Detecting Multiple Breaks in Financial Market Volatility Dynamics

describes the following Data Generating Processes: DGPl: (0.4, 0.1,0.5) and
DGP2: (0.1,0.1,0.8) which are characterized by low and high volatility per-
sistence, respectively. The sample sizes of T = 500,1000, 3000 are chosen so
as to examine not only the asymptotic behavior but also the small sample
properties of the tests. The small sample features are particularly relevant
for the sequential application of the tests in subsamples.

The model without breaks (г = 0) denotes the processes under the null
hypothesis. Under the alternative hypothesis the returns process is assumed
to exhibit breaks and four hypotheses are considered to evaluate the power
of the tests. The simulation study first examines the single change-point
followed by the multiple breaks hypothesis. In the context of (2.1) we study
a single break in the conditional variance h_t which can also be thought as
a permanent regime shift in volatility at change points ττT (π = .3,.5,.7).
Such breaks may have the following sources. : a change in the volatility
dynamics (or persistence), β_i. Hβ : a change in the intercept, ω⅛. //( :
a change in the tails of r∣(_l._l to u-^_t ~ IV(0,σ_u), (σ_u = 1.1,1.5 ) at t =
7rT + 1, ...,T. Hβ : outliers in the error, u_(jJ to u-^_t ~ N(μ_u, 1), with jump
sizes μ_u =4,5 and frequencies at given regular dates of a daily sample, Δ ∙tj,
(where Δ = 250, 500 and tj = 1,2,..., Δ∕T) and zero otherwise.⁹ The sample
sizes are T = 1000 and 3000 observations to match the empirical analysis
as well as the large samples encountered in financial asset returns series of
relatively high sampling frequency.

For the multiple breaks case we examine the alternative hypothesis that
there are two breaks (or equivalently three segments) where the change points
occur in model (2.1) as follows:

∏,t = u_ltty∣h₁'_t, h₁,_t = ω₁ + + β₁h_ljt.₁  if  1 < t ≤ [πT]

r₂,_t = u₂,_ty∣h^t, h₂,_t = ω₂ + «2^1,t-ι + β₂h₂,_t-ι  if  [πT] <t < ∖2πT]

^r3,t = ^u3,t∖∣h^t, h_s,t = ^ω3 + ^a3^ul t-1 + β3h3,t-1  if  [2τr] < t ≤ T

(2.2)

where 7Γ = 0.33 and Uj,_t ~ N(Q,σ_uj),j = 1,2,3 and uncorrelated.¹⁰ Under
the null hypothesis the simulated process (2.1) holds whereas under the al-

⁹In our experiment the above simple jump process would facilitate the evaluation of
the test’s power in the presence of controlled outliers.

¹⁰The multiple change point model can be extended to more than two breaks and some
preliminary simulation results with four breaks show that the tests share good properties,
partly because of the large sample sizes encountered in financial asset returns. Therefore,
for conciseness and comparison with other studies we report the two breaks case.

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