Capturing the Regime-Switching and Memory Properties of Interest Rates

Abstract

We propose a mean-reverting interest rate model whose mean-reverting level, speed of mean-reversion and volatility are all modulated by a weak Markov chain (WMC). This model features a simple way to capture the regime-switching evolution of the parameters as well as the memory property of the data. Concentrating on the second-order WMC framework, we derive the filters of the WMC and other auxiliary processes through a change of reference probability measure. Optimal estimates of model parameters are provided by employing the EM algorithm. The $h$-step ahead forecasts under our proposed set-up are examined and compared with those under the usual Markovian regime-switching framework. We obtain better goodness-of-fit performance based on our numerical results generated from the implementation of WMC-based filters to a 10-year dataset of weekly short-term-maturity Canadian yield rates. Some statistical inference issues of the proposed modelling approach are also discussed.

Appendices

Appendix A: Proof of Proposition 3.1

1.1 Proof of Eq. (18)

From the definition of $q_k$ in Eq. (14) we have

$$\begin{aligned} q_{k+1}&= {\bar{E}}[\Lambda _{k+1}\xi (\mathbf{x }{k+1},\mathbf{x }_k)|\fancyscript{Y}_{k+1}]\\&= {\bar{E}}[\Lambda _k\lambda _{k+1}{\varvec{\Pi }}\xi (\mathbf{x }_k,\mathbf{x }_{k-1})|\fancyscript{Y}_{k+1}] \\&= \sum _{l,m,v=1}^Nb^m_{k+1}{\bar{E}}[\Lambda _k\langle {\varvec{\Pi }}\mathbf{e }_{mv},\mathbf{e }_{lm}\rangle \langle \xi (\mathbf{x }_k,\mathbf{x }_{k-1}),\mathbf{e }_{mv}\rangle |\fancyscript{Y}_k]\mathbf{e }_{lm}\\&= \sum _{l,m,v=1}^Nb^m_{k+1}\langle {\varvec{\Pi }}\mathbf{e }_{mv},\mathbf{e }_{lm}\rangle \langle \mathbf{q }_k,\mathbf{e }_{mv}\rangle \mathbf{e }_{lm}\\&= \mathbf{B }_{k+1}{\varvec{\Pi }}\mathbf{q }_k. \end{aligned}$$

1.2 Proof of Eq. (19)

Using the expressions in (10) and (11), and the definition in (19), we obtain

$$\begin{aligned}&\gamma \big (J^{rst}\xi (\mathbf{x }_{k+1},\mathbf{x }_k)\big )_{k+1}\\&\quad ={\bar{E}}[\Lambda _{k+1}J_{k+1}^{rst}\xi (\mathbf{x }_{k+1},\mathbf{x }_k)|\fancyscript{Y}_{k+1}]\\&\quad ={\bar{E}}[\Lambda _k\lambda _{k+1}(J_k^{rst}+\langle \mathbf{x }_{k+1},\mathbf{e }_r\rangle \langle \mathbf{x }_k, \mathbf{e }_s\rangle \langle \mathbf{x }_{k-1},\mathbf{e }_t\rangle ){\varvec{\Pi }}\xi (\mathbf{x }_k,\mathbf{x }_{k-1})|\fancyscript{Y}_{k+1}]\\&\quad =\sum _{l,m,v=1}^Nb^m_{k+1}{\bar{E}}[\Lambda _kJ^{rst}_k\langle {\varvec{\Pi }}\mathbf{e }_{mv}, \mathbf{e }_{lm}\rangle \langle \xi (\mathbf{x }_k,\mathbf{x }_{k-1}),\mathbf{e }_{mv}\rangle |\fancyscript{Y}_k]\mathbf{e }_{lm}\\&\quad \quad +b^r_{k+1}\bar{E}[\Lambda _k\langle {\varvec{\Pi }}\mathbf{e }_{st},\mathbf{e }_{rs}\rangle \langle \xi (\mathbf{x }_k,\mathbf{x }_{k-1}), \mathbf{e }_{rs}\rangle |\fancyscript{Y}_k]\mathbf{e }_{rs}\\&\quad =\sum _{l,m,v=1}^Nb^m_{k+1}\langle {\varvec{\Pi }}\mathbf{e }_{mv},\mathbf{e }_{lm}\rangle \langle \gamma (J^{rst}_k\xi (\mathbf{x }_k, \mathbf{x }_{k-1}))_k,\mathbf{e }_{mv}\rangle \mathbf{e }_{lm}\\&\qquad +b^r_{k+1}\langle {\varvec{\Pi }}\mathbf{e }_{st},\mathbf{e }_{rs}\rangle \langle \mathbf{q }_k,\mathbf{e }_{st} \rangle \mathbf{e }_{rs}\\&\quad =\mathbf B _{k+1}{\varvec{\Pi }}\gamma (J^{rst}\xi (\mathbf{x }_k,\mathbf{x }_{k-1}))_k+b^r_{k+1}\langle {\varvec{\Pi }}\mathbf{e }_{st},\mathbf{e }_{rs} \rangle \langle \mathbf{q }_k,\mathbf{e }_{st}\rangle \mathbf{e }_{rs}. \end{aligned}$$

