KALCVS Call

Language Reference

KALCVS Call

CALL KALCVS( sm, vsm, data, a, f, b, h, var, pred, vpred <,un, vun>);

The KALCVS call uses backward recursions to compute the smoothed estimate $z_{t| T}$ and its covariance matrix, $P_{t| T}$ , where T is the number of observations in the complete data set.

The inputs to the KALCVS subroutine are as follows:

data: is a T ×N_y matrix containing data (y₁, ... , y_T)'.
a: is an N_z ×1 vector for a time-invariant input vector in the transition equation, or a TN_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 TN_z ×N_z matrix containing T transition matrices.
b: is an N_y ×1 vector for a time-invariant input vector in the measurement equation, or a TN_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 TN_y ×N_z matrix containing T time variant H_t matrices in the measurement equation.
var: is an (N_y + N_z) ×(N_y + N_z) covariance matrix for the errors in the transition and the measurement equations, or a T(N_y + N_z) ×(N_y + N_z) matrix containing covariance matrices in the transition equation and measurement equation noises, that is, $(\eta^'_t, \epsilon^'_t)^'$ .
pred: is a T ×N_z matrix containing one-step forecasts $(z_{1|}, ... , z_{T| T-1})^'$ .
vpred: is a TN_z ×N_z matrix containing mean square error matrices of predicted state vectors $(P_{1|}, ... , P_{T| T-1})^'$ .
un: is an optional 1 ×N_z vector containing ${\mu}_T$ .The returned value is ${\mu}_0$ .
vun: is an optional N_z ×N_z matrix containing U_T. The returned value is U₀.

The KALCVS call returns the following values:

sm: is a T ×N_z matrix containing smoothed state vectors $(z_{1| T}, ... , z_{T| T})^'$ .
vsm: is a TN_z ×N_z matrix containing covariance matrices of smoothed state vectors $(P_{1| T}, ... , P_{T| T})^'$ .

When the Kalman filtering is performed in the KALCVF call, the KALCVS call computes smoothed state vectors and their covariance matrices. The fixed-interval smoothing state vector at time t is obtained by the conditional expectation given all observations.

The smoothing algorithm uses one-step forecasts and their covariance matrices, which are given in the KALCVF call. For notation, $z_{t| T}$ is the smoothed value of the state vector z_t, and the mean square error matrix is denoted $P_{t| T}$ .For smoothing,

$\hat{\epsilon}_t & = & y_t - b_t - H_t z_{t| t-1} \ D_t & = & H_t P_{t| t-1} H^... ... t-1} {\mu}_{t-1} \ P_{t| T} & = & P_{t| t-1} - P_{t| t-1} U_{t-1} P_{t| t-1}$

where t = T, T-1, ... , 1. The initial values are ${\mu}_T = 0$ and U_T = 0.

When the SSM is specified using the alternative transition equation

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

the fixed-interval smoothing is performed using the following backward recursions:

$\hat{\epsilon}_t & = & y_t - b_t - H_t z_{t| t-1} \ D_t & = & H_t P_{t| t-1} H^... ...-1} {\mu}_{t-1} \ P_{t| T} & = & P_{t| t-1} - P_{t| t-1} U_{t-1} P_{t| t-1}$

where it is assumed that G_t = 0. You can use the KALCVS call regardless of the specification of the transition equation when G_t = 0. Harvey (1989) gives the following fixed-interval smoothing formula, which produces the same smoothed value:

$z_{t| T} & = & z_{t| t} + P^*_t (z_{t+1| T} - z_{t+1| t}) \ P_{t| T} & = & P_{t| t} + P^*_t (P_{t+1| T} - P_{t+1| t}) P^{*'_t$

where

$P^*_t = P_{t| t} F^'_t P^-_{t+1| t}$

under the shifted transition equation, but

$P^*_t = P_{t| t} F^'_{t+1} P_{t+1| t}^-$

under the alternative transition equation.

The KALCVS call is accompanied by the KALCVF call, as shown in the following code. Note that you do not need to specify UN and VUN.

call kalcvf(pred,vpred,filt,vfilt,y,0,a,f,b,h,var);
call kalcvs(sm,vsm,y,a,f,b,h,var,pred,vpred);

You can also compute the smoothed estimate and its covariance matrix on an observation-by-observation basis. When the SSM is time invariant, the following example performs smoothing. In this situation, you should initialize UN and VUN as matrices of value 0.

call kalcvf(pred,vpred,filt,vfilt,y,0,a,f,b,h,var);
n = nrow(y);
nz = ncol(f);
un = j(1,nz,0);
vun = j(nz,nz,0);
do i = 1 to n;
  y_i = y[n-i+1,];
  pred_i  = pred[n-i+1,];
  vpred_i = vpred[(n-i)*nz+1:(n-i+1)*nz,];
  call kalcvs(sm_i,vsm_i,y_i,a,f,b,h,var,pred_i,vpred_i,un,vun);
  sm  = sm_i // sm;
  vsm = vsm_i // vsm;
end;

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