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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.

In the estimation of portfolios, it is natural to assume that the utility function depends on exogenous variable. From this point of view, in this paper, we develop the estimation under the utility function depending on exogenous variable. To estimate the optimal portfolio, we introduce a function of moments of the return process and cumulant between the return processes and exogenous variable, where the function means a generalized version of portfolio weight function. First, assuming that exogenous variable is a random process, we derive the asymptotic distribution of the sample version of portfolio weight function. Then, an influence of exogenous variable on the return process is illuminated when exogenous variable has a shot noise in the frequency domain. Second, assuming that exogenous variable is nonstochastic, we derive the asymptotic distribution of the sample version of portfolio weight function. Then, an influence of exogenous variable on the return process is illuminated when exogenous variable has a harmonic trend. We also evaluate the influence of exogenous variable on the return process numerically.

This monograph provides the fundamentals of statistical inference for financial engineering and covers some selected methods suitable for analyzing financial time series data. In order to describe the actual financial data, various stochastic processes, e.g. non-Gaussian linear processes, non-linear processes, long-memory processes, locally stationary processes etc. are introduced and their optimal estimation is considered as well. This book also includes several statistical approaches, e.g., discriminant analysis, the empirical likelihood method, control variate method, quantile regression, realized volatility etc., which have been recently developed and are considered to be powerful tools for analyzing the financial data, establishing a new bridge between time series and financial engineering. This book is well suited as a professional reference book on finance, statistics and statistical financial engineering. Readers are expected to have an undergraduate-level knowledge of statistics

We study the estimation of optimal portfolios for a Reserve Fund with an end-of-period target and when the returns of the assets that constitute the Reserve Fund portfolio follow two specifications. In the first one, assets are split into short memory (bonds) and long memory (equity), and the optimality of the portfolio is based on maximizing the Sharpe ratio. In the second, returns follow a conditional heteroskedasticity autoregressive nonlinear model, and we study when the distribution of the innovation vector is heavy-tailed stable. For this specification, we consider appropriate estimation methods, which include bootstrap and empirical likelihood.