Correcting estimation bias in regime switching dynamic term structure models

Abstract

This paper extends the minimum-chi-square estimation for affine term structure models to a regime switching framework, and corrects the estimation bias in the regime switching dynamic term structure model. Biases arise as a result of highly persistent bond yields, and bias correction changes the decomposition of medium- and long-term forward rates. The bias-corrected expected short rate accounts for the pronounced moves in forward rates during the 1979–1982 monetary experiment and the financial crisis. The bias-corrected term premium becomes counter-cyclical and more negatively correlated with the short-term yield. Monte Carlo simulation shows that the decomposition of forward rates is more accurate after bias correction.

Appendices

Appendix A: Bond prices

This section derives the functional form of bond prices. We consider a zero-coupon bond with a maturity of $(n+1)$ periods at time t. Given the current regime $s_t=j$, the price of this bond, $P_{n+1,t}$, is conjectured to depend on the latent factors $F_t$ as

$$\begin{aligned} P_{n+1,t}={\textrm{e}}^{-A_{n+1}^j-B_{n+1}\cdot F_t}, \end{aligned}$$

(41)

where A is a regime-dependent scalar; the factor loading B is $1\times N_f$ and does not depend on regimes. The price of this bond is also equal to the expected price at time $t+1$ under the Q-measure discounted at the one-period risk free rate, i.e., given $q_t=j$ and $q_{t+1}=k$,

$$\begin{aligned} \log P_{n+1,t}&=\log E_t^Q\Big [{\textrm{e}}^{-r_t^j}\cdot P_{n,t+1}\Big ] \nonumber \\&=-r_t^j+\log \left( \sum _{k}\pi _{jk}^Q\cdot E_t^Q\Big [{\textrm{e}}^{-A_{n}^k-B_{n}\cdot F_{t+1}}\Big |q_t=j,q_{t+1}=k\Big ]\right) \nonumber \\&=-\Big (\delta _0^j+\delta _1\cdot F_t\Big )+\log \left( \sum _{k}\pi _{jk}^Q \cdot {\textrm{e}}^{-A_{n}^k} \right) +\left[ -B_{n}\cdot \Big [\theta ^{Qj}+\rho ^{Q} \cdot \left( F_t-\theta ^{Qj}\right) \Big ]+\frac{1}{2}B_n\Sigma ^j\Sigma ^{j'} B_n^{'}\right] . \end{aligned}$$

(42)

Equating the two prices in Eqs. (41) and (42) by the constant term and the coefficient on the latent factors gives the relationships that A and B must satisfy under the assumption of no arbitrage, i.e., Eqs. (6) and (7).

Appendix B: Risk neutral bond prices

This section derives the functional form of bond prices in the risk neutral world. We consider a zero-coupon bond with a maturity of $(n+1)$ periods at time t. Given the current regime $s_t=j$, the price of this bond, ${\ddot{P}}_{n+1,t}$, is conjectured to depend on the latent factors $F_t$ as

$$\begin{aligned} {\ddot{P}}_{n+1,t}={\textrm{e}}^{-{\ddot{A}}_{n+1}^j- {\ddot{B}}_{n+1}^{j}\cdot F_t}, \end{aligned}$$

(43)

where ${\ddot{A}}$ is a regime-dependent scalar; the factor loading ${\ddot{B}}$ is $1\times N_f$ and depends on regimes as well. The price of this bond is also equal to the expected price at time $t+1$ under the P-measure discounted at the one-period risk free rate, i.e., given $q_t=j$ and $q_{t+1}=k$,

