An affine model for short rates when monetary policy is path dependent

Abstract

I propose an affine model of short rates that incorporates a random walk with stochastic drift. This framework enables my model to capture the behavior of monetary authorities in the short rate market, allowing for minor deviations while reacting strongly to deviations large enough to threaten production. Importantly, my model facilitates the derivation of closed-form bond prices, thereby providing an analytical solution for bond-option prices. I compare my model with nine standard short rate models found in the literature. Among these, five are single-factor models and four are multifactor models. Remarkably, my model outperforms all competing short rate models, including the constant elasticity of volatility, stochastic mean, and stochastic volatility models. Moreover, it yields interest rate forecasts consistent with common term structure priors and surpasses the performance of the naive random walk model. Additionally, my stochastic mean model can explain the unspanned risks documented in the literature.

Appendices

Appendix 1: Bond and option prices

This appendix provides details of the derivations related to bond and option prices under random walk mean for the short rate process.

Proof of Proposition 2.1

Consider the NSM model of the short rate introduced in Sect. 2. Under the risk-neutral measure, the short rate dynamics are given by

$$dr_{t} = - \alpha_{1} \left( {\mu_{t} - r_{t} } \right)dt + \sigma dZ_{t} ,$$

(11)

$$d\mu_{t} = \sigma_{\mu } dW_{t} .$$

It can be verified using Ito’s formula that

$$r_{t} = e^{{\alpha_{1 } }} \left[ {r_{0} - \mathop \smallint \limits_{0}^{t} \alpha \mu_{t} e^{{\alpha_{1u } }} du + \sigma \mathop \smallint \limits_{0}^{t} e^{{\alpha_{1u } }} dZ_{u} } \right]$$

is a solution to the stochastic differential equation (SDE) in (11). It can be shown using integration by parts for the second term that

$$\begin{aligned} r_{t} & = e^{{\alpha_{1} t}} \left[ {r_{0} - \mu_{t} \mathop \smallint \limits_{0}^{t} e^{{ - \alpha_{1} u}} du + \alpha_{1} \mathop \smallint \limits_{0}^{t} d\mu_{t} \left( {\mathop \smallint \limits_{0}^{t} e^{{ - \alpha_{1} t}} dt} \right)du + \sigma \mathop \smallint \limits_{0}^{t} e^{{ - \alpha_{1} u}} dZ_{u} } \right] \\ & = e^{{\alpha_{1} t}} \left[ {r_{0} + \mu_{t} \left( {e^{{ - \alpha_{1} t}} - 1} \right) - \mathop \smallint \limits_{0}^{t} \sigma_{m} dW\left( {e^{{ - \alpha_{1} t}} - 1} \right)du + \sigma \mathop \smallint \limits_{0}^{t} e^{{ - \alpha_{1} u}} dZ_{u} } \right] \\ & = e^{{\alpha_{1} t}} \left[ {r_{0} + \mu_{t} \left( {e^{{ - \alpha_{1} t}} - 1} \right) + \sigma \mathop \smallint \limits_{0}^{t} e^{{ - \alpha_{1} u}} dZ_{u} } \right]. \\ \end{aligned}$$

The expectation follows immediately from the equation given above,

$$E\left[ {r_{t} } \right] = e^{{\alpha_{1} t}} \left[ {r_{0} + E\left[ {\mu_{t} } \right]\left( {e^{{ - \alpha_{1} t}} - 1} \right) + E\left[ {\sigma \mathop \smallint \limits_{0}^{t} e^{{ - \alpha_{1} u}} dZ_{u} } \right]} \right],$$

$$E\left[ {r_{t} } \right] = e^{{\alpha_{1} t}} \left[ {r_{0} + \mu_{t} \left( {e^{{ - \alpha_{1} t}} - 1} \right)} \right].$$

Clearly, $\underset{t\to \infty }{lim}E\left[{r}_{t}\right]=E\left[{\mu }_{t}\right]$.

Proof of Proposition 2.2

Write

$${c}_{u}={r}_{u}-{\mu }_{u}$$

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where ${c}_{u}$ is a solution of the Ornstein–Uhlenbeck equation:

$$dc\left(t\right)={\alpha }_{1}c\left(t\right)+{\sigma }_{c}d{Z}_{t}$$

Applying Ito’s lemma, the ${c}_{u}$ process is given by

$${c}_{u}={e}^{{\alpha }_{1}u}\left({c}_{0}+{\int }_{0}^{u}{\sigma }_{c}{e}^{-{\alpha }_{1}s}d{Z}_{s}\right).$$

(13)

