Title

icss methods -- Methods and formulas for icss

Model and null hypothesis

Let e(t), t=1,...,T, be a zero-mean process with unconditional
variance sigma^2(t). The null hypothesis is constancy of the unconditional
variance,

H0: sigma^2(1) = sigma^2(2) = ... = sigma^2(T).

Define the cumulative sum of squares C(k) = sum_ e(t)^2 with total
C(T), and the centered statistic D(k) = C(k)/C(T) - k/T. Under H0 the
tests converge to the supremum of a standard Brownian bridge, sup_r |W*(r)|.

The three statistics

Inclán-Tiao (test=it), eq. (1):

IT = sqrt(T/2) * max_k |D(k)|.

Its limit is free of nuisance parameters only when the data are i.i.d.
and mesokurtic (fourth moment = 3*sigma^4). Otherwise the limit depends on the
kurtosis and on the persistence of the conditional variance, and the test
over-rejects.

kappa1 (test=k1), Proposition 2:

B(k) = |C(k) - k*s2|, s2 = C(T)/T, a4 = (1/T) sum e(t)^4,

kappa1 = T^(-1/2) * max_k B(k) / sqrt(a4 - s2^2).

The denominator estimates the standard deviation of e(t)^2 and removes
the dependence on the fourth moment; valid for i.i.d. data of any kurtosis.

kappa2 (test=k2), Proposition 3:

G(k) = C(k) - (k/T)*C(T),

kappa2 = T^(-1/2) * max_k |G(k)| / sqrt(omega4),

where omega4 is the long-run variance of xi(t) = e(t)^2 - s2.
Replacing the short-run scale of kappa1 by the long-run scale accounts for
autocorrelation in the squared series, i.e. conditional heteroskedasticity. This
is the statistic recommended for financial returns.

Long-run fourth moment (kappa2)

omega4 is estimated nonparametrically from the autocovariances
gamma(j) of xi(t):

omega4-hat = gamma(0) + 2 * sum_ w(j,m) * gamma(j),

with a Bartlett kernel w(j,m) = 1 - j/(m+1) (kernel(bartlett)) or a
quadratic-spectral kernel (kernel(qs), default). The bandwidth m is
either fixed (bwidth()) or chosen by the Newey-West (1994) automatic rule
seeded by binit() (default 4), exactly as in variance.src.

Critical values and p-values

Because the ICSS algorithm evaluates the statistic on sub-samples of varying
length, icss uses the paper's finite-sample response surfaces for the
5% critical value as a function of T (Table 2 / the GAUSS constants). For the
whole-sample test the reported p-value is computed from the common asymptotic
distribution K(x) = P(sup_r|W*(r)| <= x), giving tail probability
2 * sum_ (-1)^(j+1) exp(-2 j^2 x^2).

The ICSS iterative algorithm

Multiple breaks are located by the Inclán-Tiao ICSS procedure (paper, Section 5),
ported verbatim from icss.src:

1. On the current segment, find the point that maximizes the statistic;
if it exceeds the 5% critical value, record a candidate break and recurse
on the left and right sub-segments (nbrseq, bypass, selec).

2. Bracket the first and last breaks and iterate inward
(nbreaks).

3. Fine-tune: re-test each break against its two neighbours and drop it
if it is no longer significant; repeat until the break set is stable (up to 20
sweeps).

Step-to-equation / step-to-code map

computation

C(k), D(k) cumulative sums Inclán-Tiao (1994)

IT statistic, eq. (1) Mata _icss_stat, test=0

kappa1, Proposition 2 Mata _icss_stat, test=1

kappa2, Proposition 3 Mata _icss_stat, test=2

omega4-hat long-run 4th moment Mata _icss_lvar (+ _icss_lagsel)

5% critical value response surface Mata _icss_cv

asymptotic p-value Mata _icss_pval

ICSS multiple-break search Mata _icss_icss / _icss_nbreaks / _icss_nbrseq

Every block reproduces the corresponding GAUSS procedure in icss.src
and variance.src one-to-one; the kappa2 long-run estimator matches
the quadratic-spectral / Newey-West default used in the paper's empirical
application.

Reference

Sansó, A., V. Aragó, and J. L. Carrion-i-Silvestre. 2004. Testing for
changes in the unconditional variance of financial time series. Revista de
Economía Financiera 4: 32-53.

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