$$\begin{aligned} {\ddot{P}}_{n+1,t}&=E_t^P\Big [{\textrm{e}}^{-r_t^j}\cdot {\ddot{P}}_{n,t+1}\Big |q_t=j \Big ] \nonumber \\&=\sum _{k} \left( \pi _{jk}^P \cdot {\textrm{e}}^{-r_t^j} \cdot E_t^P\Big [{\ddot{P}}_{n,t+1}^k\Big |q_t=j,q_{t+1}=k \Big ]\right) \nonumber \\&=\sum _{k} \left( \pi _{jk}^P \cdot {\textrm{e}}^{-r_t^j} \cdot E_t^P\Big [{\textrm{e}}^{-{\ddot{A}}_{n}^k-{\ddot{B}}_{n}^{k}\cdot F_{t+1}}\Big |q_t=j,q_{t+1}=k \Big ]\right) \nonumber \\&=\sum _{k}\left( \pi _{jk}^P \cdot {\textrm{e}}^{-r_t^j} \cdot {\textrm{e}}^{-{\ddot{A}}_{n}^k} \cdot {\textrm{e}}^{-{\ddot{B}}_{n}^{k}\cdot \Big [\theta ^{Pj}+\rho ^{Pj} \cdot \left( F_t-\theta ^{Pj}\right) \Big ]+\frac{1}{2}{\ddot{B}}_{n}^{k}\Sigma ^j \Sigma ^{j'} {\ddot{B}}_{n}^{k'} }\right) . \end{aligned}$$

(44)

Equating the two prices in Eqs. (43) and (44) gives

$$\begin{aligned} {\textrm{e}}^{-{\ddot{A}}_{n+1}^j- {\ddot{B}}_{n+1}^{j}\cdot F_t}&=\sum _{k}\left( \pi _{jk}^P \cdot {\textrm{e}}^{-r_t^j} \cdot {\textrm{e}}^{-{\ddot{A}}_{n}^k} \cdot {\textrm{e}}^{-{\ddot{B}}_{n}^{k}\cdot \Big [\theta ^{Pj}+\rho ^{Pj} \cdot \left( F_t-\theta ^{Pj}\right) \Big ]+\frac{1}{2}{\ddot{B}}_{n}^{k}\Sigma ^j \Sigma ^{j'} {\ddot{B}}_{n}^{k'} }\right) \\ 1&=\sum _{k}\left( \pi _{jk}^P \cdot {\textrm{e}}^{-r_t^j} \cdot {\textrm{e}}^{-{\ddot{A}}_{n}^k} \cdot {\textrm{e}}^{-{\ddot{B}}_{n}^{k}\cdot \Big [\theta ^{Pj}+\rho ^{Pj} \cdot \left( F_t-\theta ^{Pj}\right) \Big ]+\frac{1}{2}{\ddot{B}}_{n}^{k}\Sigma ^j \Sigma ^{j'} {\ddot{B}}_{n}^{k'} } \cdot {\textrm{e}}^{{\ddot{A}}_{n+1}^j+ {\ddot{B}}_{n+1}^{j}\cdot F_t} \right) . \end{aligned}$$

Using the approximation ${\textrm{e}}^y\approx 1+y$ gives

$$\begin{aligned}&1\approx \sum _{k}\pi _{jk}^P \cdot \left( 1 {-r_t^j} {-{\ddot{A}}_{n}^k}-{\ddot{B}}_{n}^{k}\cdot \Big [\theta ^{Pj}+\rho ^{Pj} \cdot \left( F_t-\theta ^{Pj}\right) \Big ]+\frac{1}{2}{\ddot{B}}_{n}^{k}\Sigma ^j \Sigma ^{j'} {\ddot{B}}_{n}^{k'} +{\ddot{A}}_{n+1}^j+ {\ddot{B}}_{n+1}^{j}\cdot F_t \right) \\&{-{\ddot{A}}_{n+1}^j- {\ddot{B}}_{n+1}^{j}\cdot F_t}=\sum _{k}\pi _{jk}^P \cdot \left( {-r_t^j} {-{\ddot{A}}_{n}^k} {-{\ddot{B}}_{n}^{k}\cdot \Big [\theta ^{Pj}+\rho ^{Pj} \cdot \left( F_t-\theta ^{Pj}\right) \Big ]+\frac{1}{2}{\ddot{B}}_{n}^{k}\Sigma ^j \Sigma ^{j'} {\ddot{B}}_{n}^{k'} }\right) , \end{aligned}$$

where

$$\begin{aligned} r_t=\delta _0^j + \delta _1\cdot F_t. \end{aligned}$$

(4)

Equating both sides by the constant term and the coefficient on the latent factors gives Eqs. (11) and (12).