Using Eq. (11) I obtain:

$$\begin{aligned} Cov\left[ {c_{t} ,c_{u} } \right] & = \sigma_{c}^{2} e^{{\alpha_{1} \left( {u + t} \right)}} E\left[ {\mathop \int \limits_{0}^{t} e^{{ - \alpha_{1} s}} dZ_{s} \left[ {\mathop \int \limits_{0}^{u} e^{{ - \alpha_{1} s}} dZ_{s} } \right]} \right] \\ & = \sigma_{c}^{2} e^{{\alpha_{1} \left( {u + t} \right)}} \mathop \int \limits_{0}^{u\Lambda t} e^{{ - 2\alpha_{1} s}} ds = \frac{{\sigma_{c}^{2} }}{{ - 2\alpha_{1} }}e^{{\alpha_{1} \left( {u + t} \right)}} \left( {e^{{2\alpha_{1} \left( {u\Lambda t} \right)}} - 1} \right). \\ \end{aligned}$$

Similarly,

$$\begin{aligned} Cov\left[ {\mu_{t} ,\mu_{u} } \right] & = \sigma_{m}^{2} E\left[ {\mathop \int \limits_{0}^{t} dW_{s} \mathop \int \limits_{0}^{u} dW_{s} } \right] \\ & = \sigma_{m}^{2} \left[ {\mathop \int \limits_{0}^{t\Lambda u} ds} \right] = \sigma_{m}^{2} \left( {t\Lambda u} \right). \\ \end{aligned}$$

Consequently,

$$\begin{aligned} Var\left[ {\mathop \int \limits_{0}^{t} c_{u} du} \right] & = Cov\left( {\mathop \int \limits_{0}^{t} c_{u} du,_{{}} \mathop \int \limits_{0}^{t} c_{u} ds} \right) \\ & = \mathop \int \limits_{0}^{t} \mathop \int \limits_{0}^{t} Cov\left( {c_{u} ,c_{s} } \right)duds = \mathop \int \limits_{0}^{t} \mathop \int \limits_{0}^{t} \frac{{\sigma_{c}^{2} }}{ - 2\beta }e^{{\alpha_{1} \left( {u + s} \right)}} \left( {e^{{ - 2a\left( {u\Lambda s} \right)}} - 1} \right)^{{}} duds = \frac{{\sigma_{c}^{2} }}{{2\alpha_{1}^{3} }}\left( {2\alpha_{1} t + 3 - 4e^{{\alpha_{1} t}} + e^{{2\alpha_{1} t}} } \right). \\ \end{aligned}$$

Similarly,

$$Var\left[ {\mathop \int \limits_{0}^{t} \mu \left( u \right)du} \right] = \sigma_{m}^{2} \mathop \int \limits_{0}^{t} \mathop \int \limits_{0}^{t} \frac{{\left( {s\Lambda u} \right)^{2} }}{2}^{{}} duds = \sigma_{m}^{2} \frac{{t^{3} }}{6}.$$

From Eq. (12), I have

$$E\left[ { - \mathop \int \limits_{0}^{t} r_{u} du} \right] = E\left[ { - \mathop \int \limits_{0}^{t} \left( {c_{u} + \mu_{u} } \right)du} \right]$$

Therefore,

$$E\left[ { - \mathop \int \limits_{t}^{T} r_{u} du} \right] = \frac{{r_{t} - \mu_{t} }}{\beta }\left( {1 - e^{{\alpha_{1} \left( {T - t} \right)}} } \right) - \mu_{t} \left( {T - t} \right)$$

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Furthermore,

$$\begin{aligned} Var\left[ { - \mathop \int \limits_{t}^{T} r_{u} du} \right] & = Var\left[ {\mathop \int \limits_{t}^{T} \left( {c_{u} + \mu_{u} } \right)du} \right] \\ & = \frac{{\sigma_{c}^{2} }}{{2\alpha_{1}^{3} }}\left( {2\alpha_{1} \left( {T - t} \right) + 3 - 4e^{{\alpha_{1} \left( {T - t} \right)}} + e^{{2\alpha_{1} \left( {T - t} \right)}} } \right) + \sigma_{m}^{2} \frac{{\left( {T - t} \right)^{3} }}{6} \\ \end{aligned}$$

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Proof of Theorem 2.1

Consider the expected value and variance of short rate specified in Propositions 2.1 and 2.2. I specify the price of a zero-coupon bond with maturity T at time t, P(t, T)using the risk neutral valuation framework:

$$P\left( {t, T} \right) = E\left[ {exp\left( { - \mathop \int \limits_{t}^{T} r_{u} du} \right)|F_{t} } \right] = E\left[ {exp\left( { - \mathop \int \limits_{t}^{T} r_{u} du} \right)|r_{t} } \right]$$

where $\left\{ {F_{t} } \right\}$ is standard filtration.

