Unconditional First Moment and Conditional Second Moment Effects: Intra-Day Stock Price, Market Index and Futures Price Processes

Abstract

A theoretical framework is developed in order to consider effects of mis-specification in either first and/or second moment equations on resultant conditional volatility parameter estimates. The conditional volatility model is considered as a special case of a general stochastic volatility structure. Conditions necessary for the behaviour of the underlying asset price processes to approximate a diffusion limit are considered as the observation interval approaches 0 (d~0). The asymptotic distribution of the measurement error process may not be obtainable as d~0 if these conditions are violated. The relative impact of mis-specification of drift in mean and drift in conditional volatility is the focus. Market features such as bid/ask bounce effects in futures and stock price processes and non-synchronous trading effects in cash index processes are explored within this framework. Other mis-specifications in mean equations such as over-differencing are jointly explored. The most important effect is mis-specification of conditional volatility equations by failing to account for contemporaneous market trading and volume of trade effects. Empirical examples are provided employing Australian, U.S. and U.K. cash index, stock price and futures price data sampled from transactions records. These estimates help quantify the relative effects on conditional volatility estimates from mis-specifying the dynamics and sampling interval for these asset price mean equations. The relative importance of mis-specification of conditional volatility equations from incorrect exclusion of variables is seen to be crucial. Volume of trade, market opening/closing and other contemporaneous volatility effects in parallel processes are allowed to enter conditional volatility equations to highlight this issue.