KALDFF Call

Language Reference

KALDFF Call

CALL KALDFF( pred, vpred, initial, s2, data, lead, int, coef, var,

intd, coefd <, n0, at, mt, qt>);

The KALDFF call computes the one-step forecast of state vectors in an SSM using the diffuse Kalman filter. The call estimates the conditional expectation of z_t, and it also estimates the initial random vector, $\delta$ , and its covariance matrix.

The inputs to the KALDFF subroutine are as follows:

data: is a T ×N_y matrix containing data (y₁, ... , y_T)'.
lead: is the number of steps to forecast after the end of the data set.
int: is an $(N_y + N_z) x N_\beta$ matrix for a time-invariant fixed matrix, or a $(T+{lead})(N_y + N_z) x N_\beta$ matrix containing fixed matrices for the time-variant model in the transition equation and the measurement equation, that is, (W'_t, X'_t)'.
coef: is an (N_y + N_z) ×N_z matrix for a time-invariant coefficient, or a (T+ lead)(N_y + N_z) ×N_z matrix containing coefficients at each time in the transition equation and the measurement equation, that is, (F'_t, H'_t)'.
var: is an (N_y + N_z) ×(N_y + N_z) matrix for a time-invariant variance matrix for the error in the transition equation and the error in the measurement equation, or a (T+ lead)(N_y + N_z) ×(N_y + N_z) matrix containing covariance matrices for the error in the transition equation and the error in the measurement equation, that is, $(\eta^'_t, \epsilon^'_t)^'$ .
intd: is an $(N_z + N_\beta) x 1$ vector containing the intercept term in the equation for the initial state vector z₀ and the mean effect $\beta$ , that is, (a', b')'.
coefd: is an $(N_z + N_\beta) x N_\delta$ matrix containing coefficients for the initial state $\delta$ in the equation for the initial state vector z₀ and the mean effect $\beta$ , that is, (A', B')'.
n0: is an optional scalar including an initial denominator. If n0>0, the denominator for $\hat{\sigma}^2_t$ is n0 plus the number n_t of elements of (y₁, ... , y_t)'. If $n0 \leq 0$ or n0 is not specified, the denominator for $\hat{\sigma}^2_t$ is n_t. With $n0 \geq 0$ , the initial values, A₁, M₁, and Q₁, are assumed to be known and, hence, at, mt, and qt are used for input containing the initial values. If the value of n0 is negative or n0 is not specified, the initial values for at, mt, and qt are computed. The value of n0 is updated as max(n0,0) + n_t after the KALDFF call.
at: is an optional $kN_z x (N_\delta + 1)$ matrix. If $n0 \geq 0$ , at contains (A'₁, ... , A'_k)'. However, only the first matrix A₁ is used as input. When you specify the KALDFF call, at returns (A'_{T-k+ lead+1}, ... , A'_{T+ lead})'. If n0 is negative or the matrix A₁ contains missing values, A₁ is automatically computed.
mt: is an optional kN_z ×N_z matrix. If $n0 \geq 0$ , mt contains (M₁, ... , M_k)'. However, only the first matrix M₁ is used as input. If n0 is negative or the matrix M₁ contains missing values, mt is used for output, and it contains (M_{T-k+ lead+1}, ... , M_{T+ lead})'. Note that the matrix M₁ can be used as an input matrix if either of the off-diagonal elements is not missing. The missing element M₁(i,j) is replaced by the nonmissing element M₁(j,i).
qt: is an optional $k(N_\delta + 1) x (N_\delta + 1)$ matrix. If $n0 \geq 0$ , qt contains (Q₁, ... , Q_k)'. However, only the first matrix Q₁ is used as input. If n0 is negative or the matrix Q₁ contains missing values, qt is used for output and contains (Q_{T-k+ lead+1}, ... , Q_{T+ lead})'. The matrix Q₁ can also be used as an input matrix if either of the off-diagonal elements is not missing since the missing element Q₁(i,j) is replaced by the nonmissing element Q₁(j,i).