Combining Eqs. (11) to (15), the bond price is given by

$$\begin{aligned} P\left( {t,T,r_{t} } \right) & = \exp \left( {\left[ { - \mathop \int \limits_{t}^{T} r_{u} \left( {r_{t} } \right)du} \right] + \frac{1}{2}Var\left[ { - \mathop \int \limits_{t}^{T} r_{u} \left( {r_{t} } \right)du} \right]} \right) \\ & = Exp\left( {\frac{{r_{t} - \mu_{t} }}{\beta }\left( {1 - e^{{\alpha_{1} \left( {T - t} \right)}} } \right) - \mu_{t} \left( {T - t} \right) + \frac{{\sigma_{c}^{2} }}{{4\alpha_{1}^{3} }}\left( {2\alpha_{1} \left( {T - t} \right) + 3 - 4e^{{\alpha_{1} \left( {T - t} \right)}} + e^{{2\alpha_{1} \left( {T - t} \right)}} + \frac{{\sigma_{\mu }^{2} }}{{\sigma_{c}^{2} }}\frac{{\alpha_{1}^{3} \left( {T - t} \right)^{3} }}{3}} \right)} \right) \\ & = Exp\left( {\left( {\frac{{1 - e^{{\alpha_{1} \left( {T - t} \right)}} }}{{\alpha_{1} }}} \right)r_{t} - \left( {\frac{{1 - e^{{\alpha_{1} \left( {T - t} \right)}} }}{{\alpha_{1} }} + T - t} \right)\mu_{t} + \frac{{\sigma_{c}^{2} }}{{4\alpha_{1}^{3} }}\left( {2\alpha_{1} \left( {T - t} \right) + 3 - 4e^{{\alpha_{{1\left( {T - t} \right)}} }} + e^{{2\alpha_{{1\left( {T - t} \right)}} }} } \right) + \frac{{\sigma_{\mu }^{2} }}{{\sigma_{c}^{2} }}\frac{{\alpha_{1}^{3} \left( {T - t} \right)^{3} }}{3}} \right) \\ & = Exp\left( {A\left( {t,T} \right)r_{t} - \mu_{t} \left( {A\left( {t,T} \right) + \left( {T - t} \right)} \right) + B\left( {t,T} \right) + D\left( {t,T} \right)} \right). \\ \end{aligned}$$

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where

$$\begin{aligned} & A\left( {t,T} \right) = \left( {\frac{{1 - e^{{\alpha_{1} \left( {T - t} \right)}} }}{{ - \alpha_{1} }}} \right) \\ & B\left( {t,T} \right) = \left[ {\frac{{A\left( {t,T} \right) + \left( {T - t} \right)}}{{2\alpha_{1}^{2} }} + \frac{{A\left( {t,T} \right)^{2} }}{{4\alpha_{1} }}} \right]\sigma_{c}^{2} \\ \end{aligned}$$

and

$$D\left(t,T\right)=\frac{{\sigma }_{\mu }^{2}{\left(T-t\right)}^{3}}{12}.$$

If HP and bHP trends are used, the bond price will be estimated using the signal to noise ratio (q). The bond price is given by

$$\begin{gathered} = Exp\left( {\frac{{r_{t} - \mu _{t} }}{{\alpha _{1} }}\left( {1 - e^{{\alpha _{1} \left( {T - t} \right)}} } \right) - \mu _{t} \left( {T - t} \right) + \frac{{\sigma ^{2} }}{{4\alpha _{1}^{3} \left( {1 + q} \right)}}\left( {2\alpha _{1} \left( {T - t} \right) + 3 - 4e^{{\alpha _{1} \left( {T - t} \right)}} + e^{{2\alpha _{1} \left( {T - t} \right)}} + q\frac{{\alpha _{1}^{3} \left( {T - t} \right)^{3} }}{3}} \right)} \right) \hfill \\ = Exp\left( {\left( {\frac{{1 - e^{{\alpha _{1} \left( {T - t} \right)}} }}{{\alpha _{1} }}} \right)r_{t} - \left( {\frac{{1 - e^{{\alpha _{1} \left( {T - t} \right)}} }}{{\alpha _{1} }} + \left( {T - t} \right)} \right)\mu _{t} + \frac{{\sigma ^{2} }}{{2\alpha _{1}^{2} \left( {1 + q} \right)}}\left( {\frac{{1 - e^{{\alpha _{1} \left( {T - t} \right)}} }}{{\alpha _{1} }}} \right) + \frac{{\sigma ^{2} }}{{2\alpha _{1}^{2} \left( {1 + q} \right)}}\left( {T - t} \right)} \right. \hfill \\ \left. {\quad + \frac{{\sigma ^{2} }}{{2\alpha _{1} \left( {1 + q} \right)}}\left( {\frac{{1 - 2e^{{\alpha _{1} \left( {T - t} \right)}} + e^{{2\beta \left( {T - t} \right)}} }}{{2\alpha _{1}^{2} }}} \right) + \left( {\frac{{q\sigma ^{2} \left( {T - t} \right)^{3} }}{{12\left( {1 + q} \right)}}} \right)} \right) \hfill \\ = Exp\left( {A\left( {t,T} \right)r_{t} - \mu _{t} \left( {A\left( {t,T} \right) + \left( {T - t} \right)} \right) + \frac{{\sigma ^{2} }}{{2\alpha _{1}^{2} \left( {1 + q} \right)}}A\left( {t,T} \right) + \frac{{\sigma ^{2} }}{{2\alpha _{1}^{2} \left( {1 + q} \right)}}\left( {T - t} \right)} \right. \hfill \\ \left. {\quad + \frac{{\sigma ^{2} }}{{2\alpha _{1} \left( {1 + q} \right)}}A\left( {t,T} \right)^{2} + \left( {\frac{{q\sigma ^{2} \left( {T - t} \right)^{3} }}{{12\left( {1 + q} \right)}}} \right)} \right) \hfill \\ = Exp\left( {A\left( {t,T} \right)r_{t} - \mu _{t} \left( {A\left( {t,T} \right) + \left( {T - t} \right)} \right) + B\left( {t,T} \right) + D\left( {t,T} \right)} \right) \hfill \\ \end{gathered}$$

