Detecting the Structural Breaks in GARCH Models Based on Bayesian Method: The Case of China Share Index Rate of Return

2.1.1 GARCH(1,1) Model with t Distribution

GARCH(1,1) model is widely used in analysis of financial time series. Because the rates of return have heavy tails, in our GARCH(1,1) model we suppose that the error term follows standard student t distribution with degree of freedom v instead of standard normal distribution. This is more reasonable. Assume that we observe y = (y₁, y₂, ⋯ , y_T), {y_t: t = 1, 2, ⋯ , T} follows GARCH(1,1) model with a single change point, the error term ε_t ~ student(0, 1, v), i.e.

where student(0, 1, v) is standard student t distribution, the degree of freedom is v(v > 2), σt2 is conditional variance and is subject to structural instability.

For the conditional volatility to be strictly positive, constants must be placed on the parameters: e.g. ω_i > 0, α_i, ß_i ≥ 0, i= 1, 2. For covariance stationarity, the GARCH parameters must satisfy 0 < α_i + β_i < 1, i = 1, 2.

2.1.2 Conditional Posterior and Gibbs Sampling

Let θ = (ω₁, α₁, β₁, w₂, α₂, β₂, v, k) be the parameter vector of the model, given a sample of Τ observations {y_t : t = 1, 2,⋯, T}, let y = (y₁, y₂, ⋯ , y_T), the posterior density of θ is

where φ(θ) is the prior density of θ, and l(y|θ) is the likelihood function.

Given the change point k, the likelihood function is

where the likelihood depends on the parameter θ through σt2.

Firstly, we need to choose priors for the model parameters. Here, let us assume each of the model parameters follows a uniform distribution, which assigns equal probability to each possible value. This is called uninformative prior and is very useful in practice since we may not know much about the model parameter before performing estimation.

Given our model specification and the prior choice, it is very difficult to derive a closed form for the joint posterior distribution φ(θ|y) over the whole set of parameters. However, the Gibbs sampling procedure allows us to draw each parameter from the conditional posterior density given other parameters fixed, and after many iterations, joint distribution of the parameters will still converge to the true joint distribution.

Given the initial value θ(0)=(ω1(0),α1(0),β1(0),ω2(0),α2(0),β2(0),υ(0),k(0)), Gibbs sampling procedure is as follows:

1. Draw ω1(1)fromφ(ω1|y,α1(0),β1(0),ω2(0),α2(0),β2(0),υ(0),k(0));

2. Draw α1(1)fromφ(α1|y,ω1(1),β1(0),ω2(0),α2(0),β2(0),υ(0),k(0));

3. Draw β1(1)fromφ(β1|y,ω1(1),α1(1),ω2(0),α2(0),β2(0),υ(0),k(0));

4. Draw ω2(1)fromφ(ω2|y,ω1(1),α1(1),β1(1),α2(0),β2(0),υ(0),k(0));

5. Draw α2(1)fromφ(α2|y,ω1(1),α1(1),β1(1),ω2(1),β2(0),υ(0),k(0));

6. Draw β2(1)fromφ(β2|y,ω1(1),α1(1),β1(1),ω2(1),α2(1),υ(0),k(0));

7. Draw υ(1)fromφ(υ|y,ω1(1),α1(1),β1(1),ω2(1),α2(1),β2(1),k(0));

8. Draw k(1)fromφ(k|y,ω1(1),α1(1),β1(1),ω2(1),α2(1),β2(1),υ(1));

⋮

By cycling repeatedly through draws of each parameter conditional on the remaining ones, we obtain a Markov chain {θ^(m) : m = 1, 2, ⋯, M}, which has equilibrium distribution φ(θ|y) under quite mild conditions.

The conditional posterior density φ(ω1|y,α1(0),β1(0),ω2(0),α2(0),β2(0),υ(0),k(0)) has a kernel given by

where θ−ω1 denotes the remaining parameters except ω₁.

The kernels of the conditional posterior densities φ(ω2|y,θ−ω2),φ(αi|y,θ−αi)andφ(βi|y,θ−βi),i = 1,2bear the same expression, but subject to the stationary restrictions.

