Joint Estimation Using Quadratic Estimating Function

Abstract

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.

1. Introduction

Godambe [1] was the first to study the inference for discrete time stochastic processes using estimating function method. Thavaneswaran and Abraham [2] had studied the nonlinear time series estimation problems using linear estimating functions. Naik-Nimbalkar and Rajashi [3] and Thavaneswaran and Heyde [4] studied the filtering and prediction problems using linear estimating functions in the Bayesian context. Chandra and Taniguchi [5], Merkouris [6], and Ghahramani and Thavaneswaran [7] among others have studied the estimation problems using estimating functions. In this paper, we study the linear and quadratic martingale estimating functions and show that the quadratic estimating functions are more informative when the conditional mean and variance of the observed process depend on the same parameter of interest.

This paper is organized as follows. The rest of Section 1 presents the basics of estimating functions and information associated with estimating functions. Section 2 presents the general model for the multiparameter case and the form of the optimal quadratic estimating function. In Section 3, the theory is applied to four different models.

Suppose that {𝐲_𝑡,𝑡=1,…,𝑛} is a realization of a discrete time stochastic process, and its distribution depends on a vector parameter 𝜽 belonging to an open subset Θ of the 𝑝-dimensional Euclidean space. Let (Ω,ℱ,𝑃_𝜽) denote the underlying probability space, and let ℱ^𝑦𝑡 be the 𝜎-field generated by {𝐲₁,…,𝐲_𝑡,𝑡≥1}. Let 𝐡_𝑡=𝐡_𝑡(𝐲₁,…,𝐲_𝑡,𝜽), 1≤𝑡≤𝑛 be specified 𝑞-dimensional vectors that are martingales. We consider the class ℳ of zero mean and square integrable 𝑝-dimensional martingale estimating functions of the formℳ=𝐠_𝑛(𝜽)∶𝐠_𝑛(𝜽)=^𝑛_𝑡=1𝐚_𝑡−1𝐡_𝑡,(1.1) where 𝐚_𝑡−1 are 𝑝×𝑞 matrices depending on 𝐲₁,…,𝐲_𝑡−1, 1≤𝑡≤𝑛. The estimating functions 𝐠_𝑛(𝜽) are further assumed to be almost surely differentiable with respect to the components of 𝜽 and such that E[(𝜕𝐠_𝑛(𝜽)/𝜕𝜽)∣ℱ^𝑦𝑛−1] and E[𝐠_𝑛(𝜽)𝐠_𝑛(𝜽)^∣ℱ^𝑦𝑛−1] are nonsingular for all 𝜽∈Θ and for each 𝑛≥1. The expectations are always taken with respect to 𝑃_𝜽. Estimators of 𝜽 can be obtained by solving the estimating equation 𝐠_𝑛(𝜽)=𝟎. Furthermore, the 𝑝×𝑝 matrix E[𝐠_𝑛(𝜽)𝐠_𝑛(𝜽)^∣ℱ^𝑦𝑛−1] is assumed to be positive definite for all 𝜽∈Θ. Then, in the class of all zero mean and square integrable martingale estimating functions ℳ, the optimal estimating function 𝐠^∗𝑛(𝜽) which maximizes, in the partial order of nonnegative definite matrices, the information matrix𝐈_𝐠𝑛(𝜽)=E𝜕𝐠^𝑛(𝜽)𝜕𝜽∣ℱ^𝑦𝑛−1^E𝐠_𝑛(𝜽)𝐠_𝑛(𝜽)^∣ℱ^𝑦𝑛−1⁻¹E𝜕𝐠^𝑛(𝜽)𝜕𝜽∣ℱ^𝑦𝑛−1(1.2) is given by𝐠^∗𝑛(𝜽)=^𝑛_𝑡=1𝐚^∗𝑡−1𝐡_𝑡=^𝑛_𝑡=1E𝜕𝜕𝐡𝜽^𝑡∣ℱ^𝑦𝑡−1^E𝐡_𝑡𝐡^𝑡∣ℱ^𝑦𝑡−1⁻¹𝐡_𝑡,(1.3) and the corresponding optimal information reduces to E[𝐠^∗𝑛(𝜽)𝐠^∗𝑛(𝜽)^∣ℱ^𝑦𝑛−1].

The function 𝐠^∗𝑛(𝜽) is also called the “quasi-score” and has properties similar to those of a score function in the sense that E[𝐠^∗𝑛(𝜽)]=𝟎 and E[𝐠^∗𝑛(𝜽)𝐠^∗𝑛(𝜽)^]=−E[𝜕𝐠^∗𝑛(𝜽)/𝜕𝜽^]. This is a more general result in the sense that for its validity, we do not need to assume that the true underlying distribution belongs to the exponential family of distributions. The maximum correlation between the optimal estimating function and the true unknown score justifies the terminology “quasi-score” for 𝐠^∗𝑛(𝜽). Moreover, it follows from Lindsay [8, page 916] that if we solve an unbiased estimating equation 𝐠_𝑛(𝜽)=𝟎 to get an estimator, then the asymptotic variance of the resulting estimator is the inverse of the information 𝐈_𝐠𝑛. Hence, the estimator obtained from a more informative estimating equation is asymptotically more efficient.