where

$$A\left( {t,T} \right)\left( {\frac{{1 - e^{{\alpha_{1} \left( {T - t} \right)}} }}{{\alpha_{1} }}} \right),$$

$$B\left( {t,T} \right) \equiv \frac{{\sigma^{2} }}{{2\alpha_{1}^{2} \left( {1 + q} \right)}}\left( {A\left( {t,T} \right) + \left( {T - t} \right)} \right) + \frac{{\sigma^{2} A\left( {t,T} \right)^{2} }}{{2\beta \left( {1 + q} \right)}},$$

and

$$D\left( {t, T} \right) = \left( {\frac{{q\sigma^{2} \left( {T - t} \right)^{3} }}{{12\left( {1 + q} \right)}}} \right).$$

Appendix 2: Signal–noise ratio

This appendix provides details of the derivations related to the signal–noise ratio,

$$q = \sigma_{\eta }^{2} /\sigma_{v}^{2} = \sigma_{\eta }^{2} /\left( {1 - \rho^{2} - \mathop \sum \limits_{j = 1}^{p} \rho_{j}^{2} } \right)\sigma_{c}^{2} .$$

The resulting transitory component from (3), c_t, possesses weak dependence properties with mean zero. Thus, c_t is an AR(p) process:

$$c_{t + 1} = \rho c_{t} + \mathop \sum \limits_{j = 1}^{p} \rho_{j} c_{t - j} + v_{t + 1} , v_{t} \sim NID\left( {0,\sigma_{v}^{2} } \right).$$

The variance is

$$= E\left( {c_{t + 1}^{2} } \right) - \left( {Ec_{t + 1} } \right)^{2} .$$

It follows immediately that

$$\sigma_{c}^{2} = \frac{{\sigma_{v}^{2} }}{{1 - \rho^{2} - \mathop \sum \nolimits_{j = 1}^{p} \rho_{j}^{2} }}.$$

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Now, consider the case in which $\mu_{t}$ is a random walk process:

$$\Delta \mu_{t + 1} = \eta_{t + 1} \sim NID\left( {0,\sigma_{\eta }^{2} } \right).$$

I specify $\mu_{t}$ as a driftless random walk process:

$$\mu_{t + 1} = \mu_{t} + \eta_{t} ,$$

which can be written as

$$\mu_{t + 1} = \mu_{0} + \mathop \sum \limits_{i = 1}^{t} \eta_{t} .$$

The variance follows immediately

$$\sigma_{{\mu_{{}} }}^{2} = \left( {t + 1} \right)\sigma_{\eta }^{2} ,$$

and

$$\sigma_{{\Delta \mu_{t + 1} }}^{2} = \left( {t + 1} \right)\sigma_{\eta }^{2} + \left( t \right)\sigma_{\eta }^{2} - 2cov\left( {\mu_{t + 1} , \mu_{t} } \right) = \left( {t + 1} \right)\sigma_{\eta }^{2} + \left( t \right)\sigma_{\eta }^{2} - 2\left( {t + 1 - 1} \right)\sigma_{\eta }^{2} = \sigma_{\eta }^{2}$$

Because the HP filter assume that $W_{t}$ and $Z_{t}$ are two independent Brownian motions I have,

$$\sigma^{2} = \sigma_{\mu }^{2} + \sigma_{c}^{2} ,$$

(18)

Because $\mu_{t}$ is driftless a random walk process, it follows that

$$\sigma_{\mu }^{2} = \left( {t + 1 - t} \right)\sigma_{\eta }^{2} ,$$

and

$$\sigma_{\Delta \mu }^{2} = \sigma_{\eta }^{2}$$

.

