Application of autoregressive tail-index model to China's stock market

Figures & data

Figure 1. Daily closing prices of SSE Index from 01/05/2005 to 10/20/2020.

Figure 2. Histogram of daily maxima of negative log-returns of 87 stocks in SSE Index from 01/05/2005 to 10/20/2020.

Figure 3. Daily maxima of standardised negative log-returns of 87 stocks in SSE Index from 01/05/2005 to 10/20/2020.

Figure 4. Histogram of daily maxima of standardised negative log-returns of 87 stocks in SSE Index from 01/05/2005 to 10/20/2020.

Table 1. MLE for cross-sectional daily maxima of negative log-returns of 87 stocks in SSE Index from 01/05/2005 to 10/20/2020.

	${\gamma }_0$	${\gamma }_1$	${\gamma }_2$	${\gamma }_3$	${\beta }_0$	${\beta }_1$	${\beta }_2$	${\beta }_3$	μ
Mean	−7.548	0.794	8.297	0.007	0.579	0.804	−0.021	0.933	−17.35
S.D.	0.333	0.006	0.351	0.001	0.080	0.026	0.004	0.167	2.285

Figure 5. Recovered tail indexes $\{{\hat{\alpha }}_t\}$ (dashed), and daily maxima of standardised negative log-returns (solid) of 87 stocks in SSE Index from 01/05/2005 to 10/20/2020.

Figure 6. Recovered scale parameters $\{{\hat{\sigma }}_t\}$ of 87 stocks in SSE Index from 01/05/2005 to 10/20/2020.

Figure 7. The 5% quantile (dashed), 95% quantile (dash-dotted) of the estimated $\{Q_t\}$ and daily maxima of standardised negative log-returns (solid) of 87 stocks in SSE Index from 01/05/2005 to 10/20/2020.

Table 2. The performance of model (5.1)–(5.3) on approximation of 1-day conditional VaR for daily maxima negative log-returns of SSE Index.

$q^0$	Expected	Actual	Violation	Relative
	violation	violation	rate	error rate
0.1	133.6	110	0.082	−0.177
0.05	66.8	57	0.043	−0.147
0.01	13.4	21	0.016	0.567
0.005	6.7	15	0.011	1.239
0.001	1.3	6	0.004	3.615