Narrow Results

The stationary distribution of a GARCH(1,1) process has a power law decay, under broadly applicable conditions. We study the change in the exponent of the tail decay under temporal aggregation of parameters, with the distribution of innovations held fixed. This comparison is motivated by the fact that GARCH models are often fit to the same time series at different frequencies. The resulting models are not strictly compatible so we seek more limited properties we call forecast consistency and tail consistency. Forecast consistency is satisfied through a parameter transformation. Tail consistency leads us to derive conditions under which the tail exponent increases under temporal aggregation, and these conditions cover most relevant combinations of parameters and innovation distributions. But we also prove the existence of counterexamples near the boundary of the admissible parameter region where monotonicity fails. These counterexamples include normally distributed innovations.

Information provided by the US Department of Homeland Security regarding potential terrorist attacks significantly affects US Treasury securities markets. When the government announces heightened terror alert levels, investors' perceptions of risk increase and investors purchase 1-month and 1-year Treasury bills and 3-year, 5-year, 7-year, and 10-year US Treasuries in a "flight-to-quality" episode. Partial anticipation of increased threat level announcements is stronger than the anticipation of announcements regarding the federal funds rate during the 10 days prior to an announcement.

For economists and investors, it is necessary to understand the random and nonlinear pattern of the stock market volatility. High volatility directly affects the financial market that leads to unpredictability. China–Pakistan Economic Corridor attracts economists and investors worldwide. Therefore, predicting the volatility of the stock markets related to CPEC is important. In this study we consider the most important stock markets lying on the route of CPEC, namely KSE 100 (Pakistan), SSE 100 (China), TADAWUL (Kingdom of Saudi Arabia), KASE (Kazakhstan), KLSE (Malaysia), BIST (Turkey), MOEX (Russia), FTSE (United Kingdom) and CAC40 (France). The daily returns of stock market indices consist of 1706 observations from December 2014 to July 2021. After the confirmation from the ARCH effect test, family GARCH models are employed, among them, based on AIC and BIC criteria, GARCH (1,1), EGARCH (1,1), and GARCH-M (1,1) are found suitable to forecast the volatility. The empirical study also suggests that the out-of-sample forecast GARCH-M (1,1) model is more appropriate as it has a minimum value of MAE, MSE, RMSE, MAPE, TheilU1, and Theil U2 among all the studied GARCH models. Furthermore, it is also found that the KSE-100 and SSE-100 have moderate and slow market average returns even though both stock markets are found to be the least risk-returns markets.