Detecting Multiple Breaks in Financial Market Volatility Dynamics

inequality results for weakly as well as strongly dependent processes, the lat-
ter being а-mixing which include long memory and ARCH-type processes.⁶
The number of breaks is estimated via a penalized least-squares approach
similar to Yao (1988). In particular, Lavielle and Moulines show that an
appropriately modified version of the Schwarz criterion yields a consistent
estimator of the number of change-points.

Consider the following generic model:

X_t = μl + ε_t t*_k~₁<t<t*_k l<k<r           (1.6)

where tə = O and t*₊₁ = T, the sample size. The indices of the breakpoint
and mean values μ_k, к = 1,. . . ,r are unknown. It is worth recalling that
X_t is a generic process. In practical applications, equation (1.6) applies
to squared returns, absolute returns, high-frequency data-driven volatility
estimates, etc. The Lavielle and Moulines tests are based on the following
least-squares computation:

r÷l ⅛

Q_τ(t) = min ∑ Σ (¾ - Mfc)²           (I-?)

^=h∙∙∙^⅛=Cl₊ι

Estimation of the number of break points involves the use of the Schwarz or
Bayesian information criterion and hence a penalized criterion Qτ(t) + β_τr,
where r is the number of break points and β_τ = 4 Iog(T)/T¹~^2d.⁷ It is shown
under mild conditions that the change-point estimator is strongly consistent
with T rate of convergence. The Lavielle and Moulines simultaneously detects
multiple breaks as opposed to the sequential adaptation of the Kokoszka and
Leipus test. It is worth noting that the Kokoszka and Leipus statistic in (1.1)
can be weighted by (k(T — k)T~¹p, O ≤ 7 < 1, as suggested in Kokoszka
and Leipus (1998, equation 1.3, p. 386) which is then equivalent to the
weighted least squares objective function when 7 = 0.5. This brings out
some similarities between the objective functions of the above two tests.

Given the asymptotic nature and sequential application of change-point
tests in subsamples we may obviously end up with relatively small samples.
It will therefore be important to appraise the power and size properties of
these change-point tests in small samples via Monte Carlo simulations.

⁶The latter are J-mixing which imply «-mixing.

⁷This formula allows for the possibility of long memory, with d Hosking’s long-range
dependence parameter.

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