KALCVF Call :: SAS/IML(R) 13.1 User's Guide

KALCVF Call

CALL KALCVF (pred, vpred, filt, vfilt, data, lead, a, f, b, h, var <*>, z0 <*>, vz0 ) ;

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

The input arguments to the KALCVF subroutine are as follows:

data: is a $T \times N_ y$ matrix that contains data $(y_1, \ldots , y_ T)^{\prime }$ .
lead: is the number of steps to forecast after the end of the data.
a: is an $N_ z \times 1$ vector for a time-invariant input vector in the transition equation, or a $(T+\mbox{lead})N_ z \times 1$ vector that contains input vectors in the transition equation.
f: is an $N_ z \times N_ z$ matrix for a time-invariant transition matrix in the transition equation, or a $(T+\mbox{lead})N_ z \times N_ z$ matrix that contains transition matrices in the transition equation.
b: is an $N_ y \times 1$ vector for a time-invariant input vector in the measurement equation, or a $(T+\mbox{lead})N_ y \times 1$ vector that contains input vectors in the measurement equation.
h: is an $N_ y \times N_ z$ matrix for a time-invariant measurement matrix in the measurement equation, or a $(T+\mbox{lead})N_ y \times N_ z$ matrix that contains measurement matrices in the measurement equation.
var: is an $(N_ z + N_ y) \times (N_ z + N_ y)$ matrix for a time-invariant variance matrix for the error in the transition equation and the error in the measurement equation, or a $(T+\mbox{lead})(N_ z + N_ y) \times (N_ z + N_ y)$ matrix that contains variance matrices for the error in the transition equation and the error in the measurement equation—that is, $(\eta ^{\prime }_ t, \epsilon ^{\prime }_ t)^{\prime }$ .
z0: is an optional $1 \times N_ z$ initial state vector $z^{\prime }_{1|0}$ .
vz0: is an optional $N_ z \times N_ z$ covariance matrix of an initial state vector $P_{1|0}$ .

The KALCVF call returns the following values:

pred: is a $(T+\mbox{lead}) \times N_ z$ matrix that contains one-step predicted state vectors $(z_{1|0},\ldots , z_{T+1|T}, z_{T+2|T}, \ldots , z_{T+\mbox{lead}|T})^{\prime }$ .
vpred: is a $(T+\mbox{lead})N_ z \times N_ z$ matrix that contains mean square errors of predicted state vectors $(P_{1|0}, \ldots , P_{T+1|T}, P_{T+2|T}, \ldots , P_{T+\mbox{lead}|T})^{\prime }$ .
filt: is a $T \times N_ z$ matrix that contains filtered state vectors $(z_{1|1}, \ldots , z_{T|T})^{\prime }$ .
vfilt: is a $TN_ z \times N_ z$ matrix that contains mean square errors of filtered state vectors $(P_{1|1}, \ldots , P_{T|T})^{\prime }$ .

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 $\epsilon _ t$ , and $A^-$ 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:

$\begin{eqnarray*} \hat{\epsilon }_ t & = & y_ t - b_ t - H_ t z_{t|t-1} \\[0.05in] D_ t & = & H_ t P_{t|t-1} H^{\prime }_ t + R_ t \\[.05in] z_{t|t} & = & z_{t|t-1} + P_{t|t-1} H^{\prime }_ t D_ t^- \hat{\epsilon }_ t \\[0.05in] P_{t|t} & = & P_{t|t-1} - P_{t|t-1} H^{\prime }_ t D_ t^- H_ t P_{t|t-1} \\[0.05in] K_ t & = & (F_ t P_{t|t-1} H^{\prime }_ t + G_ t) D_ t^- \\[0.05in] z_{t+1|t} & = & a_ t + F_ t z_{t|t-1} + K_ t \hat{\epsilon }_ t \\[0.05in] P_{t+1|t} & = & F_ t P_{t|t-1}F^{\prime }_ t + V_ t - K_ t D_ t K^{\prime }_ t \end{eqnarray*}$

And for $k$ -step forecasting for $k>1$ ,

$\begin{eqnarray*} z_{t+k|t} & = & a_{t+k-1} + F_{t+k-1} z_{t+k-1|t} \\[0.05in] P_{t+k|t} & = & F_{t+k-1} P_{t+k-1|t} F^{\prime }_{t+k-1} + V_{t+k-1} \end{eqnarray*}$

