Statistical Estimation for CAPM with Long-Memory Dependence

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. 1. Introduction The CAPM is one of the typical models of risk asset’s price on equilibrium market and has been used for pricing individual stocks and portfolios. At first, Markowitz [1] did the groundwork of this model. In his research, he cast the investor’s portfolio selection problem in terms of expected return and variance. Sharpe [2] and Lintner [3] developed Markowitz’s idea for economical implication. Black [4] derived a more general version of the CAPM. In their version, the CAPM is constructed based on the excess of the return of the asset over zero-beta return , where and are the return of the th asset and the market portfolio and is the return of zero-beta portfolio of the market portfolio. Campbell et al. [5] discussed the estimation of CAPM, but in their work they did not discuss the time dimension. However, in the econometric analysis, it is necessary to investigate this model with the time dimension; that is, the model is represented as . Recently from the empirical analysis, it is known that the return of asset follows a short-memory process. But Granger [6] showed that the aggregation of short-memory processes yields long-memory dependence, and it is known that the return of the market portfolio follows a long-memory process. From this point of view, first, we show that the return of the market portfolio and the error process are long-memory dependent and correlated with each other. For the regression model, the most fundamental estimator is the ordinary least squares estimator. However, the dependence of the error process with the explanatory process makes this estimator to be inconsistent. To overcome this difficulty, the instrumental variable method is proposed by use of the instrumental variables which are uncorrelated with the error process and correlated with the explanatory variable. This method was first used by Wright [7], and many researchers developed this method (see Reiers？l [8], Geary [9], etc.). Comprehensive reviews are seen in White [10]. However, the instrumental variable method has