1.3 Proof of Eq. (20)

Using the expressions in (10) and (11), and the definition in (20), we get

$$\begin{aligned}&\gamma (O^{rs}\xi (\mathbf{x }_{k+1},\mathbf{x }_k))_{k+1}\\&\quad =\bar{E}[\Lambda _{k+1}O_k^{rs}\xi (\mathbf{x }_{k+1},\mathbf{x }_k)|\fancyscript{Y}_{k+1}]\\&\quad =\bar{E}[\Lambda _k\lambda _{k+1}(O_k^{rs}+\langle \mathbf{x }_k,\mathbf{e }_r\rangle \langle \mathbf{x }_{k-1},\mathbf{e }_s\rangle ) \xi (\mathbf{x }_{k+1},\mathbf{x }_k)|\fancyscript{Y}_{k+1}]\\&\quad =\sum _{l,m,v=1}^Nb^m_{k+1}{\bar{E}}[\Lambda _kO^{rs}_k\langle {\varvec{\Pi }}\mathbf{e }_{mv},\mathbf{e }_{lm}\rangle \langle \xi (\mathbf{x }_k,\mathbf{x }_{k-1}),\mathbf{e }_{mv}\rangle |\fancyscript{Y}_k]\mathbf{e }_{lm}\\&\qquad +\sum _{l=1}^Nb^r_{k+1}\bar{E}[\Lambda _k\langle {\varvec{\Pi }}\mathbf{e }_{rs},\mathbf{e }_{lr}\rangle \langle \xi (\mathbf{x }_k,\mathbf{x }_{k-1}),\mathbf{e }_{rs}\rangle |\fancyscript{Y}_k]\mathbf{e }_{lr}\\&\quad =\sum _{l,m,v=1}^Nb^m_{k+1}\langle {\varvec{\Pi }}\mathbf{e }_{mv},\mathbf{e }_{lm}\rangle \langle \gamma (O^{rs}_k\xi (\mathbf{x }_k,\mathbf{x }_{k-1}))_k,\mathbf{e }_{mv}\rangle \mathbf{e }_{lm}\\&\qquad +b^r_{k+1}\sum _{l=1}^N\langle {\varvec{\Pi }}\mathbf{e }_{rs},\mathbf{e }_{lr}\rangle \langle \mathbf{q }_k,\mathbf{e }_{rs}\rangle \mathbf{e }_{lr}\\&\quad =\mathbf B _{k+1}{\varvec{\Pi }}\gamma (O^{rs}\xi (\mathbf{x }_k,\mathbf{x }_{k-1}))_k+b^r_{k+1}\langle \mathbf{q }_k,\mathbf{e }_{rs}\rangle {\varvec{\Pi }}\mathbf{e }_{rs}. \end{aligned}$$

