칼만 필터를 이용한 학습 자본자산가격결정모형의 검증 (A Test of Learning CAPM with Kalman Filter)

서지정보

· 발행기관 : 한국재무학회
· 수록지 정보 : 재무연구 / 22권 / 4호 / 63 ~ 93페이지
· 저자명 : 류두진, 이창준

초록

본 논문은 Adrian and Franzoni(2005)가 개발한 학습 자본자산가격결정모형(learning CAPM)을 재고찰한다. Adrian and Franzoni(2005)의 모형은 변수오차(errors-in- variable), 생략변수(omitted-variable) 문제로 인한 모형의 내생성을 무시하였는데, 본 논문에서는 설명변수와 오차항 사이에 상관관계가 존재할 경우에 적용할 수 있는 Kim (2006)의 방법론을 대안으로 제시하여 모형 설명력의 개선 여부를 고찰한다.
본 연구의 실증분석 결과는 아래와 같다. 첫째, Hausman(1978)의 내성성 테스트에서 시장초과수익률과 오차항 사이에 뚜렷한 상관관계가 존재하므로 Kim(2006)의 방법론을 정당화한다. 둘째, 제시된 모형에서 소형-가치주는 1.62, 대형-성장주는 0.94로 시장베타의 평균이 각각 추정되어 소형-가치주에 대해 체감하는 위험이 대형-성장주에 비해 상대적으로 높았으며 이는 소형-가치주의 높은 기대수익률을 정당화한다. 셋째, 제시된 모형의 설명력이 Adrian and Franzoni(2005)의 모형에 비해 전반적으로 개선되었으며, 특히 가중가격오차(CPE)가 약 49% 감소하였다.

영어초록

This paper re-examines the learning CAPM, a new version of conditional CAPM, developed by Adrian and Franzoni (2005). While acknowledging its firm foundation, we argue that the learning CAPM is likely to be exposed to endogeneity problems. In other words, the regressors are possibly correlated with the disturbance terms due to the omitted risk factors and errors-in-variable problems from the imperfect proxy for the unobservable market portfolio. Kim (2006) shows that the conventional Kalman filter provides an econometrician with invalid inference about the parameters if one ignores the endogeneity problems and provides new approach to circumvent endogeneity problems. In this paper, we test whether the proposed model can account for the Fama and French 25 size and book-to-market sorted portfolios. To test the joint significance of pricing errors, we use the root mean squared error (RMSE) where it gives equal weight to each portfolio as well as the composite pricing error (CPE) of Campbell and Vuolteenaho (2004) in which it gives less weight to the volatile portfolio.
The main findings of this paper are summarized as follows. First, Hausman’s (1978) specification test shows that there is a substantial correlation between the market portfolio and disturbance terms, which justifies the application of Kim’s (2006) two-step approach. We reject the null hypothesis of no correlation between the regressors and the disturbance terms in seventeen out of twenty-five portfolios. Second, our specification partly explains the size and value premium. While the time trends of the estimated market betas of the one from the Kalman filter ignoring endogeneity problems and our specification are very similar, the average market betas are considerably different. For example, the estimated average market beta of small-value portfolio by Kim’s (2006) approach is 1.62, whereas it is 1.05 when inferred from the Kalman filter ignoring endogeneity problems. It means that while the market betas of small-value stocks are underestimated if we ignore the endogeneity problems, the introduction of long-run market beta correctly estimates the true riskiness of small-value stocks. In addition, with Kim’s (2006) approach, the estimated average market beta of large-growth portfolio is 0.94, while it is 1.62 for small-value portfolio. Therefore, the proposed model explains why investors' expected return from small-value stocks is higher than that from large-growth stocks. Under our framework, small and value stocks earn more than big and growth stocks just because they are riskier than big and growth stocks. Third, the overall performance of our specification is better than that of the one from the Kalman filter ignoring endogeneity problems. The RMSE is reduced by about 28%. More importantly, the CPE drops by about 49%. Thus, the learning CAPM incorporating the endogeneity problems can better account for the size and value premium. However, our specification is not fully able to explain the value premium since the pricing errors of value portfolios are still significantly positive. Therefore, it seems that we are still missing some important determinants that explain the stock return patterns.
Adrian and Franzoni (2005) develop a new type of conditional CAPM by incorporating the long-run changes in factor loadings as well as the contemporaneous changes in market betas. In their model, the long-run market beta plays a crucial role in explaining the size and value premium. Franzoni (2004) documents that the market betas of small and value stocks have decreased substantially in the second half of twentieth century. Based on the reduction in betas of small and value stocks, they argue that the high level of market betas from the past affect today’s market betas. The slow-learning of investors caused by the introduction of the long-run mean of market betas makes them larger than the OLS estimates.
However, we argue that the regressors of the learning CAPM are very likely to be correlated with the disturbance terms. It is because : (1) there might be omitted risk factors such as SMB and HML of Fama and French (1993) which can explain most of the CAPM-related anomalies(size and value premium); and (2) there might be the errors-in-variable problems from the imperfect proxy for the unobservable market portfolio. Kim (2006) documents that the correct inference of Kalman filter strongly depends on the assumption that the regressors are uncorrelated with the disturbance terms. His simulation study shows that the conventional Kalman filter provides an econometrician with invalid inference about the parameters if one ignores the endogeneity problems. He derives a Heckman-type (1976) two-step approach to circumvent endogeneity problems and shows that one can obtain the consistent estimates of the hyper-parameters and correct inference of the time-varying factor loadings with the proposed two-step approach. Therefore, we apply Kim’s (2006) two-step approach to infer the time-varying market betas under the learning CAPM of Adrian and Franzoni (2005).

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