13.6 Appendix

13.6.1 Procedure equivalence

We show that our treatment of missing values delivers the same results as the procedure proposed by Shumway and Stoffer (2000,1982). For this task, let us assume that the $(N\times1)$ vector of observations $t$

$$y^\top _t=\begin{bmatrix}y_{1,t} & . & y_{3,t} & . & y_{5,t} & \hdots & y_{N,t} \end{bmatrix}$$

has missing values. Here, observations 2 and 4 are missing. Thus, we have only $N_t<N$ observations. For Kalman filtering in XploRe , all missing values in $y_t$ and the corresponding rows and columns in the measurement matrices $d_t$, $Z_t$, and $H_t$, are deleted. Thus, the adjusted vector of observations is

$$y_{t,1}=\begin{bmatrix}y_{1,t} & y_{3,t} & y_{5,t} & \hdots & y_{N,t} \end{bmatrix}$$

where the subscript 1 indicates that this is the vector of observations used in the XploRe routines. The procedure of Shumway and Stoffer instead rearranges the vectors in such a way that the first $N_t$ entries are the observations--and thus given by $y_{t,1}$--and the last $(N-N_t)$ entries are the missing values. However, all missing values must be replaced with zeros.

For our proof, we use the following generalized formulation of the measurement equation

$$\begin{bmatrix}y_{t,1} \\ y_{t,2} \end{bmatrix} = \begin{bmatrix}... ...pha_t + \begin{bmatrix}\varepsilon^m_{t,1} \\ \varepsilon^m_{t,2} \end{bmatrix}$$

and

$$\begin{pmatrix}\varepsilon^m_{t,1} \\ \varepsilon^m_{t,2} \end{pmatrix} = \begin{bmatrix}H_{t,11} & H_{t,12}\\ H_{t,12} & H_{t,22} \end{bmatrix}\;.$$

$y_{t,1}$ contains the observations and $y_{t,2}$ the missing values. The procedure of Shumway and Stoffer employs the generalized formulation given above and sets $y_{t,2}=0$, $d_{t,2}=0$, $Z_{t,2}=0$, and $H_{t,12}=0$ (Shumway and Stoffer; 2000, p. 330). We should remark that the dimensions of these matrices also depend on $t$ via $(N-N_t)$. However, keep notation simple we do not make this time dependency explicit. It is important to mention that matrices with subscript 1 and 11 are equivalent to the adjusted matrices of XploRe 's filtering routines.

First, we show by induction that both procedures deliver the same results for the Kalman filter. Once this equivalence is established, we can conclude that the smoother also delivers identical results.

PROOF. Given $\mu$ and $\Sigma$, the terms $a_{1\vert}$ and $P_{1\vert}$ are the same for both procedures. This follows from the simple fact that the first two steps of the Kalman filter do not depend on the vector of observations (see Subsection 13.5.3).

Now, given $a_{t\vert t-1}$ and $P_{t\vert t-1}$, we have to show that also the filter recursions

$$a_t=a_{t\vert t-1} + P_{t\vert t-1} Z^\top _t F^{-1}_t v_t\;,\quad P_t = P_{t\vert t-1}- P_{t\vert t-1} Z^\top _t F^{-1}_t Z_t P_{t\vert t-1}$$ (13.13)

deliver the same results. Using ss to label the results of the Shumway and Stoffer procedure, we obtain by using

$$Z_{t,ss}\stackrel{\mathrm{def}}{=} \begin{bmatrix}Z_{t,1} \\ 0 \end{bmatrix}$$

that

$$F_{t,ss}= \begin{bmatrix}Z_{t,1} P_{t\vert t-1} Z_{t,1}^\top & 0\... ...& 0 \end{bmatrix} + \begin{bmatrix}H_{t,11} & 0\\ 0 & H_{t,22} \end{bmatrix}\;.$$

The inverse is given by (Sydsæter, Strøm and Berck; 2000, 19.49)

$$F_{t,ss}^{-1}= \begin{bmatrix}F_{t,1}^{-1} & 0 \\ 0 & H_{t,22}^{-1} \end{bmatrix}$$ (13.14)

where $F_{t,1}$ is just the covariance matrix of the innovations of XploRe 's procedure. With (13.14) we obtain that

$$Z_{t,ss}^\top F_{t,ss}^{-1}= \begin{bmatrix}Z_{t,1}^\top F_{t,1}^{-1} & 0 \end{bmatrix}$$

and accordingly for the innovations

$$v_{t,ss}= \begin{bmatrix}v_{t,1}\\ 0 \end{bmatrix}\;.$$

We obtain immediately

$$Z_{t,ss}^\top F_{t,ss}^{-1}v_{t,ss}=Z_{t,1}^\top F_{t,1}^{-1}v_{t,1}\;.$$

Plugging this expression into (13.13)--taking into account that $a_{t\vert t-1}$ and $P_{t\vert t-1}$ are identical--delivers

$$a_{t,ss} = a_{t,1} P_{t,ss} = P_{t,1}\;.$$ and

This completes the first part of our proof.

