It is well known that mathematical programs with equilibrium constraints (MPEC) violate the standard constraint qualifications such as Mangasarian-Fromovitz constraint qualification (MFCQ) and hence the usual Karush-Kuhn-Tucker conditions cannot be used as stationary conditions unless relatively strong assumptions are satisfied. This observation has led to a number of weaker stationary conditions, with Mordukhovich stationary (M-stationary) condition being the strongest among the weaker conditions. In nonlinear programming, it is known that MFCQ leads to an exact penalization. In this paper we show that MPEC GMFCQ, an MPEC variant of MFCQ, leads to a partial exact penalty where all the constraints except a simple linear complementarity constraint are moved to the objective function. The partial exact penalty function, however, is nonsmooth. By smoothing the partial exact penalty function, we design an algorithm which is shown to be globally convergent to an M-stationary point under an extended version of the MPEC GMFCQ.Keywords Mathematical program with equilibrium constraints · Mangasarian-Fromovitz constraint qualification · Partial exact penalization · Global convergence · M-stationary points Mathematics Subject Classifications (2000) 65K05 · 90C26

This paper introduces a novel generalized autoregressive conditional heteroskedasticity–mixed data sampling–extreme shocks (GARCH‐MIDAS‐ES) model for stock volatility to examine whether the importance of extreme shocks changes in different time ranges. Based on different combinations of the short‐ and long‐term effects caused by extreme events, we extend the standard GARCH‐MIDAS model to characterize the different responses of the stock market for short‐ and long‐term horizons, separately or in combination. The unique timespan of nearly 100 years of the Dow Jones Industrial Average (DJIA) daily returns allows us to understand the stock market volatility under extreme shocks from a historical perspective. The in‐sample empirical results clearly show that the DJIA stock volatility is best fitted to the GARCH‐MIDAS‐SLES model by including the short‐ and long‐term impacts of extreme shocks for all forecasting horizons. The out‐of‐sample results and robustness tests emphasize the significance of decomposing the effect of extreme shocks into short‐ and long‐term effects to improve the accuracy of the DJIA volatility forecasts.

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Extreme shocks (e.g., wars and financial crises) cause violent fluctuations in crude oil volatility. In this paper, we first propose GARCH models in the framework of MIDAS augmented to include the impacts of extreme shocks on oil price volatility. In-sample results show that extreme shocks can induce the additional volatility of crude oil. Further, the results from out-of-sample clearly indicate that the crude oil volatility is best fitted by the EGARCH-MIDAS-ES model, which incorporates asymmetric effects in the short-term component and the significant effect of extreme shocks in the long-term component. Additionally, robustness tests confirm that the augmented volatility models can produce better prediction results, both statistically and economically, than the conventional GARCH-MIDAS model. Furthermore, we verify that negative extreme shocks can cause larger volatility, whereas positive extreme shocks of the same magnitude have smaller effects. Our contribution offers fresh insights into energy price volatility forecasting by considering extreme shocks.