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A class of martingale estimating functions is convenient and plays an important role for inference for nonlinear time series models. However, when the information about the first four conditional moments of the observed process becomes available, the quadratic estimating functions are more informative. In this paper, a general framework for joint estimation of conditional mean and variance parameters in time series models using quadratic estimating functions is developed. Superiority of the approach is demonstrated by comparing the information associated with the optimal quadratic estimating function with the information associated with other estimating functions. The method is used to study the optimal quadratic estimating functions of the parameters of autoregressive conditional duration (ACD) models, random coefficient autoregressive (RCA) models, doubly stochastic models and regression models with ARCH errors. Closed-form expressions for the information gain are also discussed in some detail.

Godambe [

This paper is organized as follows. The rest of Section

Suppose that

The function

Consider a discrete time stochastic process

The optimal estimating functions based on the martingale differences

For the general model in (

the optimal estimating function is given by

the information

the gain in information

the gain in information

We choose two orthogonal martingale differences

When the conditional skewness

There is growing interest in the analysis of intraday financial data such as transaction and quote data. Such data have increasingly been made available by many stock exchanges. Unlike closing prices which are measured daily, monthly, or yearly, intra-day data or high-frequency data tend to be irregularly spaced. Furthermore, the durations between events themselves are random variables. The autoregressive conditional duration (ACD) process due to Engle and Russell [

When

In this section, we will investigate the properties of the quadratic estimating functions for the random coefficient autoregressive (RCA) time series which were first introduced by Nicholls and Quinn [

Consider the RCA model

Since

In view of the parameter

Random coefficient autoregressive models we discussed in the previous section are special cases of what Tjøstheim [

Let

The optimal estimating function and the associated information based on

Consider a regression model with ARCH (

It is of interest to note that when

In this paper, we use appropriate martingale differences and derive the general form of the optimal quadratic estimating function for the multiparameter case with dependent observations. We also show that the optimal quadratic estimating function is more informative than the estimating function used in Thavaneswaran and Abraham [