Modeling Style Rotation: Switching and Re-switching

Appendix

Proof of Proposition 1.

1. Proving part 1 of the proposition is an easy exercise as eqs. [23]–[25] are a system of three linear equations over the first three autocovariances , , and , which can be easily resolved to confirm formula [26] and derive formula [29].

2. (a) If given the auxiliary eq. [28], the general solution of eq. [27] is given by formula [30], by fitting the boundary conditions on and , we have a system of equations

[A.1]

[A.2]

which, when resolved, produces the following formulas for coefficients and :

(b) if, the auxiliary eq. [28] has only one real root and the general solution of eq. [27] is given by

[A.3]

or remembering that and fitting the boundary conditions on and , we have a system

[A.4]

[A.5]

which, when resolved, produces the following formulas for coefficients and :

(c) if , the auxiliary eq. [28] has two complex conjugate roots and the general solution is given by

[A.6]

where and are complex conjugates too (see e.g., chapter 1.2 of J.D. Hamilton “Time Series Analysis”, 1994), and

[A.7]

[A.8]

Substituting , , we have the general solution to have the form

[A.9]

which after remembering the boundary conditions on and , produces solution of the form given in eq. [32]. Coefficients and are given by

where .

Proof of Lemma 1:

Starting with eq. [37] and using the lag operator L, we have

[A.10]

which can be rewritten as

[A.11]

that is , which proves the lemma.

Proof of Proposition 2:

The proof is similar to the proof of Proposition 1, in that we have the same general eq. [41] for the forecast of excess return as we had in eq. [27] for the autocovariance, which means that the solutions to the three cases (a), (b) and (c) which correspond to three different levels of discriminant (positive, nill or negative) will have the same general form. The difference in solutions comes from different boundary (i.e., initial) conditions and is demonstrated below.

1. If the general solution is given by formula [42], with the boundary conditions provided in [39] and [40]. We have a system of equations

[A.12]

[A.13]

where is set by Lemma 1. Therefore we have a system of equations for ?and?:

[A.14]

[A.15]

which, when solved, produces the following formulas for coefficients and :

[A.16]

[A.17]

(b) If , the auxiliary eq. [28] has only one real root and the general solution of eq. [41] is given by

[A.18]

or remembering that and fitting the boundary conditions on and , we have a system

[A.19]

[A.20]

which, when resolved, produces the following formula for coefficients and?:

[A.21]

[A.22]

and ,

(c) If , the auxiliary eq. [28] has two complex conjugate roots and the general solution is given by

[A.23]

where and are complex conjugates too (see e.g., chapter 1.2 of J.D.Hamilton “Time Series Analysis”, 1994), and

[A.24]

[A.25]

Substituting , , we have the general solution to have the form

[A.26]

which after remembering the boundary conditions, produces the following formulas for coefficients and in formula [50]:

[A.27]

[A.28]

where

Proof of Lemma 2:

We are looking for a moving average representation of the kind

[47]

for the process defined by eq. [37]. Putting formula [54] into eq. [37] we obtain

[A.29]

After equating coefficients of , we find that the coefficients have to satisfy the following conditions:

[A.30]

[A.31]

[A.32]

Assuming a solution of the kind

[A.33]

we immediately derive that the solution has the form

[A.34]

with eqs. (A.30) and (A.31) serving as the boundary conditions, which we use to find the constants and . Resolving this system of two equations with two unknowns we derive that

[49]

as required.

Proof of Lemma 3:

This lemma simply summarises the conditions under which the formula [60] for the lag at which the first switch occurs is well-defined and therefore, such lag can be computed. The proof therefore is straight-forward.

Proof of Proposition 3:

The proof of this proposition immediately follows from the infinite moving average representation [47], the definition of the h-step ahead forecast:

[A.35]

and the i.i.d. assumption in relation to all .

Proof of Lemma 4:

The proof of this lemma simply follows from starting with eqs. [2] and [5], inserting the definitions of and of [3], and collecting all components with and together.

Proof of Proposition 4:

We start from eqs. [11] and [14], where we use formula [13] which is the definition of ϕ:

[11*]

[14*]

We then aggregate across all equities in each style using formulas [4] and insert formulas [62] and [63] for the aggregate demand for equities in each two styles obtained in Lemma 4, which then leads us to formula [64].