1.4 Proof of Eq. (21)

From the expressions in (10) and (11), and the definition in (21), we have

$$\begin{aligned}&\gamma (O^r\xi (\mathbf{x }_{k+1},\mathbf{x }_k))_{k+1}\\&\quad =\bar{E}[\Lambda _{k+1}O_{k+1}^r\xi (\mathbf{x }_{k+1},\mathbf{x }_k)|\fancyscript{Y}_{k+1}]\\&\quad =\bar{E}[\Lambda _k\lambda _{k+1}(O_k^r+\langle \mathbf{x }_k,\mathbf{e }_r\rangle ){\varvec{\Pi }}\xi (\mathbf{x }_{k}, \mathbf{x }_{k-1})|\fancyscript{Y}_{k+1}]\\&\quad =\sum _{l,m,v=1}^Nb^m_{k+1}{\bar{E}}[\Lambda _kO^{r}_k\langle {\varvec{\Pi }}\mathbf{e }_{mv},\mathbf{e }_{lm}\rangle \langle \xi (\mathbf{x }_k,\mathbf{x }_{k-1}),\mathbf{e }_{mv}\rangle |\fancyscript{Y}_k]\mathbf{e }_{lm}\\&\qquad +\sum _{l,m=1}^Nb^r_{k+1}{\bar{E}}[\Lambda _k\langle {\varvec{\Pi }}\mathbf{e }_{rm},\mathbf{e }_{lr}\rangle \langle \xi ( \mathbf{x }_{k},\mathbf{x }_{k-1}),\mathbf{e }_{rm}\rangle |\fancyscript{Y}_k]\mathbf{e }_{lr}\\&\quad =\sum _{l,m,v=1}^Nb^m_{k+1}\langle {\varvec{\Pi }}\mathbf{e }_{mv},\mathbf{e }_{lm}\rangle \langle \gamma (O^{r}_k\xi (\mathbf{x }_k, \mathbf{x }_{k-1}))_k,\mathbf{e }_{mv}\rangle \mathbf{e }_{lm}\\&\qquad +\sum _{l,m=1}^Nb^r_{k+1}\langle {\varvec{\Pi }}\mathbf{e }_{rm},\mathbf{e }_{lr}\rangle \langle \mathbf{q }_k,\mathbf{e }_{rm}\rangle \mathbf{e }_{lr}\\&\quad =\mathbf B _{k+1}{\varvec{\Pi }}\gamma (O^{r}\xi (\mathbf{x }_k,\mathbf{x }_{k-1}))_k+b^r_{k+1}\mathbf{V }_r{\varvec{\Pi }}\mathbf{q }_k. \end{aligned}$$

The proof of the recursive formulae (22) follows a similar argument supporting Eq. (21) by using the definition of $\lambda _l$ and evaluating the resulting conditional expectation under $\bar{P}$ .

Appendix B: Proof of Proposition 3.2

1.1 Proof of Eq. (24)

To derive an optimal estimate for $a_{rst}$ we consider a new measure $P_{\hat{\theta }}$, which is defined in (23). This means that

$$\begin{aligned} \log \left( \frac{dP_{\hat{\theta }}}{dP_{\theta }}\right)&= \sum ^k_{l=2} \sum ^N_{r,s,t=1}[\log (\hat{a}_{rst})-\log (a_{rst})]\langle \mathbf x _l, \mathbf{e }_r\rangle \langle \mathbf x _{l-1},\mathbf{e }_s\rangle \langle \mathbf x _{l-2},\mathbf{e }_t\rangle \nonumber \\&= \sum ^N_{r,s,t=1}\log {\hat{a}_{rst}}J_k^{rst}+\text {R}, \end{aligned}$$