The change point k is a discrete variable, and it takes value from 2 to Τ — 1. Since we assume that the prior of k is uniform distribution on [0, Τ — 1] (discrete version), then there is equal probability assigned to each possible value of k, p(k=t)=1T−2. The conditional posterior density for the break point k is given by

It is hard to make random draws from the above densities, due to their complex functional forms. Existing methodologies regarding making random draws from densities of any functional form generally fall into two categories, either using acceptance/rejection sampling algorithm or applying nonparametric approximations to the inverse of the cumulative distribution function (CDF). The first method involves proposing a candidate density which we can easily draw from, and using some acceptance and rejection criteria to determine whether to keep the current draw. The second approach features Griddy-Gibbs sampler proposed in [12], where the conditional posterior kernel is evaluated at prespecified grid points, and then numerical integration method can be used to approximate the CDF which can be directly used to make random draws. For example, let F(·) be the CDF and y be a draw from U(0, 1), then x = F^–1(y) is a random draw from the CDF F(·). Not like the acceptance/rejection sampler, the Griddy-Gibbs sampler does not involve any candidate density, and thus no tuning parameter is required.

Given the value of change point k, the Griddy-Gibbs sampling method for ω1,α1,β1,ω2,α2,β2 follows the work of [13] because Griddy-Gibbs sampler doesn’t need any candidate density. However, for the sampling of change point k, we adopt acceptance/rejection sampling algorithm.

The Griddy-Gibbs sampling algorithm for GARCH model:

Let us take the sampling method for ω₁ as an example. The conditional posterior density φ(ω₁|y, α₁, β₁, ω₂, α₂, β₂, v, k) has a kernel given by (4). And the conditional posterior densities for α₁, β₁, ω₂, α₂, β₂ the same form, thus the sampling methods are the same.

Step 1 Set the initial valueθ(0)=(ω1(0),α1(0),β1(0),ω2(0),α2(0),β2(0),υ(0),k(0)).

Step 2 (a) Set grid (w₁, w₂, ⋯,w_G) for ω₁. For α and β, the grids have to be confined on [0, 1] to meet the condition for the stationarity of a GARCH process. According to [13], we choose the number of grids G = 33;

(b) Compute the values G_κ = (κ₁, κ₂, ⋯ , κ_G), where κj=κ(wj|y,θ−ω1(0)), for j = 1, 2, ⋯, G;

(c) Compute the values G_Φ = (0, Φ₂, ⋯, Φ_G) by a deterministic integration rule, where

Also compute the normalized pdf value Gφ=GκΦG. Calculate E(ω1|y,θ−ω1(0)) and Var(ω1|y,θ−ω1(0)) by the same integration rule as above, and store the values;

(d) Draw u from a uniform distribution on [0, Φ_G]. Invert Φ_j by numerical interpolation to get a draw ω1(1).

For large data set, the value of κ(w1|y,θ−ω1(0)) might become a very large number or close to zero, exceeding the capability of the computer. [9] used the following modified procedure (b′) and (c′) to handle such instead of using the procedure (b) and (c):

(b′) Define LG_κ = (log κ₁, log κ₂, ⋯, log κ_G, ), and LG* = max(log κ₁, log κ₂, ⋯·log κ_G);

(c′) Calculate G_Φ = (0, Φ₂, ⋯ , Φ_G) according to the following rule:

The added constant exp(—LG*) doesn’t change the shape of the kernel. The redefined kernel is uniformly bounded, i.e. 0<exp⁡(log⁡(κ(ω1|y,θ−ω1(0)−LG∗)≤1.

Step 3 Sample α1(1),β1(1),ω2(1),α2(1),β2(1) follow the same procedure in Step 2.

Step 4 By cycling repeatedly through draws of each parameter conditional on the remaining ones, we obtain a Markov chain {θ^(m) : m = 1, 2, ⋯ , M}, which has equilibrium distribution φ{0|y) under quite mild conditions.

2.1.3 The Sampling Method of Change Point k

The change point k is a discrete variable. For the sampling of change point k, we adopt acceptance/rejection sampling algorithm. Let p_t = P(k = t). Further suppose that a discrete random variable has probability mass function qt=1T−2,t=2,3,⋯,T−1, compute c = max_t{p_t/q_t}. The sampling procedure:

Step 1 Generate random variables U₁, U₂ ~ U(0, 1) and set z = [U₁(T – 2)] + 1. If z = 1 or z = Τ, repeat Step 1;

Step 2 If U2<pzcqt, then set k = z; otherwise return to Step 1.