The KALCVF call returns the following values:

pred: is a (T+ lead) ×N_z matrix containing estimated predicted state vectors $(\hat{z}_{1|}, ... , \hat{z}_{T+1| T}, \hat{z}_{T+2| T}, ... , \hat{z}_{T+{lead}| T})^'$ .
vpred: is a (T+ lead)N_z ×N_z matrix containing estimated mean square errors of predicted state vectors $(P_{1|}, ... , P_{T+1| T}, P_{T+2| T}, ... , P_{T+{lead}| T})^'$ .
initial: is an N_d ×(N_d + 1) matrix containing an estimate and its variance for initial state $\delta$ , that is, $(\hat{\delta}_T, \hat{\Sigma}_\delta, T)$ .
s2: is a scalar containing the estimated variance $\hat{\sigma}^2_T$ .

The KALDFF call computes the one-step forecast of state vectors in an SSM using the diffuse Kalman filter. The SSM for the diffuse Kalman filter is written

$y_t & = & X_t \beta + H_t z_t + \epsilon_t \ z_{t+1} & = & W_t \beta + F_t z_t + \eta_t \ z_0 & = & a+ A\delta \ \beta & = & b+ B\delta$

where z_t is an N_z ×1 state vector, y_t is an N_y ×1 observed vector, and

$[ \eta_t \ \epsilon_t ] & \sim & N (0, \sigma^2 [ V_t & G_t \ G^'_t & R_t ] ) \\delta & \sim & N(\mu, \sigma^2\Sigma)$

It is assumed that the noise vector $(\eta^'_t, \epsilon^'_t)^'$ is independent and $\delta$ is independent of the vector $(\eta^'_t, \epsilon^'_t)^'$ .The matrices, W_t, F_t, X_t, H_t, a, A, b, B, V_t, G_t, and R_t, are assumed to be known. The KALDFF call estimates the conditional expectation of the state vector z_t given the observations. The KALDFF subroutine also produces the estimates of the initial random vector $\delta$ and its covariance matrix. For k-step forecasting where k>0, the estimated conditional expectation at time t + k is computed with observations given up to time t. The estimated k-step forecast and its estimated MSE are denoted $z_{t+k| t}$ and $P_{t+k| t}$ (for k>0). $A_{t+k(\delta)}$ and $E_{t(\delta)}$ are last-column-deleted submatrices of A_t+k and E_t, respectively. The algorithm for one-step prediction is given as follows:

$E_t & = & (X_t B, y_t - X_t b) - H_t A_t \ D_t & = & H_t M_t H^'_t + R_t \... ...a,t} & = & \hat{\sigma}^2_t S^-_t \ K_t & = & (F_t M_t H^'_t + G_t)D^-_t \$

$A_{t+1} & = & W_t(-B, b) + F_t A_t + K_t{E}_t \ M_{t+1} & = & (F_t - K_t H_t)... ...a}^2_t M_{t+1} + A_{t+1(\delta)} \hat{\Sigma}_{\delta,t} A^'_{t+1(\delta)}$

where n_t is the number of elements of (y₁, ... ,y_t)' plus max(n0,0). Unless initial values are given and $n0 \geq 0$ , initial values are set as follows:

$A_1 & = & W_1(-B,b) + F_1(-A,a) \ M_1 & = & V_1 \ Q_1 & = & 0$

For k-step forecasting where k>1,

$A_{t+k} & = & W_{t+k-1}(-B,b) + F_{t+k-1} A_{t+k-1} \ M_{t+k} & = & F_{t+k-1}... ...igma}^2_t M_{t+k} + A_{t+k(\delta)} \hat{\Sigma}_{\delta,t} A^'_{t+k(\delta)}$

Note that if there is a missing observation, the KALDFF call computes the one-step forecast for the observation following the missing observation as the two-step forecast from the previous observation.

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