(41)

where R is independent of $\hat{a}_{rst}$. Taking expectation of (41), we have

$$\begin{aligned} E\left[ \log \left( \frac{dP_{\hat{\theta }}}{dP_\theta }\right) \Big |\fancyscript{Y}_k\right] =\sum ^N_{r,s,t=1}\log {\hat{a}_{rst}}\hat{J}_k^{rst}+\text {R}. \end{aligned}$$

(42)

The optimal estimate of $a_{rst}$ is the value that maximises the log-likelihood (41) subject to the constraint $\sum ^N_{r=1}\hat{a}_{rst}=1$. We introduce the Lagrange multiplier $\omega$ via the function

$$\begin{aligned} L(\hat{a}_{rst},\omega )=\sum ^N_{r,s,t=1}\log {\hat{a}_{rst}}\hat{J}_k^{rst} +\omega (\sum ^N_{r=1}\hat{a}_{rst}-1)+\text {R}. \end{aligned}$$

(43)

Differentiating (43) with respect to $\hat{a}_{rst}$ and $\omega$, and equating the derivatives to 0, we get

$$\begin{aligned} \frac{1}{\hat{a}_{rst}}\hat{J}_k^{rst}+\omega =0 \end{aligned}$$

(44)

and

$$\begin{aligned} \sum ^N_{r=1}\hat{a}_{rst}=1. \end{aligned}$$

(45)

Rewriting (44) yields

$$\begin{aligned} \hat{a}_{rst}=-\frac{\hat{J}_k^{rst}}{\omega }. \end{aligned}$$

(46)

Consequently, from (45) and (46) we have

$$\begin{aligned} \sum ^N_{r=1}\hat{a}_{rst}=-\sum ^N_{r=1}\frac{\hat{J}_k^{rst}}{\omega } =-\frac{\hat{O}^{st}}{\omega }=1. \end{aligned}$$

Hence, the Lagrange multiplier has the value $\omega =-\hat{O}^{st}$. From (46), the optimal estimates for $\hat{a}_{rst}$ is

$$\begin{aligned} \hat{a}_{rst}=\frac{\hat{J}_k^{rst}}{\hat{O}^{st}}, \end{aligned}$$

which is what we wanted to show in (24).

1.2 Proof of Eq. (25)

Given the parameter ${\varvec{\alpha }}=(\alpha _1,\alpha _2,\ldots ,\alpha _N)^\top \in \mathbb R ^N$, we wish to update the estimates to $\hat{{\varvec{\alpha }}}=(\hat{\alpha }_1,\hat{\alpha }_2,\ldots ,\hat{\alpha }_N)^\top \in \mathbb R ^N$. Consider a new measure $P_{\hat{\theta }}$ defined by

$$\begin{aligned} \frac{dP_{\hat{\theta }}}{dP_{\theta }}\Big |_{\fancyscript{Y}_k} =\Lambda _k^{\alpha }=\prod ^k_{l=1} \lambda _l(\hat{\alpha }_{l-1},y_l), \end{aligned}$$

where

$$\begin{aligned}&\lambda _l(\hat{\alpha }_{l-1},y_l)\\&\quad =\exp \left\{ \!-\!\frac{(y_l\!-\!\hat{\alpha }(\mathbf{x }_{l-1})y_{l-1} \!-\!\eta (\mathbf{x }_{l-1}))^2\!-\!(y_l-\alpha (\mathbf{x }_{l-1}) y_{l-1} \!-\!\eta (\mathbf{x }_{l-1}))^2}{2\sigma (\mathbf{x }_{l-1})^2}\right\} . \end{aligned}$$

