KALCVF Call

Language Reference

KALCVF Call

CALL KALCVF( pred, vpred, filt, vfilt, data, lead, a, f, b, h,

var <, z0, vz0>);

The KALCVF call computes the one-step prediction $z_{t+1| t}$ and the filtered estimate $z_{t| t}$ , as well as their covariance matrices. The call uses forward recursions, and you can also use it to obtain k-step estimates.

The inputs to the KALCVF 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.
a: is an N_z ×1 vector for a time-invariant input vector in the transition equation, or a (T+ lead)N_z ×1 vector containing input vectors in the transition equation.
f: is an N_z ×N_z matrix for a time-invariant transition matrix in the transition equation, or a (T+ lead)N_z ×N_z matrix containing transition matrices in the transition equation.
b: is an N_y ×1 vector for a time-invariant input vector in the measurement equation, or a (T+ lead)N_y ×1 vector containing input vectors in the measurement equation.
h: is an N_y ×N_z matrix for a time-invariant measurement matrix in the measurement equation, or a (T+ lead)N_y ×N_z matrix containing measurement matrices in the measurement equation.
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 variance matrices for the error in the transition equation and the error in the measurement equation, that is, $(\eta^'_t, \epsilon^'_t)^'$ .
z0: is an optional 1 ×N_z initial state vector $z^'_{1|}$ .
vz0: is an optional N_z ×N_z covariance matrix of an initial state vector $P_{1|}$ .

The KALCVF call returns the following values:

pred: is a (T+ lead) ×N_z matrix containing one-step predicted state vectors $(z_{1|}, ... , z_{T+1| T}, z_{T+2| T}, ... , z_{T+{lead}| T})^'$ .
vpred: is a (T+ lead)N_z ×N_z matrix containing mean square errors of predicted state vectors $(P_{1|}, ... , P_{T+1| T}, P_{T+2| T}, ... , P_{T+{lead}| T})^'$ .
filt: is a T ×N_z matrix containing filtered state vectors $(z_{1| 1}, ... , z_{T| T})^'$ .
vfilt: is a TN_z ×N_z matrix containing mean square errors of filtered state vectors $(P_{1| 1}, ... , P_{T| T})^'$ .

The KALCVF call computes the conditional expectation of the state vector z_t given the observations, assuming that the mean and the variance of the initial state vector are known. The filtered value is the conditional expectation of the state vector z_t given the observations up to time t. For k-step forecasting where k>0, the conditional expectation at time t+k is computed given observations up to t. For notation, V_t and R_t are variances of $\eta_t$ and $\epsilon_t$ , respectively, and G_t is a covariance of $\eta_t$ and $\epsilon_t$ .A^- stands for the generalized inverse of A. The filtered value and its covariance matrix are denoted $z_{t| t}$ and $P_{t| t}$ , respectively. For k>0, $z_{t+k| t}$ and $P_{t+k| t}$ stand for the k-step forecast of z_t+k and its mean square error. The Kalman filtering algorithm for one-step prediction and filtering is given as follows:

$\hat{\epsilon}_t & = & y_t - b_t - H_t z_{t| t-1} \ D_t & = & H_t P_{t| t-1} H^... ...hat{\epsilon}_t \ P_{t+1| t} & = & F_t P_{t| t-1}F^'_t + V_t - K_t D_t K^'_t$

And for k-step forecasting for k>1,

$z_{t+k| t} & = & a_{t+k-1} + F_{t+k-1} z_{t+k-1| t} \ P_{t+k| t} & = & F_{t+k-1} P_{t+k-1| t} F^'_{t+k-1} + V_{t+k-1}$

When you use the alternative transition equation

$z_t = a_t + F_t{z}_{t-1} + \eta_t$

the forward recursion algorithm is written

$\hat{\epsilon}_t & = & y_t - b_t - H_t{z}_{t| t-1} \ D_t & = & H_t P_{t| t-1} H... ...}_t \ P_{t+1| t} & = & F_{t+1} P_{t| t-1}F^'_{t+1} + V_{t+1} - K_t D_t K^'_t$

And for k-step forecasting (k>1),

$z_{t+k| t} & = & a_{t+k} + F_{t+k}z_{t+k-1| t} \ P_{t+k| t} & = & F_{t+k} P_{t+k-1| t} F^'_{t+k} + V_{t+k}$

You can use the KALCVF call when you specify the alternative transition equation and G_t = 0.

The initial state vector and its covariance matrix of the time invariant Kalman filters are computed under the stationarity condition

$z_{1|} & = & (I- F)^- a\ P_{1|} & = & (I- F\otimes F)^- {vec}(V)$

where F and V are the time invariant transition matrix and the covariance matrix of transition equation noise, and vec(V) is an N_z² ×1 column vector that is constructed by the stacking N_z columns of matrix V. Note that all eigenvalues of the matrix F are inside the unit circle when the SSM is stationary. When the preceding formula cannot be applied, the initial state vector estimate $z_{1|}$ is set to a₁ and its covariance matrix $P_{1|}$ is given by 10⁶I. Optionally, you can specify initial values.

The KALCVF call allows missing values in observations. If there is a missing observation, the filtered state vector for the missing observation is given by the one-step forecast.

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