Therefore,

$$q = \frac{{\sigma_{\Delta \mu }^{2} }}{{\sigma_{c}^{2} }} = \frac{{\sigma_{\eta }^{2} }}{{\sigma_{c}^{2} }} = \frac{{\sigma_{\mu }^{2} }}{{\sigma_{c}^{2} }} .$$

(19)

Appendix 3: Autocorrelated error term

This appendix provides details of derivation related to the variance of the autocorrelated error term, $\pi_{t + 1} ,$, and the variance of the White noise error term, $\varepsilon_{t + 1} ,$ of the stochastic mean NSM model.

3.1 The HP filter case

From Eq. (4a) and Eq. (5a) I have

$$\mathop \sum \limits_{j = 1}^{p} \rho_{j} c_{t - j} + v_{t + 1} = c_{t + 1} - \rho c_{t}$$

$$\pi_{t + 1} = \mathop \sum \limits_{j = 1}^{p} \rho_{j} c_{t - j} + v_{t + 1} + \eta_{t + 1} = \mathop \sum \limits_{j = 1}^{p} \rho_{j} c_{t - j} + \varepsilon_{t + 1} .$$

It follows that

$$\pi_{t + 1} = c_{t + 1} - \rho c_{t} + \eta_{t + 1} ,$$

Following Hodrick and Prescott (1997) and Ravn and Uhlig (2002), I make the assumption of independence between the permanent and transitory shocks, such that $\sigma_{c,\eta }^{2} = 0$. (See Kohn & Ansley, 1987). Hence, the variance follows immediately

$$\sigma_{\pi }^{2} = \sigma_{c}^{2} + \rho^{2} \sigma_{c}^{2} - 2\rho {\text{cov}} \left( {c_{t + 1} ,c_{t} } \right) + \sigma_{\eta }^{2} .$$

$$\sigma_{\pi }^{2} = \sigma_{c}^{2} + \rho^{2} \sigma_{c}^{2} - 2\rho^{2} \sigma_{c}^{2} + \sigma_{\eta }^{2} .$$

$$\sigma_{\pi }^{2} = \left( {1 - \rho^{2} } \right)\sigma_{c}^{2} + \sigma_{\eta }^{2} .$$

Using (19) I obtain,

$$\sigma_{\pi }^{2} = \frac{{\left( {1 - \rho^{2} } \right)}}{{\left( {1 - \rho^{2} - \mathop \sum \nolimits_{j = 1}^{p} \rho_{j}^{2} } \right)}}\sigma_{v}^{2} + \sigma_{\eta }^{2} .$$

Defining $\Omega = 1 - \rho^{2}$ and $\phi = - \mathop \sum \limits_{j = 1}^{p} \rho_{j}^{2}$ we can write,

$$\sigma_{\pi }^{2} = \frac{\Omega }{\Omega + \phi }\sigma_{v}^{2} + \sigma_{\eta }^{2} ,$$

which can be written as,

$$\sigma_{\pi }^{2} = \frac{\Omega }{\Omega + \phi }\sigma_{v}^{2} + \sigma_{\eta }^{2} + \sigma_{v}^{2} - \sigma_{v}^{2}$$

$$= \left( {\frac{\Omega }{\Omega + \phi } - 1} \right)\sigma_{v}^{2} + \sigma_{\eta }^{2} + \sigma_{v}^{2} ,$$

$$= \left( {\frac{\Omega }{\Omega + \phi } - 1} \right)\sigma_{v}^{2} + \sigma_{{}}^{2} ,$$

$$= \left( {\frac{\Omega }{\Omega + \phi } - 1} \right)\sigma_{v}^{2} + \sigma_{{}}^{2} ,$$

$$\sigma_{\pi }^{2} = - \frac{\phi }{\Omega + \phi }\sigma_{v}^{2} + \sigma_{{}}^{2} ,$$

Because $\Omega = 1 - \rho^{2}$ and $\alpha_{1} = \rho - 1,$ I can write: $\Omega = \alpha_{1}^{2} - 2\alpha_{1}$, then I have,

$\sigma_{\pi }^{2} = \left( { - \frac{\varphi }{{\left( {\varphi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}}} \right)\sigma_{v}^{2} + \sigma^{2}$.