Therefore, we have

$$\begin{aligned}&E\left[ \log \frac{dP_{\hat{\theta }}}{dP_\theta }\Big |\fancyscript{Y}_k\right] = E\left[ \sum ^k_{l=1}\log \lambda _l(\alpha _{l-1},y_l)\Big |\fancyscript{Y}_k\right] \nonumber \\&\quad = E\left[ \sum ^k_{l=1} -\!\frac{\left( \hat{\alpha } (\mathbf{x }_{l-1})^2 y^2_{l-1} -2\hat{\alpha }(\mathbf{x }_{l-1})y_{l-1}y_l +\hat{\alpha }(\mathbf{x }_{l-1})\eta (\mathbf{x }_{l-1})y_{l-1} \right) }{2\sigma (\mathbf{x }_{l-1})^2}+\text {R}\Big |\fancyscript{Y}_k\right] \nonumber \\&\quad =\sum ^k_{l=1}E\left[ \sum ^N_{r=1} -\frac{\langle \mathbf{x }_{k-1},\mathbf{e }_r\rangle }{2\sigma _r^2}\left( \hat{\alpha }_r^2 y^2_{l-1} -2\hat{\alpha }_ry_{l-1}y_l+\hat{\alpha }_r\eta _ry_{l-1}\right) \Big |\fancyscript{Y}_k\right] +\text {R}\nonumber \\&\quad =\sum ^N_{r=1}E\left[ -\frac{1}{2\sigma _r^2}\left( \hat{\alpha }_r^2 T^r_k(y^2_{k-1})-2\hat{\alpha }_rT^r_k(y_{k-1}y_k)+\hat{\alpha }_r\eta _rT^r_k(y_{k-1})\right) \Big |\fancyscript{Y}_k\right] +\text {R}\nonumber \\&\quad =\sum ^N_{r=1}-\frac{1}{2\sigma _r^2}\left( \hat{\alpha }_r^2 \hat{T}^r_k(y^2_{k-1})-2\hat{\alpha }_r\hat{T}^r_k(y_{k-1}y_k)+\hat{\alpha }_r\eta _r\hat{T}^r_k(y_{k-1})\right) +\text {R}, \end{aligned}$$

(47)

where R does not involve $\hat{\alpha }$. We differentiate the above expression and set its derivative to 0. This gives the optimal choice for $\hat{\alpha }_i$ given the observation data $y_k$. We get

$$\begin{aligned} \hat{\alpha }_r=\frac{\hat{T}^r_k(y_{k-1},y_k) -\eta ^r\hat{T}^r_k(y_{k-1})}{\hat{T}^r_k(y_{k-1}^2)}. \end{aligned}$$

1.3 Proof of Eq. (26)

Given the parameter ${\varvec{\eta }}=(\eta _1,\eta _2,\ldots ,\eta _N)^\top \in \mathbb R ^N$, we wish to obtain the update $\hat{{\varvec{\eta }}}=(\hat{\eta }_1,\hat{\eta }_2,\ldots ,\hat{\eta }_N)^\top \in \mathbb R ^N$. Consider a new measure $P_{\hat{\theta }}$ defined by

$$\begin{aligned} \frac{dP_{\hat{{\varvec{\theta }}}}}{dP_{{\varvec{\theta }}}}\Big |_{\fancyscript{Y}_k}=\Lambda _k^{\alpha } =\prod ^k_{l=1}\lambda _l(\hat{\eta }_{l-1},y_l), \end{aligned}$$

where

$$\begin{aligned}&\lambda _l(\hat{\eta }_{l-1},y_l)\\&\quad =\exp \left\{ -\frac{(y_l -\eta (\mathbf{x }_{l-1})y_{l-1} -\hat{\eta }(\mathbf{x }_{l-1}))^2-(y_l-\alpha (\mathbf{x }_{l-1}) y_{l-1} -\eta (\mathbf{x }_{l-1}))^2}{2\sigma (\mathbf{x }_{l-1})^2}\right\} . \end{aligned}$$