When you use the alternative transition equation

$z_ t = a_ t + F_ t z_{t-1} + \eta _ t$

the forward recursion algorithm is written

$\begin{eqnarray*} \hat{\epsilon }_ t & = & y_ t - b_ t - H_ t z_{t|t-1} \\[0.05in] D_ t & = & H_ t P_{t|t-1} H^{\prime }_ t + H_ t \bG _ t + G^{\prime }_ t H^{\prime }_ t + \bR _ t \\[0.05in] z_{t|t} & = & z_{t|t-1} + (P_{t|t-1} H^{\prime }_ t + G_ t) D_ t^- \hat{\epsilon }_ t \\[0.05in] P_{t|t} & = & P_{t|t-1} - (P_{t|t-1} H^{\prime }_ t + G_ t) D_ t^- (H_ t P_{t|t-1} + G^{\prime }_ t) \\[0.05in] K_ t & = & (F_{t+1} P_{t|t-1} H^{\prime }_ t + G_ t)D_ t^- \\[0.05in] z_{t+1|t} & = & a_{t+1} + F_{t+1} z_{t|t-1} + K_ t \hat{\epsilon }_ t \\[0.05in] P_{t+1|t} & = & F_{t+1} P_{t|t-1}F^{\prime }_{t+1} + V_{t+1} - K_ t D_ t K^{\prime }_ t \end{eqnarray*}$

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

$\begin{eqnarray*} z_{t+k|t} & = & a_{t+k} + F_{t+k}z_{t+k-1|t} \\[0.05in] P_{t+k|t} & = & F_{t+k} P_{t+k-1|t} F^{\prime }_{t+k} + V_{t+k} \end{eqnarray*}$

You can use the KALCVF call when you specify the alternative transition equation and $\mathbf{G}_ t = \mb {0}$ .

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

$\begin{eqnarray*} z_{1|0} & = & (I - F)^- a \\[0.05in] P_{1|0} & = & (I - F \otimes F)^- \mbox{vec}(V) \end{eqnarray*}$

where $F$ and $V$ are the time-invariant transition matrix and the covariance matrix of transition equation noise, and vec $(V)$ is an $N_ z^2 \times 1$ 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 $z_{1|0}$ is set to $a_1$ and its covariance matrix $P_{1|0}$ is given by $10^6$ I. Optionally, you can specify initial values.

The KALCVF call accepts 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.

The following program gives an example of the KALCVF call:

q = 2;
p = 2;
n = 10;
lead = 3;
total = n + lead;

seed = 25735;
x = round(10*normal(j(n, p, seed)))/10;
f = round(10*normal(j(q*total, q, seed)))/10;
a = round(10*normal(j(total*q, 1, seed)))/10;
h = round(10*normal(j(p*total, q, seed)))/10;
b = round(10*normal(j(p*total, 1, seed)))/10;

do i = 1 to total;
   temp = round(10*normal(j(p+q, p+q, seed)))/10;
   var = var//(temp*temp`);
end;

call kalcvf(pred, vpred, filt, vfilt, x, lead, a, f, b, h, var);

/* default initial state and covariance */
call kalcvs(sm, vsm, x, a, f, b, h, var, pred, vpred);
print sm[format=9.4] vsm[format=9.4];

Figure 24.180: Smoothed Estimate and Covariance

sm		vsm
-1.5236	-0.1000	1.5813	-0.4779
0.3058	-0.1131	-0.4779	0.3963
-0.2593	0.2496	2.4629	0.2426
-0.5533	0.0332	0.2426	0.0944
-0.5813	0.1251	0.2023	-0.0228
-0.3017	0.7480	-0.0228	0.5799
1.1333	-0.2144	0.8615	-0.7653
1.5193	-0.6237	-0.7653	1.2334
-0.6641	-0.7770	1.0836	0.8706
0.5994	2.3333	0.8706	1.5252
		0.3677	0.2510
		0.2510	0.2051
		0.3243	-0.4093
		-0.4093	1.2287
		0.1736	-0.0712
		-0.0712	0.9048
		1.3153	0.8748
		0.8748	1.6575
		8.6650	0.1841
		0.1841	4.4770