Appendix C: Estimation of RS-DTSMs

1.1 Appendix C.1: Reduced-form models

We derive the reduced-form model of the observed bond yields $\{Y_1,Y_2\}$. According to equation (8), the bond yield vector is a function of latent factors, i.e., given the current regime $q_t=j$,

$$\begin{aligned} \begin{bmatrix} Y_{1,t} \\ Y_{2,t} \end{bmatrix} = \begin{bmatrix} {\varvec{a}}_{1}^j \\ {\varvec{a}}_{2}^j \end{bmatrix} + \begin{bmatrix} {\varvec{b}}_{1} \\ {\varvec{b}}_{2} \end{bmatrix} \cdot F_t + \begin{bmatrix} 0 \\ \Sigma _e^{j} \cdot u_{e,t} \end{bmatrix}, \end{aligned}$$

where

$$\begin{aligned} \underset{(N_1\times 1)}{{\varvec{a}}_{1}^j}= \begin{bmatrix} a_{n_1}^j \\ a_{n_2}^j \\ \vdots \\ a_{n_N}^j \end{bmatrix}, \qquad \underset{(N_1\times N_f)}{{\varvec{b}}_{1}}= \begin{bmatrix} b_{n_1} \\ b_{n_2} \\ \vdots \\ b_{n_N} \end{bmatrix}, \qquad \underset{(N_2\times 1)}{{\varvec{a}}_{2}^j}= \begin{bmatrix} a_{m_1}^j \\ a_{m_2}^j \\ \vdots \\ a_{m_M}^j \end{bmatrix}, \qquad \underset{(N_2\times N_f)}{{\varvec{b}}_{2}}= \begin{bmatrix} b_{m_1} \\ b_{m_2} \\ \vdots \\ b_{m_M} \end{bmatrix}. \end{aligned}$$

Given $q_{t+1}=k$ and $q_t=j$, the reduced-form model is

$$\begin{aligned} Y_{1,t+1}&={\varvec{a}}_1^k+{\varvec{b}}_1\cdot F_{t+1}={\varvec{a}}_1^k+{\varvec{b}}_1\cdot \Big [\theta ^{Pj}+\rho ^{Pj} \cdot \left( F_{t}-\theta ^{Pj}\right) +\Sigma ^j\cdot u_{t+1}^P\Big ] \nonumber \\&={\varvec{a}}_1^k+{\varvec{b}}_1\cdot \left[ \theta ^{Pj}+\rho ^{Pj}\cdot \Big ({\varvec{b}}_1^{-1}\cdot \left( Y_{1,t}-{\varvec{a}}_1^j\right) -\theta ^{Pj}\Big )+\Sigma ^j\cdot u_{t+1}^P\right] \nonumber \\&={\varvec{a}}_1^k+{\varvec{b}}_1\cdot \theta ^{Pj}+\underbrace{{\varvec{b}}_1\cdot \rho ^{Pj}\cdot {\varvec{b}}_1^{-1}}_{\phi _{11}^j}\cdot \Big (Y_{1,t}-{\varvec{a}}_1^j-{\varvec{b}}_1\cdot \theta ^{Pj}\Big )+\underbrace{{\varvec{b}}_1\cdot \Sigma ^j\cdot u_{t+1}^P}_{\Omega _{1}^j={\varvec{b}}_{1}\cdot \Sigma ^j\cdot \Sigma ^{j'}\cdot {\varvec{b}}_{1}^{'}}, \end{aligned}$$

(45)

and define

$$\begin{aligned} \alpha _1^{jk}= & {} {\varvec{a}}_1^k+{\varvec{b}}_1\cdot \theta ^{Pj}-\phi _{11}^j\cdot \left( {\varvec{a}}_1^j+{\varvec{b}}_1\cdot \theta ^{Pj}\right) .\nonumber \\ Y_{2,t}= & {} {\varvec{a}}_2^j+{\varvec{b}}_2\cdot F_t+\Sigma _e^j\cdot u_t^e \nonumber \\= & {} {\varvec{a}}_2^j+\underbrace{{\varvec{b}}_2\cdot {\varvec{b}}_1^{-1}}_{\phi _{21}}\cdot (Y_{1,t}-{\varvec{a}}_1^j)+\underbrace{\Sigma _e^{j} \cdot u_{e,t}}_{\Omega _{e}^j=\Sigma _e^j\cdot \Sigma _e^{j'}}, \end{aligned}$$

(46)

and define

$$\begin{aligned} \alpha _2^j={\varvec{a}}_2^j-\phi _{21}\cdot {\varvec{a}}_1^j. \end{aligned}$$

1.2 Appendix C.2: The mapping procedure and model identification

The mapping procedure with normalisations consists of the following steps.