Therefore, we have

$$\begin{aligned}&E\left[ \log \frac{dP_{\hat{\theta }}}{dP_{\theta }}\Big |\fancyscript{Y}_k\right] = E\left[ \sum ^k_{l=1}\log \lambda _l(\alpha _{l-1},y_l)\Big |\fancyscript{Y}_k\right] \nonumber \\&\quad =E\left[ \sum ^k_{l=1} -\frac{\left( \hat{\eta }(\mathbf{x }_{l-1})^2-2\hat{\eta }(\mathbf{x }_{l-1})y_l+2\alpha (\mathbf{x }_{l-1})\hat{\eta }(\mathbf{x }_{l-1})y_{l-1} \right) }{2\sigma (\mathbf{x }_{l-1})^2} +\text {R}\Big |\fancyscript{Y}_k\right] \nonumber \\&\quad =\sum ^k_{l=1}E\left[ \sum ^N_{r=1} -\frac{\langle \mathbf{x }_{k-1},\mathbf{e }_r\rangle }{2\sigma _r^2}\left( \hat{\eta }_r^2-2\hat{\eta }_ry_l+2\alpha _r\hat{\eta }_ry_{l-1} \right) \Big |\fancyscript{Y}_k\right] +\text {R}\nonumber \\&\quad =\sum ^N_{r=1}E\left[ -\frac{1}{2\sigma _r^2}\left( \hat{\eta }_r^2O^r_k-2\hat{\eta }_rT^r_k(y_k)+2\alpha _r\hat{\eta }_rT^r_k(y_{k-1}) \right) \Big |\fancyscript{Y}_k\right] +\text {R}\nonumber \\&\quad =\sum ^N_{r=1}-\frac{1}{2\sigma _r^2}\left( \hat{\eta }_r^2\hat{O}^r_k-2\hat{\eta }_r\hat{T}^r_k(y_k)+2\alpha _r\hat{\eta }_r\hat{T}^r_k(y_{k-1}) \right) +\text {R}, \end{aligned}$$

(48)

where R does not involve $\hat{\eta }$. We differentiate the above expression and set its derivative to 0. This gives the optimal choice for $\hat{\eta }_i$ given the observation data $y_k$ . We obtain

$$\begin{aligned} \hat{\eta }_r=\frac{\hat{T}^r_k(y_k) -\alpha ^r\hat{T}^r_k(y_{k-1})}{\hat{O}^r_k}. \end{aligned}$$

1.4 Proof of Eq. (27)

To perform a change from ${\varvec{\sigma }}=(\sigma _1,\sigma _2, \ldots ,\sigma _N)^\top \in \mathbb R ^N$ to $\hat{{\varvec{\sigma }}}=(\hat{\sigma }_1,\hat{\sigma }_2,\ldots ,\hat{\sigma }_N)^\top \in \mathbb R ^N$, we define the Radon–Nikodým derivative

$$\begin{aligned} \frac{dP_{\hat{\theta }}}{dP_{\theta }}\Big |_{\fancyscript{Y}_k}=\Lambda _k^{\sigma } =\prod ^k_{l=1}\lambda _l(\hat{\sigma }_{l-1}, y_l), \end{aligned}$$

where

$$\begin{aligned} \lambda _l(\hat{\sigma }_{l-1}, y_l)&= \frac{\sigma (\mathbf{x }_{l-1})}{\hat{\sigma }(\mathbf{x }_{l-1})} \exp \left\{ \frac{(y_l-\alpha (\mathbf{x }_{l-1}) y_{l-1}-\eta (\mathbf{x }_{l-1}))^2}{2\sigma (\mathbf{x }_{l-1})^2}\right. \\&\quad \left. -\frac{(y_l-\alpha (\mathbf{x }_{l-1}) y_{l-1}-\eta (\mathbf{x }_{l-1}))^2}{2\hat{\sigma }(\mathbf{x }_{l-1})^2}\right\} . \end{aligned}$$

