Abstract
We develop a uniform test for detecting and dating the integrated or
mildly explosive behaviour of a strictly stationary generalized autoregressive conditional
heteroskedasticity (GARCH) process. Namely, we test the null hypothesis of a
globally stable GARCH process with constant parameters against the alternative that
there is an “abnormal" period with changed parameter values. During this period, the
parameter-value change may lead to an integrated or mildly explosive behaviour of the
volatility process. It is assumed that both the magnitude and the timing of the breaks
are unknown. We develop a double-supreme test for the existence of breaks, and then
provide an algorithm to identify the periods of changes. Our theoretical results hold
under mild moment assumptions on the innovations of the GARCH process. Technically,
the existing properties for the quasi-maximum likelihood estimation (QMLE) in
the GARCH model need to be reinvestigated to hold uniformly over all possible periods
of change. The key results involve a uniform weak Bahadur representation for the estimated
parameters, which leads to weak convergence of the test statistic to the supreme
of a Gaussian process. In simulations we show that the test has good size and power
for reasonably long time series. We apply the test to the conventional early-warning
indicators of both the financial market and a representative of the emerging Fintech
market, i.e. the Bitcoin returns.
mildly explosive behaviour of a strictly stationary generalized autoregressive conditional
heteroskedasticity (GARCH) process. Namely, we test the null hypothesis of a
globally stable GARCH process with constant parameters against the alternative that
there is an “abnormal" period with changed parameter values. During this period, the
parameter-value change may lead to an integrated or mildly explosive behaviour of the
volatility process. It is assumed that both the magnitude and the timing of the breaks
are unknown. We develop a double-supreme test for the existence of breaks, and then
provide an algorithm to identify the periods of changes. Our theoretical results hold
under mild moment assumptions on the innovations of the GARCH process. Technically,
the existing properties for the quasi-maximum likelihood estimation (QMLE) in
the GARCH model need to be reinvestigated to hold uniformly over all possible periods
of change. The key results involve a uniform weak Bahadur representation for the estimated
parameters, which leads to weak convergence of the test statistic to the supreme
of a Gaussian process. In simulations we show that the test has good size and power
for reasonably long time series. We apply the test to the conventional early-warning
indicators of both the financial market and a representative of the emerging Fintech
market, i.e. the Bitcoin returns.
| Original language | English |
|---|---|
| Pages (from-to) | 467-491 |
| Number of pages | 25 |
| Journal | Econometrics Journal |
| Volume | 26 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 1 Sept 2023 |