Because $q = \frac{{\sigma_{\eta }^{2} }}{{\sigma_{v}^{2} }}$ we obtain,

$$\sigma_{\eta }^{2} = q\sigma_{v}^{2} ,$$

$$\sigma_{{}}^{2} = \left( {1 + q} \right)\sigma_{v}^{2} ,$$

$$\sigma_{v}^{2} = \left( {\frac{1}{1 + q}} \right)\sigma_{{}}^{2} .$$

Plug this in (19) I get,

$$\sigma_{\pi }^{2} = \left( { - \frac{\phi }{\Omega + \phi }} \right)\left( {\frac{1}{1 + q}} \right)\sigma_{{}}^{2} + \sigma_{{}}^{2} ,$$

$$\sigma_{\pi }^{2} = \left( {1 - \frac{\phi }{{\left( {\Omega + \phi } \right)\left( {1 + q} \right)}}} \right)\sigma_{{}}^{2} .$$

Because $\Omega = 1 - \rho^{2}$ and $\alpha_{1}^{{}} = \rho - 1,$ we can write: $\Omega = \alpha_{1}^{2} - 2\beta$, then we have,

$$\sigma_{\pi }^{2} = \left( {1 - \frac{\phi }{{\left( {\phi + \alpha_{1}^{2} - 2\beta } \right)\left( {1 + q} \right)}}} \right)\sigma_{{}}^{2} ,$$

$$\sigma_{\pi }^{2} = \left( {\frac{{\left( {\alpha_{1}^{2} - 2\alpha_{1} } \right)\left( {1 + q} \right) + q\phi }}{{\left( {\phi + \alpha_{1}^{2} - 2\alpha_{1} } \right)\left( {1 + q} \right)}}} \right)\sigma_{{}}^{2} ,$$

$$\sigma_{\pi }^{2} = \left( {\frac{{\left( {\alpha_{1}^{2} - 2\alpha_{1} } \right) + \frac{q}{{\left( {1 + q} \right)}}\phi }}{{\left( {\phi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}}} \right)\sigma^{2}$$

Let

$$\partial = \left( {\frac{{\left( {\alpha_{1}^{2} - 2\alpha_{1} } \right) + \frac{q}{{\left( {1 + q} \right)}}\phi }}{{\left( {\phi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}}} \right)$$

Then

$$\sigma_{\pi }^{2} = \partial \sigma^{2} .$$

3.2 The BN filter case

Because the BN filter assumes that $Corr\left({\eta }_{t},{\nu }_{t}\right)={\rho }_{\eta \upsilon }=1$, I have

$$\sigma_{\pi }^{2} = \left( { - \frac{\varphi }{{\left( {\varphi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}}} \right)\sigma_{v}^{2} + \sigma_{v}^{2} + \sigma_{\mu }^{2} + 2\sigma_{v} \sigma_{\mu }$$

$$\sigma_{\pi }^{2} = \left( {1 - \frac{\varphi }{{\left( {\varphi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}}} \right){ }\sigma_{v}^{2} + \sigma_{\mu }^{2} + 2\sigma_{v} \sigma_{\mu }$$

$$\sigma_{\pi }^{2} = \left( {1 - \frac{\varphi }{{\left( {\varphi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}}} \right){ }\sigma_{v}^{2} + \sigma^{2} - \sigma_{v}^{2}$$

$$\sigma_{\pi }^{2} = \left( { - \frac{\varphi }{{\left( {\varphi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}}} \right)\sigma_{v}^{2} + \sigma^{2}$$

$$\sigma_{\pi }^{2} = \left( {1 - \frac{\varphi }{{\left( {\varphi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}} + 2\frac{{\sigma_{\mu } }}{{\sigma_{v} }}} \right){ }\sigma_{v}^{2} + \sigma_{\mu }^{2}$$

$$\sigma_{\pi }^{2} = \left( { - \frac{\varphi }{{\left( {\varphi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}} + 2\frac{{\sigma_{\mu } }}{{\sigma_{v} }}} \right){ }\sigma_{v}^{2} + \sigma_{v}^{2} + \sigma_{\mu }^{2}$$

Define $\lambda =\frac{{\sigma }_{v}^{2}}{{\sigma }^{2}}$ I obtain,

$$\sigma_{\pi }^{2} = \left( { - \frac{\varphi }{{\left( {\varphi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}}} \right)\lambda \sigma^{2} + \sigma^{2}$$

$$\sigma_{\pi }^{2} = \left( {1 - \frac{\varphi }{{\left( {\varphi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}}\lambda } \right)\sigma^{2}$$

Appendix 4: GMM estimation

Define λ as the entire parameter vector. I have the following orthogonality conditions:

• 1. The Aït-Sahalia Model: $\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} - \alpha_{0} - \alpha_{1} r_{t} - \alpha_{2} r_{t}^{2} - \alpha_{3} r_{t}^{ - 1} } \right]$. The moment conditions are given by:

  $$h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\varepsilon_{t + 1} , \varepsilon_{t + 1} r_{t } , \varepsilon_{t + 1} r_{t}^{2} , \varepsilon_{t + 1} r_{t}^{ - 1} , \varepsilon_{t + 1}^{2} - \sigma^{2} r_{t}^{2\gamma } , \left( {\varepsilon_{t + 1}^{2} - \sigma^{2} r_{t}^{2\gamma } } \right)r_{t } } \right]$$

• 2. The CKLS model: $\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} - \alpha_{0} - \alpha_{1} r_{t} } \right]$. The moment conditions are given by:

  $$h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\varepsilon_{t + 1} , \varepsilon_{t + 1} r_{t, } \varepsilon_{t + 1}^{2} - \sigma^{2} r_{t}^{2\gamma } , \left( {\varepsilon_{t + 1}^{2} - \sigma^{2} r_{t}^{2\gamma } } \right)r_{t } } \right]$$

• 3. The AG model: $\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} - \alpha_{0} - \alpha_{1} r_{t} } \right]$. The moment conditions are given by:

  $$h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\varepsilon_{t + 1} , \varepsilon_{t + 1} r_{t, } \varepsilon_{t + 1} r_{t}^{2} , \varepsilon_{t + 1}^{2} - \sigma^{2} r_{t}^{2\gamma } , \left( {\varepsilon_{t + 1}^{2} - \sigma^{2} r_{t}^{3} } \right)r_{t } , \left( {\varepsilon_{t + 1}^{2} - \sigma^{2} r_{t}^{3} } \right)r_{t}^{3} } \right]$$

• 4. The CIR Model: $\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} - \alpha_{0} - \alpha_{1} r_{t} } \right]$. The moment conditions are given by:

  $$h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\varepsilon_{t + 1} , \varepsilon_{t + 1} r_{t, } \varepsilon_{t + 1}^{2} - \sigma^{2} r_{t} , \left( {\varepsilon_{t + 1}^{2} - \sigma^{2} r_{t } } \right)r_{t } } \right]$$

• 5. The Vasicek Model: $\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} - \alpha_{0} - \alpha_{1} r_{t} } \right]$. The moment conditions are given by:

  $$h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\varepsilon_{t + 1} , \varepsilon_{t + 1} r_{t, } \varepsilon_{t + 1}^{2} - \sigma^{2} , \left( {\varepsilon_{t + 1}^{2} - \sigma^{2} } \right)r_{t } } \right]$$

• 6. The BDF Model: $\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} + \alpha_{1} \left( {\theta_{t} - r_{t} } \right)} \right]$. The moment conditions are given by:

  $$h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\varepsilon_{t + 1} , \varepsilon_{t + 1} r_{t, } \varepsilon_{t + 1} r_{1, t } , \varepsilon_{t + 1} r_{2, t ,} \varepsilon_{t + 1}^{2} - \sigma^{2} , \left( {\varepsilon_{t + 1}^{2} - \sigma^{2} } \right)r_{t } } \right]$$

  where

  $$\theta_{t} = \alpha_{\theta ,1} \left[ {B\left( {T_{2} } \right)T_{1} r_{1,t} - B\left( {T_{1} } \right)T_{2} r_{2,T} } \right]$$
  $$B\left( T \right) = \frac{{2\left( {e^{\delta T} - 1} \right)}}{{\left( {\delta + k} \right)\left( {e^{\delta T} - 1} \right) + 2\delta }}$$
  $$\sigma^{2} = \sigma_{0}^{2} + \sigma_{1}^{2} r_{t}$$

  and

  $$\delta = \sqrt {\left( {\delta^{2} + \sigma_{1}^{2} } \right)}$$

  where $\sigma_{1} = 0$ if BDF-VAS is considered and $\sigma_{0} = 0$ if BDF-CIR is considered.

• 7. The Heston Model: $\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} - \alpha_{0} - \alpha_{1} r_{t} } \right]$ and $\varepsilon_{v, t + 1} = \left[ {v_{t + 1} - v_{t} - \alpha_{v} - \alpha_{v} v_{t} } \right]$. The moment conditions are given by:

  $$h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\varepsilon_{t + 1} , \varepsilon_{v, t + 1} , \varepsilon_{t + 1} r_{t} , \varepsilon_{v, t + 1} v_{t } , \varepsilon_{v, t + 1}^{2} - \sigma_{v}^{2} , \left( {\varepsilon_{v, t + 1}^{2} - \sigma_{v}^{2} } \right)v_{t } } \right]$$

  I follow Andersen, and Lund (1997) and specify $v_{t + 1}$ as a GARCH(1,1) model.