Hence,

$$\begin{aligned}&E\left[ \log \left( \frac{dP_{\hat{\theta }}}{dP_\theta }\right) \Big |\fancyscript{Y}_k\right] \nonumber \\&\quad =E\left[ \sum ^k_{l=1}-\log \hat{\sigma }(\mathbf{x }_{l-1}) -\frac{\left[ y_l-\alpha (\mathbf{x }_{l-1}) y_{l-1}-\eta (\mathbf{x }_{l-1})\right] ^2}{2\hat{\sigma }(\mathbf{x }_{l-1})^2}\Big |\fancyscript{Y}_k\right] +\text {R}\nonumber \\&\quad =E\left[ -\sum ^k_{l=1}\sum ^N_{r=1}\langle \mathbf{x }_{l-1}, \mathbf{e }_r\rangle \right. \nonumber \\&\quad \quad \left. \left( \log \hat{\sigma }_r\!+\!\frac{y_l^2+\alpha _r^2 y^2_{l-1} +\eta ^2_rO^r_k-2\alpha _ry_{l-1}y_l-2\eta _ry_l+2\eta _r\alpha _ry_{l-1}}{2\hat{\sigma }_r^2}\right) \Big |\fancyscript{Y}_k\right] \!+\!\text {R}\nonumber \\&\quad =\sum ^N_{r=1}\left( -\log \hat{\sigma }_r\hat{O}^r_k\right. \nonumber \\&\quad \left. \!-\!\frac{\hat{T}^r_k(y_k^2)\!+\!\alpha _r^2 \hat{T}^r_k(y^2_{k-1}) \!+\!\eta ^2_r\hat{O}^r_k\!-\!2\alpha _r \hat{T}^r_k(y_{k-1}y_k) \!-\!2\eta _r \hat{T}^r_k(y_k)\!+\!2\eta _r\alpha _r\hat{T}^r_k(y_{k-1})}{2\hat{\sigma }_r^2}\right) \nonumber \\&\quad \quad +\text {R} , \end{aligned}$$

(49)

where R is independent of $\hat{\sigma }$. We differentiate the above expression in $\hat{\sigma }_r$ and equate the result to zero. Solving the equation we get the optimal choice of $\hat{\sigma }^2$, which is

$$\begin{aligned} \hat{\sigma }^2_r\!=\!\frac{\hat{T}^r_k(y^2_k)\!+\!\alpha ^2_r\hat{T}^r_k(y_{k-1})\!+\! \eta ^2_r\hat{O}^r_k\!-\!2\alpha _r\hat{T}^r_k(y_ky_{k-1})\!-\!2\eta _r\hat{T}^r_k(y_k) \!+\!2\eta _r\alpha _r\hat{T}^r_k(y_{k-1})}{\hat{O}^r_k} \end{aligned}$$

and this agrees with Eq. (27).

Appendix C: Proof of Eq. (36)

We use mathematical induction to prove (36) for $h\ge 3$. Following Eqs. (34) and (35), we have, when $h=3$,

$$\begin{aligned} E[y_{k+3}|{\fancyscript{Y}}_k]&= \langle {\varvec{\alpha }},\mathbf{p }_{k+2}\rangle E[y_{k+2}|{\fancyscript{Y}}_k]+\langle {\varvec{\eta }},\mathbf{p }_{k+2}\rangle \\&= \langle {\varvec{\alpha }},\mathbf{p }_{k+2}\rangle \left( \langle {\varvec{\alpha }},\mathbf{p }_{k+1}\rangle E[y_{k+1}|{\fancyscript{Y}}_k] +\langle {\varvec{\eta }},\mathbf{p }_{k+1}\rangle \right) +\langle {\varvec{\eta }},\mathbf{p }_{k+2}\rangle \\&= \prod ^2_{i=1}\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{i-1}\mathbf{p }_k\rangle (\langle {\varvec{\alpha }},\hat{\mathbf{x }}_k\rangle y_k +\langle {\varvec{\eta }},\hat{\mathbf{x }}_k\rangle )\\&+\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}\mathbf{p }_k\rangle \langle {\varvec{\eta }},\mathbf{A }\mathbf{p }_k\rangle +\langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}\mathbf{p }_k\rangle . \end{aligned}$$