The Kalman smoother recursions use only system matrices that are the same for both procedures. In addition to the system matrices, the output of the filter is used as an input, see Subsection 13.5.5. But we have already shown that the filter output is identical. Thus the results of the smoother are the same for both procedures as well. $\qedsymbol$

13.6.2 Smoothed constant state variables

We want to show that the Kalman smoother produces constant estimates through time for all state variables that are constant by definition. To proof this result, we use some of the smoother recursions given in Subsection 13.5.5. First of all, we rearrange the state vector such that the last $k\leqslant K$ variables are constant. This allows the following partition of the transition matrix

$$T_{t+1}=\begin{bmatrix}T_{11,t+1} & T_{12,t+1}\\ 0 & I \end{bmatrix}$$ (13.15)

with the $k\times k$ identity matrix $I$. Furthermore, we define with the same partition

$$\tilde{P}_{t}\stackrel{\mathrm{def}}{=}T_{t+1} P_{t} T_{t+1}^\top... ...}_{11,t} & \tilde{P}_{12,t}\\ \tilde{P}_{12,t} & \tilde{P}_{22,t} \end{bmatrix}$$

The filter recursion for the covariance matrix are given as

$$P_{t+1\vert t} = T_{t+1}P_tT^\top _{t+1}+R_{t+1}$$

where the upper left part of $R_{t+1}$ contains the covariance matrix of the disturbances for the stochastic state variables. We see immediately that only the upper left part of $P_{t+1\vert T}$ is different from $\tilde{P}_{t}$.

Our goal is to show that for the recursions of the smoother holds

$$P^*_t=\begin{bmatrix}M_{11,t} & M_{12,t}\\ 0 & I \end{bmatrix}\;,$$ (13.16)

where both $M$s stand for some complicated matrices. With this result at hand, we obtain immediately

$$a^k_{t\vert T}=a^k_{t+1\vert T}=a^k_{T}$$ (13.17)

for all $t$, where $a^k_{t\vert T}$ contains the last $k$ elements of the smoothed state $a_{t\vert T}$.

Furthermore, it is possible to show with the same result that the lower right partition of $P_{t\vert T}$ is equal to the lower right partition of $P_T$ for all $t$. This lower right partition is just the covariance matrix of $a^k_{t\vert T}$. Just write the smoother recursion

$$P_{t\vert T}=P_t(I-T_{t+1}^\top P^{*\top }_t)+ P^{*}_t P_{t+1\vert T} P^{*\top }_t\;.$$

Then check with (13.15) and (13.16) that the lower-right partition of the first matrix on the right hand side is a $k\times k$ matrix of zeros. The lower-right partition of the second matrix is given by the the lower-right partition of $P_{t+1\vert T}$.

PROOF. Now we derive (13.16): We assume that the inverse of $T_{t+1}$ and $T_{11,t+1}$ exist. The inverses for our model exist because we assume that $\phi_2\neq0$. For the partitioned transition matrix (Sydsæter, Strøm and Berck; 2000, 19.48) we derive

$$T^{-1}_{t+1}=\begin{bmatrix}T^{-1}_{11,t+1} & -T^{-1}_{11,t+1} T_{12,t+1}\\ 0 & I \end{bmatrix}\;.$$ (13.18)

Now, it is easy to see that

$$P^*_t=T^{-1}_{t+1}\tilde{P}_t P_{t+1\vert t}^{-1}\;.$$ (13.19)

We have (Sydsæter, Strøm and Berck; 2000, 19.49)

$$P_{t+1\vert t}^{-1}= \begin{bmatrix}\Delta_t & -\Delta_t\tilde{P}... ...}\tilde{P}_{12,t} \Delta_t\tilde{P}_{12,t}\tilde{P}_{22,t}^{-1} \end{bmatrix}\;$$ (13.20)

with $\Delta_t$ as a known function of the partial matrices. If we multiply this matrix with the lower partition of $\tilde{P}_t$ we obtain immediately $[ 0 \ I ]$. With this result and (13.18) we derive (13.16). $\qedsymbol$