2. General Model and Method

Consider a discrete time stochastic process {𝑦_𝑡,𝑡=1,2,…} with conditional moments𝜎𝜇_𝑡2𝑡(𝜽)=E𝑦_𝑡∣ℱ^𝑦𝑡−1,𝛾_𝑡(𝜽)=Var𝑦_𝑡∣ℱ^𝑦𝑡−1,(𝜽)=1𝜎^3𝑡(𝜽)E𝑦^𝑡𝜅_𝑡−𝜇_𝑡(𝜽)³∣ℱ^𝑦𝑡−1,(𝜽)=1𝜎^4𝑡(𝜽)E𝑦^𝑡−𝜇_𝑡(𝜽)⁴∣ℱ^𝑦𝑡−1−3.(2.1) That is, we assume that the skewness and the excess kurtosis of the standardized variable 𝑦_𝑡 do not contain any additional parameters. In order to estimate the parameter 𝜽 based on the observations 𝑦₁,…,𝑦_𝑛, we consider two classes of martingale differences {𝑚_𝑡(𝜽)=𝑦_𝑡−𝜇_𝑡(𝜽),𝑡=1,…,𝑛} and {𝑠_𝑡(𝜽)=𝑚^2𝑡(𝜽)−𝜎^2𝑡(𝜽),𝑡=1,…,𝑛} such that⟨𝑠⟩_𝑡⟨𝑚⟩_𝑡=E𝑚^2𝑡∣ℱ^𝑦𝑡−1=E𝑦_𝑡−𝜇_𝑡²∣ℱ^𝑦𝑡−1=𝜎^2𝑡,⟨=𝑚E,𝑠⟩𝑠^2𝑡_𝑡∣ℱ^𝑦𝑡−1=E𝑦_𝑡−𝜇_𝑡⁴+𝜎^4𝑡−2𝜎^2𝑡𝑦_𝑡−𝜇_𝑡²∣ℱ^𝑦𝑡−1=𝜎^4𝑡𝜅_𝑡+2,=E𝑚_𝑡𝑠_𝑡∣ℱ^𝑦𝑡−1=E𝑦_𝑡−𝜇_𝑡³−𝜎^2𝑡𝑦_𝑡−𝜇_𝑡∣ℱ^𝑦𝑡−1=𝜎^3𝑡𝛾_𝑡.(2.2)

The optimal estimating functions based on the martingale differences 𝑚_𝑡 and 𝑠_𝑡 are 𝐠^∗𝑀(𝜽)=−∑^𝑛𝑡=1(𝜕𝜇_𝑡/𝜕𝜽)(𝑚_𝑡/⟨𝑚⟩_𝑡) and 𝐠^∗𝑆(𝜽)=−∑^𝑛𝑡=1(𝜕𝜎^2𝑡/𝜕𝜽)(𝑠_𝑡/⟨𝑠⟩_𝑡), respectively. Then, the information associated with 𝐠^∗𝑀(𝜽) and 𝐠^∗𝑆(𝜽) are 𝐈_𝐠^∗𝑀(𝜽)=∑^𝑛𝑡=1(𝜕𝜇_𝑡/𝜕𝜽)(𝜕𝜇_𝑡/𝜕𝜽^)(1/⟨𝑚⟩_𝑡) and 𝐈_𝐠^∗𝑆(𝜽)=∑^𝑛𝑡=1(𝜕𝜎^2𝑡/𝜕𝜽)(𝜕𝜎^2𝑡/𝜕𝜽^)(1/⟨𝑠⟩_𝑡), respectively. Crowder [9] studied the optimal quadratic estimating function with independent observations. For the discrete time stochastic process {𝑦_𝑡}, the following theorem provides optimality of the quadratic estimating function for the multiparameter case.

Theorem 2.1. For the general model in (2.1), in the class of all quadratic estimating functions of the form 𝒢_𝑄={𝐠_𝑄(𝜽)∶𝐠_𝑄(𝜽)=∑^𝑛𝑡=1(𝐚_𝑡−1𝑚_𝑡+𝐛_𝑡−1𝑠_𝑡)}, (a)the optimal estimating function is given by 𝐠^∗𝑄(𝜽)=∑^𝑛𝑡=1(𝐚^∗𝑡−1𝑚_𝑡+𝐛^∗𝑡−1𝑠_𝑡), where𝐚^∗𝑡−1=1−⟨𝑚,𝑠⟩^2𝑡⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡⁻¹−𝜕𝜕𝜇𝜽^𝑡⟨𝑚1⟩_𝑡+𝜕𝜎^2𝑡𝐛_∗𝑡−1𝜕𝜽⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡,=1−⟨𝑚,𝑠⟩^2𝑡⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡⁻¹𝜕𝜕𝜇𝜽^𝑡⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡−𝜕𝜎^2𝑡𝜕𝜽⟨𝑠1⟩_𝑡;(2.3)(b) the information 𝐈_𝑔^∗𝑄(𝜽) is given by 𝐈_𝐠^∗𝑄(𝜽)=^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹𝜕𝜕𝜇𝜽^𝑡𝜕𝜕𝜇𝜽^𝑡1⟨𝑚⟩_𝑡+𝜕𝜎^2𝑡𝜕𝜽𝜕𝜕𝜎𝜽^2𝑡1⟨𝑠⟩_𝑡−𝜕𝜇^𝑡𝜕𝜽𝜕𝜕𝜎𝜽^2𝑡+𝜕𝜎^2𝑡𝜕𝜽𝜕𝜕𝜇𝜽^𝑡⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡;(2.4)(c) the gain in information 𝐈_𝐠^∗𝑄(𝜽)−𝐈_𝐠^∗𝑀(𝜽) is given by ^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹𝜕𝜕𝜇𝜽^𝑡𝜕𝜕𝜇𝜽^𝑡⟨𝑚,𝑠⟩^2𝑡⟨𝑚⟩^2𝑡⟨𝑠⟩_𝑡+𝜕𝜎^2𝑡𝜕𝜽𝜕𝜕𝜎𝜽^2𝑡1⟨𝑠⟩_𝑡−𝜕𝜇^𝑡𝜕𝜽𝜕𝜕𝜎𝜽^2𝑡+𝜕𝜎^2𝑡𝜕𝜽𝜕𝜕𝜇𝜽^𝑡⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡;(2.5)(d) the gain in information 𝐈_𝐠^∗𝑄(𝜽)−𝐈_𝐠^∗𝑆(𝜽) is given by ^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹𝜕𝜕𝜇𝜽^𝑡𝜕𝜕𝜇𝜽^𝑡1⟨𝑚⟩_𝑡+𝜕𝜎^2𝑡𝜕𝜽𝜕𝜕𝜎𝜽^2𝑡⟨𝑚,𝑠⟩^2𝑡⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡−⟨𝑚,𝑠⟩^2𝑡−𝜕𝜇^𝑡𝜕𝜽𝜕𝜕𝜎𝜽^2𝑡+𝜕𝜎^2𝑡𝜕𝜽𝜕𝜕𝜇𝜽^𝑡⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡.(2.6)

