Language Reference: KALDFS Call :: SAS/IML(R) 9.2 User's Guide

Language Reference

KALDFS Call

computes the smoothed state vector and its mean square error matrix from the one-step forecast and mean square error matrix computed by KALDFF

CALL KALDFS( sm, vsm, data, int, coef, var, bvec, bmat, initial, at,

mt, <, un, vun>);

The inputs to the KALDFS subroutine are as follows:

data: is a matrix containing data $({y}_1, ... , {y}_t)^'$ .
int: is an $(n_y + n_z) x n_\beta$ vector for a time-invariant intercept, or a $(t+{lead})(n_y + n_z) x n_\beta$ vector 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 matrix for a time-invariant coefficient, or a $(t+{lead})(n_y + n_z) x n_z$ matrix containing coefficients at each time in the transition equation and the measurement equation - that is, $({f}^'_t, {h}^'_t)^'$ .
var: is an matrix for a time-invariant variance matrix for transition equation noise and the measurement equation noise, or a $(t+{lead})(n_y + n_z) x (n_y + n_z)$ matrix containing covariance matrices for the transition equation and measurement equation errors - that is, $(\eta^'_t, \epsilon^'_t)^'$ .
bvec: is an $n_\beta x 1$ constant vector for the intercept for the mean effect $\beta$ .
bmat: is an $n_\beta x n_\delta$ matrix for the coefficient for the mean effect $\beta$ .
initial: is an $n_\delta x (n_\delta + 1)$ matrix containing an initial random vector estimate and its covariance matrix - that is, $(\hat{\delta}_t, \hat{\sigma}_{\delta, t})$ .
at: is a $tn_z x (n_\delta + 1)$ matrix containing $({a}^'_1, ... , {a}^'_t)^'$ .
mt: is a matrix containing $({m}_1, ... , {m}_t)^'$ .
: is the estimated variance in the end of the data set, $\hat{\sigma}^2_t$ .
un: is an optional $n_z x (n_{\delta} + 1)$ matrix containing ${u}_t$ . The returned value is ${u}_0$ .
vun: is an optional matrix containing ${u}_t$ . The returned value is ${u}_0$ .

The KALDFS call returns the following values:

sm: is a matrix containing smoothed state vectors $({z}_{1| t}, ... , {z}_{t| t})^'$ .
vsm: is a matrix containing mean square error matrices of smoothed state vectors $({p}_{1| t}, ... , {p}_{t| t})^'$ .

Given the one-step forecast and mean square error matrix in the KALDFF call, the KALDFS call computes a smoothed state vector and its mean square error matrix. Then the KALDFS subroutine produces an estimate of the smoothed state vector at time

- that is, the conditional expectation of the state vector ${z}_t$ given all observations. Using the notations and results from the KALDFF section, the backward recursion algorithm for smoothing is denoted for

${e}_t & = & ({x}_t {b}, {y}_t - {x}_t {b}) - {h}_t {a}_t \ {d}_t & = & {h}_... ...}}_{t-1} {m}_t) + {c}_{t(\delta)} \hat{\sigma}_{\delta,t} {c}^'_{t(\delta)}$

where the initial values are ${u}_t = 0$ and ${u}_t = 0$ , and ${c}_{t(\delta)}$ is the last-column-deleted submatrix of ${c}_t$ . Refer to De Jong (1991) for details about smoothing in the diffuse Kalman filter.

The KALDFS call is accompanied by the KALDFF call as shown in the following code:

  
    ny = ncol(y); 
    nz = ncol(coef); 
    nb = ncol(int); 
    nd = ncol(coefd); 
    at = j(nz,nd+1,.); 
    mt = j(nz,nz,.); 
    qt = j(nd+1,nd+1,.); 
    n0 = -1; 
    call kaldff(pred,vpred,initial,s2,y,0,int,coef,var,intd,coefd, 
                n0,at,mt,qt); 
    bvec = intd[nz+1:nz+nb,]; 
    bmat = coefd[nz+1:nz+nb,]; 
    call kaldfs(sm,vsm,x,int,coef,var,bvec,bmat,initial,at,mt,s2);

You can also compute the smoothed estimate and its covariance matrix observation by observation. When the SSM is time invariant, the following code performs smoothing. You should initialize UN and VUN as matrices of value 0. Here is the code:

  
    n  = nrow(y); 
    ny = ncol(y); 
    nz = ncol(coef); 
    nb = ncol(int); 
    nd = ncol(coefd); 
    at = j(nz,nd+1,.); 
    mt = j(nz,nz,.); 
    qt = j(nd+1,nd+1,.); 
    n0 = -1; 
    call kaldff(pred,vpred,initial,s2,y,0,int,coef,var,intd,coefd, 
                n0,at,mt,qt); 
    bvec = intd[nz+1:nz+nb,]; 
    bmat = coefd[nz+1:nz+nb,]; 
    un  = j(nz,nd+1,0); 
    vun = j(nz,nz,0); 
    do i = 1 to n; 
       call kaldfs(sm_i,vsm_i,y[n-i+1],int,coef,var,bvec,bmat, 
                   initial,at,mt,s2,un,vun); 
       sm  = sm_i // sm; 
       vsm = vsm_i // vsm; 
    end;

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