• 8. The Chen Model: $\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} + \alpha_{1} \left( {\theta_{t} - r_{t} } \right)} \right],$ $\varepsilon_{v, t + 1} = \left[ {v_{t + 1} - v_{t} - \alpha_{vo} - \alpha_{v1} v_{t} } \right],$ and $\varepsilon_{\theta , t + 1} = \left[ {\theta_{t + 1} - \theta_{t} - \alpha_{\theta o} - \alpha_{\theta 1} \theta_{t} } \right].$ The moment conditions are given by: $h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\varepsilon_{t + 1} , \varepsilon_{v, t + 1} ,\varepsilon_{\theta , t + 1} , \varepsilon_{t + 1} r_{t, } \varepsilon_{v, t + 1} v_{t, } , \varepsilon_{\theta , t + 1} \theta_{t, } , \varepsilon_{v, t + 1}^{2} - \sigma_{v}^{2} , \left( {\varepsilon_{v, t + 1}^{2} - \sigma_{v}^{2} ,} \right)v_{t } , \varepsilon_{\theta , t + 1}^{2} - \sigma_{\theta }^{2} , \left( {\varepsilon_{\theta , t + 1}^{2} - \sigma_{\theta }^{2} ,} \right)\theta_{t } } \right]$. I follow Andersen, and Lund (1997) and specify $v_{t + 1}$ as a GARCH(1,1) model.

• 9. The Al-Zoubi (2019) I(2) model: $\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} - \alpha_{1} c_{t} } \right]$.

  $$h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\pi_{t + 1} , \varepsilon_{t + 1} c_{t} , \varepsilon_{t + 1}^{2} - \sigma^{2} , \left( {\varepsilon_{t + 1}^{2} - \sigma^{2} } \right)c_{t} } \right]$$

• 10. The NSM-HP and NSM-bHP Models:$\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} - \alpha_{1} c_{t} } \right]$. Letting:

  $$\partial = \left( {\frac{{\left( {\alpha_{1}^{2} - 2\alpha_{1} } \right) + \frac{q}{{\left( {1 + q} \right)}}\phi }}{{\left( {\phi + \alpha_{1}^{2} - 2\alpha_{1} } \right)}}} \right)$$

  then

  $$h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\pi_{t + 1} , \pi_{t + 1} c_{t} , \pi_{t + 1}^{2} - \partial \sigma^{2} , \left( {\pi_{t + 1}^{2} - \partial \sigma^{2} } \right)c_{t} } \right]$$

  .

• 11. The NSM-HAM Model:

  $\varepsilon_{t + 1} = \left[ {r_{t + 1} - r_{t} - \alpha_{1} \left( {r_{t} - b_{0} - b_{1} r_{t - 7} - b_{2} r_{t - 8} - b_{3} r_{t - 9} - b_{4} r_{t - 10} } \right) } \right]$. The moment conditions are given by:

  $h\left( {r_{t + 1} ,\lambda } \right) = \left[ {\varepsilon_{t + 1} , \varepsilon_{t + 1} r_{t} , \varepsilon_{t + 1}^{2} - \sigma^{2} , \left( {\varepsilon_{t + 1}^{2} - \sigma^{2} } \right)r_{t } } \right]$.

To test the validity of my model, I minimize the GMM criterion of the form,

$$\frac{1}{T}\left[ {f\left( {x_{t + 1} ,Y_{t} ,\lambda } \right)} \right]{\prime} W_{T} \frac{1}{T}\left[ {f\left( {x_{t + 1} ,Y_{t} ,\lambda } \right)} \right],$$

(18)

where W_T is a consistent estimate of $\left( {{\text{var}} \left[ {\left( {1/T} \right)\left( {f\left( {x_{t + 1} ,Y_{t} ,\lambda } \right)_{t} } \right)} \right]} \right)^{ - 1}$ and Y_t is a K-dimensional vector of instrumental variables.

Under the null hypothesis that GMM restrictions are valid, I have that:

$$\frac{1}{T}\left[ {f\left( {x_{t + 1} ,Y_{t} ,\lambda } \right)} \right]{\prime} W_{T} \frac{1}{T}\left[ {f\left( {x_{t + 1} ,Y_{t} ,\lambda } \right)} \right]\mathop \sim \limits^{a} \chi_{L - k}^{2} .$$

(19)

For my model to be robust with respect to heteroskedasticity and autocorrelation variance, I follow Inoue and Shintani (2006) and use the Parzen kernel of Gallant (1987) with two lags to calculate the moments weighting matrix.