Estimation of the empirical risk‐return relation: A generalized‐risk‐in‐mean model

Abstract

The risk–return relation is an important topic in finance. To quantify such relation, the article introduces a new heteroscedastic volatility model, called the generalized‐risk‐in‐mean model (GRM), and considers its entire statistical inference procedure. To adapt potential heavy‐tailed phenomena in financial data or an over‐parametrization problem in modeling, the self‐weighted quasi‐maximum likelihood estimation (S‐QMLE) is studied and its asymptotics is established, which is non‐standard when null volatility coefficients exist. Compared with GARCH‐in‐mean models (GARCH‐M) in the literature, asymptotics of the S‐QMLE of our model is much easier to be established under some tractable yet simple conditions. It is very difficult to obtain asymptotics of the QMLE in the GARCH‐M, which requires many over‐complicated assumptions that are hard to verify in practice. Further, the Wald, Lagrange multiplier, and quasi‐likelihood ratio tests are proposed to test for coefficients, and their limiting distributions are derived. Simulation studies are conducted to assess the finite‐sample performance of the entire statistical inference procedure and a real example is analyzed to illustrate the usefulness of the GRM.