Proof. We choose two orthogonal martingale differences 𝑚_𝑡 and 𝜓_𝑡=𝑠_𝑡−𝜎_𝑡𝛾_𝑡𝑚_𝑡, where the conditional variance of 𝜓_𝑡 is given by ⟨𝜓⟩_𝑡=(⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡−⟨𝑚,𝑠⟩^2𝑡)/⟨𝑚⟩_𝑡=𝜎^4𝑡(𝜅_𝑡+2−𝛾^2𝑡). That is, 𝑚_𝑡 and 𝜓_𝑡 are uncorrelated with conditional variance ⟨𝑚⟩_𝑡 and ⟨𝜓⟩_𝑡, respectively. Moreover, the optimal martingale estimating function and associated information based on the martingale differences 𝜓_𝑡 are 𝐠^∗Ψ(𝜽)=^𝑛_𝑡=1𝜕𝜕𝜇𝜽^𝑡⟨𝑚⟨𝑚,𝑠⟩⟩^𝑡_𝑡−𝜕𝜎^2𝑡𝜕𝜽⟨𝜓𝜓⟩^𝑡_𝑡=^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹×−𝜕𝜇^𝑡𝜕𝜽⟨𝑚⟨𝑚⟩,^2𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡+𝜕𝜎^2𝑡𝜕𝜽⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡𝑚^𝑡+𝜕𝜇^𝑡𝜕𝜽⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡−𝜕𝜎^2𝑡𝐈_𝐠∗Ψ𝜕𝜽⟨𝑠1⟩_𝑡𝑠^𝑡,(𝜽)=^𝑛_𝑡=1𝜕𝜕𝜇𝜽^𝑡⟨𝑚⟨𝑚,𝑠⟩⟩^𝑡_𝑡−𝜕𝜎^2𝑡𝜕𝜽𝜕𝜕𝜇𝜽^𝑡⟨𝑚,𝑠⟩_𝑡⟨𝑚⟩_𝑡−𝜕𝜎^2𝑡𝜕𝜽^⟨𝜓1⟩_𝑡=^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹×𝜕𝜇^𝑡𝜕𝜽𝜕𝜕𝜇𝜽^𝑡⟨𝑚,𝑠⟩^2𝑡⟨𝑚⟩^2𝑡⟨𝑠⟩_𝑡+𝜕𝜎^2𝑡𝜕𝜽𝜕𝜕𝜎𝜽^2𝑡1⟨𝑠⟩_𝑡−𝜕𝜇^𝑡𝜕𝜽𝜕𝜕𝜎𝜽^2𝑡+𝜕𝜎^2𝑡𝜕𝜽𝜕𝜕𝜇𝜽^𝑡⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡.(2.7) Then, the quadratic estimating function based on 𝑚_𝑡 and 𝜓_𝑡 becomes 𝐠^∗𝑄(𝜽)=^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹×−𝜕𝜇^𝑡𝜕𝜽⟨𝑚1⟩_𝑡+𝜕𝜎^2𝑡𝜕𝜽⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡𝑚^𝑡+𝜕𝜇^𝑡𝜕𝜽⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡−𝜕𝜎^2𝑡𝜕𝜽⟨𝑠1⟩_𝑡𝑠^𝑡(2.8) and satisfies the sufficient condition for optimality E𝜕𝐠^𝑄(𝜽)𝜕𝜽∣ℱ^𝑦𝑡−1=Cov𝐠^𝑄(𝜽),𝐠^∗𝑄(𝜽)∣ℱ^𝑦𝑡−1𝐾,∀𝐠^𝑄(𝜽)∈𝒢_𝑄,(2.9) where 𝐾 is a constant matrix. Hence, 𝐠^∗𝑄(𝜽) is optimal in the class 𝒢_𝑄, and part (a) follows. Since 𝑚_𝑡 and 𝜓_𝑡 are orthogonal, the information 𝐈_𝐠^∗𝑄(𝜽)=𝐈_𝐠^∗𝑀(𝜽)+𝐈_𝐠^∗Ψ(𝜽) and part (b) follow. Hence, for each component 𝜃_𝑖, 𝑖=1,…,𝑝, neither 𝑔^∗𝑀(𝜃_𝑖) nor 𝑔^∗𝑆(𝜃) is fully informative, that is, 𝐼_𝑔^∗𝑄(𝜃_𝑖)≥𝐼_𝑔^∗𝑀(𝜃_𝑖) and 𝐼_𝑔^∗𝑄(𝜃_𝑖)≥𝐼_𝑔^∗𝑆(𝜃_𝑖).

Corollary 2.2. When the conditional skewness 𝛾 and kurtosis 𝜅 are constants, the optimal quadratic estimating function and associated information, based on the martingale differences 𝑚_𝑡=𝑦_𝑡−𝜇_𝑡 and 𝑠_𝑡=𝑚^2𝑡−𝜎^2𝑡, are given by 𝐠^∗𝑄(𝜽)=1−𝛾²𝜅+2^𝑛−1_𝑡=11𝜎^3𝑡−𝜎^𝑡𝜕𝜇_𝑡𝜕𝜽+𝜅+𝛾2𝜕𝜕𝜎𝜽^2𝑡𝑚^𝑡+1𝜅+2𝛾𝜕𝜕𝜇𝜽^𝑡−𝜎1_𝑡𝜕𝜎^2𝑡𝐈_𝐠∗𝑄𝜕𝜽𝑠^𝑡,(𝜽)=1−𝛾²𝜅+2⁻¹𝐈^𝐠∗𝑀(𝜽)+𝐈_𝐠^∗𝑆(𝜽)−𝛾𝜅+2^𝑛_𝑡=11𝜎^3𝑡𝜕𝜕𝜇𝜽^𝑡𝜕𝜕𝜎𝜽^2𝑡+𝜕𝜎^2𝑡𝜕𝜽𝜕𝜕𝜇𝜽^𝑡.(2.10)