Therefore, the statement is true for $h=3$. Assume the statement is true for $h=m$, i.e.,

$$\begin{aligned} E[y_{k+m}|{\fancyscript{Y}}_k]&= \prod ^{m-1}_{i=1}\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{i-1}\mathbf{p }_k\rangle \left( \langle {\varvec{\alpha }},\hat{\mathbf{x }}_k\rangle y_k+\langle {\varvec{\eta }},\hat{\mathbf{x }}_k\rangle \right) \nonumber \\&~~~~+\sum _{i=1}^{m-2}\prod _{j=i}^{m-2}\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{j}\mathbf{p }_k\rangle \langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}^{i-1} \mathbf{p }_k\rangle +\langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}^{m-2}\mathbf{p }_k\rangle . \end{aligned}$$

We demonstrate that Eq. (36) is true for $h=m+1$.

$$\begin{aligned}&E[y_{k+m+1}|{\fancyscript{Y}}_k]\\&\quad =\langle {\varvec{\alpha }},\mathbf{p }_{k+m}\rangle E[y_{k+m}|{\fancyscript{Y}}_k]+\langle {\varvec{\eta }},\mathbf{p }_{k+m}\rangle \\&\quad =\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{m-1}\mathbf{p }_k\rangle \left( \prod ^{m-1}_{i=1}\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{i-1}\mathbf{p }_k\rangle \left( \langle {\varvec{\alpha }},\hat{\mathbf{x }}_k\rangle y_k+\langle {\varvec{\eta }},\hat{\mathbf{x }}_k\rangle \right) \right. \\&\qquad \quad \left. +\sum _{i=1}^{m-2}\prod _{j=i}^{m-2}\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{j}\mathbf{p }_k\rangle \langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}^{i-1}\mathbf{p }_k\rangle \right) \\&\qquad +\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{m-1}\mathbf{p }_k\rangle \langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}^{m-2}\mathbf{p }_k\rangle +\langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}^{m-1}\mathbf{p }_k\rangle \\&\quad =\prod ^{m}_{i=1}\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{i-1}\mathbf{p }_k\rangle \left( \langle {\varvec{\alpha }},\hat{\mathbf{x }}_k\rangle y_k +\langle {\varvec{\eta }},\hat{\mathbf{x }}_k\rangle \right) +\sum _{i=1}^{m-2}\prod _{j=i}^{m-1} \langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{j}\mathbf{p }_k\rangle \langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}^{i-1}\mathbf{p }_k\rangle \\&\qquad +\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{m-1}\mathbf{p }_k\rangle \langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}^{m-2}\mathbf{p }_k\rangle +\langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}^{m-1}\mathbf{p }_k\rangle \\&\quad =\prod ^{m}_{i=1}\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{i-1}\mathbf{p }_k\rangle \left( \langle {\varvec{\alpha }},\hat{\mathbf{x }}_k\rangle y_k +\langle {\varvec{\eta }},\hat{\mathbf{x }}_k\rangle \right) +\sum _{i=1}^{m-1}\prod _{j=i}^{m-1}\langle {\varvec{\alpha }},\mathbf{A }{\varvec{\Pi }}^{j} \mathbf{p }_k\rangle \langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}^{i-1}\mathbf{p }_k\rangle \\&\quad \quad +\langle {\varvec{\eta }},\mathbf{A }{\varvec{\Pi }}^{m-1}\mathbf{p }_k\rangle . \end{aligned}$$

Therefore, by the principle of mathematical induction, the statement in Eq. (36) is true for $h\ge 3$.