Generally the Likelihood Ratio statistic Λ for standard hypotheses is asymptotically χ 2 distributed, and the Bartlett adjustment improves the χ 2 approximation to its asymptotic distribution in the sense of third-order asymptotics. However, if the parameter of interest is on the boundary of the parameter space, Self and Liang (1987) show that the limiting distribution of Λ is a mixture of χ 2 distributions. For such "nonstandard setting of hypotheses", the present paper develops the third-order asymptotic theory for a class S of test statistics, which includes the Likelihood Ratio, the Wald and the Score statistic, in the case of observations generated from a general stochastic process, providing widely applicable results. In particular, it is shown that Λ is Bartlett adjustable despite its nonstandard asymptotic distribution. Although the other statistics are not Bartlett adjustable, a nonlinear adjustment is provided for them which greatly improves the χ 2 approximation to their distribution and allows a subsequent Bartlett-type adjustment. Numerical studies confirm the benefits of the adjustments on the accuracy and on the power of tests whose statistics belong to S.

CHARN model is a famous and important model in the finance, which includes many financial time series models and can be assumed as the return processes of assets. One of the most fundamental estimators for financial time series models is the conditional least squares CL estimator. However, recently, it was shown that the optimal estimating function estimator G estimator is better than CL estimator for some time series models in the sense of efficiency. In this paper, we examine efficiencies of CL and G estimators for CHARN model and derive the condition that G estimator is asymptotically optimal.

We investigate the Capital Asser Pricing Model (CAPM) with time dimension. By using time series analysis, we discuss the estimation of CAPM when market portfolio and the error process are long-memory process and correlated with each other. We give a sufficient condition for the return of assets in the CAPM to be short memory. In this setting, we propose a two-stage least squares estimator for the regression coefficient and derive the asymptotic distribution. Some numerical studies are given. They show an interesting feature of this model.

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