• 1. The mapping is between structural parameters $\left\{ \underset{(N_f\times N_f)}{\rho ^Q}, \underset{(N_f\times 1)}{\delta _1}, \underset{\frac{N_f(N_f+1)}{2}\cdot N_q}{\Sigma ^j} \right\}$ and reduced-form parameters $\left\{ \underset{(N_2\times N_1)}{\phi _{21}},\underset{\frac{N_1(N_1+1)}{2}\cdot N_q}{\Omega _1^j} \right\}$.

  Normalisations include

  $$\begin{aligned}&\Sigma ^{j=1}=I_{N} \\&\rho ^Q \text { is lower triangular}. \end{aligned}$$

  Then the number of elements in structural parameters is

  $$\begin{aligned} \rho ^Q&\qquad \frac{N_f(N_f+1)}{2} \\ \delta _1&\qquad N_f \\ \Sigma ^j&\qquad \frac{N_f(N_f+1)}{2}\cdot (N_q-1). \end{aligned}$$

  When $N_1=N_f$ and $N_2=1$, the number of elements in reduced-form parameters is

  $$\begin{aligned} \phi _{21}&\qquad N_2\times N_1=N_f \\ \Omega _1^j&\qquad \frac{N_1(N_1+1)}{2}\cdot N_q=\frac{N_f(N_f+1)}{2}\cdot N_q. \end{aligned}$$

  The mapping is just-identified with equal number of elements in structural and reduced-form parameters. The output is $\left\{ \rho ^Q, \delta _1, \Sigma ^j \right\}$ and therefore $\{{\varvec{b}}_{1},{\varvec{b}}_{2}\}$.

• 2. Given $\{{\varvec{b}}_{1},{\varvec{b}}_{2}\}$, solve $\rho ^{Pj}$ analytically from $\phi _{11}^{j}$.

• 3. Finally, the mapping is between structural parameters $\left\{ \underset{(N_f\times 1)\cdot N_q}{\theta ^{Pj}},\underset{(N_f\times 1)\cdot N_q}{\theta ^{Qj}},\underset{(1\times 1)\cdot N_q}{\delta _0},\underset{N_q\cdot (N_q-1)}{\pi ^{Q}}\right\}$ and reduced-form parameters $\left\{ \underset{(N_1\times 1)\cdot (2 N_q-1)}{\alpha _1^{kj}},\underset{(N_2\times 1)\cdot N_q}{\alpha _2^j} \right\}$⁷.

  Normalisations include

  $$\begin{aligned}&\theta ^{P1}=0_{N_f\times 1} \\&\pi _{jk}^Q \text { is fixed}, \end{aligned}$$

  where $\pi ^Q$ can be set equal to its counterpart in the physical measure, $\pi ^P$, or the constant transition probability implied by the smoothed probability if $\pi ^P$ is time-varying. Then the number of elements in structural parameters is

  $$\begin{aligned} \theta ^{Pj}&\qquad (N_f\times 1)\cdot (N_q-1) \\ \theta ^{Qj}&\qquad (N_f\times 1)\cdot N_q \\ \delta _0^j&\qquad (1\times 1)\cdot N_q. \end{aligned}$$

  When $N_1=N$ and $N_2=1$, the number of elements in reduced-form parameters is

  $$\begin{aligned} \alpha _{1}^{kj}&\qquad (N_1\times 1)\cdot (2 N_q-1)=(N_f\times 1) \cdot (2 N_q-1) \\ \alpha _{2}^j&\qquad (N_2\times 1)\cdot N_q=N_q. \end{aligned}$$

  The mapping is just-identified with equal number of elements in structural and reduced-form parameters.