3. Applications

3.1. Autoregressive Conditional Duration (ACD) Models

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 [10] had been proposed to model such durations, in order to study the dynamic structure of the adjusted durations 𝑥_𝑖, with 𝑥_𝑖=𝑡_𝑖−𝑡_𝑖−1, where 𝑡_𝑖 is the time of the 𝑖th transaction. The crucial assumption underlying the ACD model is that the time dependence is described by a function 𝜓_𝑖, where 𝜓_𝑖 is the conditional expectation of the adjusted duration between the (𝑖−1)th and the 𝑖th trades. The basic ACD model is defined as 𝜓_𝑖𝑥_𝑖=𝜓_𝑖𝜀_𝑖,=E𝑥_𝑖∣ℱ^𝑥𝑡_𝑖−1,(3.1) where 𝜀_𝑖 are the iid nonnegative random variables with density function 𝑓(⋅) and unit mean, and ℱ^𝑥𝑡_𝑖−1 is the information available at the (𝑖−1)th trade. We also assume that 𝜀_𝑖 is independent of ℱ^𝑥𝑡−1. It is clear that the types of ACD models vary according to different distributions of 𝜀_𝑖 and specifications of 𝜓_𝑖. In this paper, we will discuss a specific class of models which is known as ACD (𝑝, 𝑞) model and given by 𝜓_𝑡𝑥_𝑡=𝜓_𝑡𝜀_𝑡,=𝜔+^𝑝_𝑗=1𝑎_𝑗𝑥_𝑡−𝑗+^𝑞_𝑗=1𝑏_𝑗𝜓_𝑡−𝑗,(3.2) where 𝜔>0, 𝑎_𝑗>0, 𝑏_𝑗>0, and ∑^{max(𝑝,𝑞)𝑗=1}(𝑎_𝑗+𝑏_𝑗)<1. We assume that 𝜀_𝑡's are iid nonnegative random variables with mean 𝜇_𝜀, variance 𝜎^2𝜀, skewness 𝛾_𝜀, and excess kurtosis 𝜅_𝜀. In order to estimate the parameter vector 𝜽=(𝜔,𝑎₁,…,𝑎_𝑝,𝑏₁,…,𝑏_𝑞)^, we use the estimating function approach. For this model, the conditional moments are 𝜇_𝑡=𝜇_𝜀𝜓_𝑡, 𝜎^2𝑡=𝜎^2𝜀𝜓^2𝑡, 𝛾_𝑡=𝛾_𝜀, and 𝜅_𝑡=𝜅_𝜀. Let 𝑚_𝑡=𝑥_𝑡−𝜇_𝑡 and 𝑠_𝑡=𝑚^2𝑡−𝜎^2𝑡 be the sequences of martingale differences such that ⟨𝑚⟩_𝑡=𝜎^2𝜀𝜓^2𝑡, ⟨𝑠⟩_𝑡=𝜎^4𝜀(𝜅_𝜀+2)𝜓^4𝑡, and ⟨𝑚,𝑠⟩_𝑡=𝜎^3𝜀𝛾_𝜀𝜓^3𝑡. The optimal estimating function and associated information based on 𝑚_𝑡 are given by 𝐠^∗𝑀(𝜽)=−(𝜇_𝜀/𝜎^2𝜀)∑^𝑛𝑡=1(1/𝜓^2𝑡)(𝜕𝜓_𝑡/𝜕𝜽)𝑚_𝑡 and 𝐈_𝐠^∗𝑀(𝜽)=(𝜇^2𝜀/𝜎^2𝜀)∑^𝑛𝑡=1(1/𝜓^2𝑡)(𝜕𝜓_𝑡/𝜕𝜽)(𝜕𝜓_𝑡/𝜕𝜽^). The optimal estimating function and the associated information based on 𝑠_𝑡 are given by 𝐠^∗𝑆(𝜽)=−2/𝜎^2𝜀(𝜅_𝜀+2)∑^𝑛𝑡=1(1/𝜓^3𝑡)(𝜕𝜓_𝑡/𝜕𝜽)𝑠_𝑡 and 𝐈_𝐠^∗𝑆(𝜽)=(4/(𝜅_𝜀+2))∑^𝑛𝑡=1(1/𝜓^2𝑡)(𝜕𝜓_𝑡/𝜕𝜽)(𝜕𝜓_𝑡/𝜕𝜽^). Then, by Corollary 2.2 that the optimal quadratic estimating function and associated information are given by 𝐠^∗𝑄(𝜽)=1𝜎^2𝜀𝜅_𝜀+2−𝛾^2𝜀^𝑛_𝑡=1−𝜇^𝜀𝜅_𝜀+2+2𝜎_𝜀𝛾_𝜀𝜓^2𝑡𝜕𝜓_𝑡𝜕𝜽𝑚^𝑡+𝜇^𝜀𝛾_𝜀−2𝜎_𝜀𝜓_𝑡𝜎_𝜀𝜓^3𝑡𝜕𝜓_𝑡𝐈_𝐠∗𝑄𝜕𝜽𝑠^𝑡,(𝜽)=1−𝛾^2𝜀𝜅_𝜀+2⁻¹𝐈^𝐠∗𝑀(𝜽)+𝐈_𝐠^∗𝑆(𝜽)−4𝜇^𝜀𝛾_𝜀𝜎_𝜀𝜅_𝜀+2^𝑛_𝑡=11𝜓^2𝑡𝜕𝜓_𝑡=4𝜎_2𝜀𝜕𝜽𝜕𝜕𝜓𝜽^𝑡+𝜇^2𝜀𝜅_𝜀+2−4𝜇_𝜀𝜎_𝜀𝛾_𝜀𝜎^2𝜀𝜅_𝜀+2−𝛾^2𝜀^𝑛_𝑡=11𝜓^2𝑡𝜕𝜓_𝑡𝜕𝜽𝜕𝜕𝜓𝜽^𝑡,(3.3) the information gain in using 𝐠^∗𝑄(𝜽) over 𝐠^∗𝑀(𝜽) is2𝜎_𝜀−𝜇_𝜀𝛾_𝜀²𝜎^2𝜀𝜅_𝜀+2−𝛾^2𝜀^𝑛_𝑡=11𝜓^2𝑡𝜕𝜓_𝑡𝜕𝜽𝜕𝜕𝜓𝜽^𝑡,(3.4) and the information gain in using 𝐠^∗𝑄(𝜽) over 𝐠^∗𝑆(𝜽) is𝜇_𝜀𝜅_𝜀+2−2𝜎_𝜀𝛾_𝜀²𝜎^2𝜀𝜅_𝜀+2−𝛾^2𝜀𝜅_𝜀+2^𝑛_𝑡=11𝜓^2𝑡𝜕𝜓_𝑡𝜕𝜽𝜕𝜕𝜓𝜽^𝑡,(3.5) which are both nonnegative definite.

