_{1}

^{*}

We establish that [1]’s parameters are universally unidentified and a subset of their parameterization is over identified. As a solution to the problem with the identifiability, we propose a new representation of double-regime three-factor GDTSMs whose parameters are just-identified when the number of the pricing-with-error yields equals 2. This new parametrization has another advantage over [2] in that we can back out Q parameters and P parameters separately and make the estimation of structural parameters easier. Finally, we show that regime-switching three-factor arbitrage-free dynamic Nelson-Siegel model is a restricted special case of our model.

After [

[

But in the practical experience of those who have used DSY model are tremendous numerical challenges in estimating the necessary parameters from the data due to highly non-linear and badly behaved likelihood sur- faces. For example, [

… Even with these normalizations/constraints, the resulting maximally flexible

Another problem with DSY model is its identification. We find that DSY model parameters are universally unidentified. If there are some parameters in the model that are unidentified, then it will be wrong to make con- clusions from its parameters’ estimate, let us say about how regime-shift risks are priced.

This paper proposes solution to them and other problems with regime-switching affine term structure model of [

One implication is that the parameters of these reduced-form representations contain all the observable impli- cations of [

A second and completely separate contribution of the paper is that we propose our canonical representation of GDTSMs, which is then used in double-regime environment as a new form of regime-switching GDTSMs. Us- ing this form of representation, it is possible for the parameters of interest to be inferred directly from estimates of the reduced-form parameters themselves. This is a very useful result because the latter are often simple re- gime-switching OLS coefficients. Although translating from reduced-form parameters into structural parameters involves a mix of analytical and numerical calculations, the numerical component is far simpler than that asso- ciated with the usual approach of trying to find the maximum of the likelihood surface directly as a function of the structural parameters.

There have been several other recent efforts to use new development in GDTSMs for multi-regime considera- tion. [

The chief difference between this paper and other relevant papers is that they focus on how the re- gime-switching GDTSMs should be represented, whereas we also examine how the parameters of the regime- switching GDTSMs are to be estimated.

The rest of the paper is organized as follows. Section 2 describes [

In this section, we just briefly describe the model set by [

where

The regime-switching

The continuously compounded yield on a one-period zero-coupon bond in regime j is assumed to be the affine function of

Letting

where,

with initial conditions:

where

The market prices of factor (MPF) risks in regime j,

Given the time t + 1 regime

where

and

The regime-switching P probabilities

where j ≠ k. And then, the market price of regime-shift (MPRS) risk from

[

[

Given the time t regime

where

Inverting (2) results in

Then,

where,

The remaining

where,

The P-measure regime-switching probability

where,

Letting

gime-switching affine pricing and

regime-switching multivariate linear regression model.

However, which kind of mapping it may be is not inherent in the model but depends on the data structure used. For example, if the dimension of

[

Firstly, let us look at the flexibility of [

in their model. Consequently,

and

ers, and

unrestricted full matrix,

tion

Secondly, let us look at the total number of parameters for both models.

duce the total number of free parameters in

Due to the problems with identifiability of [

In [

there exist

with

Although, as is pointed out in [

Coefficient estimated | Number of free parameters | Coefficient estimated | Number of free parameters | Coefficient estimated | Number of free parameters |
---|---|---|---|---|---|

6 | 3 | 3 | |||

5 | 5 | 3 | |||

3 | 3 | 3 | |||

3 | 1 | 1 | |||

1 | 1 | 1 | |||

1 | 2 | 2 | |||

1 | 1 |

Coefficient estimated | Number of free parameters | Coefficient estimated | Number of free parameters | Coefficient estimated | Number of free parameters |
---|---|---|---|---|---|

3 | 3 | 1 | |||

3 | 3 | 3 | |||

9 | 1 | 1 | |||

9 | 1 |

study regime-switching three-factor GDTSMs. Next, we propose an alternative normalization in the following Theorem.

Theorem 1. Every three-factor canonical GDTSM is observationally equivalent to the three-factor canonical GDTSM with

where,

Proof:

Assuming some three-factor canonical GDTSM takes the following form:

For ease of exposition, we assume we have found

Then, letting

factor, because the mapping from

where,

Likewise, the

where,

cause we do not impose any restriction on either

Finally, we can transform the short rate as an affine function of the new state variables as follows,

where,

By Theorem 1, we will establish the reparametrization of [

Given the time t regime

where

pendent on time, and

Like [

Unlike [

be a different affine function of

Then, given the time t regime

where,

with initial conditions:

where

Given the time t regime

where

Like [

And then, the market price of regime-shift (MPRS) risk from

Like [

not set

del. Consequently,

A distinctive feature of this reparametrization is that, in estimation, there is an inherent separation between the parameters of the

As in [

Let

Given the time t regime

where

Inverting (2) results in

Then,

where,

The remaining 2 yields can be expressed as follows,

where,

The

where,

In summary, we can use the method proposed in [

Step 1. The estimate of the 6 unknowns in

Step 2. The estimate of

Step 3. The estimate of the 4 unknowns in

and,

Step 4. The estimate of

Step 5. The estimate of

that is

Step 6. The estimate of

Step7. The estimate of

is

In every step, the solving processes can be invertible, so we can also obtain

When M = 1, the situation is different. In Step 1, there are still 6 unknowns in

ations in

only 2 equations in

ti-to-one and so the parameters of our normalization are unidentified.

When

× 2 = 6 equations in

of our normalization are over identified.

The next question is how to obtain the standard error for these state-space parameters

Within [

In this section, we will show that the regime-switching extension on the AFNS model of [

By [

where

First, we let

where

Second, let

where

where

Comparing (10) with (3), we find that the regime-switching AFNS model is the constrained special case of

the our normalization with

Sometimes, we want to test if these constraints are valid. We could set regime-switching AFNS model as the null model and our representation as the alternative model, and then, under a desired statistical significance level, we compare likelihood ratio to the chi squared value with degrees of freedom equal to 5.

[

Besides, due to the tremendous numerical challenges in estimating the necessary parameters, we hope that our method will help to make these models a more effective tool for research in better describing the historical in- terest rate data.

This work is supported by Research Innovation Foundation of Shanghai University of Finance and Economics under Grant No. CXJJ-2013-321. And I am especially grateful to Professor Hong Li for his support and encou- ragement. All errors are my own.