bvartools (version 0.0.1)

kalman_dk: Durbin and Koopman Simulation Smoother

Description

An implementation of the Kalman filter and backward smoothing algorithm proposed by Durbin and Koopman (2002).

Usage

kalman_dk(y, z, sigma_u, sigma_v, B, a_init, P_init)

Arguments

y

a \(K \times T\) matrix of endogenous variables.

z

a \(KT \times M\) matrix of explanatory variables.

sigma_u

the inverse of the constant \(K \times K\) error variance-covariance matrix. For time varying variance-covariance matrices a \(KT \times K\) can be specified.

sigma_v

the inverse of the constant \(M \times M\) coefficient variance-covariance matrix. For time varying variance-covariance matrices a \(MT \times M\) can be specified.

B

an \(M \times M\) autocorrelation matrix of the transition equation.

a_init

an M-dimensional vector of initial states.

P_init

an \(M \times M\) variance-covariance matrix of the initial states.

Value

A \(M \times T+1\) matrix of state vector draws.

Details

The function uses algorithm 2 from Durbin and Koopman (2002) to produce a draw of the state vector \(a_t\) for \(t = 1,...,T\) for a state space model with measurement equation $$y_t = Z_t a_t + u_t$$ and transition equation $$a_{t + 1} = B_t a_{t} + v_t,$$ where \(u_t \sim N(0, \Sigma_{u,t})\) and \(v_t \sim N(0, \Sigma_{v,t})\). \(y_t\) is a K-dimensional vector of endogenous variables and \(Z_t = z_t^{\prime} \otimes I_K\) is a \(K \times M\) matrix of regressors with \(z_t\) as a vector of regressors.

The algorithm takes into account Jaroci<U+0144>ski (2015), where a possible missunderstanding in the implementation of the algorithm of Durbin and Koopman (2002) is pointed out. Following that note the function sets the mean of the initial state to zero in the first step of the algorithm.

References

Durbin, J., & Koopman, S. J. (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89(3), 603--615.

Jaroci<U+0144>ski, M. (2015). A note on implementing the Durbin and Koopman simulation smoother. Computational Statistics and Data Analysis, 91, 1--3. https://doi.org/10.1016/j.csda.2015.05.001

Examples

# NOT RUN {
# Prepare data
data("e1")
data <- diff(log(e1))
temp <- gen_var(data, p = 2, deterministic = "const")
y <- temp$Y
x <- temp$Z
k <- nrow(y)
z <- kronecker(t(x), diag(1, k))
t <- ncol(y)
m <- k * nrow(x)

# Priors
a_mu_prior <- matrix(0, m)
a_v_i_prior <- diag(0.1, m)

a_Q <- diag(.0001, m)

# Initial value of Sigma
sigma <- tcrossprod(y) / t
sigma_i <- solve(sigma)

# Initial values for Kalman filter
y_init <- y * 0
a_filter <- matrix(0, m, t + 1)

# Initialise the Kalman filter
for (i in 1:t) {
  y_init[, i] <- y[, i] - z[(i - 1) * k + 1:k,] %*% a_filter[, i]
}
a_init <- post_normal_sur(y = y_init, z = z, sigma_i = sigma_i,
                          a_prior = a_mu_prior, v_i_prior = a_v_i_prior)
y_filter <- y - matrix(a_init, k) %*% x

# Kalman filter and backward smoother
a_filter <- kalman_dk(y = y_filter, z = z, sigma_u = sigma,
                      sigma_v = a_Q, B = diag(1, m),
                      a_init = matrix(0, m), P_init = a_Q)
                      
a <- a_filter + matrix(a_init, m, t + 1)

# }