When 𝜀_𝑡 follows an exponential distribution, 𝜇_𝜀=1/𝜆, 𝜎^2𝜀=1/𝜆², 𝛾_𝜀=2, and 𝜅_𝜀=3. Then, 𝐈_𝐠^∗𝑀(𝜽)=∑^𝑛𝑡=1(1/𝜓^2𝑡)(𝜕𝜓_𝑡/𝜕𝜽)(𝜕𝜓_𝑡/𝜕𝜽^), 𝐈_𝐠^∗𝑆(𝜽)=(4/5)∑^𝑛𝑡=1(1/𝜓^2𝑡)(𝜕𝜓_𝑡/𝜕𝜽)(𝜕𝜓_𝑡/𝜕𝜽^), and 𝐈_𝐠^∗𝑄(𝜽)=∑^𝑛𝑡=1(1/𝜓^2𝑡)(𝜕𝜓_𝑡/𝜕𝜽)(𝜕𝜓_𝑡/𝜕𝜽^), and hence 𝐈_𝐠^∗𝑄(𝜽)=𝐈_𝐠^∗𝑀(𝜽)>𝐈_𝐠^∗𝑆(𝜽).

3.2. Random Coefficient Autoregressive Models

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 [11].

Consider the RCA model𝑦_𝑡=𝜃+𝑏_𝑡𝑦_𝑡−1+𝜀_𝑡,(3.6) where {𝑏_𝑡} and {𝜀_𝑡} are uncorrelated zero mean processes with unknown variance 𝜎^2𝑏 and variance 𝜎^2𝜀=𝜎^2𝜀(𝜃) with unknown parameter 𝜃, respectively. Further, we denote the skewness and excess kurtosis of {𝑏_𝑡} by 𝛾_𝑏, 𝜅_𝑏 which are known, and of {𝜀_𝑡} by 𝛾_𝜀(𝜃), 𝜅_𝜀(𝜃), respectively. In the model (3.6), both the parameter 𝜃 and 𝛽=𝜎^2𝑏 need to be estimated. Let 𝜽=(𝜃,𝛽)^, we will discuss the joint estimation of 𝜃 and 𝛽. In this model, the conditional mean is 𝜇_𝑡=𝑦_𝑡−1𝜃 then and the conditional variance is 𝜎^2𝑡=𝑦^2𝑡−1𝛽+𝜎^2𝜀(𝜃). The parameter 𝜃 appears simultaneously in the mean and variance. Let 𝑚_𝑡=𝑦_𝑡−𝜇_𝑡 and 𝑠_𝑡=𝑚^2𝑡−𝜎^2𝑡 such that ⟨𝑚⟩_𝑡=𝑦^2𝑡−1𝜎^2𝑏+𝜎^2𝜀, ⟨𝑠⟩_𝑡=𝑦^4𝑡−1𝜎^4𝑏(𝜅_𝑏+2)+𝜎^4𝜀(𝜅_𝜀+2)+4𝑦^2𝑡−1𝜎^2𝑏𝜎^2𝜀, ⟨𝑚,𝑠⟩_𝑡=𝑦^3𝑡−1𝜎^3𝑏𝛾_𝑏+𝜎^3𝜀𝛾_𝜀. Then the conditional skewness is 𝛾_𝑡=⟨𝑚,𝑠⟩_𝑡/𝜎^3𝑡, and the conditional excess kurtosis is 𝜅_𝑡=⟨𝑠⟩_𝑡/𝜎^4𝑡−2.

Since 𝜕𝜇_𝑡/𝜕𝜽=(𝑦_𝑡−1,0)^ and 𝜕𝜎^2𝑡/𝜕𝜽=(𝜕𝜎^2𝜀/𝜕𝜃,𝑦^2𝑡−1)^, by applying Theorem 2.1, the optimal quadratic estimating function for 𝜃 and 𝛽 based on the martingale differences 𝑚_𝑡 and 𝑠_𝑡 is given by 𝐠^∗𝑄(𝜽)=∑^𝑛𝑡=1𝐚^∗𝑡−1𝑚_𝑡+𝐛^∗𝑡−1𝑠_𝑡, where𝐚^∗𝑡−1=1−⟨𝑚,𝑠⟩^2𝑡⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡⁻¹−⟨𝑦𝑚⟩^𝑡−1_𝑡+𝜕𝜎^2𝜀𝜕𝜃⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡,𝑦^2𝑡−1⟨𝑚,𝑠⟩_𝑡⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡^𝐛_∗𝑡−1,=1−⟨𝑚,𝑠⟩^2𝑡⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡⁻¹𝑦^𝑡−1⟨𝑚,𝑠⟩_𝑡⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡−𝜕𝜎^2𝜀𝜕𝜃⟨𝑠1⟩_𝑡,−⟨𝑦𝑠⟩^2𝑡−1_𝑡^.(3.7) Hence, the component quadratic estimating function for 𝜃 is𝑔^∗𝑄(𝜃)=^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹×−𝑦^𝑡−1⟨𝑚⟩_𝑡+𝜕𝜎^2𝜀𝜕𝜃⟨𝑚⟨𝑚⟩,_𝑡𝑠⟨𝑠⟩⟩^𝑡_𝑡𝑚^𝑡+𝑦^𝑡−1⟨𝑚,𝑠⟩_𝑡⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡−𝜕𝜎^2𝜀𝜕𝜃⟨𝑠1⟩_𝑡𝑠^𝑡,(3.8) and the component quadratic estimating function for 𝛽 is𝑔^∗𝑄(𝛽)=^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹𝑦^2𝑡−1⟨𝑚,𝑠⟩_𝑡𝑚_𝑡⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡−𝑦^2𝑡−1𝑠_𝑡⟨𝑠⟩_𝑡.(3.9) Moreover, the information matrix of the optimal quadratic estimating function for 𝜃 and 𝛽 is given by𝐈_g^∗𝑄(𝜽)=𝐼^𝜃𝜃𝐼_𝜃𝛽𝐼_𝛽𝜃𝐼_𝛽𝛽,(3.10) where𝐼_𝜃𝜃=^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹⟨𝑦𝑚⟩^2𝑡−1_𝑡+𝜕𝜎^2𝜀𝜕𝜃²1⟨𝑠⟩_𝑡−2𝜕𝜎^2𝜀𝜕𝜃𝑦^𝑡−1⟨𝑚,𝑠⟩_𝑡𝐼⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡,(3.11)_𝜃𝛽=𝐼_𝛽𝜃=^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹𝜕𝜕𝜎𝜃^2𝜀⟨𝑠1⟩_𝑡−𝑦^𝑡−1⟨𝑚,𝑠⟩_𝑡⟨𝑚⟩_𝑡⟨𝑠⟩_𝑡𝑦^2𝑡−1𝐼,(3.12)_𝛽𝛽=^𝑛_𝑡=11−⟨⟨𝑚𝑚⟩,_𝑡𝑠⟨⟩𝑠⟩^2𝑡_𝑡⁻¹𝑦^4𝑡−1⟨𝑠⟩_𝑡.(3.13)

