, R, C, Q, Z, D, E, s_0 = Vector(), P_0 = Matrix(); outputs = [:loglh, :pred, :filt], Nt0 = 0)
chand_recursion(y, T, R, C, Q, Z, D, E, s_pred = Vector(), P_pred = Matrix(); allout = false, Nt0 = 0, tol = 0.0)
tempered_particle_filter(y, Φ, Ψ, F_ϵ, F_u, s_init; verbose = :high, include_presample = true, fixed_sched = [], r_star = 2, c = 0.3, accept_rate = 0.4, target = 0.4, xtol = 0, resampling_method = :systematic, N_MH = 1, n_particles = 1000, Nt0 = 0, adaptive = true, allout = true, parallel = false)
hamilton_smoother(y, T, R, C, Q, Z, D, E, s_0, P_0; Nt0 = 0)
koopman_smoother(y, T, R, C, Q, Z, D, s_0, P_0, s_pred, P_pred; Nt0 = 0)
carter_kohn_smoother(y, T, R, C, Q, Z, D, E, s_0, P_0; Nt0 = 0, draw_states = true)
durbin_koopman_smoother(y, T, R, C, Q, Z, D, E, s_0, P_0; Nt0 = 0, draw_states = true)
For more information, see the docstring for each function (e.g. enter ?kalman_filter in the REPL).
All of the provided algorithms can handle time-varying state-space systems. To do this, define regime_indices, a Vector{Range{Int64}} of length n_regimes, where regime_indices[i] indicates the range of periods t in regime i. Let T_i, R_i, etc. denote the state-space matrices in regime i. Then the state space is given by:
s_{t+1} = C_i + T_i*s_t + R_i*ϵ_t (transition equation)
y_t = D_i + Z_i*s_t + u_t (measurement equation)
ϵ_t ∼ N(0, Q_i)
u_t ∼ N(0, E_i)
Letting Ts = [T_1, ..., T_{n_regimes}], etc., we can then call the time-varying methods of the algorithms:
kalman_filter(regime_indices, y, Ts, Rs, Cs, Qs, Zs, Ds, Es, s_0 = Vector(), P_0 = Matrix(); outputs = [:loglh, :pred, :filt], Nt0 = 0)
hamilton_smoother(regime_indices, y, Ts, Rs, Cs, Qs, Zs, Ds, Es, s_0, P_0; Nt0 = 0)
koopman_smoother(regime_indices, y, Ts, Rs, Cs, Qs, Zs, Ds, s_0, P_0, s_pred, P_pred; Nt0 = 0)
carter_kohn_smoother(regime_indices, y, Ts, Rs, Cs, Qs, Zs, Ds, Es, s_0, P_0; Nt0 = 0, draw_states = true)
durbin_koopman_smoother(regime_indices, y, Ts, Rs, Cs, Qs, Zs, Ds, Es, s_0, P_0; Nt0 = 0, draw_states = true)
The tempered particle filter is a particle filtering method which can approximate the log-likelihood value implied by a general (potentially non-linear) state space system with the following representation:
s_{t+1} = Φ(s_t, ϵ_t) (transition equation)
y_t = Ψ(s_t) + u_t (measurement equation)
ϵ_t ∼ F_ϵ(∙; θ)
u_t ∼ N(0, E)
Cov(ϵ_t, u_t) = 0
- The documentation and code are located in src/filters/tempered_particle_filter.
- The example is located in docs/examples/tempered_particle_filter
- The paper proposing and analyzing the method is Herbst and Schorfheide (2019)
- For nonlinear smoothing, see LowLevelParticleFilters.jl
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