On Convergence of the QMLE for Misspecified GARCH Models

In this paper we study the behavior of GARCH(1,1) parameter estimates when data is generated by certain types of stochastic volatility models including well known models from the literature on realized volatility and mathematical finance. Our main result states that the parameter estimates (a, b) tend to (0,1) as the sampling frequency is increased thereby establishing that the stochastic sequence of QMLEs do indeed behave as the deterministic parameters considered in the literature on filtering based on misspecified ARCH models, see e.g. Nelson (1992). The convergence result is in line with the empirical finding that a GARCH model fitted to virtually any financial data set exhibits the property that a+b tends to one, a fact commonly referred to as the IGARCH effect. Hence, the paper suggests that the IGARCH effect could be caused by misspecification. An included study of simulations and empirical high frequency data is found to be in very good accordance with the mathematical results.