In view of the parameter 𝜃 only, the conditional least squares (CLS) estimating function and the associated information are directly given by 𝑔_CLS(𝜃)=∑^𝑛𝑡=1𝑦_𝑡−1𝑚_𝑡 and 𝐼_CLS(𝜃)=(∑^𝑛𝑡=1𝑦^2𝑡−1)²/∑^𝑛𝑡=1𝑦^2𝑡−1⟨𝑚⟩_𝑡. The optimal martingale estimating function and the associated information based on 𝑚_𝑡 are given by 𝑔^∗𝑀(𝜃)=−∑^𝑛𝑡=1(𝑦_𝑡−1𝑚_𝑡/⟨𝑚⟩_𝑡) and 𝐼_𝑔^∗𝑀(𝜃)=∑^𝑛𝑡=1(𝑦^2𝑡−1/⟨𝑚⟩_𝑡). Moreover, the inequality^𝑛_𝑡=1𝑦^2𝑡−1⟨𝑚⟩_𝑡^𝑛_𝑡=1𝑦^2𝑡−1⟨𝑚⟩_𝑡≥^𝑛_𝑡=1𝑦^2𝑡−1²(3.14) implies that 𝐼_CLS(𝜃)≤𝐼_𝑔^∗𝑀(𝜃). Hence the optimal estimating function is more informative than the conditional least squares one. The optimal quadratic estimating function based on the martingale differences 𝑚_𝑡 and 𝑠_𝑡 is given by (3.8) and (3.11), respectively. It is obvious to see that the information of 𝑔^∗𝑄(𝜃) is larger than that of 𝑔^∗𝑀(𝜃). Therefore, we can conclude that for the RCA model, 𝐼_CLS(𝜃)≤𝐼_𝑔^∗𝑀(𝜃)≤𝐼_𝑔^∗𝑄(𝜃), and hence, the estimate obtained by solving the optimal quadratic estimating equation is more efficient than the CLS estimate and the estimate obtained by solving the optimal linear estimating equation.

3.3. Doubly Stochastic Time Series Model

Random coefficient autoregressive models we discussed in the previous section are special cases of what Tjøstheim [12] refers to as doubly stochastic time series model. In the nonlinear case, these models are given by𝑦_𝑡=𝜃_𝑡𝑓𝑡,ℱ^𝑦𝑡−1+𝜀_𝑡,(3.15) where {𝜃+𝑏_𝑡} of (3.6) is replaced by a more general stochastic sequence {𝜃_𝑡} and 𝑦_𝑡−1 is replaced by a function of the past, ℱ^𝑦𝑡−1. Suppose that {𝜃_𝑡} is a moving average sequence of the form𝜃_𝑡=𝜃+𝑎_𝑡+𝑎_𝑡−1,(3.16) where {𝑎_𝑡} consists of square integrable independent random variables with mean zero and variance 𝜎^2𝑎. We further assume that {𝜀_𝑡} and {𝑎_𝑡} are independent, then E[𝑦_𝑡∣ℱ^𝑦𝑡−1] depends on the posterior mean 𝑢_𝑡=E[𝑎_𝑡∣ℱ^𝑦𝑡−1], and variance 𝑣_𝑡=E[(𝑎_𝑡−𝑢_𝑡)²∣ℱ^𝑦𝑡−1] of 𝑎_𝑡. Under the normality assumption of {𝜀_𝑡} and {𝑎_𝑡}, and the initial condition 𝑦₀=0, 𝑢_𝑡 and 𝑣_𝑡 satisfy the following Kalman-like recursive algorithms (see [13, page 439]):𝑢_𝑡(𝜃)=𝜎^2𝑎𝑓𝑡,ℱ^𝑦𝑡−1𝑦_𝑡−𝜃+𝑚_𝑡−1𝑓𝑡,ℱ^𝑦𝑡−1𝑣_𝑡𝜎^2𝑒(𝜃)+𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎+𝑣_𝑡−1,(𝜃)=𝜎^2𝑎−𝜎^4𝑎𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑒(𝜃)+𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎+𝑣_𝑡−1,(3.17) where 𝑢₀=0 and 𝑣₀=𝜎^2𝑎. Hence, the conditional mean and variance of 𝑦_𝑡 are given by𝜎_2𝑡𝜇_𝑡(𝜃)=𝜃+𝑢_𝑡−1(𝜃)𝑓𝑡,ℱ^𝑦𝑡−1,(𝜃)=𝜎^2𝑒(𝜃)+𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎+𝑣_𝑡−1(𝜃),(3.18) which can be computed recursively.

Let 𝑚_𝑡=𝑦_𝑡−𝜇_𝑡 and 𝑠_𝑡=𝑚^2𝑡−𝜎^2𝑡, then {𝑚_𝑡} and {𝑠_𝑡} are sequences of martingale differences. We can derive that ⟨𝑚,𝑠⟩_𝑡=0, ⟨𝑚⟩_𝑡=𝜎^2𝑒(𝜃)+𝑓²(𝑡,ℱ^𝑦𝑡−1)(𝜎^2𝑎+𝑣_𝑡−1(𝜃)), and ⟨𝑠⟩_𝑡=2𝜎^4𝑒(𝜃)+4𝑓²(𝑡,ℱ^𝑦𝑡−1)𝜎^2𝑒(𝜃)(𝜎^2𝑎+𝑣_𝑡−1(𝜃))+2𝑓⁴(𝑡,ℱ^𝑦𝑡−1)(𝜎^2𝑎+𝑣_𝑡−1(𝜃))². The optimal estimating function and associated information based on 𝑚_𝑡 are given by𝑔^∗𝑀(𝜃)=−^𝑛_𝑡=1𝑓𝑡,ℱ^𝑦𝑡−11+𝜕𝑢^𝑡−1(𝜃)𝜕𝜃⟨𝑚𝑚⟩^𝑡_𝑡𝐼_𝑔∗𝑀,(𝜃)=^𝑛_𝑡=1𝑓²𝑡,ℱ^𝑦𝑡−11+𝜕𝑢_𝑡−1(𝜃)/𝜕𝜃²⟨𝑚⟩_𝑡.(3.19) Then, the inequality ^𝑛_𝑡=1𝑓²𝑡,ℱ^𝑦𝑡−11+𝜕𝑢^𝑡−1(𝜃)𝜕𝜃²⟨𝑚⟩_𝑡^𝑛_𝑡=1𝑓²𝑡,ℱ^𝑦𝑡−11+𝜕𝑢_𝑡−1(𝜃)/𝜕𝜃²≥_𝑛⟨𝑚⟩_𝑡_𝑡=1𝑓²𝑡,ℱ^𝑦𝑡−11+𝜕𝑢^𝑡−1(𝜃)𝜕𝜃²²(3.20) implies that𝐼_CLS(𝜃)=∑^𝑛𝑡=1𝑓²𝑡,ℱ^𝑦𝑡−11+𝜕𝑢_𝑡−1(𝜃)/𝜕𝜃²²∑^𝑛𝑡=1𝑓²𝑡,ℱ^𝑦𝑡−11+𝜕𝑢_𝑡−1(𝜃)/𝜕𝜃²⟨𝑚⟩_𝑡≤𝐼_𝑔^∗𝑀(𝜃),(3.21) that is, the optimal linear estimating function 𝑔^∗𝑀(𝜃) is more informative than the conditional least squares estimating function 𝑔_CLS(𝜃).

The optimal estimating function and the associated information based on 𝑠_𝑡 are given by𝑔^∗𝑆(𝜃)=−^𝑛_𝑡=1𝜕𝜎^2𝑒(𝜃)𝜕𝜃+𝑓²𝑡,ℱ^𝑦𝑡−1𝜕𝑣^𝑡−1(𝜃)𝜕𝜃⟨𝑠𝑠⟩^𝑡_𝑡𝐼_𝑔∗𝑆,(𝜃)=^𝑛_𝑡=1𝜕𝜎^2𝑒(𝜃)𝜕𝜃+𝑓²𝑡,ℱ^𝑦𝑡−1𝜕𝑣^𝑡−1(𝜃)𝜕𝜃²1⟨𝑠⟩_𝑡.(3.22) Hence, by Theorem 2.1, the optimal quadratic estimating function is given by 𝑔^∗𝑄(𝜃)=−^𝑛_𝑡=11×𝜎𝑓^2𝑒(𝜃𝑡,)ℱ+𝑓²_𝑦𝑡−1𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎+𝑣_𝑡−1(𝜃)1+𝜕𝑢^𝑡−1(𝜃)𝜕𝜃𝑚^𝑡+𝜕𝜎^2𝑒(𝜃)/𝜕𝜃+𝑓²𝑡,ℱ^𝑦𝑡−1𝜕𝑣_𝑡−1(𝜃)/𝜕𝜃𝜎^2𝑒(𝜃)+𝑓²𝑡,ℱ^𝑦_t−1𝜎^2𝑎+𝑣_𝑡−1(𝜃)𝑠^𝑡.(3.23) And the associated information, 𝐼_𝑔^∗𝑄(𝜃)=𝐼_𝑔^∗𝑀(𝜃)+𝐼_𝑔^∗𝑆(𝜃), is given by 𝐼_𝑔^∗𝑄(𝜃)=^𝑛_𝑡=11×𝜎𝑓^2𝑒(𝜃)+𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎+𝑣_𝑡−1(𝜃)₂𝑡,ℱ^𝑦𝑡−11+𝜕𝑢^𝑡−1(𝜃)𝜕𝜃²+𝜕𝜎^2𝑒(𝜃)/𝜕𝜃+𝑓²𝑡,ℱ^𝑦𝑡−1𝜕𝑣_𝑡−1(𝜃)/𝜕𝜃²𝜎^2𝑒(𝜃)+𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎+𝑣_𝑡−1(𝜃).(3.24) It is obvious to see that the information of 𝑔^∗𝑄 is larger than that of 𝑔^∗𝑀 and 𝑔^∗𝑆, and hence, the estimate obtained by solving the optimal quadratic estimating equation is more efficient than the CLS estimate and the estimate obtained by solving the optimal linear estimating equation. Moreover, the relations 𝜕𝑢_𝑡(𝜃)𝜕𝜃=−𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎1+𝜕𝑢_𝑡−1(𝜃)/𝜕𝜃𝜎^2𝑒(𝜃)+𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎+𝑣_𝑡−1(𝜃)𝜎^2𝑒(𝜃)+𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎+𝑣_𝑡−1(𝜃)²−𝜎^2𝑎𝑦_𝑡−𝑓𝑡,ℱ^𝑦𝑡−1𝜃+𝑢_𝑡−1(𝜃)𝜕𝜎^2𝑒(𝜃)/𝜕𝜃+𝑓²𝑡,ℱ^𝑦𝑡−1𝜕𝑣_𝑡−1(𝜃)/𝜕𝜃𝜎^2𝑒(𝜃)+𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎+𝑣_𝑡−1(𝜃)²𝜕𝑣_𝑡,(𝜃)𝜕𝜃=𝜎^4𝑎𝑓²𝑡,ℱ^𝑦𝑡−1𝜕𝜎^2𝑒(𝜃)/𝜕𝜃+𝑓²𝑡,ℱ^𝑦𝑡−1𝜕𝑣_𝑡−1(𝜃)/𝜕𝜃𝜎^2𝑒(𝜃)+𝑓²𝑡,ℱ^𝑦𝑡−1𝜎^2𝑎+𝑣_𝑡−1(𝜃)²(3.25) can be applied to calculate the estimating functions and associated information recursively.

3.4. Regression Model with ARCH Errors

Consider a regression model with ARCH (𝑠) errors 𝜀_𝑡 of the form𝑦_𝑡=𝐱_𝐭𝜷+𝜀_𝑡,(3.26) such that E[𝜀_𝑡∣ℱ^𝑦𝑡−1]=0, and Var(𝜀_𝑡∣ℱ^𝑦𝑡−1)=ℎ_𝑡=𝛼₀+𝛼₁𝜀^2𝑡−1+⋯+𝛼_𝑠𝜀^2𝑡−𝑠. In this model, the conditional mean is 𝜇_𝑡=𝐱_𝐭𝜷, the conditional variance is 𝜎^2𝑡=ℎ_𝑡, and the conditional skewness and excess kurtosis are assumed to be constants 𝛾 and 𝜅, respectively. It follows form Theorem 2.1 that the optimal component quadratic estimating function for the parameter vector 𝜽=(𝛽₁,…,𝛽_𝑟,𝛼₀,…,𝛼_𝑠)^=(𝜷^,𝜶^)^ is 𝐠^∗𝑄(𝜷)=1(𝜅+2)1−𝜅𝛾+²2⁻¹×^𝑛_𝑡=11ℎ^2𝑡−ℎ^𝑡(𝜅+2)𝐱_𝐭+2ℎ^𝑡1/2𝛾^𝑠_𝑗=1𝛼_𝑗𝐱_𝐭𝜀_𝑡−𝑗𝑚^𝑡+ℎ^𝑡1/2𝛾𝐱_𝐭−2^𝑠_𝑗=1𝛼_𝑗𝐱_𝐭𝜀_𝑡−𝑗𝐠_∗𝑄𝑠^𝑡,(𝜶)=1(𝜅+2)1−𝜅𝛾+²2⁻¹×^𝑛_𝑡=11ℎ^2𝑡ℎ^𝑡1/2𝛾1,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑝^𝑚_𝑡−^𝑛_𝑡=11,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑝^𝑠_𝑡.(3.27) Moreover, the information matrix for 𝜽=(𝜷^,𝜶^)^ is given by𝐈=1−𝛾²𝜅+2⁻¹𝐈^𝜷𝜷𝐈_𝜷𝜶𝐈_𝜶𝜷𝐈_𝜶𝜶,(3.28) where𝐈_𝜷𝜷=^𝑛_𝑡=1𝐱^𝐭𝐱_𝐭ℎ_𝑡+41,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑠^1,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑠𝐈ℎ^2𝑡(𝜅+2)_𝜷𝜶,=−^𝑛_𝑡=1ℎ^𝑡1/2𝛾_𝑡𝐱_𝐭−2∑^𝑠𝑗=1𝛼_𝑗𝐱_𝐭𝜀_𝑡−𝑗1,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑠𝐈ℎ^2𝑡(𝜅+2)_𝜶𝜷,=𝐼^𝜷𝜶=−^𝑛_𝑡=11,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑠^ℎ^𝑡1/2𝛾𝐱_𝐭−2∑^𝑠𝑗=1𝛼_𝑗𝐱_𝐭𝜀_𝑡−𝑗𝐈ℎ^2𝑡(𝜅+2)_𝜶𝜶,=^𝑛_𝑡=11,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑠^1,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑠ℎ^2𝑡(𝜅+2).(3.29)

It is of interest to note that when {𝜀_𝑡} are conditionally Gaussian such that 𝛾=0, 𝜅=0,𝐸∑^𝑠𝑗=1𝛼_𝑗𝐱_𝐭𝜀_𝑡−𝑗1,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑠ℎ^2𝑡(𝜅+2)=𝟎,(3.30) the optimal quadratic estimating functions for 𝜷 and 𝜶 based on the estimating functions 𝑚_𝑡=𝑦_𝑡−𝐱_𝐭𝛽 and 𝑠_𝑡=𝑚^2𝑡−ℎ_𝑡, are, respectively, given by𝐠^∗𝑄(𝜷)=−^𝑛_𝑡=11ℎ^2𝑡ℎ^𝑡𝐱_𝐭𝑚_𝑡+^𝑛_𝑡=1^𝑠_𝑗=1𝛼_𝑗𝐱_𝐭𝜀_𝑡−𝑗𝐠_∗𝑄𝑠^𝑡,(𝜶)=−^𝑛_𝑡=11ℎ^2𝑡1,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑠^𝑠_𝑡.(3.31) Moreover, the information matrix for 𝜽=(𝜷^,𝜶^)^ in (3.28) has 𝐈_𝜷𝜶=𝐈_𝜶𝜷=𝟎,𝐈_𝜷𝜷=^𝑛_𝑡=1ℎ_𝑡𝐱_𝐭𝐱_𝐭+2∑^𝑠𝑗=1𝛼_𝑗𝐱_𝐭𝜀_𝑡−𝑗∑^𝑠𝑗=1𝛼_𝑗𝐱_𝐭𝜀_𝑡−𝑗ℎ^2𝑡𝐈_𝜶𝜶,=^𝑛_𝑡=11,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑠^1,𝜀^2𝑡−1,…,𝜀^2𝑡−𝑠2ℎ^2𝑡.(3.32)

4. Conclusions

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 [2]. Following Lindsay [8], we conclude that the resulting estimates are more efficient in general. Examples based on ACD models, RCA models, doubly stochastic models, and the regression model with ARCH errors are also discussed in some detail. For RCA models and doubly stochastic models, we have shown the superiority of the approach over the CLS method.